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# Applications of a New Element Model of Solid Bodies in Plasticity

Based on the deformations under the equivalent external forces are the same, a new element model of elastic perfectly-plastic and orthotropic solid bodies is presented, and the elastic perfectly-plastic and isotropic materials is a special subclass. Furthermore, the method for determining the displacements, the stresses and the strains of a body under the action of applied forces has been given. A new method for predicting the engineering elastic constants of a fiber-reinforced composite material is also presented. It can be found that the precision by using the new element method is good, and the limit analysis has the highest precision in all methods.

Keywords: element model, elastic perfectly-plastic, Plasticity, Elasticity, stress, truss

Jiang Ke. Applications of a New Element Model of Solid Bodies in

Plasticity. Advanced Materials Research, March, 2013, Vols.690-693:

1800-1805.

My blog also includes other related papers:

1. Jiang Ke. A New Model of Orthotropic Bodies. Applied Mechanics and Materials,August,2012, Vols.204-208: 4418-4421.

2. Jiang Ke. A New Element Model of Solid Bodies. Applied Mechanics and Materials, May, 2012, Vols.174-177: 2115-2118.

3. Jiang Ke. A new method for solving solid structures. COMPUTATIONAL MECHANICS: Proceedings of International Symposium on Computational Mechanics (ISCM 2007), July 30-August 1, 2007, Beijing,China. Tsinghua University Press & Springer.

Example:

The figures and animations of the calculation results are shown as follows：

http://blog.sina.com.cn/s/blog_4be425660101c4y2.html

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Applications of a New Element Model of Solid Bodies in Plasticity.pdf | 325.11 KB |

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## Comments

## Re.: A system of trusses as a model of a continuum solid element

Dear Jiang,

1. The basic idea:I have rapidly browsed through your three papers (this one and the preceding two posts [^] [^]).The basic idea, viz. that a system of trusses can be taken as a model of a continuum solid element, is an excellent vehicle from the pedagogical angle, whatever its merits as a new analysis method may be.

Indeed, the latter may be, prima facie, taken to be in doubt, and further studies would be necessary for a proper evaluation.

2. Why the doubt:The variational/weighted residual formalism of FEM is, frustratingly, far too wide: you can always treat FDM, FVM, and indeed every other technique as a special case of that formalism. Itisfrustrating because, for example, if someone has learnt FEM the usual way, but not FVM, he probably wouldn't even suspect that something like FVM could be built. However, once FVM is available to him, he could always turn around and say: oh well, that's nothing new, that's just such and such special case of the FEM formalism.The only way to ask

sucha guy to shut up his mouth would be take concrete cases, ask him to implement FEM the way he had always done (without first knowing FVM), then do it the FVM way, and find the concrete application classes where FVM can outperform, in some way, his version of FEM. For instance, certain applications in CFD wherein you have a structured, fine, mesh for FVM and let him run wild with his FEM with unstructured mesh.Chances are, a really "FEM-committed" guy could argue that you could treat even Monte Carlo, SPH and LBM methods, even MD (molecular dynamics) as falling within some kind of a suitable FE (variational/WR) scheme. You could arguably do that. Yet, these methods, do stand as separately recognizable methods because they emphasize certain combinations of certain physically relevant elements, as the defining part of their formulation.

So, if you could do something similar for your approach, it would be good to see.

Also, some studies and comparisons from the specifically computational angle would be helpful. Perhaps, the new method does not offer any significant advantages from the computational costs point of view.

These concerns, essentially, are why I said that at least prima facie, the particular or distinctive merits of this approach, taken as a new analysis method, may be taken to be not clear at this stage.

