# Journal Club Theme of December 2009: Impact Behaviour of Materials with Cellular Structures

Welcome to the forum! Discussion topics were suggested initially as follow**:**

**Metal foams, cell deformation (bending, buckling, plasticity and fracture), constitutive stress-strain behaviour of cellular materials, energy absorption, hypervelocity impact, shock wave behaviour, 1D shock modelling, shock attenuation, shock enhancement, Material Point Method (MPM) simulation and microscopic tomography experimental observation;**

and later were extended to many related:

**Natural cellular materials, sunflower stem, negative Poisson’s ratio, nanoscale deformation in nacre, mesh-free methods, eXtended Element Free Method (XEFG), eXtended Finite Element Method (XFEM), simulating discontinuities, hyperelastic model, poroelasticity, loading rate effect, fluid-structure interaction, structural optimization, iMorph software, foam geometry, pattern formation, Kelvin problem, quasicrystal, trabecular bone, gyroid, bulk metallic glass foam, peridynamics.**

=================== initial topics ==========================

1. Cellular materials, such as metal foams, are used as impact energy absorbers in crash and blast protection due to their unique constitutive behaviour. Three stages can be identified for the stress-strain curves of the uniaxial compression of metal foams: Stage I: The deformation is, in general, reversible. For closed-cell metal foam this is in the form of bending of the cell walls and edges. At the end of this stage some cells suffer collapse. This may be due to elastic buckling, plastic deformation or fracture. Stage II: The almost constant compressive stress, plateau stress, appears in a wide range of strain. Buckling and plastic collapse occur successively until all cells are collapsed. The deformation in this stage is unrecoverable. Stage III: Cell walls and edges contact each other and are crushed; giving rise to a steeply rising stress.

2. Energy absorbers for crush and blast protection are chosen so that the plateau stress is just below the stress that will cause damage to the packaged object; the best choice is then the one which has the longest plateau, and therefore absorbs the most energy.

3. Impact velocities can vary from a few meters per second to some tens of kilometers per second (hypervelocity). Hypervelocity is relevant to the field of space exploration such as the impact by space debris. Using of metal foams in this field is still under investigation.

4. Analytical solutions of shock waves in cellular materials. For 1D analysis, Reid and Peng (1997) firstly treated cellular materials subject to uniaxial compression using a simplified rigid, perfectly-plastic, locking (RPPL) model. In the RPPL model, the constitutive behaviour in stage-I is simplified as rigid; stage-II is treated as perfect plastic at the yielding plateau stress, and the second stage ends at the locking (densification) strain; stage-III is again idealized as rigid. Radford et al. (2005) used RPPL model to study the shock behaviour in a metal foam projectiles. Harrigan et al. (2009) compared RPPL with other analytical approaches on modelling shock behaviours.

5. Shock waves propagating in cellular materials will, in general, be attenuated by cell collapse at low impact speed. However, shock enhancement will occur during high speed impact; this will affect the structure design. The group led by Professor Han Zhao at the Universit´e Paris VI investigated the shock enhancement both numerically and experimentally.

6. Simulating impact behaviour of cellular materials using the Material Point Method (MPM). MPM was adopted by the U.S. Department of Energy’s ASCI (Accelerated Strategic Computing Initiative) Center for the Simulation of Accidental Fires and Explosions in simulating high speed impact on plastic bonded explosives. The group led by Prof. Hongbing Lu at the Oklahoma State University worked on MPM for many years; his group recently studied the cell-wall buckling, shear-band formation and collapse-wave propagation using MPM simulations (Daphalapurkar et al., 2008). The microstructure of the foam was determined using μ-CT and was converted to material points. The properties of the cell-walls were determined from nanoindentation on the wall of the foam. Features of the microstructures from simulations were compared qualitatively with the in-situ observations of the foam under compression using μ-CT.

Daphalapurkar, N.P., Hanan, J.C., Phelps, N.B., Bale, H., Lu, H., 2008. Tomography and simulation of microstructure evolution of a closed-cell polymer foam in compression. Mech. Adv. Mater. Struct. 15, 594-611.

Harrigan, J.J., Reid, S.R., Yaghoubi, A.S., in press. The correct analysis of shocks in a cellular material. Int. J. Impact Eng..

Pattofatto, S., Elnasri, I., Zhao, H., Tsitsiris, H., Hild, F., Girard, Y., 2007. Shock enhancement of cellular structures under impact loading: Part II analysis . J. Mech. Phys. Solids 55, 2672–2686.

Radford, D.D., Deshpande, V.S., Fleck, N.A., 2005. The use of metal foam projectiles to simulate shock loading . Int. J. Impact Eng. 31, 1152–1171.

Reid, S.R., Peng, C., 1997. Dynamic uniaxial crushing of wood. Int. J. Impact Eng. 19, 531-570.

## reference books

Some reference books that I have read:

J. D. Achenbach. Wave propagation in elastic solids, North Holland, 1973.

L. J. Gibson and M.F. Ashby. Cellular solids: structure and properties, 2nd edn. , Cambridge University Press, 1997.

M.F. Ashby, A.G. Evans, N.A. Fleck, L.J. Gibson, J.W. Hutchinson and H.N.G. Wadley. Metal foams: a design guide , Butterworth-Heinemann, 2000.

H.P. Degischer and B. Kriszt (eds). Handbook of cellular metals: production, processing, applications , Wiley-VCH, 2002.

S. Li and W.K. Liu, Meshfree particle methods, Springer, 2007.

C.R. Ethier and C. Simmons. Introductory biomechanics: from cells to organisms, Cambridge University Press, 2007

H. Zhao and N.A. Fleck (eds). IUTAM symposium on mechanical properties of cellular materials , Springer Science, 2007.

S.J. Hiermaier, Structures under crash and impact: continuum mechanics, discretization and experimental characterization, Springer, 2008

## question about geometric non-linearities

Hello

I apologize for my ignorance in that matter.

Could someone write how important is analysis involving large strains at cell (micro) level of observation. Do strains in cell walls are large? It is clear to me that geometrical non-linearities like large rotations are important in order to describe buckling and post buckling response.

Answer to my question probably depend how thick are cell walls. But how important is material, what is in case of metal foams? Can we neglect large strains and other geometric nonlinearities in cell walls made from brittle material?

Kind regards,

Lukasz

## Cell walls are relatively thin compared with edges

Dear Lukasz,

Cell walls are relatively thin compared with edges in a closed-cell foam. For those foams formed from a liquid phase, the cell formation is determined by surface tension which concentrates material into the edges. Thin walls lead to larger bending strains, and lower critical compression for the occurrence of buckling.

An interesting paper on the average cell wall thicknesses and cell edge lengthes for the closed-cell foams (ALPORAS) of high, medium and low relative densities:

T. Mukaia, T. Miyoshi, S. Nakano, H. Somekawa and K. Higashi, 2006.Compressive response of a closed-cell aluminum foam at high strain rate. Scripta Materialia 54, 533-537.

There are competitions and coupling among buckling, plastic deformation and fracture of the cell walls, which depend on the material properties and geometry (such as thickness). It is interesting to simulate such complex behavious of cellular materials under impact.

## Dear Dr.Tan, Recently,

Dear Dr.Tan,

Recently, one group (ICACS) found negative Poisson ratio in sunflower stem (not published now). The sunflower stem has a honeycomb structure, but is more diffcult than the common structure we know, i.e., hexagon in the center and stretched hexagon in the periphery. The special structure results into anisotrophic material,radical modulus is larger than tangential modulus,thus it is understood eaily by mechanisians.

## negative Poisson’s ratio

Dear Kongdong,

I am very interest in this work. Compared with the conventional foam material, the impact energy absorption can be very high for foam material with negative Poisson’s ratio.

Regards,

Henry.

note: ICCAS stands for Institute of Chemistry, Chinese Academy of Sciences

It seems to me that negative Poisson's ratio comes from two reasons:

(1) microstructure, hexagon in the center and stretched hexagon in the periphery; and

(2) anisotropic properites, radical modulus is larger than tangential modulus.

## origin of energy absorption AND negative Poisson ratio

Dear Dr.Tan,

I am new to the impact of honeycomb structures, but interested in this phenomenon. As my understanding, the energy absorption is due to the large deformation or structural collapse which absorbs impact energy. For the classical negative Poisson ratio, in addtition to structural collapse, energy is needed to conquer energy barrier between different conformations, is it this reason for "the impact energy absorption can be very high for foam material with negative Poisson’s ratio"?

However, if one material has negative Poisson ratio, but no conformation transition as the classical negative Poisson ratio material, its energy absorption is restricted to "structural collapse". These days, I calculated mechaical response of sunflower stem structure for that group, and testified the negative Poisson ratio in that material.

## structural collapse

Dear Kongdong,

I am very interested to see how you calculate the “structural collapse” for the sunflower stem structure.

Henry.

Note: What does the "structural collapse" includes? honeycomb rotation, elastic - plastic transition, buckling, fracture, or more?

## biologically inspired cellular materials

Dear Kongdong,

Thanks for the article you email to me. It is very helpful.

The topic on biologically inspired cellular materials for impact protection can be very interesting for discussion in this forum.

Henry.

## Trabecular bone

Here is a sample of a biologically inspired cellular material. The sample has been built using a rapid prototyping equipment. Porosity varies from 0 to 100% throughout the whole sample (0 on the top, fully dense material at the base).

http://people.bath.ac.uk/rg247/swansea/gyroidal_bone.html

-- Ruggero Gabbrielli

## parameters

Very interesting work. Ruggero.

What are the parameters you used for prototyping those foams?

## Nodal surfaces

The surface is an offset of the approximation of the minimal surface called gyroid. For the modelling I used triply periodic trigonometric implicit functions, also called nodal surfaces. Parameters can be chosen as desired, affecting primarily volume fraction and shape factor. The latter needs to be introduced when very low volume fractions are required because pinch-off points are reached before the volume fraction goes down to zero when the first parameter is varied. Here is a 3D model and its mathematical expression. Porosity is 90%.

http://people.bath.ac.uk/rg247/swansea/javaview/gyroid_10percent.html

The parameter is only one in this case, k. Click on the 0.01 volume fraction version to see the 2-parameter function I used.

A brief explanation is contained in this presentation, slides 7-14 (the pdf version unfortunately does not contain the animations):

http://people.bath.ac.uk/rg247/bioceramics20.pps

http://people.bath.ac.uk/rg247/bioceramics20.pdf

For details and references read this article:

http://linkinghub.elsevier.com/retrieve/pii/S0009261400014184

-- Ruggero Gabbrielli

## physical realization of the mathematical foam structures

Dear Ruggero,

Thank you for the information and nice websites!

You mentioned the physical realization of the mathematical foam structures using the Selective Laser Sintering. Can you elaborate the details or point to a reference? Thanks!

Regards,

Henry.

## Details and references

The reference for this specific work can be found here:

http://people.bath.ac.uk/rg247/swansea/development.pdf

Solid Free Forming techniques are various and many of them can be used to build samples with complex topology, such as the gyroid. In particular, the Selective Laser Sintering process, which builds part layer by layer out of a polymer powder, is able to reproduce overhangs and any feature otherwise impossible on a CNC machine. The first work I am aware of in the bio area is the following:

http://www.springerlink.com/content/0xjmr8xka2wb168j/

Resolution today can go as high as 50 micron. With a different process, called Stereolithography (SLA), which uses a curing resing under a UV light, resolution can be even better, about 10 micron. For cheaper but coarser models, a method called 3D printing is used, where polymer particles are stuck together with a binder. The videos below explain how these techniques work:

selective laser sintering: http://www.youtube.com/watch?v=gLxve3ZOmvc

stereolithography: http://www.youtube.com/watch?v=eT-OIz-Jt3w

3D printing: http://www.youtube.com/watch?v=R-JOJ91p9Wc

There is no need for support structures as mentioned in the video. Now SLA machines project light from the bottom and the part lies on a glass vat.

