iMechanica - fracture - Comments

excellent

Farid Touaiti — Fri, 29 Aug 2008 06:12:24 -0400

thank you very much.

SED

Hui-Hui Dai — Fri, 20 Jun 2008 05:00:55 -0400

In a recent paper (International Journal of Solids and Structures 45 (2008) 2613–2628), we have used SED (or DSED) as a fracture criterion. The reason is that we manage to construct the analytical solution for the localization in a three-dimensional slender cylinder which gives the point-wise strain energy distribution and then it is easy to use the DSED. I thought that the work is related to the discussions on this topic so I provide the title and abstract below.

On constructing the analytical solutions for localizations in a slender cylinder composed of an incompressible hyperelastic material

Hui-Hui Dai a,*, Yanhong Hao b, Zhen Chen c

a Department of Mathematics and Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, 83 TatChee Avenue,

Kowloon Tong, Hong Kong

b Department of Mathematics, City University of Hong Kong, 83 TatChee Avenue, Kowloon Tong, Hong Kong

c Department of Civil and Environmental Engineering, University of Missouri-Columbia, Columbia, MO 65211-2200, USA

Abstract

In this paper, we study the localization phenomena in a slender cylinder composed of an incompressible hyperelastic material subjected to axial tension. We aim to construct the analytical solutions based on a three-dimensional setting and use the analytical results to describe the key features observed in the experiments by others. Using a novel approach of coupled series-asymptotic expansions, we derive the normal form equation of the original governing nonlinear partial differential equations. By writing the normal form equation into a first-order dynamical system and with the help of the phase plane, we manage to solve two boundary-value problems analytically. The explicit solution expressions (in terms of integrals) are obtained. By analyzing the solutions, we find that the width of the localization zone depends on the material parameters but remains almost unchanged for the same material in the post-peak region. Also, it is found that when the radius–length ratio is relatively small there is a snap-back phenomenon. These results are well in agreement with the experimental observations. Through an energy analysis, we also deduce the preferred configuration and give a prediction when a snap-through can happen. Finally, based on the maximum-energy-distortion theory, an analytical criterion for the onset of material failure is provided.

I am mainly an applied mathematician working on "Understanding and Exploiting Nonlinearity" in solids (nonlinear waves, instabilities and phase transitions, etc.). I am not an expert on fracture or damage mechanics and this is my first piece of work related to this area. However, I would certainly like to do more work in this area, in particular, fracture/damage caused by post-bifurcation behaviour (I view the damage caused by localization in a cylinder subject to tension/extension as one of such problems).

Im interested in extending

raulito — Sat, 07 Jun 2008 09:08:31 -0400

Im interested in extending the work to include HD Creep and other slip mechanisms

Google Group

Martin J. Gross — Fri, 06 Jun 2008 03:50:47 -0400

Dear Stephane,

could you please give a link to the mentioned google group. I can't find it. Thank you.

By the way, I tried to download the openXFEM++ from your homepage, but it doesn't work.

Best regards from Munich/Germany

Martin J. Gross

shock capturing

Charlie.liang — Thu, 05 Jun 2008 03:41:36 -0400

Any application of XFEM to capture compressible flow shocks?

Any suggestions will be appreciated.

XFEM Codes Matlab and C++

Stephane Bordas — Fri, 23 May 2008 06:51:07 -0400

Thanks for your interest in these codes, I am happy that we finally got communicating through regular email and that you found these codes useful.

Do not hesitate to get back to me, I strongly believe in sharing codes to facilitate work, and will be happy to help you, you can also visit computational_mechanics_discussion on google group,

Dr Stephane Bordas

http://people.civil.gla.ac.uk/~bordas

XFEM Matlab code

AdyDesh — Tue, 20 May 2008 14:03:47 -0400

Hello,

I am doing MS in Civil Engineering Dept. in the Indian Institute of Science, Bangalore.

My problem involves crack propagation under fatigue loading, for which I am going to implement the XFEM method.

For that I was trying to download the Extended FEM Matlab code from http://people.civil.gla.ac.uk/~bordas/phu.html, but was unable to download.

I am studying all the XFEM related papers, to start coding the method, I thought it would be easier if I go through a code for a basic XFEM problem.

If anybody has the code, please send it to me at aditya[at]civil[dot]iisc[dot]ernet[dot]in

Thank You.

