iMechanica - strain
https://www.imechanica.org/taxonomy/term/132
enOverview of solid mechanics
https://www.imechanica.org/node/21672
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/179">solid mechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/131">stress</a></div><div class="field-item even"><a href="/taxonomy/term/132">strain</a></div><div class="field-item odd"><a href="/taxonomy/term/9000">equilibrium</a></div><div class="field-item even"><a href="/taxonomy/term/11815">kinematics. constitutive relations</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p> </p>
<p>This video (<a href="https://www.youtube.com/watch?v=dOysposp9LY">LINK</a>) attempts to give the big picture of solid mechanics along with the brief explanation of the constituent topics.</p>
<p>----------------------------------------------------------------------------------------------------------------------</p>
<p><span>This is an educational outreach initiative targeted at all engineers interested in mechanics and seek simpler explanations. Kindly share if you learn something out of this !</span></p>
<p> </p>
<p>Thanks,</p>
<p>Prithivi</p>
</div></div></div>Sun, 08 Oct 2017 06:07:14 +0000rajan_prithivi21672 at https://www.imechanica.orghttps://www.imechanica.org/node/21672#commentshttps://www.imechanica.org/crss/node/21672Strain calculation in ABAQUS
https://www.imechanica.org/node/19166
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>Hello,</p>
<p>I have some troubles in calculating the strain in ABAQUS. I have the following Model with one element (see picture) and a linear elastic isotropic Material (Steel), which is sheared in the x-direction. The analytical result of the strain tensor gives me in xz-direction 0.25 (see picture). The result in ABAQUS is 0.5. I ran the same simulation in CalculiX CrunchiX and got the expected strain of 0.25 in the xz-direction</p>
<p>Can anyone tell me how ABAQUS calculates the strain for a linear static analysis? Why is the ABAQUS result exactly 50% higher?</p>
<p>Thanks and regards</p>
<p>Christof</p>
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</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/75">mechanician</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/132">strain</a></div><div class="field-item odd"><a href="/taxonomy/term/289">ABAQUS</a></div><div class="field-item even"><a href="/taxonomy/term/10887">linear elastic</a></div></div></div>Tue, 24 Nov 2015 16:12:01 +0000cclemen19166 at https://www.imechanica.orghttps://www.imechanica.org/node/19166#commentshttps://www.imechanica.org/crss/node/19166How obtain tension strain concrete(σt0) in damaged plasticity model in ABAQUS?
https://www.imechanica.org/node/17610
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>How obtain tension strain concrete(σt0) in damaged plasticity model in ABAQUS?</p>
<p>hello engineers</p>
<p>i need help about concrete damaged plasticity</p>
<p>fo example :</p>
<p>if Compressive stress (Fu)= 51.2 MPa</p>
<p>and tension stress (σ t)=0.3 x 51.2^(2/3)=4.136</p>
<p>1)why (σ t) is 2.36 in picture(uploaded)?</p>
<p>2)what is tension strain ? and how? i cant understand.</p>
<p>i cant obtain εt = total tensile strain.</p>
<p>how obtain εcr ?</p>
<p>please help</p>
<p> </p>
</div></div></div><div class="field field-name-upload field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><table class="sticky-enabled">
<thead><tr><th>Attachment</th><th>Size</th> </tr></thead>
<tbody>
<tr class="odd"><td><span class="file"><img class="file-icon" alt="Image icon" title="image/jpeg" src="/modules/file/icons/image-x-generic.png" /> <a href="https://www.imechanica.org/files/Ashampoo_Snap_2014.12.06_15h25m38s_003_.jpg" type="image/jpeg; length=69410">Ashampoo_Snap_2014.12.06_15h25m38s_003_.jpg</a></span></td><td>67.78 KB</td> </tr>
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</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/962">software</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/169">Plasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/289">ABAQUS</a></div><div class="field-item even"><a href="/taxonomy/term/3182">concrete</a></div><div class="field-item odd"><a href="/taxonomy/term/4503">concrete damage plasticity</a></div><div class="field-item even"><a href="/taxonomy/term/132">strain</a></div><div class="field-item odd"><a href="/taxonomy/term/131">stress</a></div><div class="field-item even"><a href="/taxonomy/term/10247">totallstress</a></div><div class="field-item odd"><a href="/taxonomy/term/2374">engineer</a></div></div></div>Sat, 06 Dec 2014 23:18:23 +0000siren30017610 at https://www.imechanica.orghttps://www.imechanica.org/node/17610#commentshttps://www.imechanica.org/crss/node/17610New Ebook on Elastic Solids at Amazon
https://www.imechanica.org/node/14877
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/464">bending</a></div><div class="field-item odd"><a href="/taxonomy/term/469">torsion</a></div><div class="field-item even"><a href="/taxonomy/term/934">Composites</a></div><div class="field-item odd"><a href="/taxonomy/term/963">stability</a></div><div class="field-item even"><a href="/taxonomy/term/1112">thermal</a></div><div class="field-item odd"><a href="/taxonomy/term/1552">computation</a></div><div class="field-item even"><a href="/taxonomy/term/3952">plates</a></div><div class="field-item odd"><a href="/taxonomy/term/5612">shells</a></div><div class="field-item even"><a href="/taxonomy/term/8862">constitutive</a></div><div class="field-item odd"><a href="/taxonomy/term/8863">axial</a></div><div class="field-item even"><a href="/taxonomy/term/8864">pressure vessels</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
This treatise provides a broad overview of the definitions of<br />
fundamental quantities and methods of analysis for the use of solid materials<br />
in structural components. The presentation is limited to the linear elastic<br />
range of material behavior where there is a one to one relationship between<br />
load and displacement. <strong> </strong>Fundamental<br />
methods of analysis and typical results for structures made of elastic solid materials<br />
subjected to axial, bending, torsion, thermal, and internal pressure loading;<br />
basics concepts of stability, plates, shells, finite element method, and mechanics<br />
of fibrous composite materials.
</p>
<p>
The treatise is intended as an introduction for specialist in fields<br />
such as aerospace engineering, mechanical engineering, materials science, and<br />
civil engineering. It is also intended<br />
to serve as an overview, and possibly the only formal study of the subject for<br />
specialists in other fields of engineering and science. For a course of study<br />
at the college or university level, it is expected that it would, at most, be<br />
equivalent to a one hour semester course. Finally, it is intended as an<br />
introductory overview for those in secondary science education and those<br />
teaching at the secondary level, as well as an introductory course for those<br />
studying in the arts and sciences. As the emphasis on STEM (Science,<br />
Technology, Engineering and Mathematics) education in the United States has increased<br />
in recent years, this treatise is a contribution to that effort. The treatise<br />
is written from the perspective of an engineer, one with more than forty years<br />
of experience as an engineering professor.