Regarding the geometry, if one wishes to check the original references then I would recommend to take a look at the work of the people who first discovered the surfaces as mathematical entities: Schwarz (P and D surfaces) and Schoen (G surface).

1. H.A.Schwarz. Gesammelte Mathematische Abhandlungen. Springer-Verlag, Berlin, 1890.

which is obviously in German. The original document can be found here:

http://openlibrary.org/b/OL23301458M/Gesammelte_mathematische_abhandlungen.

2. A.H.Schoen. Infinite periodic minimal-surfaces without self-intersections. NASA Technical Note, D-5541:1-70, 1970.

which can be downloaded directly from the archive at NASA.

http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700020472_1970020472.pdf

-- Ruggero Gabbrielli

## Infinite periodic minimal surfaces

The shapes taken by soap films are minimal surfaces, very interesting.

Infinite periodic minimal surfaces can be used to describe materials with complex cellular structures.

## Inverse problem

Dear Ruggero,

Given a foam structure, is it possible (then how) to find a mathematical formula to approximate the surfaces?

Regards,

Henry.

## Which structure?

Dear Henry,

what is a foam structure? This post was about the gyroid, which little has to do with foams (at least, topologically). If you mean a real foam then the answer is maybe yes, but the effort required for such an operation probably overcomes the actual benefit.

Mathematical formulae can be used for many applications and they could potentially approximate every geometrical system we could think of (with Fourier series, for example). All this comes at a cost for the user and for the specific applications it is intended for. I am not aware of any theory used for this scope at the moment, but I wouldn't be surprised if anybody looked at such a thing and developed a theory to deal with them. Please let me know if you know of any.

-- Ruggero Gabbrielli

## PhD thesis

Dear Ruggero,

I think I am getting a better understanding of your PhD thesis now. So you have two parts, one is about foam geometry (Kelvin problem), the other is on porous materials (gyroid).

Regards,

Henry.

## origin of energy absorption AND negative Poisson ratio

Dear kongdong,

the energy absorption in negative Poisson ratio materials seems to be due to the rotation of their internal components. As shown by Bertoldi, Reis et alt., simple squared honeycomb structures can easily give a negative Poisson ratio.

https://bertoldi.seas.harvard.edu/publications/2009/Adv-Mat-2009.pdf

Is this the structural collapse you are referring to? The animation below shows how 2D periodic materials undergo to a process in which their internal components rotate, this way producing high negative Poisson ratios.

http://people.bath.ac.uk/rg247/swansea/javaview/auxetic.html

-- Ruggero Gabbrielli

## auxeticity

Dear Ruggero,

Thanks a lot for the Java demonstrations. The mathematical calculations for the negative Poisson’s ratios of square and hexagonal lattices are quite interesting.

The Java animations show quite similar pictures in the paper by Xiaodong Li et al., 2006. He posted his papers in this forum during the 1st week. His observations under the atomic force microscope showed that nano-grain rotation and deformation, which cause the negative Poisson’s ratio, are the major mechanisms contributing to energy dissipation in nacre.

Using a micromechanics model, such as the Mori-Tanaka method, your mathematical calculations can be incorporated into Xiaodong’s nanoscale experimental observations, and thus give quantitative descriptions of the auxetic behaviour and energy dissipation of nacre. We can develop a quite nice journal article together with Xiaodong, I think.

Regards,

Henry.

## auxetics

Henry,

this sounds like a brilliant idea. I am reading the papers you suggested.

-- Ruggero Gabbrielli

## modelling

Ruggero,

I am considering the following:

(1) To simplify the modelling, let's study a 2D deformation under uniaxial tension.

(2) For the first step approximation we can

treat the grains as rigid, and thus focus more on the grain-rotation.(3) Then the problem is very simply, it is a 2D web of linked hexagonal stones (suppose we treat each nanograin as a rigid hexagonal stone) that subjects to uniaxial tension!

(4) If assume the uniform distribution (honeycomb lattice) of stones, the Poisson's ratio can be calculated directly.

(5) For the above calculation, the major difficulty is in modelling the cohesion between the hexogonal stones. We can assume it to be linear in the beginning. Later, when we deal with the energy absorbing and failure of the nacre, we consider a nonlinear cohesion law between nanograins.

(6) Further, we consider a non-uniform (random) distribution of hexagonal stones.

(7) For the 2nd step approximation, we may further consider the

deformation of the nanograins. There are several ways to model the problem, either through numerical simulation, or analytical solution using Mori-Tanaka method. I think Mori-Tanaka method is especially suitable for this problem of a material with many grains.## modelling

I'm not sure I understand what the system is exactly. Is it a tessellation of rigid hexagons? If it is, how can this system show a negative Poisson ratio?

-- Ruggero Gabbrielli

## initial shapes should be irregular

You are right! If the system is initially tessellated with hexagonal stones, there will be no fattening after tension (in certain directions).

Therefore, to model the auxetic behaviour, the initial shapes of the stones should be irregular.

Do I interpret correctly?

## Shape regularity

Shape regularity does not affect global material behaviour directly. Auxetic behaviour can be modelled using irregular or regular grains. It's the way they pack that determines their auxetic behaviour.

-- Ruggero Gabbrielli

## definition of the problem

Therefore, the problem for solving can be defined as:

For certain shapes and packing of 2D rigid grains, which subject to uniaxial tension/compression, find the deformation in the transverse direction.

Since grains are assumed to be rigid, the deformation comes from the debonding between the grains at the grain interfaces.

## a micromechanical model for nacre

A micromechanical model for nacre is reported in this paper.

K. Bertoldi, D. Bigoni, W.J. Drugan, 2008. Nacre: An orthotropic and bimodular elastic material. Compos. Sci. Techno. Vol, 68, 1363–1375

## did not show negative Poisson’s ratio for nacre

It seems that the model in this paper by Bertoldi et al. (Compos. Sci. Techno., 2008) did not show negative Poisson’s ratio for nacre?

## Unidirectional auxetic behaviour?

Dear Ruggero,

The system studied in the paper by K. Bertoldi et al. (Adv. Mater. 2009, Vol. 21, 1–6) comprises a square lattice of circular holes in an elastomeric matrix which was subjected to uniaxial compression. The system shows

unidirectional (compression)negative Poisson’s ratio behaviour.The nacre studied by X. Li et al. (Nano Lett., 2006, Vol. 6, 2301-2304) shows the negative Poisson’s ratio when the nacre is under

tension. Do you think nacre also show negative Poisson’s ratio when subjected tocompression?Regards,

Henry.

## directionality

It all depends on the structure of nacre. If the grains are already completely packed, it would be difficult to have a negative Poisson's ratio in compression. From the images in the paper by X. Li et al. (2006) it seems that this is the case. The article by K. Bertoldi also shows positive Poisson's ratio for nacre. A recent work by X. Li and Z. Huang (2009) that has been already posted here shows an interesting mechanism adopted by nature to form the structure of nacre. It seems to me that screw dislocation has something to do with aperiodic arrangement.

Regarding the directionality, it is possible to create a material showing auxetic behaviour in both the loading directions, tension and compression. Simply pierce a thin film with a square lattice of alternatively oriented ellipses.

-- Ruggero Gabbrielli

## piercing a thin film

Why piercing a thin film with a square lattice of alternatively oriented ellipses can create negative Poisson’s ration for both uniaxial tension and compression?

## Alternatively orthogonal ellipses

Because they are very similar to the idealized model for auxetic material I have shown previously. I would invite you to take a closer look at it, if you haven't already. Sometimes images explain a lot better - and quicker - a mechanism of action than a thousand words can do.

http://people.bath.ac.uk/rg247/swansea/javaview/auxetic.html

This time consider the configration half way through the animation as the initial, unloaded state. It is not hard to see that holes are alternatively orthogonal rhombi. If you round their corners you can get the ellipses I mentioned. Now imagine applying uniaxial tension to the specimen. It doesn't matter if this is done through x or y, however it has to be a lattice direction.

Then do the same with a uniaxial compression. In either cases the material has negative Poisson's ratio.

-- Ruggero Gabbrielli

## positive moduli

In your idealized model of a system of thin, rigidly connected squares under uniaxial in-plane loading, the moduli in x and y directions are all zeros. How to modify the model in order to get positive moduli?

## Elastic moduli

Why are the elastic moduli zero? If the squares are rigidly connected the stress is nonzero where they touch each other. Also the strain is not zero, as it can be easily seen by the size variation of the unit cell.

Or maybe you are considering the model a mechanism of hinged rigid squares? In this case simply add a torsional spring at each hinge to get a positive modulus.

--

Ruggero Gabbrielli

## critical load

If so, there will be a critical load Pc.

When the applied load is lower than Pc, the Poisson’s ratio is positive;

When the applied load is higher than Pc, the Poisson’s ratio is negative.

## Critical load?

Why? Poisson's ratio is always negative in this model, as long as the load is applied in one of the lattice (in this case also principal) directions. The rotation of the squares causes the specimen to laterally shrink when subject to compression and to laterally expand when subject to tension.

--

Ruggero Gabbrielli

## I was wrong

Ruggero, you are right, the Poisson's ratio is always negative in this model. I was thinking about a wrong model which does not apply to this case.

## quasi-crystallization and periodic/aperiodic tiling

Dear Ruggero,

Regarding the paper by X. Li and Z. Huang (2009) which shows screw dislocation and amorphous aggregation as two dominant mechanisms during nacre’s biomineralization process, don't you think these two mechanisms can be related to something of quasi-crystallization, and periodic/aperiodic tiling?

Regards,

Henry.

## order in nacre

Yes, it could surely be. Unfortunately it is not easy to say. Do you have additional references about aperiodic order in this (and other) natural materials?

-- Ruggero Gabbrielli

## Re: Negative Poisson's ratio

Thanks for the message. Recently, we found negative Poisson's ratio in nacre. The experiemntal results may help. For details, please see the following papers.

1. X.D. Li, Z.H. Xu and R. Z. Wang, "In Situ Observation of Nanograin Rotation and Deformation in Nacre," Nano Letters, 6 (2006) 2301-2304.

2. X. D. Li, W. C. Chang, Y. J. Chao, R. Z. Wang and M. Chang, "Nanoscale Structural and Mechanical Characterization of a Natural Nanocomposite Material - the Shell of Red Abalone," Nano Letters, 4 (2004) 613-617 .

3. X. D. Li and Z. W. Huang, "Unveiling the Formation Mechanism of Pseudo-Single Crystal Aragonite Platelets in Nacre," Physical Review Letters, 102 (2009) 075502.

## Nacre has a unique structure for energy absorbing

Very interesting research and nice results!

Nacre has a unique structure for energy absorbing. I will look at this in details.

Any thoughts on the nano-grain rotation when the loading is dynamic under high speed impact, such as the case of struck by an object at speed of hundreds or thousands of meters per second?

Just asking for fun.

## Nature has its own way for impact protection

Nature has its way for impact protection. Snail shells have a unique hard-soft-hard sandwich layered structure, as Dr. Haimin Yao , a MIT postdoc reported.

"The outer hard layer contains small, grain-like particles. When under attack, these granules help to dispel the energy of the blow, spreading it out across the outer region. Any fractures that occur will disperse along jagged lines guided by the granules, forming fissures in the top layer." -- An article on Fox News.

The image is from http://seastarboats.us/, a website for sea boating.

## CISM lecture note

I delivered lectures on “Impact of Cellular Materials” for a one-week course (28 Sept - 2 Oct, 2009) “Cellular and Porous Materials in Structures and Processes ” at the International Centre for Mechanical Sciences (CISM ), Udine, Italy. Attached is the lecture note with Professor Shaoxing Qu at the Zhejiang University, written when I was in Manchester and during the summer visit in Hangzhou, China.

