Morton E. Gurtin

A continuum theory of elastic material surfaces

A mathematical framework is developed to study the mechanical behavior of material surfaces. The tensorial nature of surface stress is established using the force and moment balance laws. Bodies whose boundaries are material surfaces are discussed and the relation between surface and body stress examined. Elastic surfaces are defined and a linear theory with non-vanishing residual stress derived. The free-surface problem is posed within the linear theory and uniqueness of solution demonstrated. Predictions of the linear theory are noted and compared with the corresponding classical results. A note on frame-indifference and symmetry for material surfaces is appended.

THERMODYNAMICS WITH INTERNAL STATE VARIABLES.

This is a study of the thermodynamics of nonlinear materials with internal state variables whose temporal evolution is governed by ordinary differential equations. After employing a method developed by Coleman and Noll to find the general restrictions which the Clausius—Duhem inequality places on response functions, we analyze various types of dynamical stability that can be exhibited by solutions of the internal evolution equations. We also discuss integral dissipation inequalities, conditions under which temperatures can be associated with internal states, and the forms taken by response functions when the material is a fluid.

The Linear Theory of Elasticity

Linear elasticity is one of the more successful theories of mathematical physics. Its pragmatic success in describing the small deformations of many materials is uncontested. The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson. The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others. These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations. In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness.

Surface stress in solids

Abstract In a recent paper [6] a general theory of surface stress was presented. Here we discuss several simple solutions within this theory.

A general theory of heat conduction with finite wave speeds

> 0 is a constant. This equation, which is parabolic, has a very unpleasant feature: a thermal disturbance at any point in the body is felt instantly at every other point; or in terms more suggestive than precise, the speed of propagation of disturbances is infinite. In this paper we develop a general theory of heat conduction for nonlinear materials with memory, a theory which has associated with it finite propagation speeds. In Section 3 we determine the restrictions that thermodynamics places on our constitutive relations. We show that our theory differs f rom other theories of heat conduction in that the heat-flux, like the entropy, is determined by the functional for the free-energy. In Section 6 we study the propagation of certain types of weak discontinuities. We show that in certain circumstances waves travelling in the direction of the heat-flux vector propagate faster than waves travelling in the opposite direction. In Section 7 we deduce the linearized theory appropriate to infinitesimal temperature gradients. We show that the linearized constitutive equation for the heat-flux q has the form: 1

A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations

This study develops a gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations. The theory is based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems. The field equations consist of the yield conditions coupled to the standard macroscopic force balance; these are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. As an aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. To make contact with classical dislocation theory, the microstresses are shown to represent counterparts of the Peach–Koehler force on a single dislocation.

Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance

Abstract A unified framework for equations of Ginzburg-Landau and Cahn-Hilliard type is developed using, as a basis, a balance law for microforces in conjunction with constitutive equations consistent with a mechanical version of the second law.

Non-linear age-dependent population dynamics

where P( t ) is the total population at time t and 5 is the growth modulus. This law is clearly inapplicable to situations in which the population competes for resources (e.g., space and food), for in these situations 5 should depend on the size of the population: the larger the population, the slower should be its rate of growth. To overcome this deficiency in the Malthusian law, VERHULST [1845, 1847] assumed that f i= (6o-co0 P ) P (5o, COo=constant). (1.2)

On a theory of heat conduction involving two temperatures

ZusammenfassungIn dieser Arbeit entwickeln wir eine neue Theorie der Wärmeleitung. Diese Theorie, in der zwei Temperaturen auftreten, beseitigt einige Pathologien der klassischen Theorie.

On the plasticity of single crystals: free energy, microforces, plastic-strain gradients

This study develops a general theory of crystalline plasticity based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on plastic strain-gradients. The microforce balances are shown to be equivalent to yield conditions for the individual slip systems, conditions that account for variations in free energy due to slip. When this energy is the sum of an elastic strain energy and a defect energy quadratic in the plastic-strain gradients, the resulting theory has a form identical to classical crystalline plasticity except that the yield conditions contain an additional term involving the Laplacian of the plastic strain. The field equations consist of a system of PDEs that represent the nonlocal yield conditions coupled to the classical PDE that represents the standard force balance. These are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. A viscoplastic regularization of the basic equations that obviates the need to determine the active slip systems is developed. As a second aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. As an application of the theory, the special case of single slip is discussed. Specific solutions are presented: one a single shear band connecting constant slip-states; one a periodic array of shear bands.