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Stress or strain: which one is more fundamental?

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?

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 seems 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.

Of course, to begin with, my position is that you can never directly measure stress.

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.

The above argument, of course, is sound. Yet, it does not quite settle the issue by itself. This issue is somewhat (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.

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.

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.)

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.

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.

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 not to drop out the torques-related part. However, a discussion on the so-called couple stresses would be a digression here.

In short, since both are second order symmetric tensors, stress and strain tensors do seem completely similar.

But are they?

There is that "displacement<->the gradient tensor<->deformation" relation on the strain side. What is its parallel on the stress side?

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 field 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.

Deformation, in contrast, can be defined at a point. We don't have to refer to a geometric element in order to define what this concept means.

It is this particular difference which makes it impossible to have a direct analog of deformation on the force/stress side.

Consequently, any quantity on the force/stress side must always be defined in reference to some or the other external 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.

The rest then follows.


Sometimes, it is said that it is very obvious 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.

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.


As an aside: the difference about a quantity defined at 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.)


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.


If there are deeply thought out and interesting possibilities running counter to the arguments presented above, I would like to know about them.



Ajit, though your question is somewhat philosophical, I know what you're getting at and often think along the same lines as what you have written.

I agree that force and stress can never be directly measured. They are abstract quantities posited within theories that seek to explain motions and deformations. Even the most primitive measurement of force - Atwood's machine - requires us to accept a theory that the heavier body will fall and the lighter one will rise. All we can observe directly are the motions. Our theories don't define the heavier body as the one that falls, they state that the heavier body does fall (and exhibits many other behaviors that have nothing to do with Atwood's machine).

Personally, I have tried to define what I mean by a "fundamental quantity". I define it as a quantity that can be observed without recourse to any physical theory. For example, force is not fundamental because we have to transduce it into displacement or motion in order to observe it, and the transducer's operation can only be understood if we accept a certain theory about its operation. One can carry this to extremes and say that it is only a theory that when we perceive something to have moved, such motion is discernable to any other observer. On those grounds, even motion is not fundamental.

I also reflected on why I even care about the issue. The best answer I can give is that understanding the theoretical status of the different quantities makes us better able to apply the theory correctly. I go into more detail on my own page here.

Dear Grant, 

By "fundamental" I mean that concept (or principle, idea, etc.) on which all, or the greatest number of other concepts (or principles etc.) depend, in a the theoretical structure of a given field of knowledge.

The word "fundamental" does not always mean the same as: the primitive, the primary, the irreducible, that which is directly given in perception, the ostensively defined, etc. All these ideas are closely related to each other, sure, but they are not exactly synonymous.

A concept could be a pretty high-level abstraction, and still serve as a fundamental in some area of knowledge or study.

Thus, a concept might be fundamental in one area of knowledge but not so in another. For instance, strain and stress are fundamental to mechanics of solids and fluids. But they are not the most fundamental concepts of physics. Neither is a primitive, primary or an ostensively defined concept. I would suppose that physics is a science sufficiently basic that its fundamental concepts would need to be ostensively defined (in contrast to the mechanics of materials).

Despite the fact that force has a higher level (mathemtical) definition (given by Newton: F = ma), "force" still could argubably be taken to a notion which is basic enough to be a first-level abstraction. In other words, at least in a certain basic and qualitative sense, force could be taken to be a primary concept that is not dependent for its meaning on other concepts or theories. One could at least argue that way.  (There was someone at Cambridge in 1980s or so who wrote a PhD thesis on the meaning of the terms "Forces and Fields" throughout history, and later on published a more readable version as a book for the layman. The book made for a very absorbing and interesting reading. Somehow, despite searching Amazon, I can't be sure which one it is--I read it in Univ. of Alabama library in early 1990s. There, the author discusses arguments for and against the idea that force is a primitive concept.)

Thanks for your comments! 

[BTW, also see below a clarification on the mistakes/confusions I made in my original post.]

Gopinath Venkatesan's picture

Ajit and Grant,

Allow my student inputs here.

<< What is its parallel on the stress side?

Like Displacement -> Deformation on the strain side, we have Force -> Pressure (stress) on the stress side. And if we consider instantaneous (true) strain measure, we can eliminate the multi-valued functions even if we plot strain in Y axis (I guess, you can shed light into this). Also Force (and hence stress) we measure is with respect to some reference value and it is not exact. But strain measure is exact.

But I go with the Strain (or deformation). Because we could explain the linear behavior of materials with Stress (S-N curves), and both linear and non linear with Strain life calculations.

When we consider Force as an action that causes deformation, we are carving out a system to cover only the specific entity we wanted to analyze, and think that force is primary. But energy is needed and pre-exist before we could actually think of force.

One can store energy (in a cantilever beam) by pushing it down, as energy is transferred from us, and when we see it, we consider only that force as the action, and deformation as the result. When we release it, we have the deformation as the action, force (if any thing restraining or connected to) is the result. But according to Mathematicians, we have energy defined in terms of stress and strain, and both may be independent variables. And zero is not null.

One question I would like to ask others is that, I heard about stress singularities, but did not come across strain singularities -- which means we take strain as a fundamental measure (leaving out energy from the discussion). 




Thanks for your well thought out comments. Well thought out, they indeed are, though I think it is not possible to equate pressure and stress. The two notions differ in that even if you take the stress traction--a vector--it still is not necessarily directed normal to the surface. Another thing. Pressure is a scalar. Even if you take it in the directed sense, it acts along a single sense of direction--outwards if positive and inwards if negative. This is not how the stress traction vector behaves. It necessarily consists of a *pair* of two forces acting in opposite directional senses across the same surface.

But, I appreciated your accepting that strain is more fundamental--and the reason you forward in your own terms. I appreciated that. 

I also really appreciated your question as to why none talks of singularities in strain! This is something that I had not thought of and the question is very acute. I am sure it will lead people to think more deeply about this. After all, even if none explicitly says so, everyone *does* seem to assume that stress is more fundamental than strain is--which is a mistake. Your question really asks people to examine the assumptions from yet another angle.

Also see a common clarification below. I am correcting my own confusions!

Zhigang Suo's picture

I enjoyed reading this thread of discussion, as well as an earlier thread started by Hanry Tan on whether stress is a measurable quantity.  Thank you, Ajit, for writing so clearly about the issues. 

Gopinath raised the question about stress singularity.  I assume that he had singularity at crack tip in mind.  Here are some notes:

  • The singular stress field is an outcome of a mathematical model.  The crack is assumed to be a mathematical plane, the crack front is assumed to be a mathematical line,  and the material is assumed to be linearly elastic.  Of course, none of these assumptions are correct.  Thus, the stress is really non-singular at crack tip.
  • What is interesting is that Irwin made this wrong solution useful for us.   I have tried to summarize his idea elsewhere (G. Bao and Z. Suo, " Remarks on crack-bridging concepts," Applied Mechanics Review. 45, 355-366 (1992)).
  • If we take the stress singularity at its face value, then strain is also singular.   The stress is linear in strain, to be consistent with the model, however wrong it might be.
  • If we relax the assumptions of the model, allowing the material to deform plastically, and allowing the crack tip to blunt, the stress becomes bounded, and the strain remians very high, possibly still singular.  This was the work of McMkeeking (JMPS, 1977).
Gopinath Venkatesan's picture

Dr. Suo

Thanks for sharing your thoughts here. I accept that when I talk about stress singularity, I had crack-tip in mind, and that when I mention singularity, I knew before hand, that stress cannot go beyond its yield stress and the material at that point (where it exceeds sys) yields to make stress a reasonable value. Also by the term singular, I stick to only mathematical definition rather than its physical existence. Analogy is to see the fraction S=A/B, when B -> 0, even though B!=0. So whatever I discussed, it is subject to your quote: "The singular stress field is an outcome of a mathematical model".

If in a plot of stress vs. strain, if I see that the curve goes fairly horizontal, I stop there, and cry stress is singular, and if the plot goes fairly vertical, I cry strain is singular. And I am tempted to say, that in the plots I have seen so far, stress to be singular in all stress vs (engr) strain curves, and I have not seen much of stress vs (true) strain curves to comment. [These are added after original post (to be specific, ln(1+e) - what happens when 1+e is growing). I should not have said singular in the above statements -- My bad..sometimes when we look back at the comments we made, we laugh at ourselves.]

The last reference in your reply, work of McMkeeking (JMPS, 1977), Is that a hypothetical study to allow deformation of material only if load is applied? Because at one point, we want the material to deform without even adding any more load to it. Does his work show strain a singular property? Thats very interesting.

I am yet to read your article. I will read it sometime soon.



yoursdhruly's picture

Continuing from where Prof. Suo left off, here are my thoughts specifically on the strain singularity at a crack tip, kindly correct me if I am mistaken in my conclusions:

1. From LEFM, with a stress singularity emerging at the end of the mathematical analysis, this implies that there must be a strain singularity as well, as Prof. Suo points out above.

2. For deformation plasticity, using the HRR approach, the strain is singular in elastic-plastic materials as well (but the order of the singularity is determined by the hardening exponent in the Ramberg-Osgood law) 

3. McMeeking's work has looked at crack tip stresses and strains in ductile materials (with incremental plasticity) for a BLUNTING crack tip using FE analyses. The blunting of the tip nullifies the stress singularity, but the singular behavior of plastic strain is still observed.

4. Finally, on a related but slightly different note, Prof. C.T. Sun's recent papers (2005, 2006) argue that even CZMs do not eliminate crack tip singularity (for LEFM, this argument is good for stress and strain) unless the correct traction-separation laws are employed. 