## synopsis of book contents

Invited by Professor Andreas Öchsner, Editor-in-chief of the Springer book series on "Advanced Structured Materials", I am writing the book “Impact Behaviour of Materials with Cellular Structures”. Below is a synopsis of the book contents:

1. Cellular materials for impact protection

1.1 Natural and man-made cellular structures

1.2 Kelvin problem

1.3 Constitutive models

1.4 Energy absorbing

1.5 Auxetics

2. Impact dynamics

2.1 Wave dynamics

2.2 Plastic shock waves

2.3 Poroelasticity

2.4 Hypervelocity impact

3. Simulating impact behaviour of cellular materials

3.1 Material Point Method

3.1.1 MPM for simulating solids

3.1.2 MPM for simulating fluids

3.1.3 MPM for simulating cellular materials

3.2 Peridynamics

3.3 Smoothed particle hydrodynamics

3.4 Molecular dynamics for simulating hypervelocity impact

4. Behaviour of shock waves in metal foams

4.1 Structures of shock front

4.2 Cell collapse and shock attenuation

4.3 Shock enhancement

4.4 Shock arrest

4.5 Propagating instabilities

References:

Cell, Cell membrane , Foam , Weaire–Phelan structure, Honeycomb, Hyperelastic material, Auxetics

Impact, Wave, Poromechanics, Hypervelocity

Material Point Method , Smoothed particle hydrodynamics, Molecular dynamics, Peridynamics

Shock wave

## Material Point Method and Smoothed Particle Method

Chapter 3 is on simulation of hypervelocity impact of cellular materials. Both Material Point Method and Smoothed Particle Method will be introduced and discussed in step by step details.

A very interesting paper on comparison of MPM with the SPH is by Shang Ma, Professor Xiong Zhang and Professor Xinming Qiu at the Tsinghua University. The title of the paper is “Comparison study of MPM and SPH in modelling hypervelocity impact problems” published on the International Journal of Impact Engineering, 2009, 272-282.

## choice of numerical method

Henry: Thanks for providing link to this paper! Quite excited to hear the conclusion from their results!

## simulating discontinuities using mesh-free methods

Any comments on simulating discontinuities using mesh-free methods?

Marginal note:

The future of meshless methods

discontinuities in mesh free methods

## XEFG

Dear Dr Henry,

I have not worked on enrich EFG (XEFG) , but I think the method work well for discontinuities . The idea I think is similar to eXtended Finite Element Method (XFEM) where enrich functions are add to basis function to introduce discontinuities or to improve stress field at crack tip!

Regards,

Canh Le

## weak form

Fundamentally, Galerkin method and finite element method are all based on converting a set of continuous differential equations into a weak form.

If the problem is discontinuous itself (like for dynamic cracking, fragmentation, shock wave, etc.), i.e., the differentiation does not exist at the discontinuous surface, there is no such weak form.

Then what all those methods are trying to do is to find ways to fix this no weak form problem.

## discontinuity from a shock front

Dear Canh,

What do you think if the discontinuity is from a shock front that separates the wave behaviour before and after? Will the eXtend Element Free Galerkin or the eXtended Finite Element Method work well for this problem? What kind of enrich functions will be needed?

Regards,

Henry.

## Strong discontinuities!

Dear Henry,

For these strong discontinuities, I think the XEFG still works well. You may find useful information in the following papers:

1. T. Rabczuk and G. Zi, A meshfree method based on the local partition of unity for cohesive cracks ,

Computational Mechanics39(2006), pp. 743–7602. Development of discrete cracks in concrete loaded by shock waves

Level set techniques is often used to trace the crack paths, see in [1] above!

Regards,

Canh

## hyperbolicity condition

Dear Canh,

Many thanks for the paper. Right now I need to check what the hyperbolicity condition (as the criterion for crack initiation and propagation used in those two papers) is. I guess I may also need a criterion for the initiation and propagation of the shock front.

## discontinuities in mesh free methods

Just noticed a very nice blog on discontinuities in mesh free methods:

http://imechanica.org/node/742

However, there is no discussion on the discontinuities at the shock wave front.

Marginal note

Level set method

## moving meshless methods

Moving meshless methods are numerical methods for unsteady partial differential equations that have shock, high gradient region or high oscillatory region.

I am simulating shock wave generated by high speed impact. Any comments on moving meshless methods?

## Peridynamics

Peridynamics may be useful in simulating the discontinuities at the shock wave front.

In Stewart.A.Silling‘s J-Club forum and other Blogs, it seems to me that most of the example applications of the peridynamics are for the discontinuities in cracks, and some for phase boundaries. No example for discontinuities in shock waves.

## Peridynamics for shock wave front.

Hello Henry,

Although I am not familiar with shock wave dynamics, these two references might be useful:

1) Viscoplasticity using peridynamics

J. T. Foster , S. A. Silling , W. W. Chen

International Journal for Numerical Methods in Engineering,

Early View2) An approach to modeling extreme loading of structures using peridynamics

Paul N. Demmie & Stewart A. Silling,

Journal of Mechanics of Materials and Structures, Vol. 2 (2007), No. 10, 1921-1945

Regards,

Erkan.

## interactions between material points in peridynamic theory

Dear Erkan,

Thanks for the information. Just a general question about peridynamic theory:

Considering a linear elastic body with a surface S that contains discontinuities in displacement, how will the material points P1 and P2 interact with each other when they reside

(1) in the same side of the surface S, or

(2) in the opposite side of the surface S?

Regards,

Henry.

## Re:interactions between material points in peridynamic theory

Dear Henry,

In peridynamic theory, all material points interact with each other if there is no discontinuity in the body, such as a crack. As in your question, let's assume that there is a surface S in the body which contains a crack as a discontinuity. So,

(1) The material points which are on the same side of the surface S will not be affected from the discontinuity, so they'll continue to interact with each other.

(2) For the material points which are on the opposite sides of the the surface S, depending on the location of the material points with respect to the crack, they may be affected. If the interaction between material points P1 and P2 crosses the crack surface, then the interaction between these material points disappears. So, they'll no longer interact with each other. If this is not the case, they'll continue to interact with each other although they are on the opposite sides of the surface S.

Regards,

Erkan.

## see each other

Assume that S is the discontinuity surface. Do you mean that if P1 can see P2 (not blocked by S) then the interaction between P1 and P2 is nonzero? There is no interaction between P1 and P2 if they cannot see each other?

## Re: see each other

Dear Henry,

Yes, what you said is correct. I tried to upload a descriptive image of this process, but it didn't work. Is it OK if I e-mail the image to you so you can upload it here?

Regards,

Erkan.

## Thanks, Erkan. Please

Thanks, Erkan.

Please email me the image, and let me find a way to upload it.

## Erkan, This is your image

Erkan, This is your image. Thanks!

Please describe it.

## Interaction of Material Points If There is a Crack in PD Theory

Henry,

Thank you very much for uploading the image.

Peridynamics is a non-local version of continuum mechanics. As we all know that in continuum mechanics, the body is composed of infinite number of material points.In the figure above, just for illustration purposes, only four of the all material points in the structure are shown explicitly. These material points are named as P1, P2, P3 and P4. As I mentioned before, according to peridynamic theory, all material points are interacting with each other. These interactions are called as "bonds". However, if there is a discontinuity in the structure, such as a crack, some of these interactions will disappear due to the discontinuity. As you can see in the figure, bonds that cross the crack surface S, i.e. red dashed lines between material points P1 and P2, and between material points P3 and P4, are broken which means that there is no longer interaction between these material points due to the presence of the crack. On the other hand, for the bonds which do not cross the crack surface will not be affected from the crack surface and these interactions are shown with green dashed lines in the figure.

Regards,

Erkan.

## need to keep a record of the discontinuity surface?

Does this mean that during the peridynamic simulation, you need to keep a geometrical record of the discontinuity surface S for its change of location, shape, orientation and etc?

## Re: need to keep a record of the discontinuity surface?

Hello Henry,

No, we do not need to keep the record of the discontinuity surface. Instead, we need to keep the record of the status of the interactions between material points. So, if a bond is broken between two material points, then these two material points are not going to interact with each other again in the future. The definition of the disconitunity surfaces can be done at the beginning of the analysis and based on these definitions, the affected bonds are broken.

Regards,

Erkan.

## bonding condition may change

Erkan,

How about the peridynamic simulation for the collision of two bodies (A and B), then? Consider body A and B that are separated in the beginning; they move towards each other and make a collision.

Initially, material points in body A will have no interaction with material points in body B. However, during the collision they make contact and there should be interaction between them.

For collision simulation, how are you going to modify the algorithm?

Similar case may be in the contact of the cracked surfaces. Are you going to modify the algorithm: "

if a bond is broken between two material points, then these two material points are not going to interact with each other again in the future?"## Re: bonding condition may change

Dear Henry,

If we bring two bodies closer to each other, they still cannot interact with each other through peridynamic bonds, even if they make a contact .Because peridynamic bonds corresponds to internal forces, not external forces that one body is exerting on another body. In other words, material points in body A can only interact with material points in body A via peridynamic bonds. They cannot interact with material points in body B with peridynamic bonds. But, a material point in body A can interact with a material point in body B by using short range forces.

In peridynamic theory, for collision or contact problems, shortrange forces are utilized in order to prevent unphysical penetration of bodies within each other. So, at a particular time, if two material points are close to each other less than some specified distance, the shortrange forces start to activate between material points. And the task of these shortrange forces is pushing two material points away from each other.

Short range forces are also valid for material points which belongs to the same body.

For more information about short-range forces please see the 3. section of "R. W. Macek and S. A. Silling, "Peridynamics via finite element analysis,"

Finite Elements in Analysis and Design, Vol. 43, Issue 15, (2007) 1169-1178. DOI: 10.1016/j.finel.2007.08.012 "

Regards,

Erkan.

## different bodies

Erkan,

When writing a code, things can be quite complicated.

You said that: "when two material points P1 abd P2 are close to each other less than some specified distance d, the short range forces start to activate between those two material points."

However, when P1 and P2 are adjacent two points within a continum body, there interaction can be bond rather than short range separating force.

The code needs to make difference of the two cases.

## Re: different bodies

Henry,

Writing a peridynamic code is not very difficult, but understanding theory might take some time. Peridynamic theory is a continuum theory, so we cannot allow any material points to share the same location because this violates the continuum concept. That's why we need those short range forces between material points even if they are neighbors. These short range forces are in addition to bond forces and activates if the material points are very close to each other.

Regards,

Erkan.

P.S For an introduction for Peridynamic theory , I recommend these two papers:

S. A. Silling and E. Askari, "A Meshfree Method Based on the Peridynamic Model

of Solid Mechanics," Computers and Structures, Vol. 83 (2005) 1526-1535. DOI:10.1016/j.compstruc.2004.11.026

R. W. Macek and S. A. Silling, "Peridynamics via finite element analysis,"

Finite Elements in Analysis and Design, Vol. 43, Issue 15, (2007) 1169-1178. DOI: 10.1016/j.finel.2007.08.012

## pairwise force function

I am reading the papers suggested and is getting a much better understanding of the theory.

To my understanding, the pairwise force function between

x'andxin the equationshould depend not only on the vector difference of the displacement

uand positionx, but also onthe state, such as strain, at both

x'andx.## Re: pairwise force function

Hello Henry,

Can you explain your statement in more detail: "should depend not only on the vector difference of the displacement

uand positionx, but also onthe state, such as strain, at both

x'andx."Regards,

Erkan.

## local state

Erkan,

Consider the bond between material points A and B.

The bond between A and B with A locally at elastic state is different from that with A locally at plastic state.

Regards,

Henry.

## Re:local state

Hello Henry,

The formulation in the two references that I suggested is based on the original formulation of peridynamics. Although the original formulation is straightforward and eaiser to understand, it has some limitations. For example, Poisson's ratio of the material has to be 0.25 because of the central interactions between material points. In order to overcome these limitations, a more general form of the peridynamic theory, called "state based peridynamic theory", was introduced in 2007.

