Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyper-elasticity. However, in its original form the theory of elasticity does not assume the existence of a strain energy function. In this case, called Cauchy elasticity, stresses are directly related to the strains. Since the emergence of modern elasticity in the 1940s, research on Cauchy elasticity has been relatively limited. One possible reason is that for Cauchy materials, the net work performed by stress along a closed path in the strain space may be nonzero. Therefore, such materials may require access to both energy sources and sinks. This characteristic has led some mechanicians to question the viability of Cauchy elasticity as a physically plausible theory of elasticity. In this paper, motivated by its relevance to recent applications, such as the modeling of active solids, we revisit Cauchy elasticity in a modern form. First, we show that in the general theory of anisotropic Cauchy elasticity, stress can be expressed in terms of six functions, that we call \emph{Edelen-Darboux potentials}. For isotropic Cauchy materials, this number reduces to three, while for incompressible isotropic Cauchy elasticity, only two such potentials are required. Second, we show that in Cauchy elasticity, the link between balance laws and symmetries is lost, in general, since Noether's theorem does not apply. In particular, we show that, unlike hyperleasticity, objectivity is not equivalent to the balance of angular momentum. Third, we formulate the balance laws of Cauchy elasticity covariantly and derive a generalized Doyle-Ericksen formula. Fourth, the material symmetry and work theorems of Cauchy elasticity are revisited, based on the stress-work 1-form that emerges as a fundamental quantity in Cauchy elasticity. The stress-work 1-form allows for a classification via Darboux’s theorem that leads to a classification of Cauchy elastic solids based on their generalized energy functions. Fifth, we discuss the relevance of Carath\'eodory's theorem on accessibility property of Pfaffian equations. Sixth, we show that Cauchy elasticity has an intrinsic geometric hystresis, which is the net work of stress in cyclic deformations. If the orientation of a cyclic deformation is reversed, the sign of the net work of stress changes, from which we conclude that stress in Cauchy elasticity is neither dissipative nor conservative. Seventh, we establish connections between Cauchy elasticity and the existing constitutive equations for active solids. Eighth, linear anisotropic Cauchy elasticity is examined in detail, and simple displacement-control loadings are proposed for each symmetry class to characterize the corresponding antisymmetric elastic constants.
Ninth, we discuss both isotropic and anisotropic Cauchy anelasticity and show that the existing solutions for stress fields of distributed eigenstrains (and particularly defects) in hyperelastic solids can be readily extended to Cauchy elasticity. Tenth, we introduce Cosserat-Cauchy materials and demonstrate that an anisotropic three-dimensional Cosserat-Cauchy elastic solid has at most twenty four generalized energy functions.
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concept of stress
I am thinking of the definition of stress. Is it necessary to modify the limit process, such as not reaching a certain point or considering other effects of stress? May I get some ideas from the article or from the other articles? It's just an instinct triggered by the concept of stress.
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In reply to concept of stress by yangaohu
Stress is a continuum concept
Stress is a continuum concept. However, it is independent of constitutive equations. In other words, one first establishes the existence of a stress tensor (Cauchy’s Theorem). Constitutive equations come next. The definition of stress does not require the existence of an energy function. In both hyperelasticity and Cauchy elasticity, there is a well-defined notion of stress. In hyperelasticity, stress is derived from an energy function and related to strain through it. In Cauchy elasticity, an energy function does not necessarily exist; stress is simply a pointwise function of strain.
Regards,
Arash
In reply to Stress is a continuum concept by arash_yavari
Thank you very much for your
Thank you very much for your helpful introduction.
Best regards,
Yangao
Variational Principle(s) for Cauchy Elasticity
Hello Arash,
While what you write is true about Cauchy elasticity vis-a-vis Balance laws and symmetries being lost since Noether's theorem does not apply, I believe there is a question here of in what variables. For general Cauchy elasticity there is no strain energy density function that can be constructed - consequently no energy functional whose E-L equations give the equations of elastostatics, say. However, with (a family of) appropriate changes of variables the balance equations of nonlinear elastostatics and elastodynamics for a Cauchy elastic material do admit variational principles: see last line of Introduction and sec. 6.1 of
An action for nonlinear dislocation dynamics: https://www.sciencedirect.com/science/article/pii/S0022509622000291?via…
and
A hidden convexity of nonlinear elasticity
https://link.springer.com/article/10.1007/s10659-024-10081-w
Clearly, the balance laws remain the same with the Dual-to-Primal mapping substituted as the change of variables (this is the defining idea of the scheme). And with the defined family of dual functionals - being bona-fide variational principles - there would be a link between symmetries of those functionals and the resulting Euler-Lagrange equations - which are the equations of elastostatics and dynamics, just stated in terms of different, adapted, variables.
