## Abstract

In the literature on pseudo-rigid bodies and their applications, it is generally assumed that these bodies can undergo only a restricted class of motions, without questioning how this restriction is to be strictly enforced. In 2004, I proposed in these *Proceedings* that such a restriction may be regarded as a ‘global constraint’ on a deformable continuum, and influenced by ideas of Antman & Marlow from the early 1990s, I assumed that the constraint is enforced by a field of reactive stresses, and I constructed a mathematical model that idealizes pseudo-rigid bodies as globally constrained continua of finite size. In a recent article in *Proceedings of the Royal Society A*, the validity of this model was challenged. Essentially, the controversy revolves around the issue of *working definitions* versus *idealized mathematical models* of pseudo-rigid bodies.

## 1. Introduction

All real solids deform under applied loads and, if the loads are sufficiently large, fail. Nonetheless, the elastic continuum and the rigid body are useful mathematical models for describing aspects of the physical behaviour of solid materials. The ideal elastic pseudo-rigid body is a novel mathematical model that occupies an intermediate role between the rigid model and the nonlinearly elastic model, being closer in spirit to the former.1 Whereas the ideal rigid body can undergo only motions in which the mutual distances between its particles are preserved, despite the magnitudes and distribution of the applied forces, the ideal pseudo-rigid body can undergo only homogeneous (or affine) motions no matter what forces are applied to it.2 A rigid body has six degrees of freedom, a pseudo-rigid body has 12, and an elastic continuum has infinitely many.

One often conceives of a rigid body as a discrete system of mass points connected by inextensible massless rods. The rods supply reactive forces that maintain the rigidity of the system under all external influences and, in general, are indeterminate. The model of a *rigid continuum* is also commonplace. But, strictly speaking, it is not mathematically equivalent to a system of mass points. In the rigid continuum, the action of rods is replaced by a field of Cauchy stress tensors, which, in general, are indeterminate. Despite the differences between the rigid continuum and the rigid system of mass points as abstract representations, it is obvious that both of these models will yield approximate dynamical information for real solids that are experiencing small deformations under loads of limited magnitude.

The following is an elementary example of a planar elastic pseudo-rigid body. Take four massless rigid rods, freely jointed to form a square, place a mass point at each corner, and attach a spring across a diagonal. Under arbitrary applied forces, this body can undergo only homogeneous motions; constraint reactions are supplied by the rigid rods. Similarly, rigid rods can be linked to form a lattice of parallelograms or parallelepipeds that deform homogeneously. As another type of example, take the square mentioned above, and replace two opposite sides each by a rigid rod sliding inside a rigid tube. More generally, one can construct an elastic pseudo-rigid system of mass points by linking the masses together in any way that keeps their motions affine, and by adding massless springs to provide elasticity. Dissipative elements may be added to produce inelasticity.

In the introduction to Casey (2004), by way of motivation for later assumptions, I explained how the deformation in a particular inhomogeneously deformed elastic continuum could be adjusted to achieve homogeneity by attaching an adjustable constraining system (see fig. 1 of Casey (2004)). Reactive forces are developed in the constraints and restore homogeneity of the deformation. The applied traction is supported in part by the elastic response of the material and in part by the reactive stresses. This particular elastic pseudo-rigid body is not just a piece of elastic material, but rather the combination of the elastic material together with the constraining system.

If one starts out with a nonlinearly elastic continuum and considers only homogeneous motions, a straightforward procedure, which is sketched in §2, leads to the equations that are employed in the literature on pseudo-rigid bodies. However, the elastic continuum itself is certainly not a pseudo-rigid body, because it can experience non-homogeneous motions as well as homogeneous ones.3 To create a mathematical model of a *pseudo-rigid continuum*—either elastic or more general—I proposed in Casey (2004) that the condition that the motion be homogeneous may be regarded as a ‘global constraint’, meaning that this condition is a restriction on the set of motions of a deformable continuum rather than on the set of deformation gradient tensors, which is the case for classical internal constraints, such as incompressibility, for instance. The conception of global constraints for continua is due to Antman & Marlow (1991).4 These constraints differ in a subtle way from the usual constraints, and require a different theoretical framework, as will be indicated in §3. However, a field of reactive stresses is still required to ensure that the global constraint equation is satisfied under arbitrary applied loads.5

In a recent article, Steigmann (2006) criticized my model of a pseudo-rigid continuum. In Steigmann's view, the condition that the motions of a continuum be homogeneous does not ‘constitute the kind of constraint that gives rise to reactive stresses’, and he concludes that the stresses in the continuum should be determined (entirely) by constitutive equations. I disagree and will elucidate the reasons why.