XFEM- Matrix of extended shape functions derivatives

Marco A. Fdz — Fri, 02 May 2008 00:54:37 -0400

Hello, I am interesting on the implementation of XFEM for fracture problems, especially for quasi-static crack growth analysis. (Currently)I have been reading all around the X-FEM (papers published, programs, etc), also I am exploring the XFEM matlab code developed by Nguyen Vinh Phu and Dr. Stephane Bordas (http://people.civil.gla.ac.uk/~bordas). However some basic queries regarding the evaluation of enriched shape functions and how to form the associated matrix of extended shape functions derivatives [B], launch me to consult your help. I noticed that in some earlier publications, the evaluation of each enrichment function is performed only at simple sample points (i.e. gauss points) w.r.t the crack, and in others like in your job it is performed on two different points at the same time (i.e. gauss point and the node to enrich) also w.r.t. the crack.

For the last, Why did you do a subtraction of both evaluations? Are there specific reasons?, and what can you recommend me? I know that my question could be very obviously, but as soon as I noticed, the doubt arise.

To be more explicit for example in the former, the following about the build of matrix [B] is reported as: for a H(x) enrichment type Bia =[(Ni*H),x 0

0 (Ni*H),y

(Ni*H),y (Ni*H),x]

for a B(x) enrichment type Bibalpha=[(Ni*PHIa),x 0

0 (Ni*PHIa),y

(Ni*PHIa),y (Ni*PHIa),x]

And in your code a subtraction between the evaluation of two different points are done,

for a H(x) enrichment type Bia = [(Ni*(H-H(xi))),x 0

0 (Ni*(H-H(xi))),y

(Ni*(H-H(xi))),y (Ni*(H-H(xi))),x]

for a B(x) enrichment type Bbalpha= [(Ni*(PHIa-PHIa(xk))),x 0 0 (Ni*(PHIa-PHIa(xk))),y (Ni*(PHIa-PHIa(xk))),y (Ni*(PHIa-PHIa(xk))),x] ___________________________________________________________________

Thanks in advance for your support. Marco

atomistic simulation of cracks in ferroelectrics

Arash_Yavari — Wed, 30 Apr 2008 19:35:25 -0400

Dear Chad:

The best known and most common interatomic potentials for
ferroelectrics are shell potentials (here "shell" is very different
from what we usually understand as a shell in mechanics). These
potentials were introduced in the sixties by Dick and Overhauser:

Dick, B. G. and A. W., Overhauser [1964], Theory of the dielectric
constants of alkali halide crystals, Physical Review 112: 90-103.

In these potentials, each atom has a core (nucleus and inner electrons)
and a massless shell (valance electrons). The shell is assumed to be
spherical and uniformly charged. What enters into the energy expression
is the position of the center of the shell. Total energy has the
following three parts:

1) Short-range energy: Only shells contribute to this part and for the
most part this is a repulsive energy that prevents shells to get too
close to one another (Pauli repulsion).

2) Core-shell energy: In a given atoms core and shell interact usually
by a nonlinear spring (a fourth order polynomial in terms of the
relative distance). This prevents collapse of core and shell in the
same atom.

3) Electrostatic energy: This is the classical Coloumb potential. For a
given atoms shell and core do not interact electrostatically. This is
the troublesome part as long range interactions have to be very
carefully treated as the lattice sums representing energy and force are
conditionally convergent (Ewald summation method is usually used for
periodic systems).

In this model, spontaneous polarization (for example in the tetragonal
phase of PbTiO3) is mainly due to relative shifts of cores and shells
in the direction of polarization.

These potentials have been used by several groups in understanding the structure of domain walls, free surfaces, etc.

I have done some calculations for domain walls (surfaces on which
polarization is discontinuous, i.e. boundaries between different
energetically-equivalent variants in a single crystal). My experience
with these potentials is that they are very sensitive and the resulting
stiffness matrices could be close to ill-conditioned.

There have been analytic/semi-analytic lattice calculations for cracks
in the past mostly for idealized 2D lattices (an interesting observed
phenomenon was lattice trapping in early seventies). The main
contributers are Slepyan and Marder. However, as far as I can see it,
their methods cannot be used for analysis of something complicated like
PbTiO3 with its fairly complicated interatomic potential. My guess is
that the best approach would be MD.

Regards,
Arash

Re: Modeling R-Curve behaviour using CZM

Zhigang Suo — Wed, 30 Apr 2008 14:41:00 -0400

Here is one such paper:

Tvergaard, V., Hutchinson, J.W.,"
The relation between crack growth resistance and fracture process parameters in elastic-plastic solids." J. Mech. Phys. Solids 40, 1377-1397(1992).

Modeling R-Curve behaviour using CZM

Siva P V Nadimpalli — Wed, 30 Apr 2008 12:57:45 -0400

Hi All,

Thanks for your valuable comments and explanations about this topic. I am relatively new to this (Cohesive Zone Model) field. From a limited amount of reading and discussions with collegues I understood this topic to certain extent. From what I understand, CZM requires a traction separation law for modeling, which is obtained based on fracture energy.