I am interested in these issues and would appreciate any comments.


Gopinath Venkatesan's picture


Thanks for sharing your thoughts on this.

I learnt a new stuff today from Dr. Suo's and your comments.


Zhigang Suo's picture

Dear Dhruv:  The following concerns your point 4.

  • Sometimes, a singularity is deliberatly left at the crack tip, even when a cohesive zone is used.  The singularity models a short-range interaction, and the cohesive zone models a long-range interaction.  For example, in a ceramic-matrix composite, the toughness of the matrix (atomic bonds) can be modeled by a singularity, and the bridging fibers can be modeled by a cohesive zone.
  • When the traction-separation starts at zero traction at zero separation, and goes up smoothly, the cohesive zone will not eliminate the singularity.  However, one can model the material as two half spaces joined on an entire plane by a set of nonlinear springs.  Then, singularity will not appear.
  • The above issues were discussed in Z. Suo, S. Ho and X. Gong, "
    Notch ductile-to-brittle transition due to localized inelastic band
    ," ASME J. Engng. Mater. Tech.
    319-326 (1993).
yoursdhruly's picture

Dear Prof. Suo,

Thank you very much for your insight and direction: I will read your paper in earnest.


De Xie's picture

Dear Zhigang,

 As to this discussion, I have exactly the same thinking about CZM as your said above.  So,  I totally agree with you.  I don't konw the details of Prof. Sun's papers but suspect that statement.  ( I should read them soon).

I did model the fracture of braided composites by uisng CZM with FEA.  In this case, no "clear" and "clean" crack can be observed or defined (due to bridging and ... as you mentioned above.  So, I think we'd better use "fracture" to replace "crack" to leave "crack" as something sharp).  Therefore, we resorted to CZM.

However, how to implement CZM with FEA in an efficient manner comes as another big issue.  Actually, many versions exist with different names.  However, the center is the same: assume a strain potential energy function and implement it as conventional continuum elemenet.  With this approach, the CZM is treated as a continuum complient layer.  We (Prof. Waas from UMich and I) refer to this as the Continnum Cohesive Zone Model (CCZM).

 Prof. Waas and I proposed our own version of implementation.  We called it Discrete Cohesive Zone Model (DCZM).  The center is the nonlinear spring as you said above.  With this approach, the CZM is treated as a spring foundation.  Besides its simplicity, the most amazing thing is that it is NOT sensitive to the FEA mesh size.  Therefore, it should be very attractive to engineers, who do not want to be invovled too much to arrange the proper mesh size.

 I have two papers on this topic.  One is addressing the methodology and the other is about the application to braided composites.  I appreciate it if you would like to read them and critisize.

1.   Xie D and Waas AM, Discrete cohesive zone model for mixed-mode fracture using finite element analysis, Engineering Fracture Mechanics, 73(2006): 1783-1796.
2.  Xie D, Salvi AG, Sun C, Waas AM, Caliskan A, Discrete cohesive zone model to study static fracture in a plate of textile braided carbon fiber composites, Journal of Composite Materials, 40(2006): 2025-2046.

In short, I consider CZM as a promising tool in fracture mechanics.



Dear Zhigang,

0. It was wonderful to receive complement from you!

1. Re. your AMR paper (30.pdf). Excerpts from your paper are in blue.

>> "Irwin's LEFM erects upon a single premise: At the onset of fracture, the material is elastic over the whole component, except for a damage zone localized around the crack tip, whose size L is much smaller than crack size a..."

Of course. ("...elastic over whole component..." That's a very nice, terse way to put it.)

>> "The condition is satisfied by either a large crack, or a brittle solid suffering little diffused damage upon fracture."

IMHO, the size of the *crack* is not the most important consideration. Instead, the *relative* smallness of the damage zone as compared to the overall region of analysis is. Practically speaking, the specimen size is a far more important consideration than crack size. More on this, in a separate point below.

>> "The elastic field in a component is analyzed as if the crack-tip--or the tiny inelastic zone--were a mathematical point with no physical structure, an idea analogous to the boundary layer approach in fluid mechanics. Such stress field is square root singular"

Ummm... The crack-tip zone may be large in absolute terms and yet, may not significantly affect either the existence of singularity or its strength as such. IMHO, it doesn't have to be a mathematical point. See the explanation below.

2. Some more thoughts about singularity at the crack-tip.

K is a global parameter. A lot of insight is to be gained by thinking about this one characteristic of K. In particular, K is not a field quantity: it does not vary from point to point, but instead refers to a cracked specimen as a whole. The defining equation for K is: stress at the crack-tip = K / sqrt(2 pi r) X f(theta), where the K term has absorbed both the remote applied stress and the flaw size.

By definition, K refers to how stress is distributed ahead of the crack tip, but *not* to the field values *at* the tip. This is a subtle but extremely important point.

The actual referent (or meaning) of the concept of K is the suggestion of the inverse-square-root singularity in the overall stress distribution as one starts from the bulk of the specimen and moves towards the crack tip.

Please note where we begin from--we do not begin from the crack-tip at all, as is usually imagined (and discussed).

Accordingly, contrary to what is generally believed, the following two issues are not at all relevant in the discussion of the singularity: (i) Whether the crack tip actually has zero radius or not, and (ii) whether infinitely large local stresses actually occur at the crack tip or not. The two issues are really speaking insignifant.

What matters is whether the particular manner in which the local stresses elsewhere in the KIC test component display closeness to analytical solution or not.

(Too much thinking in the past has been led astray by failure to appreciate this subtle point. BTW, it is not just engineering mechanician who repeatedly makes this mistake. The theoretical physicist is no exception either. Go through any interpretation of singularity in the astrophysical context: black/brown/grey/warm/white holes etc.)

3. I think the problem with the analytical solutions (e.g. Inglis') is that they are *incomplete* (and to some extent *irrelevant*) rather than being plain wrong.

The real problem with the analytical solution for the cracked specimen is not that the limiting case it deals with cannot be observed in reality--the case wherein the ellipse stretches so much that the crack tip radius would drop down to zero.

The real problem is that the analysis is conducted for an infinitely extended plane (or body)--not for a finite-sized specimen.

Thus, the real problem is that the issue of the relative *sizes* of: (A) the crack, (B) the crack-tip zone, and (C) the rest of the volume, *cannot* at all be discussed within the framework of any analytical solution.

Yet, since the actual specimens *are* finite-sized, the ratios of (B) to (C), (A) to (C) and even the ratio (B) to (A), are all important. It is these ratios which together indicate whether the rest of the field is sufficiently close to the projected mathematical field exhibiting a crack-tip singularity.

The proportion of (C) to (B) becomes immediately relevant for plastic materials. The ratio (B) to (A) may be significant if the inhomogeneous microstructural entities are large enough.

As seen above, the singularity issue is really settled by the kind of stress distribution there is in the bulk of the specimen. It is for this reason that Irwin could at all become successful with the plastically deforming materials such as high strength alloy steels. Even if the crack-tip blunts, you still have a reliable material property if the plastic zone is small enough to get hidden in the errors in the experimental measurements conducted over the bulk of the specimen. This can happen if the specimen is large enough. The procedure has the obvious limitation: For very ductile materials, the "specimen" must be made as large as buildings!

4. I believe that a lot of confusion will go away if we start calling it the "cracked-region singular stress field" rather than "crack-tip singular stress field."

5. The issue of fracture singularity is, in a sense, similar to that of ideal gas law (IGL). (And the various positions taken by people are also epistemologically so similar!)

IGL *is* an abstract law. Yet, it does not mean that it is floatingly abstract--it does have a real basis. The ideal gas law is a projection of what real gases do behave like, over a great range of conditions of temperature and pressure.

Of course, the real gas behavior deviates from the ideal gas near low temperatures, high pressures, etc. But this does not invalidate IGL--it only delimits the scope of its proper application.

Similarly, the fact of size effects does not invalidate the idea of KIC at its roots. It only delimits the scope of application of the idea.

6. Comments on Dhruv's excellent post here (and also a bit on CZM) will follow soon... For the time being, I also invite all to express your thoughts regarding the SI units of fracture toughness here

0. I think my original post above (appearing top-most in this thread) needs to be modified. I knew a modification was needed but I kept on procrastinating...

1. One modification is needed here:

>> "There is that "displacement<->the gradient tensor<->deformation" relation on the strain side"

In retrospect, I think this statement of mine confuses more than it clarifies. It should be something like:

"Displacements <--> Relative Displacements (or Deformation) <--> Relative Displacements Gradient Tensor <-> Strain Tensor"

When I originally wrote the above statement, I was trying to highlight the fact that strain is but a part of the relative displacement gradient tensor. But, by mistake, I didn't write the word "strain" at all.

It is true that in the history of solid mechanics, the term "deformation" has been used in many senses--as relative displacement, as relative displacements gradient tensor, as rotation, and as strain. I simply continued the confusions though I had begun on a better note, by defining the sense of that term above.

2. Another confusion: I wrote the following:

>> "Deformation, in contrast, can be defined at a point. We don't have to refer to a geometric element in order to define what this concept means."

Instead, it should be read this way:

>> "Displacement, in contrast, can be defined at a point. We don't have to refer to a geometric element in order to define what this concept means."

Actually, the argument surrounding this quote is a little weaker than what appears in the original post and on the first reading. It is valid, but weaker. Let me clarify.

Displacements *are* point phenomena. But relative displacements are not. One has to refer to a *line* element to define or measure the relative displacement. Therefore, one really can't say that deformations are point phenomena.