## 2. A necessary global dynamical equation for any deformable continuum

In this section, I consider an arbitrary deformable three-dimensional continuum , and supposing that is experiencing an arbitrary homogeneous motion, I will sketch a standard derivation of an important dynamical equation that must satisfy.

Employing the same notation as in Casey (2004), let be the position vector of a particle in a fixed occupiable reference configuration of in some Newtonian frame of reference, and let be the position vector of *X* in the current configuration of at time *t*. Every homogeneous motion of has a representation of the form(2.1)where is an invertible second-order tensor-valued function of time with positive determinant and is an arbitrary vector-valued function of time. The velocity and acceleration gradients are(2.2)where a superposed dot signifies material time differentiation.

Let and be the regions occupied by in its reference and current configuration, respectively, and let (resp. ) be the boundary of (resp. ). The mass of is given by , where , *ρ* are mass densities, and , are volume elements. Let and denote the position vectors of the mass centres of in the reference and current configurations, respectively, and let , . Then, in view of (2.1) and a similar equation for the motion of the mass centre, we have(2.3)where use has also been made of (2.2)_{1,2}. The Euler tensors of , taken with respect to the mass centre in the reference and current configurations of , are(2.4)where denotes the tensor product of vectors. , are symmetric and positive definite, and is constant. From (2.3)_{1} and (2.4)_{1,2}, it is clear that for homogeneous motions of ,(2.5)

Let be the body force per unit mass acting on at time *t*, and let and be the Cauchy traction vector and the Cauchy stress tensor, respectively. The resultant external force acting on at time *t* is(2.6)where is the area element. Likewise, the resultant external torque about the mass centre is(2.7)Euler's laws for the balance of linear and angular momentum may be stated as(2.8)for every motion of . And, Euler's laws imply Cauchy's laws(2.9)which again hold for all motions of .

Partial global information may be obtained from Cauchy's laws by means of a technique due to Signorini (Truesdell & Toupin 1960, §§216–220). Thus, by taking the tensor product of both sides of (2.9)_{1} with and integrating over , we arrive at the formula(2.10)where *V* is the current volume of , is the mean Cauchy stress tensor, i.e.(2.11)and(2.12)is called the Möbius tensor. The result (2.10) holds for all motions of , and in view of (2.2)_{2} and (2.5), for homogeneous motions it becomes(2.13)Thus, for every homogeneous motion of a deformable continuum , it is necessary that (2.13) should hold.

If constitutive equations are given such that is determined by some functional of the motion (Truesdell & Noll 1965, §26), then, by virtue of (2.11), will also be determined.

Ideally, pseudo-rigid continua can experience only homogeneous motions, and hence the dynamical equation (2.13) must hold for them, and is universally accepted in the literature (e.g. Cohen & Muncaster 1988). Almost all authors have assumed implicitly that such tractions are applied as are consistent with the assumed motion (2.1) of an elastic continuum, or, at least in applications, that (2.1) and (2.13) are approximately satisfied. Some authors regard pseudo-rigid bodies as being ‘small’ in some loose sense. In other words, authors have relied on working definitions.

Now, one can repeat the foregoing derivation for rigid motions of a deformable elastic continuum and arrive at(2.14)where the tensor is a rotation. Thus, rigid bodies must necessarily satisfy (2.14), but that does not furnish a *definition* of the ideal rigid body.6

Similarly, a precise definition is needed for the ideal pseudo-rigid body. Casey (2004) raised the question: how can a pseudo-rigid continuum keep its motion homogeneous in the face of arbitrary applied external forces? For example, if a homogeneous elastic continuum is subjected to a homogeneous deformation, it can be maintained in equilibrium by a traction field applied to its boundary. However, if this traction field is replaced by an arbitrary equipollent traction field, then, in general, the deformation will be changed from a homogeneous to a non-homogeneous one. As mentioned before, I suggested that a homogeneous elastic continuum could be made into a pseudo-rigid continuum by incorporating a constraining system that redistributes the applied loads in a manner which keeps the deformation homogeneous. This constraining system does not restrict the value of the deformation gradient at a point of the body, and hence is fundamentally different from the usual type of internal constraint.

