On pseudo-rigid bodies

David J Steigmann

Abstract

The concept of the pseudo-rigid body, a model of hypothetical bodies constrained to deform homogeneously, is discussed critically. An analysis is given of a recent attempt, published in this journal, to establish this model on the basis of continuum mechanics.

Keywords:

1. Introduction

In this note, the ideas underlying the pseudo-rigid body concept are examined critically. In §2, a conjecture is offered about the motivation one might have for studying a model of this kind. Later, the idea put forth in Casey (2004), to the effect that pseudo-rigid bodies are continua endowed with certain local reactive stresses associated with a global constraint on possible motions, is examined. To this end the classical Lagrange multiplier rule, which forms the basis of the usual decomposition of the local stress into constitutive and reactive parts in the presence of local constraints, is reviewed in §3. This in turn sets the stage for the discussion, in §4, of the unjustified use of Lagrange multipliers to support a similar decomposition of stress as part of the definition of pseudo-rigid bodies proposed in Casey (2004). In §5, it is demonstrated that the model yields conclusions which, in the opinion of the present writer, are untenable. The note concludes with a brief discussion of related work on the mechanics of thin bodies.

2. Motivation

The central feature of the pseudo-rigid body model is the stipulation that bodies of finite size deform homogeneously even when subjected to localized forces due, for example, to contact with other bodies. This means that bodies deform in such a way that the strain distributions within them remain uniform during an interval of time which may span an impact event, for example. This, of course, is contrary to everyday experience.

This divergence of reality and the restrictions that would be imposed on it by promoters of the model leads naturally to the question of motivation. The simple answer would appear to be that all motions allowed by the model are expressible in the formEmbedded Image(2.1)where x is the position at time t of a material point of the body, X is its position in some reference shape, F(t) is the gradient of the deformation function Χ and c(t) is a vector function of time. The problem of determining the motion is, thus, reduced to one of integrating ordinary differential equations with time as the independent variable. These are, of course, far more tractable than the equations one normally encounters in any conventional model of the motions of deformable bodies.

The subject of ordinary differential equations is highly developed and in recent times have produced results of great import in dynamical systems theory; in particular, those emerging from the study of bifurcations and chaos. Some may have been inspired to adopt the pseudo-rigid body as a showcase for the application and exhibition of these developments. However, it appears that the fallacy of the pseudo-rigid body concept has been demonstrated in Papadopoulos (2001). This is explained in §4.

3. Constrained elasticity

Casey (2004) suggests that the pseudo-rigid body model may be based on the mechanics of continuous media. Thus, the body is conceived as a continuum subject to the global constraint that its motions be of the form (2.1). The experienced student of continuum mechanics has come to associate the word constraint with the concept of a reactive stress (Truesdell & Noll 1965), a contribution to the net stress supported by the material of the body which cannot be directly related to the motion that the body undergoes. That is, the reactive stress is not determined by any constitutive equation for the material. Instead, it is determined, conventionally, by adding it to the list of unknowns to be found by solving the initial-boundary-value problem at hand, a system which includes the constraint equations themselves. The so-called constraint (2.1) does not conform to this program. To illustrate the point, attention is directed here to nonlinearly elastic bodies in which the active stress—that part of the stress which is determined constitutively—is a function of the current value of the deformation gradient. Such materials are offered in Casey (2004) as prototypical examples of pseudo-rigid bodies.

In the conventional theory of elasticity, the constitutive response is described by a stored-energy function W(F). This is a scalar field over the reference shape of the body. The Piola stress P(F) is work-conjugate to F in the sense thatEmbedded Image(3.1)where the dot product is the conventional Euclidean inner product of nine-dimensional vectors and the superposed dots represent the derivative with respect to a parameter in an arbitrary parametrized set of configurations. In the absence of constraints, it is necessary and sufficient thatEmbedded Image(3.2)the gradient of W with respect to F. This expression for the stress follows whether or not the deformation is homogeneous. This is because the constitutive response of the material is assumed to be determined, as in Casey (2004), by the local values of the deformation gradient at the material point in question. Therefore, the restriction to motions with deformation gradients that are independent of X, while entailing no loss of generality insofar as constitutive response is concerned, is unnecessary. It is this fact which allows the theory of elasticity to be used, when appropriate, to describe motions that are not necessarily homogeneous.

