Effects of couple-stresses in linear elasticity

Publisher Summary The fundamental assumption of all theories of plasticity—that of time independence of the equations of state—makes simultaneous description of the plastic and rheologic properties of a material impossible. I t is well-known that in many practical problems, the actual behaviour of a material is governed by plastic as well as by rheologic effects. It can even be said that for many important structural materials, rheologic effects are more pronounced after the plastic state has been reached. Every material shows more or less pronounced viscous properties. In some problems, the influence of viscous properties of the material may be negligible, while in others, it may be essential. Both sciences—plasticity and rheology—are concerned with the description of important mechanical properties of structural materials. Each of them has created its own methods of investigation and has developed within the framework of certain assumptions which, unfortunately, cannot always be satisfied in reality. The results of rheology are confined to cases where plastic strain is of no decisive importance.

Fundamental Problems in Viscoplasticity

This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.

Dissipative dynamical systems Part II: Linear systems with quadratic supply rates

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

The Linear Theory of Elasticity

By the term ‘acceleration wave’ is meant an isolated geometric surface that moves relative to the material and across which the acceleration (but not the velocity) is discontinuous. The class of materials considered is characterized by a homogeneous linear tensor relation between stress-rate and strain-rate, arbitrary except for a symmetry restriction on the coefficients. A general matrix equation is obtained for the possible wave speeds and polarizations (with modifications when the material is incompressible). Calculations are carried out in detail for a wide range of elastic/plastic solids. Other topics include stationary discontinuities; the relation with vibration analysis; a physical interpretation for the matrix equation; and connexions between the theories of waves, stability, and uniqueness.

Acceleration waves in solids

On the linear theory of viscoelasticity

: This paper deals with the quasi-static analysis of transient thermal stresses in the linear theory of homogeneous and isotropic viscoelastic solids with temperature-dependent physical characteristics The underlying constitutive law rests on the temperature-time equivalence hypothesis according to which a uniform temperature change affects the viscoelastic response of the material through a corresponding uniform shift of the logarithmic time scale If the temperature varies with time, the preceding hypothesis implies a dependence of the instantaneous stresses upon the entire temperature history Following an exposition of the theoretical background, exact solutions are deduced to two specific problems The first concerns the transient thermal stresses in a slab of infinite extent, generated by a temperature field that depends arbitrarily (Author-PL)

On Transient Thermal Stresses in Viscoelastic Materials With Temperature-Dependent Properties

On the integration of the equations of motion in the classical theory of elasticity

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On the Completeness of the Boussinesq-Papkovich Stress Functions

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Eli Sternberg

Papers

On the linear theory of viscoelasticity

On Transient Thermal Stresses in Viscoelastic Materials With Temperature-Dependent Properties

On the integration of the equations of motion in the classical theory of elasticity

On the Completeness of the Boussinesq-Papkovich Stress Functions

On Saint-Venant’s principle