## 3. Local and global material constraints

Two different types of material constraints (or internal constraints) have been considered in the literature on continuum mechanics. The more familiar type, which may be referred to as ‘local constraints’ are restrictions on the values of the deformation gradient, and have been treated by various methods. The other type of material constraint, called ‘global constraints’, are restrictions on the whole deformation field at each time *t*, and require a novel treatment.

### (a) Local constraints

In the context of a purely mechanical development, a satisfactory treatment of local internal constraints is given by Truesdell & Noll (1965, §30), the main assumptions of which are worth recalling here.

Thus, consider a constraint of the form(3.1)where *ϕ* is a scalar-valued function, with . Equation (3.1) restricts the set of deformation gradients that can be accessed by each particle of the deformable body. Applying a standard objectivity argument, one finds that the function *ϕ* depends only on the right stretch tensor ** U**,7 and hence can be written as a different function of the right Cauchy–Green tensor :(3.2)The latter equation defines a five-dimensional hypersurface, the constraint manifold, in the six-dimensional Euclidean space of symmetric second-order tensors. For every motion satisfying the constraint, it follows from (3.2) that(3.3)or equivalently,(3.4)where is the symmetric part of the velocity gradient .

Truesdell & Noll (1965, §30) assume that the Cauchy stress tensor at each particle of the locally constrained continuum is given as a sum(3.5)where is prescribed by a constitutive equation, and is indeterminate, independent of rates, and, most importantly, is workless in any motion that satisfies the constraint, i.e.(3.6)for all satisfying (3.4). A standard mathematical argument then yields the result that(3.7)where *λ* is a Lagrange multiplier. We may call the ‘active stress’ and the ‘constraint response’ or ‘reactive stress’. Each additional local constraint, up to a maximum of six, will produce an additional hypersurface, and an additional Lagrange multiplier.

For a rigid continuum,(3.8)which is equivalent to six constraints of the type (3.2), and (3.7) then implies that there are six components of reactive stress. The active stress tensor is found by evaluating the constitutive function of the beginning unconstrained material at ; if the reference configuration of the unconstrained material is stress-free, and the stress tensor is then completely reactive.

It is important to bear in mind that even local constraints require a *theory* for handling them, and, in particular, that the existence of indeterminate stresses is assumed by Truesdell & Noll (1965).

A second point to be made is that a locally constrained material cannot be a member of the class of unconstrained materials with which it is associated, i.e. they are new theoretical materials. For instance, an *incompressible elastic material* cannot be a member of the set of elastic materials. For, if it were, would be completely determined by an elastic constitutive equation, applied to isochoric motions, whereas, for incompressible materials, is determined only up to an indeterminate pressure. Locally constrained materials may, however, be considered as limits of unconstrained materials.8

### (b) Global constraints

Let be any motion of a deformable body , and let be the set of all motions of . A global constraint is a condition placed on that restricts it to belong to some proper subset of . A theory of such constraints was given by Antman & Marlow (1991).9

The condition (2.1) restricts to belong to the set of homogeneous motions, a proper subset of , and is a global constraint. It places no restriction on the value that itself may have, and therefore it cannot be viewed at all as a local constraint.10

A material constraint cannot be maintained in the presence of arbitrary loads unless reactive stresses can by generated, and Antman & Marlow (1991) *assume* that whenever a continuum is globally constrained, there exists a field of reactive stresses satisfying a decomposition of the form (3.5).

Since the condition (2.1) does not correspond to any local constraint, the argument leading up to (3.7) cannot be applied to it. A new argument is needed. For general global constraints, Antman & Marlow (1991) proposed a *global constraint principle*, which involves an integral statement of the worklessness of the reactive stresses. In the case of pseudo-rigid bodies, I was able to adopt a different approach.

### (c) A simple example of a global constraint

Consider a prismatic bar of homogeneous elastic material *E* (figure 1). Let it be constrained axially ( direction) by a system of threaded rigid rods *AA*, with adjustable nuts *B*, and, for simplicity, consider tensile loading *T* only.11 Let the bar be constrained in the and directions by rigid rods (not shown in figure 1), and suppose also that the cross-section cannot be sheared. The constraints on the bar may be expressed in global form as(3.9)where is the length of the bar in its stress-free reference configuration, and *l* is the length of threaded rod between the nuts, and can be controlled by adjusting them. Clearly, for each fixed value of *l*, if the tension *T* applied to the ends of the bar is exactly that corresponding to the elastic constitutive equation, no reactive stresses will develop, whereas if it is greater, then reactive tensions will develop in the threaded rigid rods.