S. A. Silling, M. Epton, O. Weckner, J. Xu and E. Askari, "Peridynamic States

and Constitutive Modeling," Journal of Elasticity, Vol. 88 (2007) 151-184. DOI: 10.1007/s10659-007-9125-1

I think what you are mentioning in your statement can be done within the "state based peridynamics" framework.

Regards,

Erkan.

## damage

Erkan,

Thank you for pointing the reference.

The modified peridynamics in the above paper incorporates the feature of damage through the influence function that exclude damaged bonds. In this approach the bond can be in only two states, either damaged or undamaged, not in between.

However, by physical intuition, debonding between points should be a continuous process from 0 (complete debonding) to 1 (complete bonding).

Nevertheless, the modified peridynamic theory can still describe damage evolution since bonds are not necessarily broken at the same time.

The question is, what is the physical meaning of a single bond between two points A and B? Is it an averaged summation of all the inter-atomic forces between the atom-aggregate localized around A and the atom-aggregate localized around B? If so, the bond should not be just in two states, damaged and undamaged, but changes continuously with time evolution.

Regards,

Henry.

## microelastic material

Equation (7) in the paper,

S. A. Silling and E. Askari, "A Meshfree Method Based on the Peridynamic Model

of Solid Mechanics," Computers and Structures, Vol. 83 (2005) 1526-1535 .

is said to be the definition of a

microelasticmaterial. I do not agree with this definition of "elasticity".## Re: microelastic material

Henry,

For the proof of this expression, please see the Section 4 of the following paper:

Silling, S. A. (2000). "Reformulation of Elasticity Theory for Discontinuities and

Long-Range Forces".

Journal of the Mechanics and Physics of Solids48: 175–209. doi:10.1016/S0022-5096(99)00029-0. http://www.ingentaconnect.com/content/els/00225096/2000/00000048/00000001/art00029.This paper is the first peridynamics paper appeared in the literature.

Regards,

Erkan.

## continuum version of molecular dynamics

The peridynamic theory may be thought of as a continuum version of molecular dynamics, and the major difference between molecular dynamics and peridynnamics are:

(1) The inter-particle force in molecular dynamics is not history dependent; while the

inter-particle force in peridynamics is history dependent: once it is broken it is always broken, as described by the mu function in Silling and Askari 's paper (2005):

(2) In molecular dynamics, the interparticle force depends on the

distancebetween two particles; in peridynamics, this depends onstretch, which I think is a brilliant idea.## Debonding is a gradual process

The statement that bond is either broken or not broken seems does not reflect the real situlation. Debonding is a gradual process from complete bonding, through partial debonding, to comple debonding (broken).

## Re: Debonding is a gradual process

Hello Henry,

Also in the peridynamic theory, bonds are not broken at the same time, and it will take some time to create a new crack surface. So, I think, peridynamic theory is reflecting the real situation. Yes, I mentioned in my previous e-mail that we are breaking bonds if it crosses a discontinuity, but this is only valid if we have an initial crack before starting our analysis.

Regards,

Erkan.

## bond parameters

What does the bond depend on?

## Re: bond parameters

Bond force depends on bond constant (material constant) and the strain of the peridynamic bond (for an isotropic material).

## definition of strain in peridynamics

What is the definition of strain in peridynamics?

## Re: definition of strain in peridynamics

Hello Henry,

I am back again. First, I want to apologize for my long silence, but I was really busy. You asked about the strain in peridynamics. Peridynamic theory is not a classical theory. However, terms like stress and strain are used in the classical continuum theory. So, in peridynamic theory (PD), we don't need to define stress or strain of a material point to obtain the solution of the problem.The important thing in PD theory is the interaction of the material points, that we name as peridynamic bond.

But, if we still want to obtain the strain field, I think we can obtain by an approximation technique using the displacement field.

Regards,

Erkan.

P.S. I'll try to answer your other questions related with the PD theory soon.

## bond definition in peridynamics

Dear Erkan,

I have doubt about the current peridynamic bond definition. See the deformation of a continuum body shown in my figure below: before deformation, material points A and B have no interaction (bond=0); therefore, after deformation material points A and B should still have no interaction according to your definition. This seems awkward for this case.

Regards,

Henry.

## Re: bond definition in peridynamics

Dear Henry,

This is a good question. As I mentioned before, all material points in the structure should interact with each other as long as there is no discontinuity in the structure. And if there is some type of discontinuity, some of the bonds are permanently broken, so associated material points will no longer interact with each other. At this point, we need to think about why these bonds are broken. Is the reason geometrical or physical? I am sorry that I was not very clear on this issue in my previous messages. Yes, we break the bonds which cross the discontinuity surface and from this definition, the reason looks like something to do with the geometry. And as you showed in your figures, if we follow this logic, these bonds might interact after the deformation. Actually this is not the case. Because we are breaking the bonds due to physical reasons. When we apply an external load to the structure, some of the energy due to external forces will be converted strain energy. So, when a particular bond deforms from its original form, it will gain some strain energy. But, if the bond is broken for some reason, then this strain energy is used to create crack surfaces. So, we already converted the strain energy of the bond in creating a crack surface and we cannot gain this energy back. That's why when a bond is broken, it is broken permanently.

Regards,

Erkan.

## does bond breaking need time?

When crack propagates, the newly fractured surfaces may break the visibility of many material points instantly.

In the peridynamic theory, all those bonds that cross the newly formed crack surface are suddenly broken, no matter how far they are away from. I have a question: does bond breaking need time?

## Re: does bond breaking need time?

Hello Henry,

Sorry for my late response. I think I need to correct yor statement. Peridynamic theory is a continuum theory like the classical continuum theory. In peridynamic theory (PD), we do not need to have a discontinuity in the structure. So, PD works very well even if there is no discontinuity in the structure. Its advantage over the classical continuum theory is that it has an integral form instead of the spatial partial derivatives as in the classical continuum mechanics and it works without any problem if there is a discontinuity in the structure or not. So, if there is a discontinuity in the structure, we do not need to use any special criteria to propagate the crack, find the crack direction, orientation etc. And during the analysis we do not need to check if there is a discontinuity in the structure or not. So, we don't break the bonds which cross the discontinuity in the structure during the analysis, because we do not need to check if there is any discontinuity in the structure. Then, how can bonds be broken during the analysis? If the bond strain exceeds some critical value (which depends on the energy release rate of the material), then we are breaking the bonds. And if there is a crack in the body, because the strain will be higher around the crack tips, broken bonds will initiate around that region and the crack will start to propagate. But as I mentioned before, we do not force any criteria on when or how the crack propagates. Yes, in my previous e-mails I said that we need to break the bonds which cross the crack surface, but this check is done only at the beginning of the analysis.

Regards,

Erkan.

## verification

"Peridynamics theory works very well even if there is no discontinuity in the structure."

Are there any papers published so far that verify the peridynamics theory with analytical solutions exist for comparison?

## Deafening Silence

Have there been any responses elsewhere to Henry's post asking for verification problems/papers for peridynamics?

Matt Lewis

Los Alamos, New Mexico

## Re: verification of peridynamics.

Dear Henry and Matt,

If you are asking about verification of peridynamic theory for some basic problems such as a 1-Dimensional bar subjected to tension loading, or a beam subjected to a transverse point load, etc, I myself verified PD theory for many problems like these. Both the analytical solutions and peridynamic solutions agree very well. But, if you are asking about a publication, it looks like there are not many comparisons against the analytical solutions.If you have an access to this PhD thesis, you can find many basic problems solved by PD theory:

Bahattin Kilic, "Peridynamic Theory for Progressive Failure Prediction in Homogeneous and Heterogeneous Materials", The University of Arizona, 2008.

Another publication of Kilic and Madenci, "Structural stability and failure analysis using peridynamic theory", International Journal of Non-linear Mechanics, 44, 845-854, 2009, demonstrates the application of peridynamic theory for structural stability analysis. Buckling loads for a bar subjected to different types of boundary conditions were successfully captured.

Regards,

Erkan.

## General Constitutive Model for anisotropic foam

Dear Henry,

We recently tested a transversely isotropic polymeric closed-cell foam. We found that there is no constitutive model to simulate its 3D mechanical response. Do you have any suggestion on that?

Zaoyang

## Dear Zaoyang, A variety

Dear Zaoyang,

A variety of models have been developed for the transversely isotropic foams. What type of mechanical response (plastic deformation, failure, impact) are you simulating? What kind of difficulties are you facing for a 3D modelling?

Regards,

Henry.

## Uniaxial plus Poisson Ratio

Dear Henry,

Currently we have uniaxial compression data (in preferred direction and transverse direction) and Poisson ratio data. But we cannot find a hyperelastic model to fit them. Do you know any references?

Best regards,

Zaoyang

## transversely isotropic foams

You may try the elastic-plastic constitutive model for transversely isotropic foams developed in this paper:

V.L. Tagarielli, V.S. Deshpande, N.A. Fleck and C. Chen, 2005. A constitutive model for transversely isotropic foams, and its application to the indentation of balsa wood. Int. J. Mech. Sci. 47, 666–686.

The parameters in that model can be derived through curve fitting of your uniaxial and Poisson’s ratio data.

## Thanks a lot

Dear Henry,

Thanks a lot, I will read the paper.

Best regards,

Zaoyang

## Is small strain measure still valid?

Dear Henry,

In that paper, they still use small strain measure though the nominal compressive strain can be 80-90%. Is that still valid? My original plan is to use a hyperelastic model to consider loading only.

Best,

Zaoyang

## Dear Zaoyang, I have the

Dear Zaoyang,

I have the same doubt about the validity of small strain measure used.

What kinds of hyperelastic models have you tried?

Regards,

Henry.

## Dear Zaoyang, I would be

Dear Zaoyang,

I would be happy to pass you a fortran subroutine to give your model a try. If you wish, get in touch on the email, vito.tagarielli@eng.ox.ac.uk. Best,

Vito

## response of lightweight structures to intense blast and impact

Dear Vito,

Your work on structural dynamics, with particular emphasis on the response of lightweight structures to intense blast and impact loading, fits the theme of the forum quite well. Could you introduce to us some of your work?

Regards,

Henry

## Dear Henry et al, please

Dear Henry et al,

please have a look at this recently accepted paper. Here we try and simulate the response of cellular solids (PVC foams and balsa wood) in sandwich configurations, subject to intense dynamic loading (blast). Creep potentials are derived for the cellular materials in order to introduce strain rate sensitivity through a visco-plastic model. This seems to be sufficient to capture the propagation of plastic shock-waves through the foams.

Prediction of the dynamic response of composite sandwich beams under shock loading

International Journal of Impact Engineering,,In Press, Corrected ProofAvailable online 1 December 2009V.L. Tagarielli, V.S. Deshpande, N.A. Fleck

## discontinuities at the shock front

Dear Vito,

Thank you for the nice article!

In your finite element simulations, how did you deal with the discontinuities, such as those of stresses, velocities and densities at the shock wave front?

Regards,

Henry.

## No hyperelastic models suitable for anisotropic foam

Dear Henry,

As far as we know, there is no anisotropic compressible hyperelastic model for foam. We are trying to develop one but it is very difficult. By the way, after uniaxial loading and unloading (up to 90%), only less than 10% of the nominal strain cannot be recovered. So plastic model is not good for polymeric foam in this case.

Are you in Scotland now? Maybe we should arrange a meeting :)

Best regards,

Zaoyang

## Zaoyang, I can see the

Zaoyang,

I can see the difficulties of the problem you are working on.

Yes, I have moved to Aberdeen; may visit you in Glasgow later.

Regards,

Henry.

## Summary of the 1st week discussions

Here is a summary of the discussions in the 1st week of December. The discussions relate to the three major research focuses in the area of impact of cellular materials (hypervelocity impact, natural cellular materials and constitutive modelling of foam materials).

