iMechanica - measurement - Comments

Re: strain is the basic mesurable physical quantity

Mike Graham — Fri, 30 May 2008 00:59:22 -0400

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.

strain is the basic mesurable physical quantity

chao — Thu, 29 May 2008 21:04:40 -0400

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

Another nice paper on soft tissue strain measurement

MichelleLOyen — Thu, 03 Apr 2008 08:39:01 -0400

H. Lu et al did some beautiful work on higher order effects in Digital Image Correlation, see

Deformation Measurements by Digital Image Correlation: Implementation of a Second-order Displacement Gradient

by H. Lu and P.D. Cary, Experimental Mechanics 40 (2000) 393.

Most algorithms for strain measurement are direct and do not require an assumed functional form to the constitutive model.

My concern is in the time-dependent mechanical behavior of soft tissues in this context. If you deform the sample rapidly and use these data to examine local strains, you might get a very different picture than if you examine the equilibrium configuration, where physical rearrangement of the tissue microstructure might be possible.

egg vs. hen

Zaoyang Guo — Wed, 02 Apr 2008 20:08:00 -0400

Dear Tammy,

Very interest topic. One of my research interests is to develop hyperelastic constitutive model for soft tissue. If the strain measurement requires the constitutive model to be pre-determined, then the experiment vs. model is like egg vs. hen. Which one should come first?

I cannot access the papers you mentioned currently, can you introduce more about the resolution of the strain measurement?

Many thanks,

Zaoyang

Reply to Ajit - force, motion, energy, momentum

Grant Henson — Fri, 30 Nov 2007 13:44:33 -0500

Ajit,

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
elastodynamics.

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.

Reply to Grant Henson: Expanding on "Static" and "Dynamic"

Ajit R. Jadhav — Fri, 30 Nov 2007 07:53:48 -0500

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).

OK.

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?

Different forces in different theories

Grant Henson — Thu, 29 Nov 2007 15:56:56 -0500

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.

Re: Duals, Energy, Variation, Invariance, and Fundamentality

Ajit R. Jadhav — Thu, 29 Nov 2007 06:05:39 -0500

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.

work conjugacy

Amit Acharya — Sat, 20 Oct 2007 22:39:19 -0400

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.

Force itself can be an ambiguous notion

Zhigang Suo — Sat, 20 Oct 2007 16:22:21 -0400

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.

Stress and strain are duals

Xiaobo Yu — Sat, 20 Oct 2007 10:10:49 -0400

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 www.gams.com 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?

Stress observability and the relaxation modulus

Grant Henson — Fri, 21 Sep 2007 12:52:54 -0400

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.

JMPS and APL papers on deformable dielectrics

Zhigang Suo — Tue, 04 Sep 2007 07:39:22 -0400

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. http://dx.doi.org/10.1016/j.jmps.2007.05.021
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.

Ajit R. Jadhav — Tue, 04 Sep 2007 03:30:49 -0400

I disagree.

Why is Zhigang's theory an effective theory?

Biswajit Banerjee — Mon, 03 Sep 2007 23:23:17 -0400

Zhigang,

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?

Biswajit