</p>
<p>Normal<br />
0<br />
false<br />
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EN-US<br />
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X-NONE</p>
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</div></div></div>Fri, 21 Jun 2013 19:07:46 +0000Carl T. Herakovich14877 at https://www.imechanica.orghttps://www.imechanica.org/node/14877#commentshttps://www.imechanica.org/crss/node/14877Stress/strain of a body performing a translation
https://www.imechanica.org/node/10570
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/6438">idea</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hello all,
</p>
<p>
I am having task to determine the strength analysis (stress/strain) of a translating machine part (body). The part is driven by a set of gears, placed on the top of the body. The input is a constant acceleration value to the propultion engine. Firstly I should do a 2D analysis with a retangle representing the machine part.
</p>
<p>
I am new to this subject, so any idea, approach, <span class="Apple-style-span"> advice or helping material is welcomed. </span>
</p>
<p>
Thank you in advance!
</p>
<p>
Ljupco Poposki
</p>
<p>
</p>
</div></div></div>Wed, 13 Jul 2011 10:05:30 +0000ljpoposki10570 at https://www.imechanica.orghttps://www.imechanica.org/node/10570#commentshttps://www.imechanica.org/crss/node/10570How to supply a visualization for the displacement gradient tensor
https://www.imechanica.org/node/9159
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/973">software</a></div><div class="field-item odd"><a href="/taxonomy/term/5262">Tensor Visualization</a></div><div class="field-item even"><a href="/taxonomy/term/5263">Visualization</a></div><div class="field-item odd"><a href="/taxonomy/term/5550">displacement</a></div><div class="field-item even"><a href="/taxonomy/term/5689">toy</a></div><div class="field-item odd"><a href="/taxonomy/term/5690">displacement gradient</a></div><div class="field-item even"><a href="/taxonomy/term/5691">rotation</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Hi all,
</p>
<p>
[Warning: The writing is long, as is usually the case with my posts :)]
</p>
<p>
It all began with a paper that I proposed for an upcoming conference in India. The extended abstract got accepted, of course, but my work is still in progress, and today I am not sure if I can meet the deadline. So, I may perhaps withdraw it, and then submit a longer version of it to a journal, later.
</p>
<p>
Anyway, here is a gist of the idea behind the paper. I am building a very small pedagogical software called "toyDNS." DNS stands for <strong>D</strong>isplacement, strai<strong>N</strong>, and stre<strong>S</strong>s, and the order of the letters in the acronymn emphasizes what I (now) believe is the correct hierarchical order for the three concepts. Anyway, let's keep the hierarchical order aside and look into what the software does---which I guess could be more interesting.
</p>
<p>
The sofware is very very small and simple. It begins by showing the user a regular 2D grid (i.e. squares). The user distorts the grid using the mouse (somewhat similar to the action of an image-warping software). The software then, immediately (in real time, without using menus etc.) computes and shows the following fields in the adjacent windows: (i) the displacement vector field, (ii) the displacement gradient tensor field, (iii) the rotation field, (iv) the strain field, (v) and the stress field. The software assumes plane-stress, linear elasticity, and uses static configuration data for material properties like nu and E. The software also shows the boundary tractions (forces) that would be required to maintain the displacement field that the user has specified.
</p>
<p>
Basically, the idea is that the beginning undergraduate student encountering the mechanics of materials for the first time, gets to see the importance of the rotation field (which is usually not emphasized in textbooks or courses), and thereby is able to directly appreciate the reason why an arbitrary displacement field does uniquely determines the corresponding stress fields but why the converse is not true---why an arbitrary stress/strain field cannot uniquely determine a corresponding displacement field. To illustrate this point (call it the compatibility issue if you wish) is the whole rationale behind this toy software.
</p>
<p>
Now, when it comes to visualizing the fields, I can always use arrows for showing the vector fields of displacements and forces. For strains and stresses, I can use Lame's ellipse (in 2D). In fact, since the strain and stress fields are symmetric, in <em>2D</em>, they each have only 3 components, which means that the symmetric tensor object as a whole can directly map onto an RGB (or HLS) color-space, and so, I can also show a single, full-color field plot for the strain (or stress) field.
</p>
<p>
Ok. So far, so good.
</p>
<p>
The problem is with the displacement gradient tensor (DG for short here). Since the displacement field is arbitrary, there is no symmetry to the DG tensor. Hence, even in 2D, there are 4 independent components to it---i.e. one component too many than what can be accomodated in the three-component color-space. So, a direct depiction of the tensor object taken as a whole is not possible, and something else has to be done. So, I thought of the following idea.
</p>
<p>
First, the notation. Assume that the DG tensor is being described thus:
</p>
<p>
DG11 DG12<br />
DG21 DG22
</p>
<p>
=
</p>
<p>
du/dx du/dy<br />
dv/dx dv/dy
</p>
<p>
where DGij are the components of the DG tensor, u and v are the x- and y-components of the displacement field, and the d's represent the <em>partial</em> differentation. (Also imagine as if the square brackets of the matrix notation are placed around the components listing above.)
</p>
<p>
Consider that DGij can be taken to represent a component of a vector that refers to the i-th face and j-th direction. Understanding this scheme is easier to do for the stress tensor. For the stress tensor, Sij is the component of the traction vector acting across the i-the face and pointing in the j-th direction. For instance, in fig. 2.3 here: <a href="http://en.wikipedia.org/wiki/Stress_(mechanics">http://en.wikipedia.org/wiki/Stress_(mechanics</a>), T^{e_1} is the vector acting across the face normal to the 1-axis.
</p>
<p>
Even if the DG tensor is not symmetric, the basic idea would still apply, wouldn't it?
</p>
<p>
Thus, each row in the DG tensor represents a vector: the first row is a vector acting on the face normal to the x-axis, and the second is another vector (which, for DG, is completely indpendent of the first) acting on the face normal to the y-axis. For 2D, subsitute "line" in place of "face."
</p>
<p>
If I now show these two vectors, they would completely describe the DG tensor. This representation would be somewhat similar to the "cross-bars" visualization commonly used in engineering software for the stress tensor, wherein the tensor field is shown using periodically cross-bars---very convenient if the grid is regular and uniform and has square elements.
</p>
<p>
Notice a salient difference, however. Since the DG tensor is <em>asymmetric</em>, the two vectors will not in general lie at right-angles to each other. The latter is the case only with the symmetric tensors such as the strain and stress tensors.