1) Hypervelocity impactRunQiang Chi , a student from the Harbin Institute of Technology, China, mentioned that cellular materials are widely used as cabin walls in spacecrafts which are under threat of hypervelocity impact of orbital debris. The near-earth space environment is cluttered with man-made debris and naturally occurring meteoroids. Hypervelocity impacts between any spacecraft and this particulate environment can lead to catastrophic failure. Researches on hypervelocity impact performance of foam materials was conducted in NASA. The Hypervelocity Impact Research Centre at the Harbin Institute of Technology is building up reputations in researches on spacecraft system, high velocity impacts, shock wave, penetration and catastrophic failure; they developed a 3D Smoothed-Particle Hydrodynamics (SPH) code to simulate hypervelocity impact.

2) Natural cellular materialsKongdong discussed the negative Poisson’s ratio of the sunflower honeycomb structure. The impact energy absorption can be very high for foam materials with negative Poisson’s ratio. The energy absorption is due to the large deformation or structural collapse which absorbs impact energy; extra energy is needed to overcome energy barrier between different conformations. Kongdong is supervised by Professor Baohua Ji at the Institute of Biomechanics and Biomedical Engineering, Tsinghua University, and Professor Lanhong Dai, deputy director of the State Key Laboratory of Nonlinear Mechanics, Chinese Academy of Sciences.

When talking about the negative Poisson’s ratio of the cellular materials, Professor Xiaodong Li at the University of South Carolina, USA, referred to his recent work on natural nanoscale composite materials. Dr. Li gives talks in several US conferences on the formation mechanism of pseudo single-crystal aragonite platelets in nacre, and on the deformation and toughening mechanisms of natural biological nanocomposites. His observations under the atomic force microscope showed that nano-grain rotation and deformation are the two prominent mechanisms contributing to energy dissipation in nacre. Information on the nanoscale behaviours of natural cellular materials can be very useful for a biomimetic design for impact protection.

Marginal note:Materials with negative Poisson's ratios (auxetics) get fatter when stretched and thinner when compressed. An interesting paper discussing a mechanism for achieving auxetic behaviour in foam cellular materials:

J.N. Grima, R. Gatt, N. Ravirala, A. Alderson and K.E. Evans, 2006. Negative Poisson's ratios in cellular foam materials. Mat. Sci. Eng. A 423, 214-218.

The dynamic crushing performance for auxetic foam is significantly superior to the normal foam, see paper:

F. Scarpa, L.G. Ciffo and J.R. Yates, 2004. Dynamic properties of high structural integrity auxetic open cell foam. Smart Mater. Struct. 13, 49-56.

3) Constitutive modelling of foam materialsDr. Zaoyang Guo, a Lecturer at the University of Glasgow, UK, is looking for an anisotropic compressible hyperelastic model for foam materials. The large densification of foams can be modelled using either a physical description of the molecular interplay or by a phenomenological approach. In the phenomenological approach, the material is treated as a continuum and several material parameters are usually needed to reflect the nonlinearity in the load-compression relationships. The large densification strain of foam materials presents challenges for constitutive modelling. Dr. Guo obtained his PhD from the Northwestern University, USA, under the supervision of Professor Zdenek Bazant.

## Re: Foam impact

Understanding the shear and tensile behavior of foams under moderate to high rates of loading is important for many engineering applications. Can you point me to sources that deal with these issues?

-- Biswajit

## Re: Foam impact

Dear Biswajit,

The two subjects (shearing and tension compared with compression of foams, rates effects) and their coupling are difficult, and depend on different foam materials.

The loading rate controls the deformation and failure modes of foam cells; and the rate sensitivity of cellular materials comes from various sources. When loaded slowly, the gas filled in the closed cell foam can move out slowly without resistance; under impact, the gas entrapped in cells may have no time to move out. During the quasi-static loading, the deformation is localized in collapsing bands of cells. This motion is deterred by micro-inertia effect due to dynamic loading; the result is that collapse is more distributed rather than localized (Lee et al., 2006).

The micro-scale deformation during tensile loading is different from during compressive loading which involves extensive bending and buckling. For tension and shearing, Wang et al. (2009) showed that the slope of the initial loading portion of the curve is lower than that of the unloading curve, which means that the Young's modulus and shear modulus are different for loading and unloading. For high-rate dynamic loading, these differences may be reduced due to inertia effects.

Lee, S., Barthelat, F., Moldovan, N., Espinosa, H.D., Wadley, H.N.G., 2006. Deformation rate effects on failure modes of open-cell Al foams and textile cellular materials . Int. J. Solids Struct. 43, 53–73.

Wang, X.Z., Wu, L.Z., Wang, S.X., 2009. Tensile and shear properties of aluminium foam. Materials Technology: Advanced Performance Materials, 24, 161-165.

## Poroelasticity for impact behaviour of cellular materials

The loading rate effect on the behaviours of cellular materials, as mentioned in the above post, relates to something called poroelasticity, see Professor Zhigang Suo’s blog (Poroelasticity, or diffusion in elastic solids) and Dr. Michelle Oyen's blog (Poroelasticity references).

## Re: Poroelasticity for impact behaviour of cellular materials

Dear Henry: I'm intrigued by your remark of the connection between impact of a cellular material and poroelasticity. Could you please elaborate, or provide a reference? Thank you. Zhigang

## impact behaviours and poroelasticity properties

Dear Zhigang,

I am trying to formulate the connection between the impact behaviours and the poroelasticity properties of a cellular material.

Let’s consider the behaviour of closed-cell metal foam. When loaded slowly, the air filled in the cell can move out slowly without resistance; under impact, the air entrapped in cells may have no time to move out. Thus the kinetics of the air affects the strain-rate effect for the constitutive behaviour of closed-cell foam.

Consider another case of the impact of a bone. The bulk modulus of the matrix of bone is usually several times higher than that of the fluid in the bone. Thus the pore fluid does not share much of the mechanical loading when loaded slowly. However, this is not true in the case of dynamic loading - the transient stiffness can be much higher than the stiffness of the drained bone.

We may further design metal foams by filling the cells with a shear-thickening fluid. In this way we can adaptively control the energy-absorption and stiffness of the foam during impact.

## Mandel-Cryer effect

Transient mechanical responses of a fluid-saturated poroelastic material are interesting because of peculiar interactions between elastic deformation of its skeleton and diffusion of pore fluid.

One of such interactions is the Mandel-Cryer effect. For a sudden loading on the poroelastic body, the pore fluid pressure deep in the body rises to a peak higher than its initial value, then declines and finally vanishes, in contrast to a monotonous decrease of pressure from the initial value in conventional diffusion phenomena.

## simulation

Dear Biswajit,

I remember that you simulated the dynamic compression of metal foams several years ago. Is there any further progress?

## Re; Dynamic compression of closed-cell foams

Hi Henry,

Thanks for the pointers. Can you send me a copy of the paper on the tensile and shear properties of metal foams?

I was unable to get funcding for my proposal on dynamic fluid-structure interactions in closed-cell foams. So that investigation was not pursued beyond what I presented in 2005. I hope others will be able to take the ideas further.

-- Biswajit

## I do not have the PDF file.

I do not have the PDF file. I have sent an email to the author of that paper, and invited him for a visit of the forum.

## compressive loading rate

What was the compressive loading rate in your simulations trying to find the shear band instability?

## Research groups

Cellular Solids, Professor Lorna J. Gibson, Massachusetts Institute of technology

Cambridge Centre for Micromechanics, University of Cambridge

Crashworthiness, Impact & Structural Mechanics Group, Cranfield University.

CELPACT: Cellular and structures for impact performance, Project Supported by the European Commission

## Porosity gradients effects?

Manufacturing advances offer the possibility to control porosity gradients in cellular materials (foam, hollow spheres).

## Cellular concretes

Cellular concretes are lightweight materials produced by blending a cementitious slurry with a stable, three-dimensional pre-formed foam.

On impact sound. Hitting a wall of convenional concrete with a hammer will let you feel the full force on the other side, whilst the air embedded in cellular concrete will not allow the blow to pass through.

As cellular concrete is compressed during impact, resistance increases and kinetic energy is absorbed.

## Summary of the 2nd week discussions

Here is a summary of the discussions in the 2nd week of December. The discussion topics are mainly about modelling and simulation related to

micro-structural evolutionduring the impact of cellular materials.Simulating the micro-structural evolution during the impact of metal foams, which involves cell-wall buckling, shear-band formation and fluid-structural interactions, is challenging. Dr. Biswajit Banerjee investigated the dynamic fluid-structure interactions in closed-cell metal foam (2005). It is a pity that the work did not continue because of the lack of funding support. Dr. Nitin Daphalapurkar and others (2008) used the Material Point Method to simulate the microstructure evolution of a closed-cell polymer foam in compression. He is happy to know that MPM has many advantages over Smoothed Particle Hydrodynamics in simulating hypervelocity impact.

Simulating the motion discontinuities, such as those at the shock front or at the crack tip, is important. Canh Le of the Computational Mechanics and Design Research Group at the Department of Civil and Structural Engineering, University of Sheffield, felt that eXtended Element-Free Galerkin method (XEFG) and eXtended Finite Element Method (XFEM) will work well for discontinuities. In these methods, enriching functions are add to the basis functions to introduce discontinuities.

Poroelasticity is considered for modelling loading rate effect on the behaviour of cellular materials. Professor Zhigang Suo at the Harvard University showed his interest on the connection between the impact of a cellular material and poroelasticity. Poroelasticity was taught in his course Advanced Elasticity. Consider the behaviour of a sponge: when immersed in water, the sponge absorbs water; when a saturated sponge is squeezed, water will come out; when the squeezing is fast, the behaviour relates to that during the impact of a cellular material.

On the

constitutive scale, Dr. Vito Tagarielli has a Fortran subroutine for a hyperelastic model for foams. He is the first author of the paper on a constitutive model for transversely isotropic foams, and its application to the indentation of balsa wood, published in the International Journal of Mechanical Sciences, 2005. Dr. Vito Tagarielli is a Departmental Lecturer in Engineering Science at the University of Oxford.## Hyperelastic Model for Foams

For cellular rubber, hyperelastic models are absolutely appropriate. Is there any more information about Tagarielli's model?

I have a paper currently under consideration for publication that describes a hyperelastic foam model. I have coded it as an ABAQUS UMAT subroutine and used it for large deformation problems, including combined compression and torsion of disks. The model is based on some simple empirical strain energy functions to get after the "buckling" stage of compressive deformation (the plateau-like stress strain behavior) + lock-up terms that are strain energy functions derived from the deformation of a spherical shell composed of an incompressible Mooney-Rivlin material. This really simple (and isotropic) unit cell gives great results. I'm wondering if using an ellipsoidal unit shell might provide a useful model for anisotropic foams. The fitting process is relatively easy and mostly involves linear least squares fitting of coefficients.

I will review the publisher's requirements regarding dissemination during this review stage and try to make available what I am permitted to.

Regards to all,

Matt

Matt Lewis

Los Alamos, New Mexico

## predict buckling

Dear Matt,

Can your model predict buckling of the spherical shell when the compressive load is high enough?

## Buckling

Henry,

I wish it were that clever. No, the spherical shell is primarily used to give an incompressible response as the porosity is eliminated. It also provides a coupling between isochoric and volumetric deformation that is straightforward.

Volumetric buckling is accounted for with a strain energy defined in terms of J-1 that is quadratic initially but goes linear at a volumetric buckling strain.

This model was a compromise between an empirical fit and a micromechanically-based model. It has parameters that can be considered statistically, as one can evaluate the effects of variation of moduli, buckling strains, and initial porosity.

I am writing to the journal today to make sure I can post some form of the paper here. I will post the UMAT source this week.

Matt Lewis

Los Alamos, New Mexico

## Volumetric buckling

What is the volumetric buckling you mentioned?