It's an interesting question to follow up on.
- Amit
In reply to Variational Principle(s) for Cauchy Elasticity by Amit Acharya
Re: Variational Principle(s) for Cauchy Elasticity
Thanks, Amit! This is interesting. I wonder if these new variables have any physical meaning or interpretation?
Not sure if this is completely related, but Michael Berry and his collaborators have done a lot of work on “curl forces.” In this paper (https://royalsocietypublishing.org/doi/10.1098/rspa.2015.0002), they show that a subset of curl forces admit Hamiltonians that are of the form: an anisotropic kinetic energy plus a scalar potential. There may be a connection between that and what you’re doing. It’s definitely worth exploring.
Regards,
Arash
Re^2: VP for Cauchy Elasticity
Thanks, Arash, for the link to Berry's paper, which I was not aware of. In the context of Newtonian particle mechanics, we have carried out this program - and for any type of forces, conservative or dissipative, along with accounting for nonlinear constraints:
Action principles for dissipative, non-holonomic Newtonian mechanics
So this is more general than what is considered in that paper and shows a way to construct an action for any type of given force system (including the curl forces mentioned).
As for your question about the physical interpretation of the new variables in the dual formalism - in my thinking, all the physics is in the primal system of PDE/ODE/constraints, and the dual variational principle and its associated dual variables (Lagrange multipliers) are just a mathematical device - a very strong one because of the dual principles' guaranteed convexity, regardless of the convexity/monotonicity status on the primal side - for solving the primal problem; we have exploited this in a series of works, both rigorous and computational and more is to come. Of course, one knows from experience that Lagrange multipliers often have physical meaning, so may be these do as well - however, here there is an infinite set parametrized by the choice of the auxiliary potential H. So I think the scheme associates a certain type of gauge invariance with the primal system in the way it works....I am happy to devise/construct strong primal physics, and have a robust mathematical scheme to go after the often non-standard nonlinear equations that show up in the primal physics I think about.
Dissipative vs. Non-Conservative Forces
Thanks, Amit. Just a quick comment: “Curl forces” are non-conservative but also non-dissipative, and so is Cauchy elasticity. I don’t think this affects your variational formulation, but it’s worth emphasizing that non-conservative forces are not necessarily dissipative.
Regards.
Arash
Non conservative forces ….
Yes Arash, of course :)
the surprise (for me) really is that even something that does not conserve energy has a dual variational principle, and if the forces were time-independent, the corresponding dual Hamiltonian would be conserved along the flow yielding a conserved quantity (over a short period of time at least, in which the primal problem would be dissipating energy).
It is discussed in the context of the Lorenz problem.
Re: Non conservative forces ….
Hi Amit,
We’ve observed something similar for curl forces. One can define an auxiliary conservative force that admits a Hamiltonian. A curl force, in general, does not conserve energy but still has a conserved quantity, though it’s clearly not the physical energy. You can find the details in another paper I just posted: The Darboux Classification of Curl Forces.
Re: non conservative, non dissipative.....
Hello Arash,
The dual Lagrangian (rather than the conserved Hamiltonian) is extremely useful in existence proofs for very difficult primal problems which, when they arise from a primal variational principle like nonlinear hyperelasticity, may not have existence of minimizers - see the existence proof for variational dual solution for the SVK material, in the 'Hidden convexity of nonlinear elasticity paper' link I gave earlier and also an interesting, stable elastodynamic computation with softening. Also, existence proof for Chern Simons theory whose primal energy has a cubic nonlinearity - so no existence of minimizers or maximizers. More has been done for the Euler equations, the Navier Stokes equations, the Pontryagin maximum principle for controls etc. - this is all recorded in the work of my group and with collaborators and there should be a reasonable record on imechanica.
So it is extremely useful once one gets to the business of actually solving nonlinear PDE/ODE (with or without approximation), beyond formulating theories of continuum mechanics (which is essential, and of which I have done a few! and solving which provided the motivation for the scheme).
Of course, one never knows - even the Hamiltonian of the dual Lagrangian is probably going to be useful - in understanding qualitative features of the dual dynamics which then directly map back to the primal dynamics (or statics).
Another aspect is a path integral formulation of primal problems that have no variational principle.
Yet another is the calculation of periodic orbits of a chaotic system.
Thanks for your other paper - it is intersting, I'll take a look. The dual scheme, although in different variables, produces dual Lagrangians on demand (which have conserved Hamiltonians under the usual assumption to go from a Lagrangian to a Hamiltonian, see the Lorenz example worked out in my Newtonian dynamics paper)... so many conserved Hamiltonians, in principle, in dual variables....
I'll write up my findings on dual formulation for particle dynamics with curl forces at some point.
I am posting here our offline correspondence on the discussion on non-conservative/non-dissipative materials. I think it is useful for someone trying to follow and tie up loose ends.