</p>
<p>
My question is this: Do you see any issues with this kind of visualization for the DG tensor? Is there any loss of generality by following this scheme of visualization? I mean, I read some literature on visualization of asymmetric tensors, and noticed that they sometimes worry about the eigenvalues being complex, not real. I think that complex eigenvalues would not be a consideration for the above kind of depiction of the DG tensor---the rotation part will be separately shown in a separate window anyway. But, still, I wanted to have the generality aspect cross-checked. Hence this post. Am I missing something? assuming too much? What are the other things, if any, that I need to consider? Also: Would you be "intuitively" comfortable with this scheme? Can you think of or suggest any alternatives?
</p>
<p>
Comments are welcome.
</p>
<p>
--Ajit
</p>
<p>
[E&OE]
</p>
</div></div></div>Mon, 25 Oct 2010 08:06:29 +0000Ajit R. Jadhav9159 at https://www.imechanica.orghttps://www.imechanica.org/node/9159#commentshttps://www.imechanica.org/crss/node/9159What is stress? Who has ever seen stress? Is stress a physical quantity?
https://www.imechanica.org/node/8872
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/3085">configurational forces</a></div><div class="field-item odd"><a href="/taxonomy/term/5550">displacement</a></div><div class="field-item even"><a href="/taxonomy/term/5551">invariant integral</a></div><div class="field-item odd"><a href="/taxonomy/term/5552">pseudo physical quantity</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong><span>What is stress? Who has ever seen stress? Is stress a physical quantity?</span></strong> </p>
<p class="MsoNormal">
<span>Professor Yi-Heng Chen, Xi’an Jiaotong University, 710049, P.R.China</span>
</p>
<p class="MsoNormal">
<span>e-mail: <a href="mailto:yhchen2@mail.xjtu.edu.cn">yhchen2@mail.xjtu.edu.cn</a></span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>In fact, this question has been bothering the present author for more than 15 years.</span>
</p>
<p class="MsoNormal">
<span>As well-known, all previous and present researchers who were/are majoring in solid mechanics in mechanical engineering or aerospace engineering always used the classical concept of stress as they used other physical quantities: displacement or strain etc.</span>
</p>
<p class="MsoNormal">
<span>However, there is no one in the world, who has ever seen stress by using any tool!</span>
</p>
<p class="MsoNormal">
<span>Moreover, no one really understood what the physical meaning of stress is!</span>
</p>
<p class="MsoNormal">
<span>Or instead, no one recognized the detailed fact whether stress is a real physical quantity?</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>In is well-known that for a given geometric coordinate system the displacements of a mass point of a solid are actually physical quantities without any doubt! This is because the displacements of a mass point could be directly seen or measured by using eyes, some advanced optic instruments, or even the electron microscope. </span>
</p>
<p class="MsoNormal">
<span>Thus their grads are also physical quantity at the same point where (or around a small region) the continuous first order differentials exist. </span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>The major difficulty is what the stress is?</span>
</p>
<p class="MsoNormal">
<span>Obviously, the concept of stress is quite different from the strain and displacements at least due to the following three viewpoints:</span>
</p>
<p class="MsoNormal">
<span>(1) First, the stress is invisible or un-measurable by using any existing instruments including all optic instruments and electrical instruments, whereas the displacements or strains, as well known, are measurable. For example, some new optic instruments could be used to clarify mere 0.01 micro meter of displacements and 1 micro strain as GOM or other corporations reported. Moreover, some advanced electric microscope has 0.18 nano meter resolving power!<span> </span></span>
</p>
<p class="MsoNormal">
<span>(2) Second, the classical concept of stress is based on the generalized Hook law in elasticity and then it is extended to treat some structural problems in plasticity such as the plastic deformation theory or plastic fluid theory. More recently, this concept is extended to micro mechanics, damage mechanics, even nano mechanics. However, strictly speaking, this concept is not yielded from experimental observations but from the man’s brain! Thus, it is not a real physical quantity or, in other words, it is a pseudo physical quantity, just an imagined physical quantity! This question is easy to be proved because no one in the literature who claimed that he has seen the stress or he has measured the stress! </span>
</p>
<p class="MsoNormal">
<span>(3) Third, in non-linear and inhomogeneous materials under some loadings, there are many crystals (metal) or the particles (ceramic) with the size scale of several micros. Generally speaking, the stress field in an inhomogeneous material is not uniform or even not unique although the strain field or displacement field in the same inhomogeneous material is unique (the strain field could uniquely deduced from the measured displacement field). Due to micro defects nucleation, growth, coalescence etc (a nonreversible thermodynamic process), the measured displacement field varies and the deduced strain field varies as well. But each strain/displacement field might lead to several different stress fields, depending on the different constitutive relations established by researchers (or from researches’ brain). In other words, each constitutive relation would only yield a special stress field. Many constitutive relations would yield many different stress fields but researchers actually don’t see or measure their stress fields. </span>
</p>
<p class="MsoNormal">
<span>The things become clear! </span>
</p>
<p class="MsoNormal">
<span>The stress concept is actually established from the man’s brain rather than established from real observations! </span>
</p>
<p class="MsoNormal">
<span>This is a major obstacle at 21 century in advanced solid mechanics for modern materials because the advanced technology promotes the instruments becoming smaller and smaller such as MEMS or NEMS, and then researchers majoring in solid mechanics face on some challenge to study small scale mechanics such as micro mechanics or nano mechanics. However, no one could tell us whether the stress (as a macroscopic and pseudo physical quantity) concept is still valid in micomechanics or nano mechanics with defects? </span>
</p>
<p class="MsoNormal">
<span>If so, he should tell us as why?</span>
</p>
<p class="MsoNormal">
<span>If not so, he should tell us what is the possible and alternative physical quantity instead of the stress?</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>This question is very clear as shown in the following figures (attached from websites).</span>
</p>
<p class="MsoNormal">
<span>Figure 1 shows a microscope photo. There are many solid particulars in the photo in the inhomogeneous material. The question is what is the stress on the surface of each particular? How large the stress is? Is this valuable to find the detailed distribution of the stress field as the displacement field or strain field? More importantly, from the micro scale viewpoint, whether the imagined stress field, if it exists, could be used to introduce some phenomenological parameters to evaluate the material failure?</span>
</p>
<p class="MsoNormal">
</p>
<p class="MsoNormal">
<span></span>
</p>
<p class="MsoNormal">
<span>Figure 1. The first example of microscope photo attached from website.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span></span>
</p>
<p class="MsoNormal">
<span>Figure 2. The second example of microscope photo attached from website.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Figure 2 also shows a detailed displacement field as well as the strain field but no one could obtain the detailed stress field although the detailed strain-displacement field could be measured.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span></span>