It is also true that deformation is still somewhat simpler. It refers to a line element. Deformation is the difference between the local displacements undergone at the two end-points of a line segment. But stress refers to a surface element. So, even if my original argument has now weakened, stress still remains a more complicated concept. Its definition still involves assumptions as to how it varies over the surface--such assumptions are still not in the same way involved while defining deformation from displacements. So, overall, even in the revised argument, there still isn't any parallel to deformation on the force/stress side.

3. Really speaking, I believe that one of the two most important points of my above post is this passage:
>> "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 not to drop out the torques-related part."

The other is:
>>For field quantities, force necessarily arises only in the context of a geometric element of nonzero side 

Oftentimes, introductory texts "prove" complementarity property of shear by appeal to moments balance. What they ought to clearly mention is that moments balance doesn't necessarily lead to complementarity of shear stresses--because couple-stresses can exist.

The real reason to define stress in analogy to strain is, actually, three-folds. One, because the theoretical structure remains similar. Two, because a linear form of constitutional law can thereby be provided--which simplifies analytical mathematics. Three, because the linear model does work at least in the small deformation and small strain limits.

4. Hope with the present clarification, my above post becomes clearer. I will modify it to reflect the corrections after our current exchange is over. (For the time being, I am keeping it as is, to allow easier referencing in our communications.)

5. I regret the errors.

But if you ask me why so many errors and typos creeped in here in the first place, the reason (or at least a part of it) is that at the time of posting this article, the Drupal editor would often go dead while typing or crash in the middle of writing or posting.

Gopinath Venkatesan's picture


Yes, it should be Force -> Stress, instead of Pressure. Thanks for correcting me.


This refers to the query Mahendra generated in the thread on scalars and vectors; see here

Mahendra's concern, in his own words is the following: I have some questions in mind regarding which is more fundamental strain or stress. Consider a metal rod which is heated in between two rigid supports and constrained laterally. It tries to expand.There is no observable deformation. However, there is an intutive feeling of some kind of push acting on the rigid  supports and  some kind of compression on the metal rod.

My answer is that Mahendra himself gives away the clue to the answer. Notice the one word he uses: "constrained." Constraining is nothing but imposing displacement boundary conditions--even if there is no observable deformation. So, here, the relative displacement is the one between the configuration at high temperature without constraints and that with the particularly noted constraints. The difference between these two configuration is what gives rise to strain here, and therefore, to stress. 

Mahendra, the particular (intuitive) feel you mention is right on target, but notice that we are sticking to explicit reasoning.

Another point. The argument for fundamentality of strain does not depend on whether motion is more fundamental to force or not. The former is the issue in mechanics of materials; the latter is in general physics.

Observe that the moment you say "stress" (or "strain"), you already are in a field-based description. My (novel) assertion is that in any fields-based description, the quantity of force necessarily arises only in the context of a geometrical element of non-zero size--which is not the case for the usual (Newtonian) mechanics where the Earth and the Moon can be imagined to be point masses. 

In other words, I assert, there is a fundamental difference between the meaning of the term "force" as it is used in the usual (Newtonian or relativistic) mechanics, and in the mechanics of materials. For mechanics of materials, the more apt term is "internal resistive force."

This thread of discussion goes to the core of mechanics.  Here I'd like to mention a specific phenomenon that speaks to aspects under discussion.

Consider a parallel-plate capacitor, i.e., a layer of a vacuum between two metal plates.  When a votage is applied between the two metal plates, the two plates attract each other.  We know about the attraction because we need to apply a force to keep the two plates apart.  Without the applied force, the two plates will fly toward each other. 

We also know that the two plates have electric charges of oposite signs.  We know that because we can measure how much current flows through the external circuit.  It all seems to make  sense.  A positive charge attracts a negative charge.

Now, replace the vacuum by a solid dielectric.  If the dielectric is an elastomer, upon the application of the voltage, the elastomer will become thinner.  In this case, no external force need to be applied to maintain equilibrium.  Many people would like to say that the voltage applies a compressive stress to the elastomer.  Indeed, they even give a name to the stress:  they call it the Maxwell stress.

If the dielectric is some other materials, upon the application of the voltage, the dielectric may either become thinner or thicker.  In the literature, you can find experimental data of how much thinner or thicker a given material will become.  If a material indeed becomes thicker, would you be willing to say that the voltage applies a tensile stress to the dielectric?  Slick people do not say that.  They give a different name to the phenomenon.  They call it electrostriction!

My coworkers and I find the practice unacceptible.  In both cases, thinner or thicker, if no external force is applied, we would simply say that the stress in the dielectric is zero.  After all, thinner or thicker is just an observation of strain, and says nothing about stress.  For example, we would never call thermal expansion a tensile stress.

In a recent paper, entitled a nonliner field theory of deformable dielectrics, we have formulated a theory consistent with this simple point of view.  We believe that we have resolved a long-standing basic issue in electromechanics.  We really like the paper, and hope you will, too.


Rui Huang's picture


Your statement of zero stress in deformed dielectrics (thinner or thicker) made me think again. At first, I cannot accept it. Then, I thought of thermal expansion, where no stress is generated if the material is free of any constraint. To make the analogy, there must be a quantity in your theory of deformable dielectrics taking the place of temperature as a state variable  for thermal expansion.  I will have to read your paper again to see if such a quantity exists and then to see what material properties are in analogy to the coefficient of thermal expansion.

Another interesting situation to be considered is as follows. If the electrodes (e.g., two parallel metal plates) are separated from the dielectrics by two layers of vacuum such that no direct contact between the electrode plates and the deformable dielectrics. Upon the application of voltage, the electrodes would attract each other. To prevent contact between the electrodes and the dielectrics, an external force must be applied to the electrodes, similar to the case with no dielectrics in between. In such a situation, would the dielectrics still deform? What would the theory predict? Is there any experimental evidence? Of course, we are not talking about piezoelectric materials or any other materials with intrinsic electro-mechanical coupling (i.e, apparent microstructural change under an electrical field).


Zhigang Suo's picture

  1. Another state variable is electric displacement.
  2. If an external force is applied, stress is force divided by area.  This stress exists in the dielectric.  In our theory, as well as in Maxwell's original theory for vacuum, the same stress also exists in the vaccum.
  3. The behavior of the multilayer can be calculated once a material law for the dielectric is prescribed.  In a follow-up paper, we prescribed a material law.  We showed that the Maxwell stress emerges from a very special material law.
Rui Huang's picture

  1. The electric displacement is linearly related to strain for piezoelectric materials due to intrinsic electromechanical coupling (i.e., constitutive relation in continuum mechanics). For nonpiezoelectric materials, a higher-order electromechanical coupling gives nonlinear relationship between electrical displacement and strain (e.g., electrostriction), which also can be characterized in terms of constitutive behavior of the material although the intrinsic EM coupling mechanism is less obvious to me. What are the differences between these conventional electromechanical theories and the newly developed nonlinear theory? 
  2. With vacuum layers in between, the external force is applied directly onto the electrodes (parallel plates), and this force (mechanical) is balanced by the electrostatic force between the charged plates. Thus, stress exists in the electrodes (a different topic).  I am not sure how one applies force onto a layer of vacuum and what it means by stress in vacuum. In the situation I described above, no contact between the electrodes and dielectrics thus no external force is applied directly to the dielectrics, unless the vacuum layer is capable of transferring any force from the plates to the dielectrics. It is hard for me to understand this.
  3. Does this material law contain any coupling coefficients between mechanical (strain or stress) and electrical (electrical displacement or electrical field intesnity) in addition to elastic modulus and dielectric permititivities? Or, the coupling is simply through a combination of pure elastic and pure dielectric properties.



Wei Hong's picture


 1.  There is always electromechanical coupling.
Primary school physics told us that ther are electrostatic forces between charged plates, whatever material in between, there will be deformation (except for rigid material, which does not exist.) The electrostatic energy turns into strain energy.  So in this sense, there is always electromechanical coupling.

 2. Again from highschool physics, one tends to think that the electrostatic interaction between the electrodes are far-riching (and instantaneous) , especially when there is nothing or vaccum in between.  A little bit more electromagnetics tells us that such an interaction actually propagates as a wave through a medium, and it can be blocked or shielded.  For EM wave or electrostatic field, the vacuum is one special type of material, just as steel for strain field.  In all situition we are interested, the result of such an interaction is the same as that of the stress we are familiar with.  So why don't we just call it stress? Vacuum is special only because it has 0 stiffness, but that does not prevent us from having stress, as stress is no longer stiffnes times strain.  (You can think about the thermal strain of a rigid material as an analogy.)

 3. The coupling is inevitable, unless for fake materials like rigidbody or so.



Konstantin Volokh's picture

The discussion reminds me of childhood: who is stronger elephant or whale?

Seriously, the stress is a superfluous concept because the theory of elasticity can be formulated without even noting it. The concept of stress, however, can be useful occasionally...

Dear Dr. Volokh,

I appreciate your pointing out the fact that stress and strain are intimately related. Yet, I differ from your opinion. 

Of course, in mathematical manipulation, it often does seem that we could use just one of the two symbols: sigma or epsilon. Since the constitutive relations are linear and reversible in the theory of elasticity, it is in elasticity where this proposal to eliminate one of the two symbols seems most attractive.

Yet, the concept of stress is not superfluous. In fact, it is a fundamental concept. Its usefulness is immediately apparent in any more advanced theory such as plasticity, fracture, fatigue, etc.

Apart from the utility, if one seeks to know the basic meaning of the concept, one has to look into the context such as what is given in my reply to Zhigang below. The idea of the test tells why, even if the theoretical structure shown by the definition of the stress *tensor* parallels that of the strain tensor, the basic meaning of the term "stress," nevertheless, *does* include a reference to forces and the mathematical cut.