--------- Forwarded message ---------
From: Yavari, Arash ay34 [at] gatech.edu>
Date: Sun, Jun 22, 2025 at 7:45 PM
Subject: Re: question
To: Amit Acharya acharyaamit [at] cmu.edu>
Hi Amit,
In the paper we haven’t explicitly defined this (we should have).
A Cauchy elastic solid is non-conservative if d\Omega \neq 0.
And I completely agree with your comment regarding dissipative forces.
In reply to Re: non conservative, non dissipative..... by Amit Acharya
Re: non conservative, non dissipative......
Thanks Amit, for posting our offline correspondence.
It’s interesting how powerful the "right" change of variables can be in mechanics, and more generally in physics.
Regards,
Arash
N-particle chain with non-conservative forces
Arash,
Here is a paper about the dual formulation of a mechanical system with and without damping allowing up to quadratic, full interactions anongst particles, which I think allows for a force analogous to 'curl forces' in 2 or 3 space dimensions, in the sense that the forces allowed for the 1-d particle chain cannot be written as the gradient of a potential when the matrix A is non-symmetric (and of course the matrix B is also general so can contribute to non-conservativity, so when not symmetric in first and second or third index). The force field can be formulated without the barred fields in it as well, w.l.o.g.
In this sense, when the damping parameter d = 0, and A nonsymmetric, this is a non-conservative system. The dual scheme is formulated for this and there are remarks about the corresponding Hamiltonians.
Variational principle for a damped, quadratically interacting particle chain with nonconservative forcing
(the use of non-conservative in the title wasn't because we thought that non-conservative forces was a big deal in itself, but it is covered in retrospect.)
https://link.springer.com/chapter/10.1007/978-3-031-58665-1_15
imechanica should have an accessible version of the paper as well.
The at most quadratic nonlinearity is to be able to write out everything explicitly, but the idea is general - e.g. as shown for Saint-venant Kirchhoff nonlinear elastic material, rigorously.
In reply to N-particle chain with non-conservative forces by Amit Acharya
Re: N-particle chain with non-conservative forces
Thanks, Amit! It would be interesting to see how your method works out in one of the curl force examples studied by Michael Berry or us.
Regards,
Arash
Re: curl force f(xy^2,x^3) by dual scheme
Hello Arash,
I did the Berry example which does not admit an anisotropic Hamiltonian (which you and AG adapted for sign) by the duality scheme and produced for it an explicit dual Lagrangian (which of course required to produce an explicit Dual to Primal mapping). When the Lagrangian is strictly convex in the dual velocities, it can be Legendre transformed and that gives the Hamiltonian, which will be conserved, as long as the base states used do not depend on time - for the pain-in-the-behind explicit calculation, I took the base state = 0.
At any rate, the important thing is that the E-L equations of the dual Lagrangian gives exactly the primal system with the DtP mapping (explicitly determined) stuck in. So one can solve the dual problem (a well posed elliptic boundary value problem in time!) and recover the primal solution through this mapping. And developing the mapping is simply an algebraic solve.
This is very different from what I gathered in your approach. First of all you have to solve the PDE for the potentials. Then you produce the auxiliary dynamics from the scaled auxiliary Hamiltonian. Ok, so presumably you solve the Hamiltonian system as that affords advantages - but it is not at all clear (or is it to you?) that you can recover the primal solution given the solution of the auxiliary Hamiltonian dynamics, e.g., if I look at your equations 2.50, 2.51, or 2.53, 2.54 (the other way around is ok). So apart from being able to say that your Hamiltonian is conserved (under lots of assumptions, e.g. V(x) \neq 0) are you able to say anything more in terms of it being useful?
With regard to our scheme, computing with it would be no different than whether the primal system is conservative, has curl forces or it is dissipative. To see one such computation in the ODE context you can check out
Sec. 5.3 of Hidden convexity in the heat, linear transport, and Euler’s rigid body equations: A computational approach, Quarterly of Applied Mathematics, 82, 673-703, https://arxiv.org/abs/2304.09418
which does Euler's equations for rigid body rotation (nonlinear). One intersting thing to note is that without using any symplectic anything but only the plain vanilla dual scheme we get energy to be conserved (in the conservative case, of course) Fig. 10
I'll try to write up this curl force example when I find some time, but if you want to see the computation (handwritten, ipad) let me know.
You asked me if the duality scheme works out in one of your examples - it does, but since our paper preceded yours (including the CMDS conference one) and you were not apparently aware of them, maybe I should be the one asking you how your scheme works out for the quadratic nonlinearity case in R^N, not 2 and 3, including non-conservativity and/or dissipation? :)
Jokes apart, this has been a fun exchange.
- Amit