3. An excellent vehicle for pedagogy:Yet, as I noted, this approach is an excellent vehicle to explain some of the very foundational ideas concerning continuum mechanics. And, you could treat both isotropic and nonisotropic materials very easily.4. A bit personal:I myself had tried to think of this very idea (a system of trusses as an element) along somewhat similar lines at the beginning stages of my PhD (say around 2004--2006, or roughly around the same time that your first paper came out in 2006), and of course completely independently of you. Indeed, the seed of the idea goes back at least to my UAB student days (1990--93) if not also to my MTech days at IIT Madras (1985--87). However, I had never pursued the idea even halfway through. The most I came to work on it was during 2005--2006 times, but left it aside after a while.In particular, I left it aside after implementing an FEM truss program, and modeling not just a cantilever beam but also a (2D) KIC fracture toughness specimen, via such a simplest system of trusses, and then left it at that. My PhD thesis does contain a couple of pictures depicting the (exaggerated) deformations of the cantilever and the KIC specimen, but that was that. All in all, as I said, I left it without pursuing it anywhere near the extent you actually have.

One of the reasons I kept this idea aside was that I had to put in place some other stuff (viz. the 2nd order PDEs) as primary objectives for my PhD. Another thing was that I had become apprehensive whether people would welcome such an idea if I could not already show its computational superiority over FEM, and, at that time, I had not yet had some basic theoretical issues of FEM itself become sufficiently clear to me. Of course, this kind of an apprehension isn't really called for, I now realize.

Still, whatever be the particular reasons, I would say, I had not yet come even half-way to what you have

alreadypresented via these papers. Yes, looking at the stage of completion to which you took it right back in 2006, certainly, youhad alreadygone ahead of me right at that time. ...... So, you can see that I would be happy to see this idea reaching a definite stage of

completion. So, accept my very special congratulations! Good job!(However, you still need to put it in the wider context, emphasizing the conceptual aspects where necessary, apart from the considerations of taking it as a computational method; some such a context, I have already indicated.)

5. Might as well share this:Ok, coming back to something more public. Those who have read my PhD thesis (or its extended abstract, here[^]) would know that I had made a conjecture to the effect that it should be possible to create a technique like the random walk for modeling also the stress/strain fields of solid mechanics.I am now happy to declare that this conjecture may now be taken as proved, at least in a certain special but essential case: all that you have to do is to suitably randomize the truss-system element. And, the latter part is very, very easy to do. (A most comprehensive proof would involve QM.)

6. One last point:Oh yes, I should have mentioned this above, but forgot. This approach (the truss-system as a continuum element) also provides a very good frame-work to uniformly treat the materials with positive and negative Poisson's ratios.In view of (i) its definite potential at least of improving pedagogy concerning the foundational ideas of continuum mechanics (or, at least in solid mechanics), and (ii) its potential of providing some possible advantages as a computational analysis approach (though probably not so, if seen from the viewpoint of only the computational costs/complexity), and also, (iii) its relevance in countering the previously published work (in reputed/top international journals) mathematically "proving" the impossibility of using a random walk-like technique for modeling of7.And, onemorepoint!: A request to the iMechanica admins:3Dsolid mechanics problems, please promote this post to your main page for a wider dissemination; thanks in advance.--Ajit

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[E&OE]

## Micro-truss model for concrete fracture

We had followed a similar idea (micro truss model) to simulate fracture in concrete (probably without much mathematical rigour). The links to two of our papers are given below. The basic assumption we used was that at the microscopic level, the fracture of brittle materials like concrete is caused by local tensile stresses. The tensile stresses in a truss member is readily available, unlike the first principal stress in continuum elements (which also is not very difficult to find in FEM), and we used the element KILL option in ANSYS to track the crack pattern. Reasonable levels of matching with the experimental results were obtained. Also, the simulation of inhomogenities in concrete was done quite easily.

1.

Application of micro truss and strut and tie model for analysis and design of reinforced concrete, Praveen Nagarajan, U.B. Jayadeep and T.M.Madhavan Pillaistructural elements

http://rdo.psu.ac.th/sjstweb/journal/cover-31-6-online.php

2.

Mesoscopic numerical analysis of reinforced concrete beams using a modified micro truss modelPraveen Nagarajan, U.B. Jayadeep and T.M. Madhavan Pillai

http://www.techno-press.org/?page=container&journal=imm&volume=3&num=1

We wanted to follow up these 2-D studies with 3-D studie, but couldn't find the time. Probably, we will do it in near future.

Regards,

Jayadeep

## upload a few examples of the new element method

A day later I will upload a few examples.