In other words, it is true that stress is a *secondary* concept, as compared to strain. But this does not mean that all its referents are fully given by the concept of strain alone. Therefore, one cannot really describe it as superfluous or redundant, even as in relation to strain.

If you do not like long replies, then, here is a one-liner: In the study of springs, is the idea of force superfluous? (BTW, here, this "you" is to be taken in a general sense!)  

I often write at length because I have found that most one-liners--mine at least--neglect too many crucials. The crucial neglected here is: the fact that stress is an internal, and imaginary or abstract variable--not a directly measured or external quantity. In my one-liner in the above paragraph, there is an equivocation on these two types of variables, which is tantamount to an invitation to treat as if force and stress were equivalent concepts. In the abstract 1D representation of a spring, they seem to be equivalent, though, of course, you sure know the truth to be otherwise!

Konstantin Volokh's picture

Dear Ajit,

Nonlinear elasticity does not require stresses either: a nonlinear potential is constructed whose variation provides the equilibrium conditions without stresses. I believe that theories including structural evolution (plasticity, damage, etc) can also be formulated without stresses after some effort...



Agreed, that would be the way to construct unique-valued stresses under constitutive nonlinearity.

But the engineer within me is getting uncomfortable. And there are two sides to that discomfort. (i) Why would I want to build such a theory? With what purpose in mind? What end? What use? (ii) With such a force-less (or stress-less) theory as my tool, how would I go about designing something--for instance, a bridge in a Minnesota?

The moment I say something like: This structural member will take the load, I am very much relying on a concept like stress--whether kept implicit via a potential formulation or otherwise. If so, might as well make it explicit. If so, that involves definition. Which involves the discussion above... Which leads to the realization that it is fundamental--though strain is even more fundamental.



Dear Zhigang,

Thanks for pointing out the very interesting discussion in the thread on deformable dielectrics. I have just downloaded your papers using a cybercafe (because the dial-up at my home now crawls at barely 4 kbps instead of the advertised 57.6 kbps). I will study your papers later. ... A word about that very interesting thread: I may have to correct myself later, but I find it reasonable that there is no need to have electric body forces. I will post a comment in that thread later on...


Now, coming to your above post and the current thread... You present an intriguing argument (in blue):

My coworkers and I find the practice unacceptible. In both cases, thinner or thicker, if no external force is applied, we would simply say that the stress in the dielectric is zero. After all, thinner or thicker is just an observation of strain, and says nothing about stress. For example, we would never call thermal expansion a tensile stress.

I disagree.

Stress is the *internal* resistive force-like quantity; a quantity imagined to exist at the point(s) internal to a region of space and defined with respect to the area of a mathematical cross-section.

The test to see if a nonzero stress exists at a point or not follows. Take a mathematical cut, mentally hold the two parts, but then, let go one of them suddenly, and see if the other part has a tendency to fly away from (or into or towards) the mathematical cut. (Taking the cut integrates the stress distribution over the area of the cut, and so transforms the internal and abstract quantity of stress into the physically observable quantity of force, and it is the latter which, at least in the imagination, produces acceleration.)

Apply this test to the region of the elastomer. Take a mathematical cut parallel to the two plates and then, let go only one of the cut-away parts. What would happen? ... Obviously, there is stress. (Or so, I presume!) 

So, whether an actual force (dp/dt) is applied to the *plates* or not is not at all the issue. The issue is: What is the state of the elastomer?

As an aside: We do not call thermal expansion a stress. Yet, observe that we do ascribe a state of stress if that expansion is *constrained*. Constraining something is the same as forcing it--only the nuance of the words, the subtle shade of meaning, differs. For instance, the soil merely constrains the foundation of a building. Yet, we do speak of something like support forces.  


This leads to an interesting question. Can't we even hypothetically think of a situation whereby there is strain but no stress?

According to the viewpoint expressed by Prof. Volokh (pl. see the comment below), such a possibility cannot at all exist.

Actually, this is a matter of how you interpret the definitions. If you can imagine a body force that, somehow, happens to have a tensorial nature--not vectorial--then we could arrange to have that force distributed in such a manner that a state of strain would not necessarily imply a state of stress. But of course, none knows of any form of such a body force. And, as I said, it then becomes a matter of definitions: Why would we call such a body force a body force, why wouldn't we call it stress?

So, leaving arbitrary ideas aside, strain should be taken as always accompanied by stress (and vice versa)--whether the constitutive relation is linear, elastic, non-dissipative, reversible, etc., or not. In all cases, strain and stress always go together--you can't have one but not the other.

Zhigang Suo's picture

Dear Ajit and Rui:

Thank you so much for raising all kinds of concerns and objections.  Similar thoughts occured to us when we started to work on the problem.  By now I believe that we have resolved these issues to our own satisfaction.  Our object has been to formulate a theory that can describe observable phenomena.  Within our theory, here is how we analyze the free-body diagrams.

Case 1, a parallel-capacitor with a vacuum gap.  When a voltage is applied between the two electrodes, you have to apply a pair of forces to the two electrodes to maintain equilibrium.  In this case, our theory (and Maxwell's theory) says that the vacuum is in a state of stress.  If you regard the top electrode as a free body, the force you applied will balance the force due to the stress in the vacuum.

Case 2, a parallel-capacitor with a solid dielectric.  When a voltage is applied between the two electrodes, you do not need to apply a pair of forces to the two electrodes to maintain equilibrium.  Let's say you don't apply forces.  If you regard the top electrode as a free body, because you don't apply a force to the electrode, there is no stress in the dielectric.  In this case, our theory says the stress in the dielectric is zero.

My class notes on deformation and polarization contain additional examples. 

Dear Zhigang,

As it often happens, I realized the main issue only *after* I had already posted my earlier reply!! 

As I see it, the main issue seems to be: What mechanical supports for the plates do you assume--yielding (i.e. displacing) or non-yielding (i.e. staying put where they are, all the time).

Let me explain at length.

To take out the complication due to gravity, assume the plates to be vertical. So, let us say, there is a left plate and a right plate (instead of a top and a bottom one). They are arranged like the books kept in a neatly arranged library shelf. The left plate is permanently fixed in space, say to the laboratory table--it does not at all move, so, it enjoys what is technically known as the perfectly non-yielding support. This leaves only the right plate to move or to stay put where it was.

The left plate carries A charge and the right carries B charge where A and B have algebraically opposite signs. So, the plates attract.

Case 1. Vacuum (i.e. nothing material) in between the plates. The Coulombian attraction between the plates implies that to keep the right hand side (RHS) plate where it is, a force acting towards RHS must be applied to it. Assuming the external force applied to the right plate balances the Coulombian attraction, there is no net movement of the right plate. So, the right plate, again, is a non-yielding support.

However, since the externally applied, outwards acting forces are being transmitted through the volume (i.e. the region of material-free space) lying between the plates, we must ascribe a state of stress in that volume. To not have stress, we must stop applying the external force, in which case the RHS plate will simply bang on to the LHS plate and discharge both the plates. But, this is not the case; this circumstance is being avoided using a net pair of forces acting on the plates.

Therefore, if a mechanical abstraction of this situation is to be isolated, we would have to assign a state of tensile stress to the material-free space between the two plates.

Case 2. We have elastomer in between the plates. The electrostatic field between the plates is now a superposition of two fields. (i) The Coulombian attraction between the net extra charges on the two plates. (ii) The electric field within the volume of the elastomer. (We neglect the edge and boundary effects of the elastomer, esp. those near the plates. BTW, when we say the phrase "electric field within the elastomer," we actually mean: the *modification* to the local equilibrium field. The local equilibrium field has no net charge. The absolute values of such an electric field is not our concern. We take it as a base-line, and are interested only in the changes brought about to such a field. The changes is what we mean when we say "the electric field in the elastomer.")

The electric field within the elastomer balances the Coulombian attraction between the charged plates. But does the elastomeric field come without any mechanical effect? Of course not!! The elastomer *thins*.

So, the crucial issue is: What kind of plate do you have on the right hand side? Mechanically speaking, does it yield or not?

Case 2-Y: The RHS plate yields and so moves towards left because the elastomer has thinned. In this case, obviously, there is no mechanical stress. But note, in this case, there is no mechanical strain either. It's like the free contraction due to a drop in temperature. (I have dropped a misleading statemetn here in this revision.) If you take a cut through the elastomer, nothing happens by way of displacement of the RHS plate. That is the ultimate test to determine if there is stress or not.

Case 2-NY: The RHS plate does not yield. In this case, you will have mechanical strain (in the horizontal direction). This strain will, in turn, serve to further modify the electric field existing within the elastomer. Such a further modified field will no longer exactly balance the Coulombic attraction. So, any finite sub-portion of the region of the elastomer (say a mathematical cube) will experience tractions on its bounding surfaces. So, there will be mechanical stress. The situation is analogous to cooling a metal and then mechanically stretching it again to bring to its original volume. Now, if you take a vertical cut through the elastomer (parallel to the plates), the RHS portion of the elastomer flies to the right. Presuming the elastomer to be glued to the plate, the RHS plate too flies to the right.

Thus, whether you will have stress or not is being determined by the kind of supports you have. Whether you have elastomer or not is a secondary issue.

Do I miss (or misinterpret) something?

Xuanhe Zhao's picture

Dear Dr Ajit R. Jadhav,

Thank you for the initiation of this thread of very interesting
discussions. I think you have exactly pointed out the attitude we adopted in
developing the nonlinear field theory of deformable dielectrics: working with the
measurable quantities.