</p>
<p class="MsoNormal">
<span>Figure 3. The third example of microscope photo attached from website.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Figure 3 show another displacement field with some surface cracks. However, it is still unclear as what is the stress field although the strain-displacement field could be measured. </span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Moreover, some famous experts have claimed that the stress in their analyses at the nano scale is as large as 100GPa! This stress is over the material elastic modulus? This result privates an evidence that the stress concept is not realistic in nano mechanics with defects.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>Of course, it is not fair to overthrow the previous contributions based on the so-called stress analyses. Indeed, all classical strength theories including the fracture mechanics were established from the stress analyses which yielded a large amount of fortune and received significant attention from researches!</span>
</p>
<p class="MsoNormal">
<span>Also, the author does not wish to simply throw over this concept. This might upside down! </span>
</p>
<p class="MsoNormal">
<span>The goal of the present paper is to motivate the further discussions with other researchers who are interested in the topic and to introduce an alternative physical quantity to replace the stress concept, especially in inhomogeneous mechanics, micro mechanics and nano mechanics.</span>
</p>
<p class="MsoNormal">
<span>From the physical view point, the configurational force and the associated invariant integrals might be a useful choice instead of working in stress analyses. This is because these concepts are induced directly from Eshelby’s force with defects, which is based on energy balance viewpoint. In fact, the author has some initial attempts at this research direction [1-17] including one book [11] summarizing the potentials applications of the projected conservation laws of Jk-vector and the M-integral and the L-integral.</span>
</p>
<p class="MsoNormal">
<span>Other advances in this topic such as the Fatigue Damage Driving Force (FDDF) for a cloud of micro-defects will be reported in the author’s subsequent papers.</span>
</p>
<p><span> </span> </p>
<p class="MsoNormal">
<span>[1] Chen Yi-Heng., On the contribution of discontinuities in a near-tip stress field to the J-integral, <em>International Journal of Engineering Science</em>, Vol. 34, 819-829(1996).</span>
</p>
<p class="MsoNormal">
<span>[2] Han J. J, and Chen Yi-Heng., On the contribution of a micro-hole in the near-tip stress field to the J-integral, <em>International Journal of Fracture</em>, Vol. 85, 169-183(1997).</span>
</p>
<p class="MsoNormal">
<span>[3] Zhao L. G, and Chen Yi-Heng., On the contribution of subinterface microcracks near the tip of an interface crack to the J-integral in bimaterial solids, <em>International Journal of Engineering Science</em>, Vol. 35, 387-407(1997).</span>
</p>
<p class="MsoNormal">
<span>[4] Chen Yi-Heng., and Zuo Hong, Investigation of macrocrack-microcrack interaction problems in anisotropic elastic solids-Part I. General solution to the problem and application of the J-integral, <em>International Journal of Fracture, </em>Vol. 91, 61-82(1998).</span>
</p>
<p class="MsoNormal">
<span>[5] Chen Yi-Heng., and Han J. J.<span> </span>Macrocrack-microcrack interaction in piezoelectric materials, Part I. Basic formulations and J-analysis, <em>ASME Journal of Applied Mechanics</em>, Vol. 66, No. 2, 514-521 (1999).</span>
</p>
<p><span>[6] Chen Yi-Heng</span><span>.,<span> and Han J.J. Macrocrack-microcrack interaction in piezoelectric materials, Part II. Numerical results and Discussions, <em>ASME Journal of Applied Mechanics</em>, Vol. 66, No. 2, 522-527(1999).</span></span><span>[7] Tian W.Y. and Chen Yi-Heng., A semi-infinite interface crack interacting with subinterface matrix cracks in dissimilar anisotropic materials, Part I, Fundamental formulations and the J-integral analysis, <em>International Journal of Solids and Structures,</em> Vol. 37, 7717-7730 (2000). </span><span>[8] Chen Yi-Heng., and Tian W.Y. A semi-infinite interface crack interacting with subinterface matrix cracks in dissimilar anisotropic materials, Part II, Numerical results and discussions, <em>International Journal of Solids and Structures</em>, Vol.37, 7731-7742 (2000). </span><span>[9] Chen Yi-Heng</span><span>., <span>M-integral analysis for two-dimensional solids with strongly interacting cracks, Part I. In an infinite brittle sold, <em>International Journal of Solids and Structures</em>., Vol. 38/18, 3193-3212 (2001). </span></span></p>
<p class="MsoNormal">
<span>[10] Chen Yi-Heng., M-integral analysis for two-dimensional solids with strongly interacting cracks, Part II. In the brittle phase of an infinite metal/ceramic bimaterial, <span> </span><em>International Journal of Solids and Structures.</em>, Vol. 38/18, 3213-3232 (2001).</span>
</p>
<p><span>[11] </span><em><span>Books: Advances in conservation laws and energy release rates, Kluwer Academic Publishers,</span></em><em><span> </span></em><em><span>The Netherlands (ISBN 1402005008)</span></em><em><span>, 2002</span></em><em><span>.</span></em><span>[</span><span>12</span><span>] Chen, Y.H., and Lu, T.J., (2003) </span><span>Recent developments and applications in invariant integrals, </span><em><span>ASME Applied Mechanics Reviews, </span></em><span>Vol. 56, 515-552<em>.</em></span> </p>
<p class="MsoNormal">
<span>[13] Li Q, Chen YH. (2008) Surface effect and size dependence on the energy release due to a<span> </span>nanosized hole expansion in plane elastic materials, <em>ASME Journal Applied Mechanics, </em>Vol. 75, Novermber.</span>
</p>
<p class="MsoNormal">
<span>[14] Hu Y.F., and Chen Yi-Heng, (2009), M-integral description for a strip with two holes before and after coalescence, <em>Acta Mechanica</em>, Vol. 204(1), 109-.</span>
</p>
<p class="MsoNormal">
<span>[15] Hu Y.F., and Chen Yi-Heng, (2009), M-integral description for a strip with two cracks before and after coalescence, <em>ASME Journal Applied Mechaics,</em> Vol.76, November, 061017-1-12.</span>
</p>
<p class="MsoNormal">
<span>[16] Hui T., and Chen Yi-Heng, (2010), The M-integral analysis for a nano-inclusion in plane elastic materials under uni-axial or bi-axial loadings, ASME Journal of Applied Mechanics, Vol. 77.</span><strong><span> 021019-1-9.</span></strong>
</p>
<p class="MsoNormal">
<span>[17] Hui T., and Chen Yi-Heng, (2010), The two state M-integral for a nano inclusion in plane elastic materials, ASME Journal of Applied Mechanics, Vol. 77.</span><strong><span> 024505-1-5.</span></strong>
</p>
<p><span> </span><span> </span><span> </span><span> </span><span> </span><span> </span> </p>
<p class="MsoNormal">
</p>
</div></div></div><div class="field field-name-upload field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><table class="sticky-enabled">
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<tbody>
<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://www.imechanica.org/files/Waht_is_stress_figures.pdf" type="application/pdf; length=98392" title="Waht_is_stress_figures.pdf">Waht_is_stress_figures.pdf</a></span></td><td>96.09 KB</td> </tr>
<tr class="even"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://www.imechanica.org/files/What%20is%20stress_second__1.pdf" type="application/pdf; length=405100" title="What is stress_second_.pdf">What is stress_second_.pdf</a></span></td><td>395.61 KB</td> </tr>
</tbody>
</table>
</div></div></div>Fri, 10 Sep 2010 11:44:38 +0000Yi-Heng Chen8872 at https://www.imechanica.orghttps://www.imechanica.org/node/8872#commentshttps://www.imechanica.org/crss/node/8872Mohr's Circle---When Was the Last Time You Used It in Your Professional Engineering Work?