The constraint equations (3.9) may be written in terms of the components of the right Cauchy–Green tensor as(3.10)Equation (3.10)_{1} defines a special one-parameter *family* of local constraints, and correspondingly, a one-parameter family of constraint hyperplanes; at each fixed value of *l*, the theory in §3*a* applies. There is an active and a reactive component of stress in the axial direction. Equations (3.10)_{2,3} define five local constraints, giving rise to five components of reactive stress.

## 4. The ideal homogeneous pseudo-rigid continuum

In Casey (2004, §3), I assumed the existence of idealized homogeneous pseudo-rigid continua characterized by the following conditions.

They can experience only homogeneous motions (2.1).

The Cauchy stress tensor field can be decomposed as in (3.5), where the active stresses are specified by a constitutive equation, and the reactive stresses are indeterminate (and are supplied by a constraining system belonging to the continuum).

The response function (or functional) for the active stress tensor is that of a homogeneous material (i.e. it is independent of ).

The constraining system ensures that at each point of the continuum, the active stress is adjusted to be equal to the mean stress, i.e.(4.1)

The following results hold for idealized homogeneous pseudo-rigid continua:(4.2)Here, and are the traction vectors associated with and , respectively (and ), is the Möbius tensor corresponding to the field , is defined by (2.12), and is the Möbius tensor generated by and .

In accordance with (4.2)_{1} and (4.2)_{4}, in an ideal homogeneous pseudo-rigid body, the active stresses form an equilibrated system, while the reactive stresses play exactly the same role as in rigid body dynamics. The integral worklessness condition (4.2)_{6} holds, which should be contrasted with the local condition (3.6) that holds for local constraints. A set of nine Lagrange equations may be obtained from the equation (Casey 2004, §4). It is to be emphasized that the theory in Casey (2004) is consistent with the equations (2.1), (2.8)_{1} and (2.13) that are employed in the literature on pseudo-rigid bodies. The essence of the new development is that a definition is given for an ideal homogeneous pseudo-rigid continuum, which had been missing from the literature.

Steigmann (2006) does not recognize the condition (2.1) as constituting a ‘genuine’ constraint, and asserts that it does not give rise to any reactive stresses.12 Since he cannot treat (2.1) by the standard theory of local internal constraints, he rejects it entirely as a constraint. Moreover, he accuses Casey (2004) of an unjustified use of Lagrange multipliers, whereas, in fact, no use at all of a Lagrange multiplier rule was made in Casey's paper. Rather, the existence of reactive stresses was a primitive assumption of his theory.

## 5. Applications

The theory of elastic pseudo-rigid bodies, employing just equations (2.1), (2.8)_{1} and (2.13), has been applied to a variety of problems.13 Thus, motions of elastic pseudo-rigid pendulums (Cohen & Sun 1988), vibrations of rectangular parallelepipeds (Rubin 1986), roto-deformations (Cohen & Muncaster 1984), deformable satellites (O'Reilly & Thoma 2003) and finite-element approximations (Solberg & Papadopoulos 1999) have been studied. Further, given the value of the classical rigid-body theory of impact (e.g. Synge 1960, §59), it is natural that the theory of pseudo-rigid bodies also should be utilized for impact studies (Cohen & MacSithigh 1991, 1994; Varadi *et al*. 1999; Solberg & Papadopoulos 2000; Kanso & Papadapoulos 2004).

Papadopoulos (2001) discussed an extension of the theory of elastic pseudo-rigid bodies to allow for deformations, which, instead of (2.1), are of the form(5.1)where is a third-order tensor-valued function of *t* that acts bilinearly on . After deriving the equations of the theory by a procedure of the type indicated in §2, he considered steady planar roto-deformations of a cylinder of a special isotropic nonlinearly elastic material. In figure 1 of his paper, he plotted an area-weighted mean stretch against a parameter *κ* for both the extended theory and the original theory of pseudo-rigid bodies.14 For values of *κ* less than about 0.7, the curves for the two models coincide. As *κ* increases beyond 0.7, the curves diverge smoothly from one another, the higher-order theory giving larger values from than the original theory.