For a parallel capacitor with vacuum or elastomer in between,
it’s the force applied on it that we can measure instead of the stress inside.
Based on this force, we define stress as force over area.

Many other theories on electroelastics tends to
differentiate various stresses in dielectrics, e.g. Maxwell stress, mechanical
stress, and thermal stress et al. While we refrain to say these theories are
wrong; however, we can’t see any physical significance in differentiating the

I think this may, to some extend, answer Dr R Huang's question "What are the differences between these conventional electromechanical theories and the newly developed nonlinear theory?"

Dear Xuanhe,

Thank you (and others too) for your compliments.

I am not sure what you mean about measurable quantities. At least in the context of quantum mechanics, I think they have a very wrong notion that a theory cannot at all include anything that is not measurable. (But even QM would be a digression here, let alone philosophy of QM.)

I would welcome your general point that if there is no physical significance in differentiating the stresses, then such sub-types need not be continued in a theory. However, I am not too well conversant with the specific researches in electroelastics, really speaking. But just the way the Maxwellian synthesis made it so easy to see how the individual laws of EM (Colomb's, Faraday's, Ampere's, Lorentz', etc.) relate to each other, similarly, if you can have a theory that can simplify understanding how these individual types of stresses are really related to each other (as a consequence of your theory, not as its own structure), then that's well and good too.

As an aside, I would be curious to know (perhaps on a separate thread) if there are any interesting consequences of the nonlinearity you have built into your theory.

Wei Hong's picture

Hi Jadhav,

 In your Case 2-Y:

We (and many people as well)  define strain as the deformation caused by whatever field, stress is not a must for the existence of a strain.  For no phylosophical reason, simply because it is measurable.  In absense of any knowledge of the material inside, strain is the easiest quantity to measure.  Our naked eye (or with some modern aid) is a quite good tool.  From the limited number of textbooks I have read, free contraction due to a drop in temperature is named "thermal strain", with the simple relation (for 1D)

ε = σ/E + α ΔT

Please let me know ff you have an alternative way of defining a measurable strain which excludes a thermal expansion.

BTW, we agree three is no stress inside the elastomer in this case, simply using the "ultimate test"

In your Case 2-NY:

I think you are refering to the case when the electrodes have rigid supports.  Your interpretation agrees with our theory.

Let me try to rephrase your conclusion to resolve any residue concern:
 Whether or not we have stress is being determined by the boundary conditions, (which includes loading and support), as this special example is statically determinant.  This is a result from Newton's law, our first principal.
But  how this stress relates to the strain and the electric field depends on the material behavior. Just like Hooke's Law, you can think of it as secondary if you like.


Dear Wei,

Thanks for your comments. I am not quite clear about your first paragraph, esp., the first two statements. Could you please elaborate?

In the meanwhile, with respect to your first point (your comments on Case 2-Y), here are my general clarifications:

The Vernier callipers (or the laser-based measuring equipment) measure displacements. So might the naked eye, to a physically more limited extent. But displacements can mean the rigid-body translation as well as the deformation. It is only the latter which contains strain. (The other thing it contains is the rigid body rotation). Thus, strain refers to gradients in the displacements. This is by definition. Since the eye cannot on its own delete the rotational part, the eye (or the Vernier, etc.) cannot at all be a tool for measuring strain--you can't see the strain, so to speak.

Strain is a mechanical concept, not thermal (or chemical, etc.).

The free thermal expansions or contractions are mere displacement fields which, by themselves, do not necessarily qualify as strains. They can imply strains if and only if suitable constraints are supposed to have been applied to restrict their occurrance. The constraints serve to bring into picture forces. (They do so at least implicitly.) It is this last step which makes the description mechanical. Note, there is no mechanics without forces.

One may derive some more clarity to the logic here by imagining expansion through non-thermal means, say, via hydration.

For instance, to cut the granite rocks in ancient times (while building temples), they didn't have diamond saws. And though explosion was known, it was far too uncontrollable for using it to cut a rock precisely. Then, how did they cut the giant rocks so cleanly?

Simple. They made small notch-like holes, each only a few inches deep, on the surface of the rock to be cut. All these holes would be aligned in a line. They then press-fit all these holes with dry wooden pegs. They then poured water over the pegs. Over a period of time, the wood, upon absorbing moisture, would try to expand, and so, tend to open up the notch in which it was fit. Several such notches aligned in a line would build sufficient local stress concentrations in the granite that it would actually open a crack connecting all these shallow holes. The wooden pegs would continue to load the rock in the Mode I. Since granite is a brittle material, the continued loading by the expanding wood would send the crack propagating through the entire volume of granite. A few inches of peg could easily cut giant monoliths of granite. The process would be, initially, slow, but definite.

Now, what I say is this: If you take a wooden peg and keep it freely hanging from the ceiling of your laboratory, and arrange to have water dropping over it, it would surely swell. Assuming a continuum in the place of the wood, can we call such a swelling as straining? The answer is "no." Why? Because, in *this* abstraction (call it a "model" if you wish), no force was deemed necessary to have caused the swelling.

Yet, the same peg, when constrained by the rock material, is taken to be strained. Why? Because, through constaining, forces do come into picture. (Further, as seen above, the rock also gets strained as a simultaneous effect because neither the wood nor the rock is perfectly rigid.) So, whether there is strain or not, depends on the abstraction being used.

If you wish to model the wood as a composite material, then you could perhaps treat the cellulose material (the cell-walls) on the same lines as what the granite rock was like in the above example. In such a case, the water molecules can be taken to act like rigid pegs. (Their rigidity is an assumption.) If an abstraction is built in this way you could *now* say that there is a strain to accompany the swelling. You could say that the cell-walls are undergoing strains. But note, you can call it a strain only if your abstraction (i) explicitly ascribes a structure to the wood and (ii) includes the cell-walls to act to place *constraints*. In contrast, in the more gross continuum description of a freely hanging peg, the same physical act of swelling cannot be taken as straining. And, in case the swelling is actively constrained, as in the case of the rock, there is strain even in the gross continuum description. 

If you accept that there can be multiple models of the same physical phenomenon, then it is easy to see that not every swelling or shrinking ought to be seen as straining. To demand that every displacement field must automatically imply mechanical strains would be a bit too rationalistic.

I suppose the case of thermal expansion/contraction is simpler--it involves only one kind of abstraction--it's always a homogeneous continuum.

Finally, I suppose that the context in the text-books surrounding the equation you quote would (or ought to) make it clear that thermal expansion is presumed to be constrained.

Wei Hong's picture

Dear Jadhav,

 I totally agree with your strain definition. (Although I assumed that human eyes are capable of detecting displacement gradient.)

However, according to this definition, I think strain is merely a geometric (or kinematic) concept, not mechanical, nor thermal or chemical.

In your wooden peg example, we still think there is strain, using the same definition as you proposed: the (symmetric part of) gradient of displacement is not 0.
I don't think force is a necessity for strain to exist, nor is stress.

We do accept that there can be multiple models for a same phenomenon, so we are free to define strain in a measurable and self-consisitent way, in which we demand every nonzero gradient
of displacement (excluding rigid body rotation) implies strain.  Strain is strain, we never call it mechanical strain.

On the contrary, the context of the equation I quote clearly states that it is free thermal expansion (plus a uniaxial tension).  On the other hand, if someone wants to define some nonphysical (but might be useful practically though) term of "thermal stress", then he ought to presume a constraint.


Dear Wei,

Fine. The question still remains, why do you need a complicated quantity like strain if you are not going to use it in a mechanical context--i.e. in regards to measuring the various aspects of responses to forces? 

You see, the definition of strain is not at all simple. You wouldn't be able to derive it except in reference to the specifically mechanical thinking.

Throughout the history of science, a comparatively so complicated quantity as strain has been found to be relevant only in relation to stress, i.e. only in relation to the response of materials to applied forces. It took about 150 years of intense thought (from Galileo to Cauchy) just to get as much right as the very definition of strain. ... Thoughts on things like, the fact that the lateral quantities (on an elemental cube) matter when forces are applied. Or, the realization that the rotational components should be subtracted.

That entire thinking was in the context of reaction to forces.

Now, at the end of all that thinking, you have got this concept. So, once you do have this complex, higher-level abstraction, you want to strip it of all its defining context, is it? Why do you want to take the concept outside of its proper context? Does a single natural phenomenon outside of the mechanical effects justify doing this?

The same concern can be raised another way. Why do you at all insist on using the idea of strain in your work? That is to say, why not describe your model with displacements alone? After all, that kind of description would be perfectly "kinematic" or "geometric" too--if that is what you were on the look out for.


I know that there has been this philosophical program one of whose implications includes reduction of mechanics to geometry. But it is obvious that, basically, such a program involves simple context-dropping.


Sorry. Unless you have additional, inductive evidence to show the meaning and applicability of the concept of strain outside the scope of mechanics (i.e. outside the context of forces and their effects), your demand simply involves the logical fallacy of concept-stealing.


I think you have a good theoretical initiative as far as describing the dielectric behavior is concerned. But I also think that you are clearly committing a mistake in calling the free displacements by the name strain (and could be confusing the issue by not clearly identifying whether the supports yield or not).

I do not wish to be cynical, but if I were to be, here, I would have asked: "What next? Associating strain with scaling?"

As far as I am concerned, the argument rests here. Any more on this aspect will be only a repetition--long, or otherwise.

Wei Hong's picture

Dear Jadhav,

 You are right, we really don't need a quantity like strain.  In fact, through out our paper, we really did not use it.