https://www.imechanica.org/node/8341
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/846">FEM</a></div><div class="field-item odd"><a href="/taxonomy/term/5261">Mohr's Circle</a></div><div class="field-item even"><a href="/taxonomy/term/5262">Tensor Visualization</a></div><div class="field-item odd"><a href="/taxonomy/term/5263">Visualization</a></div><div class="field-item even"><a href="/taxonomy/term/5264">Post-Processor</a></div><div class="field-item odd"><a href="/taxonomy/term/5265">Professional Engineering</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
As a consultant in computational mechanics, I currently help write some FEM-related code, and while doing this job, an episode from a recent past came to my mind. Let me go right on to the technical issue, keeping aside the (not so good) particulars of that episode. (In case you are curious: it happened outside of my current job, during a job interview.)</p>
<p>If you are a design engineer, FE analyst, researcher, or any professional dealing with stress analysis in your work, I seek answers to a couple of questions from you:</p>
<p>
<strong>Question 1:<br /></strong>
</p>
<p>
When was the last time you used Mohr's circle of strain/stress in your professional work? Was it a week ago? a month? a year? five years? ten years? longer? In what kind of an application or research context?</p>
<p>Please note, I do not mean to ask whether you directly or indirectly used the coordinate transformation equations---the basis for constructing Mohr's circle---to find the principal quantities. The question is: whether you spoke of Mohr's circle itself---and not of the transformation equations---in a direct manner, in a professional activity of yours (apart from teaching Mohr's circles). In other words, whether, in the late 20th and early 21st century, there was any occasion to plot the circle (by hand or using a software) in the practice of engineering, did it directly illuminate something/anything in your work.</p>
<p>In case you are curious, my own answer to this question is: No, never. I would like to know yours.
</p>
<p>
<br /><strong>Question 2:<br /></strong>
</p>
<p>
The second question just pursues one of the lines indicated in the first.</p>
<p>In a modern FEM postprocessor, visualizations of stress/strain patterns are provided, usually via field plots and contour lines.</p>
<p>For instance, they show field plots of individual stress tensor components, one at a time.</p>
<p>Recently, there also have been some attempts to try to directly show tensor quantities in full directly, via systematically arranged ellipsoids of appropriate sizes and orientations. The view you get is in a way analogous to the arrow plots for visualizing vector fields in those CFD and EM software packages. Other techniques for tensor visualization are not, IMHO, as successful as the ellipsoids. Mostly, all such techniques still are at the research stage and have not yet made to the commercial offerings.</p>
<p>Some convenience can be had by showing some scalar measures of the tensors such as the von Mises measure, in the usual field/contour plots.</p>
<p>The questions here are:</p>
<p><strong>(2.a)</strong> Would you like to see an ellipsoids kind of visualization in your engineering FEM software? If yes, would this feature be a "killer" one? Would you consider it to be a decisive kind of advantage?
</p>
<p>
<strong>(2.b)</strong> Would a simpler, colored cross-bars kind of visualization do? That is, two arrows aligned with the principal directions. The colors and the lengths of the arrows help ascertain the strength of the principal quantities.
</p>
<p>
<strong>(2.c)</strong> Would you like to see Mohr's circles being drawn for visualization or any other purposes in such a context? If yes, please indicate the specific way in which it would help you.</p>
<p>My own answers to question 2 are: (a) Ellipsoids would be "nice to have" but not "killer." I wouldn't be very insistent on them. Having them is not a decisive adavantage. (b) For 2D, this feature should be provided. (c) Not at all.</p>
<p>Please note, the questions are directed rather at experienced professionals, even engineering managers, but not so much at students as such. The reason is that the ability to buy is an important consideration here, apart from the willingness. Of course, experienced or advanced PhD students and post-docs may also feel free to share their experiences, thoughts and expectations.</p>
<p>Thanks in advance for your comments.</p>
<p>PS: Also posted in my other, personal blog here [<a href="http://ajitjadhav.wordpress.com/2010/06/03/mohrs-circle-when-was-the-last-time-you-used-it-in-your-professional-engineering-work/" target="_blank">^</a>]
</p>
<p>
[E&OE]
</p>
</div></div></div>Thu, 03 Jun 2010 17:01:13 +0000Ajit R. Jadhav8341 at https://www.imechanica.orghttps://www.imechanica.org/node/8341#commentshttps://www.imechanica.org/crss/node/8341Writing User Subroutines with ABAQUS
https://www.imechanica.org/node/7576
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/169">Plasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/289">ABAQUS</a></div><div class="field-item even"><a href="/taxonomy/term/347">elasticity</a></div><div class="field-item odd"><a href="/taxonomy/term/459">hyperelasticity</a></div><div class="field-item even"><a href="/taxonomy/term/1236">User subroutines in ABAQUS</a></div><div class="field-item odd"><a href="/taxonomy/term/1587">UMAT</a></div><div class="field-item even"><a href="/taxonomy/term/1588">VUMAT</a></div><div class="field-item odd"><a href="/taxonomy/term/4819">Neo-Hookean</a></div><div class="field-item even"><a href="/taxonomy/term/4820">Kinematic Hardening</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p align="left">
Dear All,
</p>
<p align="left">
I think that many students are looking for some tutorials about writing a UMAT in ABAQUS.
</p>
<p align="left">
You can find a comprehensive tutorial for elastic problems.