## Institute for Multiscale Materials Studies, Los Alamos

Dear Matt,

I noticed that you are involved in two collaborative projects:

(1) The Effects of Topology on the Crushing and Energy Absorption of Elastomeric Unit Cells, and

(2) Models for Electroactive Elastomers for Actuators in Polymeric Foam Systems

of the Institute for Multiscale Materials Studies of the Los Alamos National Laboratory and the University of California Santa Barbara. Can you introduce the works in those projects?

Regards,

Henry.

## using LS-DYNA to simulate impact

Is there anybody here who used LS-DYNA to simulate impact behaviours? Are the results reliable?

## Robert Hooke

Cellular materials considered here are materials with small structural unit. Robert Hooke (1635-1703, Hooke's Law of elasticity is in his name) described his observations on a thin slice of cork in his book, Micrographia, published in 1665:

“I could exceedingly plainly perceive it to be all perforated and porous, much like a Honey-comb, but that the pores of it were not regular [..] these pores, or cells, [..] were indeed the first microscopical pores I ever saw, and perhaps, that were ever seen, for I had not met with any Writer or Person, that had made any mention of them before this…”

Hooke also reported seeing similar structures in wood and in other plants.

## Dear Henry In your

Dear Henry

In your opinion, what is the optimized structure for the maximum energy absorption?

## Structural optimization

Dear Bin:

This is a very interesting and practical question.

A mixture of cells with different sizes can increase the energy absorption. Hierarchical structure of bones gives this conjecture a biological hint.

Rubbers are effective impact-protection materials and have long been used as impact energy absorbers. A combination of rubber with metal foams may provide improved energy absorption capacity. This design should also consider the temperature at the impact point.

Energy absorbing also depends on how cells are distributed, randomly, with gradient or periodic. If the structural topology is specified a priori, the optimization can be derived from Finite Element simulation, I guess.

Structure design must also consider loading conditions such as impact velocities and masses, mode of impacting (uniaxial, multi-axial, etc.), as well as the duration of the impact.

## Finite Element simulation

In my opinion, FEM may be the most efficient way to study mechanical behavior of cellular solids, for that analytical method is difficult to describe such a complex structure. Of course, cell wall or a single cell can be understood at the analytical method. Broadly speaking, all solids are 'cellular materials', because there doesn't exist extreme densed material. I learn a lot from this topic, and have a question:

1. During compression on cellular strucutre, the collapse begins at the center, and propagates from center to periphery. Why?

2. At the point of intersection of cell walls, there exists a triangle (or like a hole) hinge, does the special strucutre play some roles in the mechanical behavior? Or it has some roles for its hosts to perform some functions.

## To the first question, what

To the first question, what is the geometry of the cellular structure and how is the load applied?

## Geometry

What is the geometry of foams, exactly (either open or close cell)? Is there an ideal model for this? What is the physical principle that makes foams to have that particular geometric configuration?

-- Ruggero Gabbrielli

## PhD thesis

Dear Ruggero,

I am reading your PhD thesis with great interest. Can you introduce to us your work in this field, especially on the foam geometry and structure?

Regards,

Henry.

## Foam geometry

Thanks Henry for your interest in my work.

It all started from a need I had of working with a unique reference model for the geometry of foams. I thought that such a model would have allowed people doing FEA to work on something that closely corresponds to a real foam, without the need of importing a 3D scan of real foams all the times. Basically, I was after a

mathematical formulafor a foam.It didn't take long to find out that the physical principle that drives foam formation was surface tension, and that foams tend to minimize their surface area. Foams can be thought as aggregates of soap bubbles. If we add the additional constraint of having bubbles of exactly the same volume, then the problem is uniquely defined.

Lord Kelvin proposed a brilliant idea for the solution to this problem. He used a single cell to solve the puzzle, a truncated octahedron. Shame that such a shape does not contain pentagons, whereas foams have loads. Only 16 years ago his conjecture was proven wrong, since a new periodic cell that contained two different kinds of polyhedra was found with even lower surface area than that of the truncated octahedron. We all saw the Water Cube in Beijing last year, didn't we? That building is based on the Weaire-Phelan structure, the current answer to this surface minimization problem.

You can watch a 3D model of this foam here:

http://people.bath.ac.uk/rg247/swansea/javaview/wp-foam.html

Last year I found a new structure that seems to reopen the debate on whether the solution to the problem is a crystaline or a completely disordered arrangement of shapes (like what we see in real, monodisperse, foams). Apparently there is a lot more to know before we start to ask mathematicians to work on a formal proof. Let's not forget that the proof to the Honeycomb conjecture, the same problem in 2D, has only been given a few years ago.

-- Ruggero Gabbrielli

## simulation based on the Newton's law for particle movement

I am thinking the following, any comments?

We can use some methods, such as molecular dynamics, to physically simulate the bubble formation process according to the Newton’s law. Macroscopically the Newton’s law will finally (the system reaches the force balanced configuration) give the geometrical evolution result that corresponds to that mathematically derived according to the requirement for the minimization of the surface area.

## Newton's law

What do you mean by Newton's law? Are you talking about a number of points enclosed in a region? Did you mean a potential between the particles?

I have been using methods from pattern formation to produce my results.

-- Ruggero Gabbrielli

## Kelvin problem

Dear Ruggero,

Now I understand much better about the pattern formation method you used. Thanks!

You mentioned the debate on whether the solution to Kelvin problem is a crystaline or a disordered arrangement of shapes. What is the current status? What is your view on this?

Regards,

Henry.

## Kelvin problem status

The current best arrangement is the Weaire-Phelan partition, which is a crystalline arrangement. Many believe that this is the actual solution to the problem. I'm here to help mathematicians to get a formal result, regardless of what the answer might be. Personally, I think that randomness could play a major role in energy minimization in 3D.

Additionally, the solution might not be periodic neither disordered, but a quasicrystal.

-- Ruggero Gabbrielli

## 2D Kelvin problem

How about the 2D version of the Kelvin problem? Is honeycomb cellular structure the solution?

## The Honeycomb Conjecture

Yes, hexagons do it best. Thomas Hales proved it formally in 1999.

http://xxx.lanl.gov/abs/math.MG/9906042

-- Ruggero Gabbrielli

## Dear Ruggero, Thank you

Dear Ruggero,

Thank you for providing the link.

I have a question. Why the 3D Kelvin’s problem is so hard that it takes more than one century for finding out the solution, and still the current one cannot be proved to be the final answer?

## Good question

Also the 2D problem took quite a while to be proved. Pappus of Alexandria in the fifth century BC already knew about the honeycomb being the ideal solution. And it was obvious to many for centuries. But it took about 2,500 years and the work of the brightest minds on Earth to get a formal proof.

For the 3D case things got even worse. In 2D each cell has 6 sides, which is an integer number. In 3D this number is not known yet. All what we know is that it seems to be close to 13.5 (if not exactly this value). This doesn't allow a single shape to represent the solution, since one bubble can only have either 13 or 14 faces.

To answer your question, this problem is an optimization problem where the function lies in a multidimensional space. It is easy to find a minimum but it is not possible to prove it is an absolute minimum. All what we know so far is that the solution exists:

http://arxiv.org/ftp/arxiv/papers/0711/0711.4228.pdf

Numerically speaking, algorithms have been implemented able to search for new configurations and many have been found already. These below were obtained using the Delaney symbols, which is a way to write periodic tilings into a string of integers, hence allowing sorting and enumeration, based on an algorithm developed by Olaf Delgado-Friedrichs:

http://people.bath.ac.uk/rg247/javaview/start.html

The method is powerful, but unfortunately it seems to take large computing resources for a solution to come in a reasonable time. Calculations started last year on two HPCs and are still running while I am writing.

The most interesting results (P42a) were found using pattern forming PDEs, like the Swift-Hohenberg equation. A 2D applet showing the formation of an hexagonal pattern starting from a random configuration can be found here:

http://www.cmp.caltech.edu/~mcc/Patterns/Demo4_2.html

The new P42a structure, discovered with a similar solver that works in 3D, can be watched here, together with the well-known truncated octahedron (conjectured to be the solution by Lord Kelvin) and the Weaire-Phelan structure (the current minimizer):

http://people.bath.ac.uk/rg247/swansea/javaview/p42a_comparison.html

-- Ruggero Gabbrielli

## Swift-Hohenberg equation

What is the physics behind it for being able to using the Swift-Hohenberg equation to simulate the pattern formation of foams?

## pattern formation

Ruggero,

2D pattern formation caused by surface stresses was studied by some iMechanicians here, for example:

Yaoyu Pang and Rui Huang at the University of Texas, 2009:

Effect of Elastic Anisotropy on Surface Pattern Evolution of Epitaxial Thin Films

Wei Lu and Zhigang Suo many years ago:

Dynamics of nanoscale pattern formation of an epitaxial monolayer

Now you used 3D pattern formation to study the foam formation. Interesting and elegant works!

## Distributing points in space

Dear Henry,

many thanks for the references.

The aim is to homogeneously distribute points in a given space.

Consider a monodimensional sine wave. The maxima of the function are equally spaced. The same happens in 2D. For certain values of the parameters in the equation the system evolves towards two-dimensional waves whose maxima are equally spaced (all the neighbours are equidistant from a given point) throughout the plane and tightly packed, thus giving birth to a hexagonal pattern. Things can be easily extended to 3D and to higher dimensions, in these cases becoming useful tools for the modelling of foams (tetrahedrally closed packed structures) and periodic point sets in arbitrary dimensions, with important applications in data compression and quantization.

Unfortunately this methods only produce periodic patterns, since the calculations on the PDEs are carried out inside a torus.

-- Ruggero Gabbrielli

## Cell wall thickness

In the real case of foam formation, the thickness of the cell walls is important. Cell wall thickness is not considered in the Kelvin problem; how will this affect the application of the mathematical model?

## Cell walls

I would imagine each wall to be modelled as a shell of constant thickness, bounded by n linear edges. With the opportune boundary conditions, the problem could be solved numerically.

The Kelvin problem is only looking at the topology of the net, which is valid for both the open and close-cell foams. Bending and buckling will of course affect stress distribution and element deformation in foams under applied load. Parameters such as strut diameter for the open-cell foams and strut diameter and wall thickness for the close-cell foams will determine the behaviour of the material.

In my opinion an analytical approach could also give interesting results.

-- Ruggero Gabbrielli

## impact behaviour of bubbles

Treating foams as aggregates of soap bubbles, we probably can develop analytical solutions for the impact behaviour (stress, strain, wave propagation and bubble breaking) of these kinds of foams.

## randomness

To your post “Kelvin problem status”, what is your view of randomness in foam geometry?

The idea about quasicrystal for Kelvin’s solution is interesting; why you think so?

## Randomness

What is your definition of randomness? A random system might indeed be the solution, but this is hard to prove, simply because there is no rigorous definition for randomness. A few years ago Andrew Kraynik tried to study large periodic cells in an attempt to reproduce randomness. But his results didn't even get close to the truncated octahedron conjectured by Kelvin.

http://smu.edu/math/research/rmfoam.pdf

Aperiodic tilings on the other hand constitute a formal description for partitions of space with exact topology, yet no periodic order, and in this sense they are closer to random systems. They always require more than one shape to fill space. Also the kind of order nature comes up with for many systems is often far from periodic. Yet these systems possess some kind of order.