We did not define strain either.  The geometric quantity we used is deformation gradient, or gradient of coordinates.

I learned from textbooks (which you might not agree with) that the symmetricalized deforamtion gradient is a measure of strain.

There is nothing initiative we have done in terms of the definition of strain.

There are infinite ways of defining strain.  Personally, I think as long as it is self-consistent, the definition is fine.
Our definition and derivation are self-consistent, and I don't see any clearly committed logical mistake.
We are just trying to define thing in a simple enough way so that at least we ouselves could understand.

Before saying concept-stealing, could you please give us a definition of strain which you think is the standard?


Dear Wei,

Mathematically, there may be multiple definitions--such as the naive, so-called, engineering definitions, the true definitions, the differences arising due to the initial vs final configurations taken as the reference, etc. But the imporant issue here was, despite their differences, whether these definitions keep the broader context that this all is a matter within mechanics or not. On that count, even the engineering definition keeps the context, though the definition may not be very sound for the bi-axial and tri-axial situations. In fact, on that count, all definitions of strain given in the texts keep the context, to the best of my knowledge.

Personally, the definition that comes quickest to my mind is what I described above: take the displacements gradient tensor and drop the rotation from it. But if you want a specific equation by way of definition, I suppose I could cite one from Dieter, Riley and Dally, Malvern, Solecki & Conant, or Saad, or even Love, but I don't think it is really necessary.

If you really do not need a quantity like strain, then that's well and good. I suppose that one fact would have been enough to end the argument.

But if you still think that meeting only the requirement of self-consistency would be enough, I would ask you to re-check how sound that policy would be, epistemologically. As far as I can see, there always are many more considerations to any concept, esp. the fundamental ones. The considerations include things like: the referents (i.e. scope or the kind of units the concept subsumes), the measurement standards (not experimental, but, rather, the conceptual measurement standard), the contrasts (i.e. the conceptual common denominator), the salient attributes (i.e. the distinguishing characteristics), hierarchical antecedents, etc. Why, even special notes such as the border-line cases. A whole gamut of them, best put in one word as "context." And they are relevant not just here, but in any physical science--in fact, in any field of knowledge.

Wei Hong's picture

Dear Jadhav, 

From the definition you described here, which is exactly what we have used and I have been stating,  I can not see any reason why the free thermal expansion of a block is not a strain.

BTW, previously when saying self-consistency would be enough, I meant your definition of strain. 'Cause I was curious to know if there is any alternative but self-consistent definition of strain that excludes thermal expansion. 

Zhigang Suo's picture

Dear Ajit and Wei:

I'm a little lost in the conversation between you.  Not sure if you are still talking about our JMPS paper on deformable dielectrics.  If you are, or at least talking about things related to the paper, maybe the following notes are helpful.

  1. In Section 2, we formulated a theory without talking about strain.  This theory is not a field theory, but a more premitive theory, applicable for situations where a field theory is not really helpful.  For example, if the body is just a molecule or a cell, perhaps the body is too heterogeneous for a continuum theory to be of much value.  Even for such a system, however, we can still talk about force, and about elongation.  Section 2 is merely a restatement of thermodynamics of an electromechanical system, reminding us what may be measured, and what are work conjugates.
  2. In Sections 4 and 5, we formulated a field theory.  Strain is invoked, between Equations (5.4) and (5.5). 
Wei Hong's picture

Dear Zhigang,

We are not talking about the paper.

Ajit disagrees that a free thermal expansion (or similar deformation caused by electric field in our paper) can be called "strain".

I am trying to talk him out of this. Smile


Wei Hong's picture

Hi Ajit,

The following simple example might help to clarify:

A rod (of unknown material) of initial length 1m is under tension.  We can measure the force and thus calculate the stress, no problem.
But at the same time, the temperature increases by  20°C.  As a consequence of both the force and the temperature change, the rod extends by 5%.
For simplicity, there is no constraint involved.

What is the strain?

If you agree that the strain is 5%.  Then let us repeat the same experiment, but reduce the force to 0 (or almost 0), now the rod extends by 2%.
This 2% is what we usually called thermal expansion or thermal strain, you don't agree that it should be called "strain"? What is the difference between the 5% and 2%?

If you disagree that even the 5% can be called strain, then what is the strain in this case?
(Remember that the material is unkown and might be nonlinear, so please don't try to define strain with youngs modulous.)

Hope this helps.



I'll jump in here without having read the entire exchange - so please pardon any repetation of previously discussed issues on my part.

You wrote

" A rod ... is under tension.  We
can measure the force ... , no problem. ..... there is no constraint involved."

There are some issues here :

1) If there are no constraints and the forces are balanced, there is no way you can tell whether a tensile force has been applied.  So you cannot measure that force and there is a problem :)

2) If the forces are unbalanced then, of course, the rod will move/deform in the direction of the larger force. 

The only way you can tell whether the rod is in a state of tension is to measure the deformation with respect to a reference placement.  The same is true for a temperature change you need a reference temperature and a reference placement.

Strain is defined independently of what causes it but needs knowledge of at least two placements.


Wei Hong's picture


Yes, you are right.  We need some proper constraint to remove rigid body motion.

You got exactly the point I want to make, strain is defined independently of what causes it, but needs only displacement.

Thank you!


There was some discussion in this thread about the concept of stress in a vacuum between two parallel plates.  I believe Zhigang's theory is a continuum theory which implies that stress is considered a point quantity. 

In that case, an analogy may be made with the mechanical theory of a material with voids - think foams, poroelasticity, etc.  We use an effective theory for such situations assuming that the scales are sufficiently separated that the continuum assumption can be made.  However, even for such materials the stresses and displacements inside the voids are not well defined quantities at the microscale even though they are well defined at the effective scale.



Zhigang Suo's picture

  1. When an electric field is present in an vacuum, a stress field may be assigned to the vacuum.   This stress field is known as the Maxwell stress, which was derived in his Treatise on Electricity and Magnetism.  The Maxwell stress has been used to calculate forces and torgues on conductors in the vacuum.
  2. We now believe that the Maxwell stress has NO theretical significance in solid dielectrics.  Rather, the effect of electric field on stress in the dielectrics is derivable from a free-energy function, as a part of material law.
  3. For very special materials, such as a vacuum (if you consider it as a material), our theory recovers the Maxwell stress. 


I feel that your theory is an effective theory because of  you develop it with a two-component example, i.e., two plates and a vacuum/dielectric in between.   Could you explain (just briefly) how you would develop the theory with just a dielectric (i.e., one component) in an electromagnetic field?


Zhigang Suo's picture

Dear Biswajit: 

I'm not sure what you are asking.  I'm sending you pdf files of two papers by email.

  • Zhigang Suo, Xuanhe Zhao and William H. Greene, A nonlinear field theory of deformable dielectrics.  Journal of the Mechanics and Physics of Solids.
  • Xuanhe Zhao, Z. Suo, A method to analyze electromechanical stability of dielectric elastomers. Applied Physics Letters 91, 061921 (2007).

The JMPS paper gives a full 3D formulation.  The APL gives an example of application, and perhaps is easier to read.


I disagree.

I have skimmed the posts in this thread with great interest although I have not contributed in a month or so. Last night I was reading Lakes's textbook on viscoelastic solids, and then I started to fall asleep (no fault of the book, I was tired.) While in a semiconscious stupor I realized the following two things:

1. It is possible to relate stress directly to objectively observable quantities other than strain. There used to be a brand of special paper on the market that would change color when you squeezed it. I don't know how it worked (tiny crushable capsules of dye?) and I've never seen it in the US (I saw it in Japan). They used it to map the contact stress in bolted joints. Probably there is a kind of material that would change color with no motion required - a chemical reaction that requires stress to activate it. So...while it still may not be possible to directly observe stress, it can be observed through means other than strain. One can envision a constitutive equation for the paper directly relating stress to color.

2. Without this "stress paper" or a similar material, the relaxation modulus in viscoelasticty is unobservable, or at least has an uncertainty principle attached to it. The relaxation modulus is defined as the time history of stress when a step strain is applied to a body and held. But how is the stress measured? Only by measuring the deflection of another body, like a load cell, that the viscoelastic body is pulling on. But such a deflection cannot occur without violating the definition of relaxation modulus as the stress when strain is held constant. In practice, the load cell is very stiff compared to the viscoelastic material, so the strain is small compared to the initial step strain. In such a case the error introduced ought to be negligible, but it can never go away completely unless stress is measured in a way that requires no deflection to be observed.

I read through the posts with great interests and could not help but to register a user name for comments.</p> <p>In my opinion, the virtual work principle is probably one of the most fundamental rules in the framework of mechanics. Stress and strain are duals in the sense of energy or work. Definitions are arbitrary and the only rule one need to obey is the correctness of energy calculation for each pair of stress and strain definition. The ABAQUS theory manual has a section describing different definitions of strain and stress.

In the sense of energy calculation, the stress definition can be deduced from the strain definition, and vice versa. There is no certainty which one should come first. When deformation method is used, it is convenient to define strains or general strains --- this is the approach adopted in many FEM analyses. Nevertheless, when force method is used, for example in finite element limit analysis following equilibrium approach, only stresses are defined explicitly. The dual of each equilibrium constraint (see for more details) may have physical means and can be used to plot deformation field.

If one would like consider more fundamental than the abstract level of mechanics, then I would think strain in general means indicates the state of an object when compared to its reference state. Meanwhile, stress in general means indicates the interaction between two objects. The change of states is resulted from stress or force – does this means force (or stress) is more fundamental? – I do not know.

When material’s property is concerned, the constitutive relationship describes how the strain is to be altered as a response to the stress.