</p>
<p align="left">
This file contains:
</p>
<p align="left">
• Motivation
</p>
<p align="left">
• Steps Required in Writing a UMAT or VUMAT
</p>
<p align="left">
• UMAT Interface
</p>
<p align="left">
• <strong>Examples</strong>
</p>
<p>
Example 1: UMAT for Isotropic Isothermal Elasticity
</p>
<p>
Example 2: UMAT for Non-Isothermal Elasticity
</p>
<p>
Example 3: UMAT for Neo-Hookean Hyperelasticity
</p>
<p>
Example 4: UMAT for Kinematic Hardening Plasticity
</p>
<p>
Example 5: UMAT for Isotropic Hardening Plasticity
</p>
<p align="left">
• VUMAT Interface
</p>
<p>
•<strong> Examples</strong>
</p>
<p>
Example 6: VUMAT for Kinematic Hardening<br />
Example 7: VUMAT for Isotropic Hardening
</p>
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</tbody>
</table>
</div></div></div>Thu, 11 Feb 2010 09:26:29 +0000Ahmad Rafsanjani7576 at https://www.imechanica.orghttps://www.imechanica.org/node/7576#commentshttps://www.imechanica.org/crss/node/7576Strain Effects on the Optical Properties of Metal Nanoparticles
https://www.imechanica.org/node/7566
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/132">strain</a></div><div class="field-item odd"><a href="/taxonomy/term/4813">plasmon resonance</a></div><div class="field-item even"><a href="/taxonomy/term/4814">surfac enhanced raman scattering</a></div><div class="field-item odd"><a href="/taxonomy/term/4815">gold nanoparticles</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
We have recently been studying the effects of strain on the optical properties of metal nanoparticles, which have become of significant interest to the materials, physics, biology and chemistry communities due to the fact that they exhibit unique optical properties, specifically surface plasmon resonance and surface enhanced raman scattering, which are being used primarily for optical sensors at the single molecule level, but for many other applications, including photothermal cancer treatment and optical imaging. While strain engineering of bandstructure in semiconductors is a well-established and important area, similar types of studies on metals have not been performed despite the immense potential of metal nanoparticles. We have performed such fundamental studies of strain effects on gold nanospheres, with the results having been accepted for publication in Journal of the Mechanics and Physics of Solids (<a href="http://dx.doi.org/10.1016/j.jmps.2009.12.001">http://dx.doi.org/10.1016/j.jmps.2009.12.001</a>). The basic finding is that both the plasmon resonance wavelength, as well as the magnitudes of the plasmon resonance and surface enhanced raman scattering, can all be tuned and enhanced using mechanical strain.
</p>
</div></div></div><div class="field field-name-upload field-type-file field-label-hidden"><div class="field-items"><div class="field-item even"><table class="sticky-enabled">
<thead><tr><th>Attachment</th><th>Size</th> </tr></thead>
<tbody>
<tr class="odd"><td><span class="file"><img class="file-icon" alt="PDF icon" title="application/pdf" src="/modules/file/icons/application-pdf.png" /> <a href="https://www.imechanica.org/files/qianJMPS2010.pdf" type="application/pdf; length=1291161" title="qianJMPS2010.pdf">qianJMPS2010.pdf</a></span></td><td>1.23 MB</td> </tr>
</tbody>
</table>
</div></div></div>Wed, 10 Feb 2010 13:24:41 +0000Harold S. Park7566 at https://www.imechanica.orghttps://www.imechanica.org/node/7566#commentshttps://www.imechanica.org/crss/node/7566Strain Energy Derivation
https://www.imechanica.org/node/5228
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Dear Sirs,
</p>
<p>
I'm trying to derive an strain energy in a small volume 2a x 2b x 2t.
</p>
<p>
I can derive "The rest terms are ommited" by myself .
</p>
<p>
By the way, I can't derive the DT2 term in the 2nd row of the figure.
</p>
<p>
Please help with reference or tips.
</p>
<p>
Thanks,
</p>
<p>
Jin Kim
</p>
</div></div></div><div class="field field-name-taxonomyextra field-type-taxonomy-term-reference field-label-above"><div class="field-label">Taxonomy upgrade extras: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/277">energy</a></div><div class="field-item odd"><a href="/taxonomy/term/2903">thermoelasticity</a></div></div></div>Wed, 08 Apr 2009 12:51:53 +0000jintting5228 at https://www.imechanica.orghttps://www.imechanica.org/node/5228#commentshttps://www.imechanica.org/crss/node/5228Producing Stress Analysis from Strain data
https://www.imechanica.org/node/4804
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>I am doing research on the stresses that are produced when a retainer (a thermoplastic sheet) is placed on the teeth. We've designed the project so that we take an initial scan of the sheet and a final "shifted" scan of the sheet. We'd like to compare, find the strain, and calculate the stress neccessary to produce this strain. </p>
<p>
I was hoping to use FEA for this...is it possible? I have access to Abaqus and Ansys, and where can i find the commands that allow me to do this.
</p>
</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/109">Ask iMechanica</a></div></div></div><div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/370">finite element methods</a></div><div class="field-item odd"><a href="/taxonomy/term/935">FEA</a></div></div></div>Sun, 15 Feb 2009 23:36:13 +0000G Rezk4804 at https://www.imechanica.orghttps://www.imechanica.org/node/4804#commentshttps://www.imechanica.org/crss/node/4804Journal Club Forum for April 1st: Strain measurement in soft tissues
https://www.imechanica.org/node/2878
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
Many musculoskeletal soft tissues, such as tendons, ligaments, meniscus and cartilage are inhomogeneous. Hence, during mechanical loading it is likely that a nonuniform strain pattern occurs within the tissue. These nonuniform strain patterns may assist in successful load transmission and minimize rupture of the tissue during physiological loading. Determination of local material properties will likely be important for successful function and design of tissue engineered replacements. In the late 1980’s uniaxial tensile tests were conducted using a video camera in conjunction with surface markers to document local strain distributions on the surface of ligaments. Photoelasticity has also been used to document local strain patterns.
</p>
<p>
As Magnetic Resonance Imaging (MRI) technology has improved in the recent decade, its utility in determining local strains, noninvasively, within soft tissues has evolved. MRI will likely influence the biomechanics community with the capability to assess <em>in vivo</em>, 3-dimensional tissue strains.
</p>
<p>
One such MRI-based technique is DENSE-FSE or displacement encoding with stimulated echoes and a fast spin echo readout. The goal of this work was to acquire images with high spatial resolution in reasonable imaging times. This approach requires that the sample first be loaded to reach steady-state to avoid motion artifacts. Images are collected while the sample is cyclically loaded requiring a repeatable loading cycle to allow for sufficient time to collect the MR images. Hence, for a single, fast displacement test that is not repeatable, this approach may not work. Reducing the required resolution would shorten the imaging time.
</p>
<p>
Neu, C. P. and J. H. Walton (2008). "Displacement encoding for the measurement of artilage deformation." Magn Reson Med 59(1): 149-55.
</p>
<p>
Gilchrist et al., presents a texture correlation algorithm using first-order displacement mapping terms with MR images. This approach eliminates the need for "tags" or "markers". However, the technique is heavily dependent on image contrast/texture and thus is sensitive to noise.
</p>
<p>
Gilchrist, C. L., J. Q. Xia, et al. (2004). "High-resolution determination of soft tissue deformations using MRI and first-order texture correlation." IEEE Trans Med Imaging 23, 546-53.