An interesting example is the 2D Penrose tiling, which is non-periodic and is made of only two rhombi (green and light blue in the figure).

http://en.wikipedia.org/wiki/Penrose_tiling

If you take the Voronoi diagram of the vertices of this tiling, another tiling (dual) will show up. Since the original tiling had 7 kinds of vertex, its dual tiling is made of 7 different tiles (the transitivity of a dual tiling can be obtained by reading that of the tiling back to front). It is interesting to note that these tiles are only pentagons, hexagons and heptagons, and that they actually constitute a foam, also known as 'simple tiling'. In a 2D simple tiling 3 edges always meet at a vertex, and 2 tiles always meet at an edge. The average number of sides per tile, weighted on the frequency with which each tile appear, is 6. Follow the white lines to visualize the dual (black lines represent the Penrose tiling).

http://ww2.lafayette.edu/~reiterc/misc_frac/penrose.html

There is much more to know on the 3D Penrose tiling. If you are interested in the technical details consider reading this article.

http://prola.aps.org/abstract/PRB/v34/i2/p797_1

If you can't be bothered with that but you still wish to see what a Voronoi diagram of a quasicrystal looks like, here is a cluster of the 3D Penrose tiling, centred on a dodecahedral vertex (which is the equivalent of the regular pentagon in the 2D case: have you spotted any at the link above?). You should be able to see a regular dodecahedron at the centre of the cluster.

http://people.bath.ac.uk/rg247/swansea/javaview/pt2.html

If you can't, try this one. It only contains the dodecahedron surrounded by twelve non-simple 13-hedra.

http://people.bath.ac.uk/rg247/swansea/javaview/pt10.html

-- Ruggero Gabbrielli

## Re: geometry and collapse from the center and then propagate

http://www.doitpoms.ac.uk/tlplib/deformation/compression.php

Please see the above link, and click 'start' on the second figure. I think it a very attractive illustration for the compression on honeycomb structure. It is clearly shown that the collapse begins at the center, and then propagates from center to periphery. The modulus of cell wall determines the collapse modes, i.e. the wall buckling is not dominant here.

ps:

I find this website is very excellent, and it may help some of us.

http://www.doitpoms.ac.uk/tlplib/index.php## Dear Kongdong,Thank you

Dear Kongdong,

Thank you for pointing to the nice website at the Cambridge.

For the compression on honeycomb structure, the phenomenon is similar to the necking during the uniaxial tension of a uniform circular rod. The necking usually occurs at the centre of the rod. This has something to do with the bifurcation I guess.

Collapse begins at the center, and then propagates from center to the periphery. This occurs only during quasi-static loading. If the honeycomb is under impact, the picture will be totally different, the collapse starts from the impact end and spread.

Regards,

Henry.

## cell wall bending and buckling

Dear Kongdong,

To the second question, the triangle hinge (for a 2D cellular structure) at the intersection points of cell walls will affect the cell wall bending and buckling.

## Image processing software

First of all thanks for the invitation to participate to the discussion

For the energy absorption i can't answer.

During my phd (entitled "From 3d image of structure ti the study of diffusive properties of cellular media"), I focused on diffusive properties and structural characterization of cellular media. Numerical simulations was carried at the pore scale on the real geometry and a fine geometry analysis of structures were also carried out. My phd is on the iMorph website . Unhappily it is written in french, but feel free to ask me questions.

I take full advantage of this post to advertise for my software.

We developped an open source software for 3D geometrical characterization of porous media (tortuosity, pore segmentations, rodes shell and plate segmentation...). The software is named iMorph and is free for non commercial projects. We tried to make the most intuitive possible interface because the soft is designed for non image science experts. But if your are willing, feel free to contribute...

Thanks again for the invitation.

Emmanuel Brun

## morphologies

Dear Emmanuel,

Welcome to iMechanica and thank you for introducing the iMorph. Can the software be used to describe the morphologies for both closed- and open cell foams?

Regards,

Henry.

## Cellular media

Dear Henry

The software was firstly designed for open cell foam geometrical characterization because we were interested in them.

But, in my opinion, the algorithms are enough efficient to characterize closed cell foams or any kind of cellular media. We even already characterized non cellular porous media formixing properties. Briefly the only limiting thing is to provide tomographic images.

Moreover as it is an open source project, people can developp their own code on the iMorph basis.

Regards

Emmanuel

## Windows 7

Dear Emmanuel,

I downloaded the code, seems like it does not run in Windows 7. Should I recompile the code?

Regarsds,

Henry

## Summary of the 3rd week discussions

Here is a summary of the discussions in the 3rd week of December. The discussion topics were mainly centred on

foam morphology.(1) Professor Bin Liu at the Tsinghua University, China, kicked off the discussions by asking what the optimized structure for the maximum energy absorption is. Microscale cellular structure optimization is one of the final goals for the researches in this field. Bin’s PhD supervisors were Professor Keh-Chih Hwang and Professor Dai-Ning Fang at the Tsinghua University.

(2) Dr. Ruggero Gabbrielli was interested in the physical principle that makes foams of the particular geometric configuration. Ruggero studied the foam geometry mathematically by solving the Kelvin problem: how to divide space into equal volume cells with minimum partitional area. He also has a Blog for those interested in the geometry of light frameworks and models for extremely porous materials. Ruggero is a researcher at the Swansea University, UK. He received his PhD recently from the University of Bath, under the supervision of Dr. Irene Turner and Dr. Chris Bowen at the Centre for Orthopaedic Biomechanics at the University of Bath, UK.

(3) Dr.

Emmanuel Brunintroduced the softwareiMorphthat they have developed for 3D geometrical characterization of cellular materials using data such as those from the X-Ray µCT images. Emmanuel was supervised by Professor Jerome Vicente and Rene occelli at the Laboratoire IUSTI, Université de Provence Marseille, France. Emmanuel is new in iMechanica, Welcome!(4)

Kongdongof the Chinese Academy of Sciences concerned about the pattern of cell collapse during the quasi-static loading. He also showed a link to a simulation at the Cambridge University. Thanks!(5) Dr. Lukasz Kaczmarczyk considered the geometric nonlinearities caused by large strains in cell walls which will have important effects in buckling and post buckling response. Lukasz is a Research Associate at the University of Glasgow.

Another thread of discussion was by Dr. Matt Lewis, an engineer of the Los Alamos National Laboratory, USA, on

hyperelastic modelling. This thread was initiated by Dr. Zaoyang Guo at the University of Glasgow, UK, in the 1st week, and followed by Dr. Vito Tagarielli at the University of Oxford, UK, in the 2nd week. Matt developed a strain-energy based model for foams.## shock front structure of a cellular material

Simulations by a group of researchers at the University of Manchester, UK,

Z. Zou, S.R. Reid, P.J. Tan, S. Li and J.J. Harrigan, 2009. Dynamic crushing of honeycombs and features of shock fronts. Int. J. Impact. Eng. 36, 165–176

showed the shock front structure during the impact of a honeycombs cellular material.

Progressive cell crushing was observed to propagate through the material in a ‘shock’ like manner when the crushing velocity exceeds a critical value. There exists a zone at the shock front across which there are essentially discontinuities in the material ‘particle velocity’, ‘stress’ and ‘strain’.

## Summary of the 4th week discussions

Here is a summary of the discussions in the 4th week of December. Dr. Ruggero Gabbrielli at the Swansea University saw foams from a

mathematical point of viewand brought new ideas to this field.(1) Quasicrystal is a relative newcomer to the field of materials science that exists in a form between amorphous and crystalline, see Nature, March 2008, and Scientific American, October 2009. The uniqueness lies in the fact that the ordered arrangement is not periodic. Ruggero thought that randomness could play a major role in energy minimization in 3D foams. The solution to the Kelvin problem might be neither periodic nor disordered, but in a form like a quasicrystal.

(2) Bones are generally classified into two types: cortical bone (also known as compact bone) with porosity ranging between 5% and 10%, and trabecular bone (also known as cancellous or spongy bone) with porosity ranging between 50% to 90%. Ruggero used gyroid surfaces to model the cellular materials with different porosities and porosity gradients.

## Dear Henry I heard

Dear Henry

I heard that NASA proposed a workshop to fabricate and research bulk metallic glasses foams, expecting their use in spacecraft to decrease its weight and improve its impact properties under space debris with high velocity. How the present evolution and troublesome issues of these kinds of foams are? Could you give me some information and tips? Thanks!

Yours sincerely

Min Yi

## bulk metallic glass foam

Dear Min,

Here are some information.

Professor Bill Johnson and his team at the California Institute of Technology are studying bulk metallic glass foam that has an extremely high strength to weight ratio. Their research consists of experiments on the ground and during NASA Space Shuttle flights. For the experiments conducted at the NASA's International Space Station, the absence of gravity facilitates the creation of more uniform metallic glass foams. These foams have potential applications for use in future moon or Mars space structures as well as for potential shielding against space debris impacts on spacecraft.

During the fabrication of the bulk metallic glass foam, the cellular structure evolves by growth of randomly distributed spherical bubbles towards polyhedral-like cells separated by microscopic intracellular membranes. The homogeneous evolution is attributed to the high viscosity of the amorphous metal, which dampens the foaming dynamics and enhances foaming controllability.

In the following paper, metallic glass foams with porosities as high as 86% were fabricated by researchers from the Caltech and the Oklahoma State University. The foams inherit the strength of the parent metallic glass and are able to deform heavily toward full densification absorbing high amounts of energy.

M.D. Demetriou, J.P. Schramm, C. Veazey, W.L. Johnson, J.C. Hanan and N.B. Phelps, 2007. High porosity metallic glass foam: A powder metallurgy route. Appl. Phys. Lett. 91, 161903

How are the researches on bulk metallic glass foam in China? The Sixth International Conference on Bulk Metallic Glasses (2008) was held in China and Beihang (your university?) was the organizer.

Regards,

Henry.

## Thanks

Yours sincerely

Min Yi

## Thanks

Dear Henry

Thanks for your detailed information! Sorry that I'd like to ask suggestions about my personal questions little to do with the topic you have launched.

I am just a fresh P.h.D in Beihang University, 4th grade in undergraduate stage with the qualification to P.h.D in Engineering Mechanics directly. I heard that international conference held in Xi’an in 2008 and Beihang was one of the hosts, with a speech of Professor Tao Zhang. But as far as I know, most study is focused on plasticity behavior and laboratory fabrications of BMGs and investigations about BMGs’ mechanical behavior are popular, like free volume model on shear band behavior, serrated flow accompanied shear band operation and so on. Even in that international conference these aspects are protagonists. I know little about BMGs foams in China. I majored in engineering mechanics in Beihang and have interests in materials. Just this summer I stepped into BMGs and my mentor gave me a subject about BMGs as contents of P.h.D thesis and research aspects, while my mentor knows little about BMGs and he is just a holder of a subproject of a “973” project in China whose chief scientist is Professor Tao Zhang. Due to my interests in materials and confidence in my mechanics knowledge utilized in materials, I accepted this subject willingly and started to read literatures and find research progress about BMGs. Short time ago. I happened to know the news about BMGs foams research launched by NASA in 2007 and some new ideas about BMGs come to me. But as for the experiments and investigations conditions of BMGs foams, it’s imaginary in Beihang even in China and the single research of BMGs’ mechanical behaviors is difficult in Beihang as my concerned. While I saw your discerning views about new materials in your blog, I ask you about BMGs foams in the USA and intent to gain some ideas and further learn that it’s a long way for me to put forward new ideas and put them into practice, thus skimming experts or professors’ blogs and views gaining more knowledge about BMGs’ development.

Thanks for your help! Could you give me some tips about my study career? I hope you will not mind if I come to you for more tips!

Yours sincerely Min Yi

## participate regularly

Dear Min,

I would encourage you to participate regularly in our discussions. The forum will be useful during your PhD studies. I am quite confident that later on more and more experts, and students like you, will join the forum.

Regards,

Henry.

## Summary of the 5th week discussions

Here is a summary of the discussions in the 5th week since the launching of this forum.

Daily discussions during the Christmas / New Year holidays with Dr. Ruggero Gabbrielli at the Swansea University involved the following:

(1) Simulating the cellular foam creation (2D and 3D) using the pattern formation method;

(2) Building cellular samples with complex topology such as the gyroid;

(3) Aperiodic tilings for partitions of space;

(4) Effect of the cell wall thickness on the behaviour of foams;

(5) Auxetics behaviours in engineering materials (foam) and biological materials (nacre), their origin and modelling;

Min Yi, a student from the Beijing University of Aerospace and Astronautics (Beihang University) raised an enquiry about bulk metallic glasses foams for spacecraft applications.## Sorry! After posting comment, contents vanish,why?

