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

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

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.

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

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 impact

RunQiang 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 materials

Kongdong 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 materials

Dr. 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.

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

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

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.

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 evolution during 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.

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

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.

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 Brun introduced the software iMorph that 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) Kongdong of 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.

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’.

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 view and 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.

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.

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 Gabbrielli pointed 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.

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.

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 .

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.

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

.

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.
 

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/

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.