Abstract For elastic–plastic materials that can undergo large deformations, the general theory of Green and Naghdi is supplemented by incorporating a novel three-factor decomposition of the deformation gradient. A certain unique right stretch tensor emerges as an appropriate variable for describing finite elastic behaviour at large plastic strains. Convenient objective forms of response functions for strain energy and stress are presented.

In the context of a purely mechanical development for materials that possess some degree of elasticity, two-factor and three-factor multiplicative decompositions of the deformation gradient, as well as related strain measures, are discussed in detail. Different factors in the decompositions (and their polar subfactors) have different degrees of rotational non-uniqueness. Moreover, different deformation measures generally behave differently under superposed rigid motions, which has important implications for the roles that they are allowed to play in constitutive equations. A certain unique right stretch tensor emerges, which yields strain measures suitable for describing anisotropic elastic responses of solids that have evolving stress-free local configurations.

For many purposes in continuum mechanics it has been found useful to decompose the deformation gradient into two factors, which result from some elastic process through which destressing is achieved at a material point. There is an essential rotational non-uniqueness in these factors; however, some subfactors are uniquely defined. In particular, a certain unique right stretch tensor is identified, which serves as a convenient independent variable for describing the anisotropic elastic response of a solid from its evolving stress-free intermediate configurations.

After reviewing the derivation of the formula for areal velocity, it is shown that the paper by Spolter [Phys. Essays 24, 260 (2011)] has a basic mathematical error which leads the author to infer the wrong physical conclusion. (C) 2017 Physics Essays Publication.

It is argued that the concept of body, as it is usually employed in continuum mechanics, is somewhat too general. Specifically, it is standard practice to associate a set of dynamical processes with an abstract body. However, many of these processes cannot be experienced by any physical body. Yet, Euler's laws for the balance of linear momentum and angular momentum are asserted to hold for such abstract bodies. It is suggested here that Euler's laws should be postulated only for abstract bodies that are composed of ideal materials belonging to some large collection.

The concept of parallelism along a surface curve, which was introduced by Levi-Civita in the context of n-dimensional Riemannian manifolds, is re-examined from a kinematical viewpoint. A special type of frame, whose angular velocity is determined by the rate at which the tangent plane turns as one moves along a surface curve, is defined and is called a Levi-Civita frame. The surface may be orientable or not. Vectors and tensors fixed on Levi-Civita frames are parallel transported. Covariant differentiation of vectors and tensors along a surface curve can be expressed in terms of the corresponding corotational rates measured on Levi-Civita frames. Relevant results on ruled surfaces are also included.

An argument is presented to justify the application of the principle of angular momentum to systems of point masses that are constrained by massless internal connections.

The well-known method for obtaining objective response functions in continuum mechanics, due to W. Noll, is reviewed and an objection to its logic, raised by R.S. Rivlin and G.F. Smith, is evaluated.

For motion of a material point along a space curve, a kinematical decomposition, discovered by Siacci, expresses the acceleration vector as the sum of two special oblique components in the osculating plane to the curve. A new proof of Siacci’s theorem is presented.

A general approach to continuum thermodynamics that was advocated by R.S. Rivlin is carried out for thermoelastic materials which can also depend on strain rate. An entropy function is constructed (rather than assumed to exist). A method for treating thermomechanical internal constraints for such materials is also presented. In this method, the properties of a constrained material are inherited from those of a related equivalence class of unconstrained materials.

For continua which contain surfaces of discontinuity, such as shock waves or acceleration waves, the integral balance laws imply certain thermomechanical jump conditions, which may be derived in various ways. In the present expository article, a form of Reynolds' transport theorem is obtained that is particularly useful for deriving these jump conditions.

Explicit formulas are derived for the Fréchet differentials and Fréchet derivatives of the right and left stretch tensors and the rotation tensor, regarded as functions of the deformation gradient. Novel decompositions, due to Chen and Wheeler, are re-interpreted.

In a previous paper by the author, a geometrical procedure was presented for deriving Lagrange’s equations for a rigid body. The rigid body was represented by an abstract particle moving in a 12-dimensional Euclidean space, called Hertzian configuration space, the metric of which is determined by the radius of gyration of the body. The present paper focuses on the representation of the underlying rotational dynamics in Hertzian space.

It is shown how the classical geometrical and modern algebraic conceptions of angular velocity can be unified by making use of a relative rotation tensor that takes the current configuration of a rigid body as a reference configuration.

The concept of areal velocity is intrinsically and historically connected with that of angular momentum. For central force fields the area swept out in the orbital plane by a particle’s radius vector is proportional to the time. For non-planar problems, it is shown that a particle’s position vector sweeps out a general conical surface and that interesting kinematical relations hold. The areal velocity vector is always perpendicular to the conical surface and is proportional to the angular momentum of the particle. In some problems, the magnitude of the areal velocity is constant while its direction changes. In others, only a component of the areal velocity vector is constant. A moving orthonormal basis associated with the areal velocity is defined and a matrix equation for the angular velocities of the basis vectors is found to have an elegant form, involving “sweeping” and “tilting” components. Illustrative examples are provided and some historical background is included.

In the context of a purely mechanical development, the concept of a "globally constrained" continuum is employed here to construct a theory of pseudo-rigid bodies. The Cauchy stress tensor T for the pseudo-rigid body is assumed to be decomposed into an "active" part T (A), which is specified by a constitutive equation, and a "reactive" part T (R), which is called into play to maintain the global constraint. The theory generalizes one presented by the author in 2004, in which the active stress tensor was given by the same response function throughout the body, i.e., pseudo-rigid bodies were regarded there as homogeneous globally constrained continua. Material inhomogeneity is now admitted. A set of Lagrange's equations follows as before.

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.

A geometrical derivation is given for Lagrange’s equations for a system of rigid bodies subject to general holonomic and non-holonomic constraints. As in the case of a similar derivation for a system of particles, the entire system is represented by an abstract particle P moving in a higher-dimensional Euclidean space, called Hertzian space, the metric of which is determined by the radius of gyration of the physical system. The holonomic constraints confine P to move in a Riemannian manifold - the configuration manifold of the constrained system - embedded in Hertzian space. Euler’s laws of linear and angular momenta are expressed as a single balance equation in Hertzian space and Lagrange’s equations emerge as covariant components of this equation taken along the coordinate directions in the configuration manifold. No appeal is made to variational principles or to notions of virtual work.

Fields, Flows and Waves: An Introduction to Continuum Models , D. F. Parker Springer-Verlag, New York, 2003. $34.95 paper (270 pp.). ISBN 1-85233-708-7 Buy at Amazon