Yours Sincerely

Min Yi

## check this

You may check this:

Problem with posting a comment using Internet Explorer

http://imechanica.org/node/3132

## Summary of the 6th week discussions

Here is a summary of the discussions in the 6th week since the launching of this forum.

Auxetics materials can be manufactured by modifying or rearranging the microstructure of existing materials such as foams, microporous polymers, and nanoscale grain roration induced auxetic behaviour has also been observed in nacre.

Dr. Ruggero Gabbriellipointed out that by simply piercing a thin film with a square lattice of alternatively oriented ellipses one can easily create a material showing auxetic behaviour in both the loading directions, tension and compression.The homogeneous evolution of cells in bulk metallic glass foams attributed to the high viscosity of the amorphous metal which dampens the foaming dynamics.

Min Yi, a student from the Beijing University of Aerospace and Astronautics, enquired about dislocation models for amorphous alloys.## Summary of the 7th week discussions

Here is a summary of the discussions in the 7th week since the launching of this forum.

Dr. Vito Tagarielli of the Oxford University introduced his recent paper simulating the response of cellular solids subject to intense dynamic loading.

Erkan Oterkus, a student from the University of Arizona, introduced two papers on peridynamics. Peridynamics may be useful in simulating the behaviour of cellular materials at the shock wave front.

## Crash and impact on the Ferrari MilleChili project.

We are writing a small proposal for deadline Jan 30 with Ferrari MilleCHILI Lab. See my posts in imechanica about it.

We have only standard composite materials and aluminum parts.

We may use the software CAD CRASH, do you know it?

Ferrari considers this software not state-of-the art, whereas I like simple approaches more than cumbersome esplicit FEM endless solutions, so in that case it has to be me pushing in this direction. Should anyone write in 2 days a small summary proposal of how he could make optimization using that software (there are various optimizers around, some even for free, we have in

house ModeFrontier for non-linear optimization and OPTISTRUCT for linear (ALTAIR, we use many things from ALTAIR) than we could consider the 30 Jan deadline proposal for the small grant. I attach the facsimile for the proposal application. The students would be

sponsored to visit your lab. We could use skype conference if you like. My skype "micheleciavarella"

You should choose one or more of our goals for crash, more appropriate to your software/experience/plans :-

1) full vehicle crash simulation manual optimization

2) crash of composite bumpers (experimental and numerical)

3) rear and front vehicle optimization for crash (including radiator position) and multiobjective (not only energy dissipation but also

technological, mounting etc)

Regards

Mike

## Frontal crash analysis and evaluation of an electric car chassis

I supervised Vinod Kavalakkat, a stduent at the University of Manchester, on a project “Frontal crash analysis and evaluation of an electric car chassis using FEA software”. Here is a summary of that project.

The aim of the project is to evaluate the crashworthiness of a representative model of a two-seater electric car chassis using commercially available Finite Element Analysis software (ANSYS-DYNA explicit dynamic analysis software). The interest in this project has been instigated by the failure of a commercially available Electric car called the G-WIZ, to survive the collision tests; it was subjected to by the concerned test authorities at TRL (Transport Research Laboratory). At present there is no procedure for testing the impact safety of electric cars as such. This project aims to study the dynamic behaviour of a simple representative model of an electric car chassis in frontal collision, under offset loading condition at different velocities and barrier heights. This is used to represent the general behaviour of electric cars like G-WIZ and Think city under frontal impact.

The image is from GoinGreen website . GoinGreen

created the market for electric vehicles and according to Newsweek is

'the largest zero emissions auto distributor on the planet today.'

## Summary of the 8th week discussions

Here is a summary of the discussions in the 8th week since the launching of this forum.

In the peridynamic theory, all material points interact with each other, these interactions are called as "bonds". The discussions in this week are mainly about bonds with daily contributions by Erkan Oterkus, a PhD student at the University of Arizona.

Besides, Mike Ciavarella is working on crash and impact for the Ferrari MilleChili project for the next generation of sport car .

## Summary of the 9th week discussions

Here is a summary of the discussions in the 9th week since the launching of this forum.

The discussions were with Erkan Oterkus about several papers of Silling et al.

R. W. Macek and S. A. Silling, "Peridynamics via finite element analysis," Finite Elements in Analysis and Design, Vol. 43, Issue 15, (2007) 1169-1178.

S. A. Silling, M. Epton, O. Weckner, J. Xu and E. Askari, "Peridynamic States and Constitutive Modeling," Journal of Elasticity, Vol. 88 (2007) 151-184.

S. A. Silling and E. Askari, "A Meshfree Method Based on the Peridynamic Model of Solid Mechanics," Computers and Structures, Vol. 83 (2005) 1526-1535.

Silling, S. A. (2000). "Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces". Journal of the Mechanics and Physics of Solids, Vol 48: 175–209.

## magic equations

Equation:

sin(x)*cos(y)+sin(y)*cos(z)+sin(z)*cos(x) - 1.3 < 0

describes a cellular material like

Equation:

sin(x)*cos(y)+sin(y)*cos(z)+sin(z)*cos(x) - 0.3 < 0

describes another cellular material like

.

## physical phenomena described by gyroid

Seeking physical phenomena that can be described by the gyroid equations like above.

## physical phenomena

Hi Henry,

try googling 'block copolymer gyroid', you'll find plenty of articles.

An article that I consider the reference point for the gyroid is this below. It cites all the milestones in the history of that surface.

http://www3.interscience.wiley.com/journal/121393803/abstract?CRETRY=1&SRETRY=0

-- Ruggero Gabbrielli

## volume fraction

Hi, Ruggero,

Nice to see you back!

For the material inside the surface described by the equation

sin(x)cos(y)+sin(y)cos(z)+sin(z)cos(x)=k

how is k related to the volume fraction?

## Volume

It's my pleasure to post on your blog, Henry.

It is not possible to express the volume in terms of elementary functions, since the integral that comes from it is an incomplete elliptic integral of the second kind. The only way to get the value is numerically.

I should make one observation about these trigonometric implicit functions. They are not minimal surfaces. However, in the example you're citing when k=0 the surface is very close to the minimal surface gyroid. If one really wants the minimal surfaces then the Scientific Graphic Project cites how to do it (numerically):

http://www.msri.org/about/sgp/jim/geom/minimal/computation/index.html

-- Ruggero Gabbrielli

## volume integration

Dear Ruggero,

What is the expression of the integration that gives the volume?

## integrand

Henry,

the expression is obtained by making the function explicit from its implicit form:

z=acos((-sin(y)(sin(x)cos(y)-k)+cos(x)sqrt(sin^2(y)+cos^2(x)-(sin(x)cos(y)-k)^2))/(sin^2(y)+cos^2(x))

-- Ruggero Gabbrielli

## analytical homogenization

Henry,

This is very nice. Has anyone performed any analytical homogenization studies using such equations to describe the microscale?

Thanks,

-Nachiket

## mathematically defined cellular materials

Nachiket,

To my knowledge, there is no analytical homogenization study for these kinds of mathematically defined cellular materials.

The stress distribution for a single period element seems cannot be analytically obtained. This is the major difficulty that blocks the advancement.

However, at least we can do some numerical analyses on these problems as the initial try. Are you interested in doing that together?

Regards,

Henry.

## Henry, I was thinking of a

Henry,

I was thinking of a numerical analysis based on the two scale homogenization theory reviewed in Hassani and Hinton ( a three part review):

http://dx.doi.org/10.1016/S0045-7949(98)00132-1

The link points to one of the parts. I was hoping that an analytical solution could be done like the authors do in the first part for relatively simple microstructures, but it seems that FEM based homogenization is most appropriate.

-Nachiket

## mathematical theory of homogenization applies

Nachiket,

Thanks for the references of Hassani and Hinton’s three-part review.

The mathematically defined cellular materials shown in my figures above have a periodic base cube of [0,2Pi]X[0,2Pi]X[0,2Pi]. The whole domain of the cellular material comprises a uniform cell structure.

Therefore, the mathematical theory of homogenization, for the computation of effective constitutive parameters, outlined in the part I applies to the current case. FEM is needed only for the calculations in the base cube.

Regards,

Henry.

## Summary of the 10th week discussions

Here is a summary of the discussions in the 10th week since the launching of this forum.

The discussions were with Nachiket Gokhale, a Senior Engineer at Weidlinger Associates, about mathematically defined cellular materials using a numerical analysis based on the two scale homogenization theory.

## Breakthrough in bubble research

Dr. Ruggero Gabbrielli has developed a new technique for mathematically modelling the structure of foam.

Image is from the University of Bath news on Ruggero's discovery.

http://www.bath.ac.uk/news/2009/09/02/foam/

## Foam geometry and cell aggregates

Henry,

many thanks for your interest in my research.

The discovery is not only related to bubbles and foams, though. More in general it is related to how soft objects, all of the same volume, tightly pack together in space. In particular, if the objects are baloons filled with liquid and there is no friction between the baloons, the resulting structure should ideally look like the Weaire-Phelan structure. But that does not happen in the real world. One reasonable explanation would be that the minimal configurations are so many and the differences in costs are so tiny that the system settles only into one of these local minima, that in which it happens to randomly fall into due to its initial state, 'forgetting' of moving to the ground state (or, better, 'not knowing' about its existence).

The new technique I developed could be improved further. At the moment this is only looking at periodic patterns, which is a big limitation. Truly random (if this means anything) structures will never arise from this techinque. New ideas under development are currently being investigating including pattern localization and studies of finite clusters.

The project is highly interdisciplinary having direct links and/or requiring knowledge from mathematics, computer science, physics, biology, chemistry, architecture, economics and even social sciences. Things are moving quite slowly recently mainly because - no matter how strange this might sound - I don't have a group working on this yet.

I made a backup copy of my research page because my accounts at the University of Bath and at Swansea University have expired. You can find it here if interested.

http://www.ruggerogabbrielli.com/

I know that all the links I previously posted on this blog are no longer correct. Sorry for any inconvenience this may have caused to anyone.

-- Ruggero Gabbrielli

## Impact Behaviour of Materials with Cellular Structures

Impact Behaviour of Materials with Cellular Structures

IMMS Seminar: 1pm, Wednesday 17th March

Venue: Seminar Room (FN 185)

Dr Henry Tan, IMMS, School Of Engineering, University of Aberdeen

This talk is on the impact behaviour of materials with cellular structures. The presentation focuses more on the problems requiring understanding, rather than results already obtained. The topics mainly come from a discussion forum at the iMechanica, hosted at Harvard University, which has been chaired by the speaker since December of 2009. Aspects of these discussions are summarised in the talk, including: Natural cellular materials, such as trabecular bone, nacre and sunflower stem; Auxetics and energy absorbing; Foam geometry, such as cellular materials described by gyroid, and Kelvin problems; Mesh-free methods, such as the Material Point Method and peridynamics, for simulating impact.

## Talk

Dear Henry,

how did you talk go?

Reading about the topics, I was wondering if you knew about the work done by Ettore Barbieri on mesh-free methods.

-- Ruggero Gabbrielli

## interface debonding in highly packed particulate composites

Ruggero,

It has been a long time (half a year) since last post with you. The talk at the Aberdeen seminar went well, basically I put a summation of the discussions in this forum.

When working on simulating the interface debonding in highly packed particulate composites such as sedimentary rocks and plastic bonded explosives, I noticed Ettore’s recent work on mesh-free approach to cohesive modelling that can be very useful to our current efforts.

A link to Ettore Barbieri’s interesting blog in iMechanica “The Future of Meshless Methods ”.

Regadrs,

Henry.