</p>
<p>
Finite element warping has also been used to track local displacements on MR images. While this pproach eliminates the need for markers in the tissue, or MR tags, it does assume the discretized template to be a hyperelastic material in determining fiber stretch. Cine-MRI and deformable image registration uses differences in image intensities between a reference position image and a loaded position image to enerate a body force that deforms a FE representation of the template so that it matches the target. This requires a priori assumptions about material properties and constitutive behavior of the material and only works for static loading.
</p>
<p>
Phatak, N. S., Q. Sun, et al. (2007). "Noninvasive determination of ligament strain with deformable image registration." Ann Biomed Eng 35, 1175-87.
</p>
<p>
Major technical advances in MRI have allowed the above approaches to be developed for non-invasive measurement of soft tissue deformations. With continued advances in MRI technology that may increase resolution and shorten loading times, these approaches and others may enable dynamic three-dimensional collection of physiologically loaded soft tissues both <em>in vivo and in vitro</em>.
</p>
</div></div></div><div class="field field-name-taxonomy-forums field-type-taxonomy-term-reference field-label-above"><div class="field-label">Forums: </div><div class="field-items"><div class="field-item even"><a href="/forum/417">Journal Club Forum</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-above"><div class="field-label">Free Tags: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/132">strain</a></div><div class="field-item odd"><a href="/taxonomy/term/704">measurement</a></div><div class="field-item even"><a href="/taxonomy/term/821">Journal Club Forum</a></div></div></div>Fri, 14 Mar 2008 19:29:31 +0000Tammy Haut Donahue2878 at https://www.imechanica.orghttps://www.imechanica.org/node/2878#commentshttps://www.imechanica.org/crss/node/2878Stress or strain: which one is more fundamental?
https://www.imechanica.org/node/1001
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/128">education</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/157">students</a></div><div class="field-item odd"><a href="/taxonomy/term/703">fundamentality</a></div><div class="field-item even"><a href="/taxonomy/term/704">measurement</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
In between stress and strain, which one is the more fundamental physical quantity? Or is it the case that each is defined independent of the other and so nothing can be said about their order? Is this the case?
</p>
<p>
To begin with these questions, consider the fact that first we have to apply a force to an object and it is only then that the object is observed to have been deformed or strained. Accordingly, one may say that forces produce strains, and therefore, it <em>seems</em> that stress has to be more fundamental. If so, how come stress cannot be measured directly? This is the paradox I would like to address here.
</p>
<p>
Of course, to begin with, my position is that you can never directly measure stress.
</p>
<p>
I have read somewhere an argument (and forgot exactly where!) that even in photoelasticity what you really measure is strain. The argument, essentially, is this: Birefringence arises because the molecular chains in the photoelastic polymer get stretched. (In case of crystals, "stress-induced" birefringence arises if the deformation is inherently anisotropic.) So, what is important here is the relative positions of atoms in the chain--not whether the atoms were carrying any load or not.
</p>
<p>
The above argument, of course, is sound. Yet, it does not quite settle the issue by itself. This issue is <em>somewhat</em> (but not fully) similar to the hen and eggs situation. To settle the issue, we have to go beyond photoelasticity mechanisms and examine it from the viewpoint of the theoretical structure of the mechanics of solids/fluids.
</p>
<p>
What is clearly undisputed is the primacy of displacements. (Definition: Displacement is the total movement of a point with respect to a fixed coordinate frame.) Displacements can be measured directly and do not need other physical quantities to be measured. Hence, they are primary.
</p>
<p>
Further, what can also be thought of directly is deformation. (Definition: Deformation is the relative movement of a point with respect to another point in the body.)
</p>
<p>
Now, displacement can be related to deformation via the relative deformation tensor, a tensor of second order. (As usual, in the simplest analysis, one assumes infinitesimally small deformations.) Now, if you split the relative deformation tensor into its symmetric and anti-symmetric parts, and ignore the anti-symmetric part (representing rotations), what you get is the strain tensor. This is the primary way strain is defined.
</p>
<p>
For homogeneous linear elastic isotropic materials, strains and stresses are directly related. So, we should expect to find a similar theoretical structure for the concept of stress too. In a way, this does turn out to be the case--but not quite fully. Let's see how.
</p>
<p>
Stress also is a second rank tensor and it also is symmetrical--it drops out the torques/couples part. (This is in analogy with the dropping of the rotation part while defining strain.) This fact about the stress tensor is usually taught in the introductory courses as the result of having moments balance out over the infinitesimal element. But the real reason is that this way we can maintain the similarity of the theoretical structure. The fact is, one could keep moment-balance (as required for static equilibrium) and yet choose <em>not</em> to drop out the torques-related part. However, a discussion on the so-called couple stresses would be a digression here.
</p>
<p>
In short, since both are second order symmetric tensors, stress and strain tensors do seem completely similar.
</p>
<p>
But are they?
</p>
<p>
There is that "displacement<->the gradient tensor<->deformation" relation on the strain side. What is its parallel on the stress side?
</p>
<p>
Here, even in the simplest case of the linear elastic (etc.) solids, it is difficult to believe that a conceptual parallel could be derived independently. One could, of course, argue from an abstract viewpoint that such derivation is possible "mathematically". But remember: deformation is by definition a point phenomenon, whereas force is by definition related to the momentum of an object. For <em>field</em> quantities, force necessarily arises only in the context of a geometric element of nonzero side--e.g. area (as in stress components), or line (as in surface tension). You always need a geometric entity like area or line element (even if it is infinitesimally small) before quantities like stress or flux can at all be defined.
</p>
<p>
Deformation, in contrast, can be defined <em>at</em> a point. We don't have to refer to a geometric element in order to define what this concept means.
</p>
<p>
It is this particular difference which makes it impossible to have a direct analog of deformation on the force/stress side.
</p>
<p>
Consequently, any quantity on the force/stress side must always be defined in reference to some or the other <em>external</em> assumptions as to how the quantity ought to vary across the relevant geometrical element. The simplest assumption of this nature is to say that stress must conceptually remain analogous to strain.
</p>
<p>
The rest then follows.
</p>
<p>
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</p>
<p>
Sometimes, it is said that it is very <em>obvious</em> that strain must be more fundamental because in the usual stress-strain diagrams, it is strain that is taken on the x-axis, i.e. as the independent variable.
</p>
<p>
However, note that this "argument" is very superfluous. To plot stress-strain in the usual manner does not need all the above kind of thought. One need only observe that the constitutive law of metals is nonlinear and metal specimens experience necking so that there is a drop in the graph of engineering stress once the point of ultimate tensile strength is reached. In such a situation, taking strain on the x-axis avoids the possibility of having a multi-valued "function." Thus, the choice to take strains on the x-axis is more of a simpler convenience that happens to be in accord with the more fundamental reasoning discussed above.