The better-known Cauchy stress T is related to P byEmbedded Image(3.3)where Embedded Image is the cofactor of F, the superscript ‘−t’ stands for the transposed inverse.

If the motion is subject to a local constraint of the formEmbedded Image(3.4)then (3.1) does not determine the stress completely. This is due to the fact that only those Embedded Image which obeyEmbedded Image(3.5)are permissible. The stress P may now include an arbitrary multiple of GF without affecting (3.1). Thus,Embedded Image(3.6)where the scalar λ may depend on X and t. Further, the function W is to be regarded as a continuously differentiable extension of the strain-energy function from the submanifold defined by (3.4) to the enveloping nine-dimensional space. This extension may be otherwise arbitrary, as the use of one in place of another amounts to an adjustment of the (as yet undetermined) multiplier λ.

The representation (3.6) derives from the fact that the constraint (3.4) defines a differentiable submanifold of the nine-dimensional space of deformation gradients. At any point of the manifold, this space may be decomposed into the direct sum of the tangent space to the manifold and its orthogonal complement. Equation (3.1) requires that the nine-dimensional vector Embedded Image be orthogonal to the tangent space and, hence, that it belong to the orthogonal complement. The latter is spanned by the single vector GF, yielding (3.6) in which the scalar λ is (at this stage) arbitrary. As is well known, the same argument applies in the presence of several constraints Gi. If these are independent in the sense that the gradients Embedded Image are linearly independent, then the set Embedded Image spans the orthogonal complement of the tangent space to the constraint manifold ∩Gi and the representation (3.6) remains valid with λGF replaced by Embedded Image, where λi are scalars. These statements comprise the Lagrange multiplier rule, a theorem in the calculus of functions of several variables. The proof of this theorem is standard and widely known (e.g. Fleming 1977).

Thus is derived a rigorous decomposition of the Piola stress into the sum of an active constitutively determined partEmbedded Image(3.7)and a reactive part (in the case of a single constraint)Embedded Image(3.8)These in turn generate the decompositionEmbedded Image(3.9)in which the active Cauchy stress Ta and the reactive Cauchy stress Tr are related to Pa and Pr, respectively, by equations like (3.3). These active and reactive Cauchy stresses are separately symmetric if W and G are invariant under arbitrary rigid body motions superposed on the motion with local gradient Embedded Image. This invariance then yields a symmetric total Cauchy stress (Truesdell & Noll 1965). It bears mentioning that the various λs in the theory are to be regarded as actual stresses which the material of the body generates in response to the constraints. If this does not happen then the constraints are null and void and the material supplies the missing stresses through the mechanism of elasticity, or some other mechanism of a constitutive nature.

Granted the symmetry of T, the equation of motion isEmbedded Image(3.10)in which the dots now refer to material time derivatives, ρ is the mass density, b is the body force per unit mass and ‘div’ is the divergence with respect to the coordinates of x. This is augmented by the mass-conservation equation Embedded Image, where ρ0 is the (known) mass density of the body in its reference shape. Equations (3.4) and (3.10) then yield a determinate system of four equations for the four variables consisting of λ(X, t) and the three components of the motion Χ(X, t). This of course is well known. It is also well known that there can be no more than six independent invariant constraints, each of which carries a Lagrange multiplier.

Rigid bodies may be viewed as elastic bodies subject to the six local constraints Embedded Image, where I is the three-dimensional identity. This yields Embedded Image, a spatially uniform rotation determined by three functions of t. The reactive stress Tr consists of six components and the strain-energy function is fixed, yielding Ta=0 at every material point. The motion is determined by six functions of t represented by Q(t) and c(t) (cf. (2.1)). The local equation of motion (3.10) is not sufficient to determine these functions and the six Lagrange multipliers together, even if traction data are appended. Nevertheless, the motion itself is determinate from the six global equations of motion consisting of three equations expressing the balance of linear momentum and three more expressing the balance of moment-of-momentum. Thus, if the net force on the body and the net moment about some given point are known, the equations generate an associated motion and vice versa.

