The Classical Field Theories

Matter is commonly found in the form of materials. Analytical mechanics turned its back upon this fact, creating the centrally useful but abstract concepts of the mass point and the rigid body, in which matter manifests itself only through its inertia, independent of its constitution; “modern” physics likewise turns its back, since it concerns solely the small particles of matter, declining to face the problem of how a specimen made up of such particles will behave in the typical circumstances in which we meet it. Materials, however, continue to furnish the masses of matter we see and use from day to day: air, water, earth, flesh, wood, stone, steel, concrete, glass, rubber, ... All are deformable. A theory aiming to describe their mechanical behavior must take heed of their deformability and represent the definite principles it obeys.

The non-linear field theories of mechanics

Convexity conditions and existence theorems in nonlinear elasticity

The basic physical concepts of classical continuum mechanics are body, configuration of a body, and force system acting on a body. In a formal rational development of the subject, one first tries to state precisely what mathematical entities represent these physical concepts: a body is regarded to be a smooth manifold whose elements are the material points; a configuration is defined as a mapping of the body into a three-dimensional Euclidean space, and a force system is defined to be a vector-valued function defined for pairs of bodies1. Once these concepts are made precise one can proceed to the statement of general principles, such as the principle of objectivity or the law of balance of linear momentum, and to the statement of specific constitutive assumptions, such as the assertion that a force system can be resolved into body forces with a mass density and contact forces with a surface density, or the assertion that the contact forces at a material point depend on certain local properties of the configuration at the point. While the general principles are the same for all work in classical continuum mechanics, the constitutive assumptions vary with the application in mind and serve to define the material under consideration.

The Thermodynamics of Elastic Materials with Heat Conduction and Viscosity

This manuscript describes a unique class of locomotive robot: A soft robot, composed exclusively of soft materials (elastomeric polymers), which is inspired by animals (e.g., squid, starfish, worms) that do not have hard internal skeletons. Soft lithography was used to fabricate a pneumatically actuated robot capable of sophisticated locomotion (e.g., fluid movement of limbs and multiple gaits). This robot is quadrupedal; it uses no sensors, only five actuators, and a simple pneumatic valving system that operates at low pressures (< 10 psi). A combination of crawling and undulation gaits allowed this robot to navigate a difficult obstacle. This demonstration illustrates an advantage of soft robotics: They are systems in which simple types of actuation produce complex motion.

Multigait soft robot

It has been shown (see C. Truesdell (1952) for a comprehensive review of this subject) that the mechanics of a homogeneous isotropic ideally elastic material may be developed on the basis of a description of the relevant elastic properties of the material in terms of a strain-energy function W which is a single-valued function of three scalar invariants of the deformation, I 1, I 2 and I 3.

Stress-Deformation Relations for Isotropic Materials

The equilibrium of a cube of incompressible, neo-Hookean material, under the action of three pairs of equal and oppositely directed forces f 1, f 2, f 3, applied normally to, and uniformly distributed over, pairs of parallel faces of the cube, is studied. It is assumed that the only possible equilibrium states are states of pure, homogeneous deformation.

Large Elastic Deformations of Isotropic Materials

A general theory of multipolar displacement and velocity fields with corresponding multipolar body and surface forces and multipolar stresses is developed using an energy principle, an entropy production inequality and invariance conditions under superposed rigid body motions. Constitutive equations for the multipolar stresses are discussed and explicit results are given for an elastic medium. Work in a previous paper by the present authors (1964) is shown to be a special case of that given here.

Multipolar continuum mechanics

The mechanics of non-linear materials with memory has been discussed in previous papers [1, 2] and an earlier report [3]*. In these papers, it was assumed that, in a fixed coordinate system, the stress at a point of the material at any instant of time is determined by the deformation gradients at that instant and at previous instants. The limitations imposed on the form of the constitutive equation by the consideration that it is unaltered by a simultaneous rotation of the body and reference system are determined. In the present paper, we assume initially a constitutive equation in the form of implicit relations between the stress, its time derivatives and gradients of displacement, velocity, acceleration, etc. at a number of instants of time in some time interval. The limitations on the form of the constitutive equation resulting from the fact that it is unaltered by the imposition on the body of an additional arbitrary angular velocity, acceleration, etc. are discussed.

The Mechanics of Non-Linear Materials with Memory

In this paper we show that a symmetric isotropic matrix polynomial in any number of symmetric 3 × 3 matrices can be expressed as a symmetric isotropic matrix polynomial, in which each of the matrix products is formed from at most six matrices and has one of a certain number of forms which are explicitly given The significance of these results in the mechanics of isotropic continua is indicated

Ronald S. Rivlin

Papers

Stress-Deformation Relations for Isotropic Materials

Large Elastic Deformations of Isotropic Materials

Multipolar continuum mechanics

The Mechanics of Non-Linear Materials with Memory

The Theory of Matrix Polynomials and its Application to the Mechanics of Isotropic Continua