To Steigmann (2006), it appears that the results of Papadopoulos (2001) demonstrate ‘the fallacy of the pseudo-rigid body concept’. Steigmann (2006, §4) argues that if the original theory were sound, then the extended theory would produce at most a perturbation on the original results. If Steigmann's perturbation argument were applied to a single initial boundary-value problem, it would make sense. However, in Papadopoulos (2001), a semi-inverse method was adopted and two different assumed motions of the elastic cylinder were considered, one being a generalization of the other. Appropriate tractions were implicitly assumed to be supplied symmetrically on the lateral surface of the cylinder in each case. The graph merely shows that at small values of *κ*, the two problems coincide, whereas for large values of *κ*, they are significantly different problems.

## 6. Conclusions

When one attempts to model three-dimensional elastic continua by reduced theories, such as shell theories and rod theories, the issue of idealized models always arises. If one starts out with the three-dimensional field equations and the elastic constitutive equation, restrictive assumptions and approximations have to be made before arriving at the reduced theory. On the other hand, one can pursue a ‘direct’ approach and model a shell as a Cosserat surface, a rod as a Cosserat curve, and a pseudo-rigid body as a Cosserat point, each satisfying a set of postulated equations. But, then, one inevitably runs into difficulties when one attempts to apply the results of the Cosserat theory to the original three-dimensional continuum.

In Casey (2004), I strove to construct an idealized model of a homogeneous pseudo-rigid continuum of finite size. I employed the concept of a global constraint, due to Antman & Marlow (1991), to do so. The constraint necessarily implies the existence of reactive stresses, whose role I made explicit in the theory.15 It is a topic of continuing interest to examine the role played by reactive stresses in applications of ideal pseudo-rigid bodies, as well as in other globally constrained continua. A theory for inhomogeneous pseudo-rigid continua is also being studied.

## Footnotes

↵For references to the sizeable literature on pseudo-rigid bodies, see Casey (2004).

↵Detailed discussions of the properties of homogeneous motions may be found in Thomson & Tait (1867, §§155–158) and Truesdell & Toupin (1960, §§42 & 142).

↵Analogously, a fully deformable continuum that is undergoing an isochoric (i.e. volume-preserving) motion is not an incompressible body. Only by imposing the ‘internal constraint’ that all motions of the continuum be isochoric do we get an incompressible body, and as is well known, an indeterminate pressure term ensures satisfaction of the constraint.

↵The reader is also referred to Marlow (1993) and Antman (1995).

↵When shells and rods are regarded as three-dimensional bodies that can experience only certain motions, global constraints naturally appear (Antman & Marlow 1991). Many authors do not take account of the essential accompanying reactive stresses in such cases. As Marlow (1993) points out, the indeterminacy of the reactive stresses, rather than being a liability, is a powerful tool in the exact solution of three-dimensional boundary-value problems for shells and rods. See also Podio-Guidugli (1989).

↵The definition given by Love (1897, §114–118) is notable, in that explicit reference is made to the constraint forces: a rigid system is an example of a system which moves in such a way that certain geometrical relations are maintained, and the internal forces between the particles of the system are so adjusted from instant to instant that the conditions are never violated.

↵In accordance with the polar decomposition theorem,

=*F*, where*RU*is a proper orthogonal tensor (a rotation) and*R*is a symmetric positive definite tensor.*U*↵Alternatively, each constrained material may be made to correspond to an equivalence class of unconstrained materials (Casey & Krishnaswamy 1998).

↵See also Marlow (1993) and Antman (1995).

↵In contrast, the rigidity condition (3.8) can be integrated to furnish the global constraint

=*x*+*R**X*, where*c*is any rotation.*R*↵A constraining system that also resists compression can easily be imagined.

↵Logically, he must therefore not accept the global constraint theory of Antman & Marlow (1991) either, since these authors assume the existence of reactive stresses

*ab initio*.↵An equivalent set of equations based on the concept of a Cosserat point (i.e. a particle to which are attached three deformable vectors (or directors)), introduced by Rubin (1985), has been utilized in some of these applications.

↵The parameter , where

*c*depends on reference mass density and two elastic constants,*ω*is angular velocity and*R*is referential radius.↵I would like to take this opportunity to point out a minor typographical error in Casey (2004, Introduction, second line).

- Received February 8, 2006.
- Accepted March 28, 2006.

- © 2006 The Royal Society