My question is, because fracture energy is not a contstant value i.e., in case of ductile materials we have phenomenon called "R-Curve" behaviour (fracture energy changes with crack length initially), is it justifiable to use only one value of fracture energy (i.e., steady state value) to derive the traction separation law?

Please point me towards any article(s) if exist about modeling "R-Curve" behaviour using CZM.

Thanks a lot for your valuable time and suggestions.

--Siva

Reply to Arash

Chad Landis — Tue, 29 Apr 2008 14:25:08 -0400

Arash,

No I am not. I would be keenly interested in seeing something like this. I think that some of the methods that you are working with might be able to look at this problem. Perhaps you could post a reply detailing some of the issues that your methods would face while trying to address this.

Chad

atmistic simulations

Arash_Yavari — Sun, 27 Apr 2008 14:52:50 -0400

Dear Chad:

Are you aware of any atomistic calculations of fracture in ferroelectric crystals?

Arash

small cracks

Arash_Yavari — Mon, 21 Apr 2008 15:28:33 -0400

Dear Zhanqi:

I'm not familiar with fracture in functionally graded materials (it
shouldn't really be that different from classical fracture mechanics as
a functionally graded material is a material in which mechanical
properties are position dependent) but I would say when the crack
length goes to zero (in 2D) stress intensity factor (SIF) should go to
zero as well. For example, when you have a crack of length 2a in an
infinite body, SIF is \sigma\sqrt{\pi a} (\sigma is the far-filed
stress) and clearly goes to zero when "a" goes to zero.

Perhaps, you should post a couple of those papers where SIF is assumed to be "1" for small cracks.

You can also think of a crack as a special "defect" in a solid. SIF is
related to the so-called "material" or "configurational" force (energy
release rate in this case). A configurational force is, by definition,
the thermodynamic force that drives propagation of the crack. Now, when
crack length goes to zero, in the limit there is no defect and hence no
configurational force.

If you want to use SIF as a strength criterion, there are problems with
small cracks. Assuming that a crack propagates when SIF reaches a
critical value, let's say K_c, in principle, you can calculate the
corresponding critical stress or "strength". Assuming that K_c is
independent of crack length, you will end up having infinite strength
in the limit of a very small crack. Of course, this is not physically
meaningful because for very small cracks you need to consider things
like surface effects, etc. and this would mean that K_c is explicitly a
function of crack length. This (and similar problems) has been a
motivation for the so-called nonlocal failure criteria, e.g. the ones
proposed by Neuber and Novozhilov.

I hope this helps.

Regards,
Arash

Reply to Kaushik

Chad Landis — Mon, 21 Apr 2008 11:37:06 -0400

Kaushik,

The question to answer is "what does the macro constitutive law need". If we take for example macro single crystal constitutive models (which are analogous to continuum slip plasticity models) we need to know the critical level of "driving force" required to cause switching on a given "transformation system", i.e. 180 or 90 degree switching for tetragonal ferroelectrics. My paper with Yu Su studied how charge defects affect the coercive field for 180 and 90 degree switching.

http://www.ae.utexas.edu/~landis/Landis/Research_files/JMPS2007.pdf

So one element that the phase field models might be able to provide to the macro models is how the coercive field is dependent on the concentration of defects. These phase field calculations have also suggested that the critical driving force can also depend on the other non-driving stresses and electric fields. For example, due to changes in piezoelectric properties across a 180 degree domain wall, the applied driving force for motion increases with combined electric field and tensile stress along the wall, suggesting that the observed critical electric field should decrease. However, the phase field calculations indicate that the critical driving force increases faster due to the tensile stress than the applied driving force, and so the critical electric field increases. This type of calculation suggests that the macro models should have a more complicated formulation than most (I think all) of them do.

Another part of the physics that is swept under the rug with the macro models is the distribution of residual stresses and electric fields associated with incompatible domain states. For example we see pictures of needle-like domain tips all of the time, and there are complicated fields associated with these. I have not thought about how to systematically study this (some type of lamination theory a la Jiangyu Li and Kaushik Bhattacharya is probably best suited for this problem), but it would be useful to have a better picture of how the residual energy evolves with the concentrations of the different variant types. The problem here is that I do not think there is a unique answer to this question. But then, I guess this is always what will happen when you represent a continuous set of internal variables (the positions of all domain walls) with a finite discrete set (volume concentrations of variant types). My point is that there are certain components of the macro models that can be informed by this type of information.

The short answer to your question would have been that there haven't been many links made between the macroscopic models and phase field models.

Chad