Note that the presence or absence of the reactive stress is unrelated to the uniformity or otherwise of the deformation gradient. This crucial point follows from the local nature of the constitutive response. A statement to the effect that the deformation gradient is an unspecified function of t at a given point of the body is a logical tautology and does not constitute a genuine constraint on the deformation gradient at the material point in question. That is, it does not define a submanifold in the space of deformation gradients at the considered material point. The Lagrange multiplier rule does not apply and so the representation (3.9) does not follow. Therefore, in the theory of elasticity as described here, motions of the class (2.1) do not constitute the kind of constraint that gives rise to reactive stresses. In the absence of local constraints like (3.4), the stresses that arise in response to such motions are determined entirely by constitutive behaviour.

4. Refinements

Early writers on the subject of pseudo-rigid bodies (e.g. Cohen & Muncaster 1988) regarded them as conventional continua that deform in accordance with (2.1). In a recent generalization of the earlier ideas (Papadopoulos 2001), the assumption of a uniform deformation gradient is relaxed and replaced by the restriction that the gradient of the deformation gradient (the second gradient of deformation) be uniform. The associated global equations of motion are derived from the theory of elasticity and applied to the solution of a simple representative problem. The results for the mean value of strain in the body are shown to depart markedly from the uniform strain obtained by using the original pseudo-rigid model, even though the constitutive equation for the underlying continuum is unchanged. This is sufficient to overturn the original concept on theoretical grounds alone. For, if the original concept were sound, then the refinement developed in Papadopoulos (2001) would yield a mere perturbation of the mean strain.

The conventional Taylor expansion of a real-valued function f(x) suffices to illustrate the point. Thus, if the values of Embedded Image should differ markedly from those supplied by Embedded Image, at points x of interest, then neither f1(x) nor f2(x) furnishes a good estimate of the value of f(x). In other words, if f2(x)−f1(x) is not small compared to f1(x), then more terms are needed in the Taylor expansion. By the same token, if the distance Embedded Image is suitably small, then Embedded Image by virtue of the presumed differentiability of f at x0. In this case, f(x) is approximated to arbitrary accuracy by Embedded Image if ϵ—the distance between x and x0—is restricted to correspondingly small values.

Of course, the same statements apply to differentiable vector-valued functions (Fleming 1977) and so the deformation gradient may be regarded as being nearly uniform in any sufficiently small neighbourhood of a material point X0, provided that the deformation function Χ(X, t) is differentiable there. On the other hand, the qualifier ‘sufficiently small’ can be given meaning only if some length-scale L (say) is specified, and it is then understood to mean Embedded Image. In applications, any number of length-scales may be present and this effectively means that spatial uniformity will obtain, provided that any point X of the body is near X0, by the standard of the smallest available scale. For example, such a scale is typically furnished by the spatial variation of the traction distribution acting on a finite body. To answer the implied question: ‘how small is small enough?’, appeal must be made to experiments or, if this is not feasible, to precisely that theory of elasticity which has been supplanted by the model. Pending such appeal the presumptive answer is: ‘the smaller, the better’.

More explicitly, to ensure spatial uniformity of the deformation gradient in a body under general conditions of loading, it is necessary that the deformation be differentiable and that the diameter of the body be much smaller than any length-scale in the problem at hand. Logically, one then has an asymptotic theory for particles rather than bodies of finite size. This is not to say that the model is not predictive for bodies of finite size when the actual deformation is uniform or nearly so. Rather, it is merely redundant when used correctly as it is then subsumed under the theory of elasticity.

The foregoing remarks do not apply to the use of pseudo-rigid bodies as effective substitutes for conventional uniform-strain finite elements (Nadler & Rubin 2004).