</p>
<p>
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</p>
<p>
As an aside: the difference about a quantity defined <em>at</em> at point by itself (e.g. displacement of a point) and a quantity that requires an infinitesimal volume, area, or line for its definition (e.g. stress, strain, electric field vector) is a fundamental one. It marks the conceptual difference between particles and fields. It therefore plays a crucial role in many other matters such as flux-conservative laws and wave-particle duality.)
</p>
<p>
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</p>
<p>
Acknowledgment: It was the discussion on Henry Tan's blog about whether stresses can ever be measured directly or not that provided the spark to write this post.
</p>
<p>
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</p>
<p>
If there are deeply thought out and interesting possibilities running counter to the arguments presented above, I would like to know about them.
</p>
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</div></div></div>Thu, 08 Mar 2007 10:02:32 +0000Ajit R. Jadhav1001 at https://www.imechanica.orghttps://www.imechanica.org/node/1001#commentshttps://www.imechanica.org/crss/node/1001Void-induced strain localization at interfaces
https://www.imechanica.org/node/463
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/18">micromechanics</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/169">Plasticity</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><span>We published <a href="http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APPLAB000089000014141909000001&idtype=cvips&gifs=yes.">this paper in APL</a> on a study of the deformation near interfaces. It provides insight in the strain localization at the interface and its influence on the deformation in bulk metals. </span>
</p><p><strong>Abstract</strong> An optical full-field strain mapping technique has been used to provide direct evidence for the existence of a highly localized strain at the interface of stacked Nb/Nb bilayers during the compression tests loaded normal to the interface. No such strain localization is found in the bulk Nb away from the interface. The strain localization at the interfaces is due to a high void fraction resulting from the rough surfaces of Nb in contact, which prevents the extension of deformation bands in bulk Nb crossing the interface, while no distinguished feature from the stress-strain curve is detected.</p>
</div></div></div>Mon, 20 Nov 2006 15:02:52 +0000Liu463 at https://www.imechanica.orghttps://www.imechanica.org/node/463#commentshttps://www.imechanica.org/crss/node/463Dynamics of terraces on a silicon surface due to the combined action of strain and electric current
https://www.imechanica.org/node/392
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/85">suo group research</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/148">Wei Hong</a></div><div class="field-item odd"><a href="/taxonomy/term/149">Zhenyu Zhang</a></div><div class="field-item even"><a href="/taxonomy/term/163">electromigration</a></div><div class="field-item odd"><a href="/taxonomy/term/312">silicon</a></div><div class="field-item even"><a href="/taxonomy/term/313">step</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>A (001) surface of silicon consists of terraces of two variants, which have an identical atomic structure, except for a 90° rotation. We formulate a model to evolve the terraces under the combined action of electric current and applied strain. The electric current motivates adatoms to diffuse by a wind force, while the applied strain motivates adatoms to diffuse by changing the concentration of adatoms in equilibrium with each step. To promote one variant of terraces over the other, the wind force acts on the anisotropy in diffusivity, and the applied strain acts on the anisotropy in surface stress. Our model reproduces experimental observations of stationary states, in which the relative width of the two variants becomes independent of time. Our model also predicts a new instability, in which a small change in experimental variables (e.g., the applied strain and the electric current) may cause a large change in the relative width of the two variants.</p>
<p> </p>
<p> Preprint available online.</p>
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</div></div></div>Sun, 05 Nov 2006 20:41:03 +0000Wei Hong392 at https://www.imechanica.orghttps://www.imechanica.org/node/392#commentshttps://www.imechanica.org/crss/node/392Persistent step-flow growth of strained films on vicinal substrates
https://www.imechanica.org/node/313
<div class="field field-name-taxonomy-vocabulary-6 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/76">research</a></div></div></div><div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/85">suo group research</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/147">step flow</a></div><div class="field-item odd"><a href="/taxonomy/term/148">Wei Hong</a></div><div class="field-item even"><a href="/taxonomy/term/149">Zhenyu Zhang</a></div><div class="field-item odd"><a href="/taxonomy/term/274">vicinal</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>We propose a model of persistent step flow, emphasizing dominant kinetic processes and strain effects. Within this model, we construct a morphological phase diagram, delineating a regime of step flow from regimes of step bunching and island formation. In particular, we predict the existence of concurrent step bunching and island formation, a new growth mode that competes with step flow for phase space, and show that the deposition flux and temperature must be chosen within a window in order to achieve persistent step flow. The model rationalizes the diverse growth modes observed in pulsed laser deposition of SrRuO3 on SrTiO3 </p>
<p> <a href="http://link.aps.org/abstract/PRL/v95/e095501" target="_blank"><em>Physical Review Letters</em> <strong>95</strong>, 095501 (2005) </a></p>
<p>Preprint available here </p>
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</div></div></div>Tue, 17 Oct 2006 14:09:34 +0000Wei Hong313 at https://www.imechanica.orghttps://www.imechanica.org/node/313#commentshttps://www.imechanica.org/crss/node/313Augustin Louis Cauchy (August 21, 1789 – May 23, 1857)
https://www.imechanica.org/node/121
<div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p><strong>Augustin Louis Cauchy</strong> ( 21 August 1789 - 23 May 1857) was a French mathematician and mechanician. In mechanics, he in 1822 formalized the stress concept in the context of three-dimensional thoery, showed its properties as consisting of a 3 by 3 symmetric arrays of numbers that transform as a tensor, derived the equations of motion for a continuum in terms of the components of stress, and gave the specific development of the theory of linear elasticity for isotropic solids. As part of his work, Cauchy also introduced the equations which express the six components of strain, three extensinal and three shear, in terms of derivatives of displacements for the case when all those derivatives are much smaller than unity; similar expressions had been given earlier by Euler in expressing rates of straining in terms of the derivatives of the velocity field in a fluid. (cited from <em>Mechanics of Solids</em> by J.R. Rice) <a href="http://en.wikipedia.org/wiki/Cauchy" class="links">Read more...</a> </p>
</div></div></div><div class="field field-name-taxonomyextra field-type-taxonomy-term-reference field-label-above"><div class="field-label">Taxonomy upgrade extras: </div><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/75">mechanician</a></div><div class="field-item odd"><a href="/taxonomy/term/130">Cauchy</a></div><div class="field-item even"><a href="/taxonomy/term/131">stress</a></div><div class="field-item odd"><a href="/taxonomy/term/132">strain</a></div><div class="field-item even"><a href="/taxonomy/term/133">equation of motion</a></div></div></div>Sun, 10 Sep 2006 16:51:17 +0000Zhen Zhang121 at https://www.imechanica.orghttps://www.imechanica.org/node/121#commentshttps://www.imechanica.org/crss/node/121