As to the measurement, stress is usually measured based on the assumption of repeated (note: not necessarily linear) correlations between stress and strain, both in general means, and mechanical-electronic transformations. It is helpful to consider what “calibration” of a modern measurement instrument really means. I find it difficult to tell which component is more fundamental from the way of measurement. For example, when strain gauge is concerned, can we say resistance is more fundamental than strain? 

Zhigang Suo's picture

Thank you very much for putting things in perspective.  What you said is good if you have no doubt what a force is.  In a previous entry to this thread, I was pointing to an important case where force itself is an ambiguous notion.

Amit Acharya's picture

just a note related to the comment

In my opinion, the virtual work principle is probably one of the most fundamental rules in the framework of mechanics. Stress and strain are duals in the sense of energy or work. Definitions are arbitrary and the only rule one need to obey is the correctness of energy calculation for each pair of stress and strain definition. The ABAQUS theory manual has a section describing different definitions of strain and stress.

(and perhaps a bit of an aside for this thread)

Hill's "Invariance in Solid Mechanics" article in Advances in Applied Mechanics (I believe it was 1978) is the definitive source on this sometimes subtle topic. In my opinion, much for the student of continuum mechanics to learn from.

Sorry for keeping you all waiting for so long. But I am certain all of you will excuse me if you come to know that the major reason behind the delay was that I was very busy writing my PhD thesis. (Yes, I can affirm that the things given by one "Joh3n" on this URL really do come in handy!)

Coming back to the present topic.

Please hold the full context in mind: the concepts of stress and strain both are abstracted from the same set of physical facts. Essentially (and I am talking off-hand here, without much rigor), these facts may be summarized as the following: apply a set of forces to a body and see that even as the body is able take those loads, it changes its geometry in the process of sustaining them.

Do you know what is the most fundamental part in this description? It is *not* the fact that the body is able to take the loads. It is not that bodies deform, and that they do so in various ways: time-dependent or otherwise, with local rotations or otherwise, characteristized with displacement gradients and strains or otherwise. It is not even that work or energy is expended in the process or that the actual process, seen as occurring at every part of the body, carries some attributes which is captured via a variation in something somewhere sometime.

The most fundamental part is not any of these. There is something much more fundamental than all of that.

It is this: In mechanics, our primary defination of force is a dynamical one---it arises in reference to the (second order differential) changes in the position of a body with respect to time. Here, no body moves. And yet, we take it that the concept of force is at all applicable to the above description.

*That*, in my opinion, is the most funny part---the part that really sets the stage for every logically subsequent develpment such as the variational principles.

The concept of stress selectively focuses only on the force-related aspects of this basic fact; the concept of strain selectively focuses on the geometry part of the same fact.

Let me illustrate the nature of the the observational basis---the inductive basis--here, by taking a simple example. Situation 1. You take a ball and hit it with a bat. The ball flies--it suffers acceleration. We can introduce Newton's ideas here by saying that there was a force acting on that ball. Situation 2. You take the same ball and hit it with two oppositely moving bats at the same time. The ball remains where it was, more or less, though it wobbles a bit. Closer inspection (or more accurate experimentation---as done by Hooke) reveals that the different parts of ball also have suffered relative displacements (it has deformed). But the center of the mass of the body, more or less, as remained the same. Yet, we have agreed to associate a force with each moving bat separately, via our conceptualization in Situation 1. So, what has happened to the forces here? Have they disappeared? The answer is: Nope. Both of them are acting there. We thus have equivalence of a dynamic and a static situation.

(Skipping some more steps in the inductive reasoning for the sake of brevity) We can now ask: If both the forces act on the ball as a whole, they must act on each part of it as well. If so, what are the magnitudes and the directions with which an arbitray part may affect its neighbourhood? Pursuing this line of thought, we think, we could express how forces transmit within a volume. But remember, by the very nature of the abstraction process involved, stress must refer to a geometric element.

Similarly, it is on the basis of empirical observations that every load sustaining body "yields" (deforms, distorts, or undergoes some *relative* displacements) that we can define strain.

Please note, both stress and strain are inductively defined without reference to virtual work or any form of variational principle. Lesson: The ideas of stresses and strains are *more* fundamental than that of the variational principle.

Since stress and strain are abstracted from the *same* set of empirical observations, there is little wonder that they are intimately related. (Contrary to Grant Henson's comment no 5151 above, experimentally, color changes without geometric changes have never been observed.) But this does not mean the asserted "duality" here carries some kind of a fundamental relevance. Firstly, there is no duality here, not in the sense of the wave-particle duality of quantum physics anyways; and secondly, whatever it is which goes under the mathematical name of "dual" here, has no fundamental physical relevance. How come? Refer to the inductive roots of the concerned concepts---and hold the full context in mind---that's all!

Since I have already shown the inductive roots of this issue, I need not separately answer many derivative issues. These include the invariance part of it. (Easy to see at least for infinitesimal stresses and strains that some facts of invariance are just consequences of extra assumptions---e.g., as I have been highlighting above, the fact that the rotation tensor is subtracted, purely as a matter of convenience of analysis---not out of any physical compulsion about it.)
Though I won't address all the derivative or secondary issues, I would still like to at least point out a few relevant facts/points.

The concept of "energy" may be a thread running through much of physics---at least that which gets studied on a preliminary or elementary level. A lot of modern texts have this tendency to portray as if this concept of "energy" is, therefore, a valid unifying theme of physics. Especially popular in the most mystical era of physics---the 20th century, esp. after 1960s---the notion is without grounds. (Halliday and Resnick have got this one thing dead wrong.)

It is also wrong, therefore, to believe that variational principles thereby become fundamental (by way of their association through the principle of virtual work). They don't. Their inductive basis does not permit them that status.

Note, historically, even the formalization of stress as a 3X3 array of numbers (i.e. as a tensor) had occurred at least one generation earlier (in 1822) before any steam began gathering to even formulate the energy principle as a broadly applicable principle. (Helmholtz's clear formulation occurred sometime *after* mid 1800s.) Note also that even the idea of entropy had been well grasped by Sadi Carnot before the idea of energy as a conservative principle was. Obviously, inductively speaking, both entropy and stress are the concepts that are *simpler* or more elementary than that of energy.

One good way to look at it is this. Just because the monotonocity of entropy changes is a fundamental law (simpler than energy conservation), do you therefore plan to invoke it in solid mechanics? Of course not!

So, where does the popular misconception about the supposed fundamentality of virtual work and varational principles *in solid mechanics* really come from? The real answer of course is that it purely comes out of the blue! But, practically speaking, the answer is that simply because someone wrote it that way on the black-boards of class-rooms. That's why!

But still, does it mean that that particular idea has any inductive basis? An answer to that is very easy to see. Just check if the people who assert its fundamentality can cite any inductive (empirical, if you wish) evidence or not. The fact is,, regardless of whether they have won the Timoshenko Medal or not (starting from Timoshenko himself), they never have given any inductive reason for their asssertion. No basis whatsover. In fact, such a basis cannot at all be given. The case rests.


I agree there is no rational basis for saying that energy is any
more fundamental than force. I will continue to believe this until
someone shows me a problem that can be solved by energy methods but
cannot be solved (even in principle) by force methods.

However, I don't agree that stress and strain are abstracted from the same empirical
observations. I do not know of any method for directly measuring a
force or a stress. But I can easily observe motions with reference to a
stationary scale, from which you can calculate your favorite
deformation or strain tensor.

Ajit, continuing with your bat and ball example, we can calculate the force on the ball by any number of different theories of increasing complexity (analytical dynamics, linear elasticity, nonlinear elastodynamics, etc.) and we will get a different answer for each theory. In some sense, the answers are not even commensurable, because point forces do not exist in continuum theories; they have to be accepted as stress resultants. So I can't agree with you that the the static and dynamic situations are equivalent.   The forces that appear in static theories (i.e. theories with no concept of inertia) are fundamentally different from the forces that appear in dynamic theories.  Because of this, I claim that forces are theoretical constructs with no basis in mere observation. Or maybe it's the converse, and I believe that since forces are theoretical constructs, forces defined in two different theories cannot be equivalent. In any case, it is fun to think about, for a while anyway

Incidentally, I had speculated in comment 5151 that there is a
material that will change color but not shape under stress. Ajit
claims that a color change has never been observed without a
geometric change. I still think such a material is plausible, but
even if one exists, it would not change the substance of our
discussion. We would simply be replacing strain by color.

Hi Grant,

I reply in an order that is slightly different from that in your above post.


Well, Grant, you caught me right on the weakest point of my comment no. 5988 above! Looks like I have no option but to write even *more*.... Phew... BTW, by now, I have given up the idea that at least as far as this thread goes, a stranger could read only the last few comments and thereby get a good picture of what is being discussed and why. It has already become impossible. So, I will just assume that people have been following this thread well continuously, and jump right in the middle of it all...

There *is* some kind of an equivalency between the dynamic and static viewpoints, though what I directly wrote in my above description (in comment no. 5988) is not enough to see it. Therefore, my above description is also a bit misleading in a way. Let me clarify....

Consider afresh a new example. A ball rests at the center of a flat horizontal table. Consider two different scenarios, each with the ball initially at rest. (1) Manually hit the ball with a striker, and the ball moves. It is obvious that a force has been operative here. But how do we know this? Answer: by measuring the motion of the ball and concluding that it has suffered acceleration (for a very short period of time). This is a plain dynamical situation. (2) Let a horizontal spring remain touching against the ball. Holding both the base of the spring and the ball, compress the spring. Is there a force operative here? Strictly speaking, within the purview of mechanics theory alone, as yet, we do not know! Not by our dynamical definition of force anyway. Now, suddenly release the hand holding the ball. The ball flies, exactly as in (1). Now, reconsider the question: has there been a force operative here? Yes. Why? Because the ball has experienced acceleration (for a very short period of time).