5. Deus ex machina

Casey (2004) sidesteps the non-existence of a theorem such as the Lagrange multiplier rule for constraints of the form (2.1) by recourse to a definition of pseudo-rigid bodies (definition 3.1), which includes a decomposition of the form (3.9) into active and reactive Cauchy stresses. Thus, pseudo-rigid bodies are objects which undergo motions of the form (2.1) and which nevertheless support Cauchy stresses of the form (3.9), in which Ta is constitutively determined and Tr is subject to the equations of motion but is otherwise entirely arbitrary. This means that the total stress is arbitrary to the same degree. Although it is only the total stress that is accessible to measurement, Casey (2004) requires that Ta and Tr be subject to separate requirements. These and other requirements constitute his definition of a pseudo-rigid body. Among them,Embedded Image(5.1)where the overbar is used to denote the volume average over the shape of the body at time t. For elastic pseudo-rigid bodies, Ta is determined by F(t). Moreover, granted (3.9), the symmetry of T implies that of Tr.

The local distribution of the total stress is such that (3.10) is satisfied. Since Embedded Image, identically, it follows thatEmbedded Image(5.2)Given that Tr is symmetric, the number of unknowns is now 18 (the six components of Tr the nine of F and the three of c). Equations (5.2) represent nine restrictions and so this problem, like that associated with the rigid body, is indeterminate, remaining so in the presence of the three additional restrictions supplied by any traction data. Nevertheless, as shown in Casey (2004) and elsewhere (Cohen & Muncaster 1988), the motion Embedded Image is determined by 12 global equations of motion derived from the local equations (e.g. Casey 2004; eqns (2.11)1 and (2.21)1,2). By using equations (2.9)1, (3.5), (3.11)1, (3.14) and (3.15) of Casey (2004), it is possible to reduce these equations toEmbedded Image(5.3)where R is the region presently occupied by the body, n is the exterior unit normal to its bounding surface ∂R, Embedded Image is the position of its mass centre, Embedded Image is position in the body relative to the mass centre and E0 is the (purely geometric) Euler tensor relative to the mass centre with position Embedded Image in the reference configuration, wherein Embedded Image. Evidently, the constitutive part of the stress plays no role in these equations. This is already apparent from (5.2)2.

We have thus reached the extraordinary (and, in the view of the present writer, untenable) conclusion that materials, regardless of their constitution, supply such stresses as may be needed to effect any motion of the form (2.1). Casey (2004) describes a hypothetical microscopic mechanism (Casey 2004, fig. 1), driven by an unidentified technology, whereby such stresses Tr might be generated, presumably in real time during the course of the motion. Apparently, the operator of this technology actively maintains the required divergence and average for these stresses in accordance with (5.2). This of course is not possible in conventional bodies, the motions of which depend strongly on the interplay between forces and material constitution whether or not the deformation gradient is uniform.

This theory is internally consistent. To the extent that it is unrelated to the theory of elasticity, it is not subject to arguments (§4) levelled against earlier ideas about pseudo-rigid bodies. Therefore, like any theory, it must be assessed by checking its predictions against the empirical record (e.g. Timoshenko & Goodier 1970, fig. 112).

A procedure similar to that used in Casey (2004) has been the subject of recent efforts to extract theories of plates, shells and rods from the theory of elasticity (cf. Casey 2004 and references therein). In these works, the deformation is subject to a global constraint analogous to (2.1) and the integrated equations of motion are deduced. This constraint imposes a specified functional dependence of the deformation on through-thickness coordinates, just as (2.1) requires that the deformation depends affinely on the three referential coordinates. Such dependence, which (as in (2.1)) is not linked a priori to any length-scale, naturally mirrors the properties of differentiable but otherwise unconstrained motions Χ(X, t) of three-dimensional bodies that are sufficiently thin in one or two dimensions against a given scale. It is not surprising, then, that the theories which emerge are deemed to be eminently useful for analysing the deformations of thin bodies (generally, the thinner, the better), in just the same way that particles are preferred in the study of compact bodies of small size.

Footnotes

    • Received June 4, 2005.
    • Accepted September 9, 2005.

References

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