Since the motion of the ball is identical in the two cases, there is a certain equivalence to the two situations. (I take equivalence to mean: equal valence or "powers," i.e., the ability to produce equal consequences in an appropriate context.)

Now, what is the equivalence here between? It is between a moving (or dyanamic) striker, and the essentially compressed spring (whose own momentum doesn't count).

It is true that the spring is not static in the sense that its front-end does move. Yet, the essential point here is that a moving spring by itself does not produce a big enough acceleration of the ball because the mass of the spring is far too small for it to achieve that. What really matters with the spring, what really defines a spring, is its ability to deliver the punch as a consequence of its compressed state. Its compression is what matters---not its momementum when it is taken as a whole object. It is for this reason that we do not associate a dynamical momentum transfer from the spring to the ball the way we do it for the case of the striker.

Now, the question is: Can we, can we not, associate a force with the compressed spring when it is not at all moving? The answer is that yes, we can indeed associate a force.

Now, since there are no moving parts in the essentialized description, can we or can we not take the liberty to describe it as a static situation? Of course we can describe it as a static situation.

So, though the primary definition of force *is* that it is the time rate of change of momentum, it is important to realize that the momentum in question is that of the ball, not of the causal agent (making a change in the momentum of the ball).


The example of moving bats which I earlier took above, doesn't convey this idea of a certain equivalence between the static and dynamic situations very well. I should have taken one more step of introducing springs on both sides of the ball, and compressing them both. Then, it would have made the idea clear.

As a generalization, there is no way we can associate forces in a static configuration except as representing a potentiality of the corresponding dynamical situation. (Though in freshman engineering courses they teach you statics earlier in the course sequence than dynamics, the fact is, they had already taught you enough of dynamics in XII standard (or American high school) so that you can at least define what force is, in the first place.)

Another point. Note how this can get to be an epistemologically dangerous issue. If you are careless about it, it can easily seem as though a potentiality and an actuality are being asserted to be at par with each other (i.e. to be one and the same thing). If so, *that* would certainly constitute a real error. I didn't mean to commit this error, but I *was* being too brief in my reply above, skipping steps, and not organizing the write up too well, so it does seems as if I was making that kind of an error... So, Grant was quite right in catching me there. He is right in that you can't equate a potentiality and an actuality. What you *can* do is to assert the equality of the force that would be produced once the applicable potentiality is actualized. (As a philosophically interesting aside, notice, the actuality of compressing the spring precedes the potentiality of the spring releasing the force. So, contrary to what many believe, actuality *precedes* potentiality, i.e. actuality *does not follow* potentiality. This particular gem of a generalization was first observed by Aristotle.)

Anyway, to conclude now, note that the possibility of the abovementioned kind of epistemological error does not mean that there are absolutely no grounds for having an equivalence. There are: The motion of the ball in both cases remains identical.


Another point. I repeat, the set of empirical facts behind the concepts of stress and strain is the same. In principle. Let me now just illustrate one application.

You say that you can measure motions but not stresses. My question is: But how about experimentally measuring the motions of interior points? You will have to take a cut, either physically or in thought. If so, then just do the identical for stresses also. The only difference would be that in the case of strains, you will measure displacements of the parts while they were still held adjacent to each other in the forced configuration; in the case of stresses, you will measure accelerations of the cut away parts after they have been released and allowed to fly (as per the stresses operative across the cut).


About your speculation in comment 5151. Yes, I now realize it better. It's a valuable point. We would simply be replacing strain by color.

BTW, please also note, color is arguably a scalar i.e. a frequency (unless you wanted to create an abstract RGB space for it) whereas even a strain across a face would involve a set of two separate vectors---one in the plane of the face and another normal to it. So, even in field visualizations, a color scheme could be used only to indicate the scalar measures like volume changes or effective stress/strain, or, at the most, displacements (using that abstract RGB space), but not a tensor like strain.


Finally, coming to the denial of fundamentality of energy. Thank you, Grant. I am glad that you concurr that there is no rational basis for saying that energy is any more fundamental than force is.

Though it is so true, a century of relativity theory later, that statement does shock a lot of people out of their wits as soon as one says it!

Not that you would require it, Grant, but let me go ahead with another way of putting the same fact. This second way is in the form of a counter-question which I like to raise whenever the listener refuses to grant me the possibility of the fundamentality of force....

The question which I then raise is this: Can you think of any physical situation / process / transaction / interaction etc. wherein energy gets transferred but not momentum?



I am enjoying this discussion immensely. When I bring these points
up to my casual acquaintances, they slowly migrate to the other side
of the room.

Manually hit the ball with a striker, and the ball moves. It is
obvious that a force has been operative here. But how do we know
this? Answer: by measuring the motion of the ball and concluding
that it has suffered acceleration (for a very short period of time).

In order to conclude this, don't you have to accept the theory that
motions are caused by forces and not, for instance, the thoughts of
God? All I am saying is that I can measure displacements directly,
with no need to accept any kind of mechanical theory. And on that
basis I claim that motion is more fundamental than force.

This is a plain dynamical situation. (2) Let a horizontal spring
remain touching against the ball. Holding both the base of the
spring and the ball, compress the spring. Is there a force operative
here? Strictly speaking, within the purview of mechanics theory
alone, as yet, we do not know! Not by our dynamical definition of
force anyway. Now, suddenly release the hand holding the ball. The
ball flies, exactly as in (1). Now, reconsider the question: has
there been a force operative here? Yes. Why? Because the ball has
experienced acceleration (for a very short period of time).

Since the motion of the ball is identical in the two cases, there is
a certain equivalence to the two situations. (I take equivalence to
mean: equal valence or "powers," i.e., the ability to produce equal
consequences in an appropriate context.)

I agree that a force acted in both situations. However, as you have
posed the problem, we need two different theories to calculate the
force: analytical dynamics to calculate the force from the motion of
the ball, and elasticity to calculate the force that can be exerted
by the spring.

If I could calculate the force by each theory and get the same answer, I could agree that force seems to have some underlying significance independent of any particular theory. But I do not believe you would get the exact same answer, even if you had exact measurements. For example, if you calculated the force from the spring, you would really get a time-varying quantity. You could then only say that the impulse was the same, and I'm not even sure about that.

Another way to pose the thought experiment is within a single theory
that can handle both situations- for instance, elastodynamics. In
that case, we truly have the same force, but we have a priori
decided that we are observing two special cases of a general theory,
so there is no way to use these observations to tie two notions of
force together - there is only one notion of force, that found in

I will concede that the same kind of hairsplitting logic could also be applied to the motion - we have to assume the scale didn't grow or move, etc. But I can accept that more easily than I can accept the theory relating force to motion.

You say that you can measure motions but not stresses. My question
is: But how about experimentally measuring the motions of interior
points? You will have to take a cut, either physically or in
thought. If so, then just do the identical for stresses also. The
only difference would be that in the case of strains, you will
measure displacements of the parts while they were still held
adjacent to each other in the forced configuration; in the case of
stresses, you will measure accelerations of the cut away parts after
they have been released and allowed to fly (as per the stresses
operative across the cut).

But I still need a theory that relates accelerations to stresses. I
don't need a theory to measure the deformations.

The question which I then raise is this: Can you think of any
physical situation / process / transaction / interaction etc.
wherein energy gets transferred but not momentum?

I need a theory in order to be able to talk about energy and
momentum. So I'll choose the theory of classical thermodynamics. I
can transfer energy to steam by heating it up, without imparting any
momentum. (Classical thermodynamics doesn't posit any relationship
between energy and the momentum of the steam particles.)

You may argue that classical thermodynamics is a limited theory, but
so is every other known theory. A stronger objection is that
classical thermo is not only limited but has been superseded by a
better theory, statistical thermodynamics, in which internal energy
is related to the momentum of the constituent particles. In that case, I would have to think harder.

chao's picture

 This is an interesting topic.  Without reading the long contributions by so many experts here, let me just point out that in Physics 101, there are (only) three basic physical quantities, length, time and mass.   Strain is the change of length and therefore a basic physical quantity that can be measured.   Stress is derived from strain through a formula.   As such, for a  given strain (which is the deformation of a body) one can get any stress value depending on what formula is used, e.g. linear elastic, visco-elastic, plastic.   NIST has the standard to measure length, not stress !!!     In the class of Experimental Stress Analysis, one in fact does "experimentally determines the strain".

Bill Y.J. Chao

I'm not sure I agree that there are only those three physical quantities. In particular, temperature, electric current, and luminous intensity come to mind. It is certainly the case that stress can be represented by length, time and mass (ie: mass / length / time^2), but I am not sure I believe those are all that is needed in all of physics or even all of mechanics.

That being said, I believe your genral point is correct—straiin is the fundamental quantity. At least that is the one we observe. No one has ever observed a force. We hypothesise forces to explain the motions we observe.   


Furui Xiong's picture

struggle between theory and application


 I have modelled one cube concrete specimen in ansys . I want to plot stress-strain curve (static analysis , nonlinear solution )  .

can you help me(why stress-strain curve is linear) 


can u give me link to henry tans blog regarding this topic


Nithesh P

Amit Pandey's picture

The concept of yield point is very loosely defined so far..this manuscript brings a fresh look to this old debatable question.  

Just to note: I think I have reached what I think is a proper resolution of this issue in my mind.



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