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

The equations of motion, boundary conditions and stress-strain relations for a highly elastic material can be expressed in terms of the stored-energy function. This has been done in part I of this series (Rivlin 1948 a ), for both the cases of compressible and incompressible materials, following the methods given by E. & F. Cosserat for compressible materials. The stored-energy function may be defined for a particular material in terms of the invariants of strain. The form in which the equations of motion, etc., are deduced, in the previous paper, does not permit the evaluation of the forces necessary to produce a specified deformation unless the actual expression for the stored-energy function in terms of the scalar invariants of the strain is introduced. In the present paper, the equations are transformed into forms more suitable for carrying out such an explicit evaluation. As examples, the surface forces necessary to produce simple shear in a cuboid of either compressible or incompressible material and those required to produce simple torsion in a right-circular cylinder of incompressible material are derived.

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Large elastic deformations of isotropic materials IV. further developments of the general theory

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

On finite deformation elasto-plasticity

In addition to the classical governing equations in continuum mechanics, two kinds of governing equations are necessary in the solution of boundary-value problems for the stress fields in multi-phase hyperelastic media with the surface/interface energy effect. The first is the interface constitutive relation, and the second are the discontinuity conditions of the traction across the interface, namely, the Young-Laplace equations. In this paper, the interface consitutive relations are presented in terms of the interface energy in both Lagrangian and Eulerian descriptions within the framework of finite deformation, and the expressions of the interface stress for an isotropic interface are given as a special case. Then, by introducing a fictitious stress-free configuration, a new energy functional for multi-phase hyperelastic media with interface energy effect is proposed. The functional takes into account the interface energy and the interface stress-induced ``residual'' elastic field, which reflects the intrinsic physical properties of the material. All field equations, including the generalized Young-Laplace equation, can be derived from the stationary condition of this functional. The present theory is illustrated by simple examples. The results in this paper provide a theoretical framework for studying the elastostatic problems of multi-phase hyperelastic bodies that involve surface/interface energy effects at finite deformation.

A theory of hyperelasticity of multi-phase media with surface/interface energy effect

Generalized strain measure with applications to physical problems

It is shown that the use of generalized measures helps (1) to bridge the gulf between the microscopic and macroscopic description of physical phenomenon; (2) to eliminate semi-empirical conditions like Tresca and von Mises and creep strain laws like those of Norton; (3) to simplify constitutive equations by prescribing a priori the order of the measure of a deformation; (4) the elimination of additional response coefficients; (5) a greater coordination between the theoretical and experimental results.

Measure-concept in mechanics

The mathematical theory of Elasticity, as at present developed, is based on the assumption that the displacements that have to be considered in elastic solid bodies are so small that the squares and products of the first differential coefficients of m, v, w with respect to x, y, z can be neglected in comparison with their first powers. That is why we cannot use it to get a satisfactory solution of many a problem in elasticity in which the displacement is finite and the strain produced is not small enough to justify the above assumption. For example, we can bend any rectangular plate in the form of a cylinder by couples applied to the edges only. As far as I know there exists no exact solution for this simple problem. In Section III I have attempted to give its solution based on the theory of finite strain. The first step towards the solution of the type of problems we have in mind is to get the components of strain corresponding with any displacement. Like the body-stress equations, these should be referred to the actual position of a point P of the material in the strained condition, and not to the position of a point considered before strain. The importance of this point, overlooked by various authors, cannot be exaggerated. Apparently Filon and Coker were the first to notice it and stress its importance. To a first approximation it is immaterial which method of reference is adopted, for the values of the strain components in the two cases differ only in the second order terms.

Finite strain in elastic problems

THE classical macroscopic treatment of problems in plasticity, creep and relaxation has to assume semi-empirical yield conditions like those of Tresca, Von Mises and creep-strain laws like those of Norton, Odqvist, Kachanov and others1. This is a direct consequence of using linear strain measures, which neglect the non-linear transition region through which the yield occurs and the fact that creep and relaxation strains are never linear. Tresca2 pointed out that a transition state, which he called the mid-zone, existed, and Todhunter and Pearson have supported him3. In this state the whole of the material participates, and not simply a line or selected region, as is assumed by classical theories. Sir Lawrence Bragg4 expresses this view when he points out that an element of a crystal, embedded in the matrix, should be pictured as being strained together with the whole mass of the metal, and hence the slip must take place by a whole intra-atomic distance. A recent numerical investigation5 on flow and deformation theories also shows that a continuous approximation through a transition state leads to a satisfactory convergent solution.

Transition Theory of Elastic-Plastic Deformation, Creep and Relaxation

Elastic-plastic transition is obtained in current literature with the help of a semi-empirical yield condition like that of Tresca or Von Mises. The stresses are obtained from the elastic solution and then substituted in the yield condition to get the transition surface. The possibility of treating it as a transition or turning point phenomenon in finite deformation has not been explored. When the plastic state tends to set in, the stressstrain relations undergo a change. This should be reflected in our equations. The linear classical theory cannot do it. If we adopt the theory of finite deformation, the equations of equilibrium give rise to non-linear differential equations whose turning points can give the elastic-plastic transition. The asymptotic solution at these points then give the displacements and the stresses, and no semi-empirical yield condition is necessary. The yield stress can be related to the elastic coefficients in the transition range through the tension-stretch result in finite deformation. This method is applied to the plastic deformation of shells and tubes under pressure.



Den elastisch-plastischen Ubergang erhalt man in der gelaufigen Literatur mit Hilfe einer halbempirischen Fliesbedingung wie der von Tresca oder von v. Mises. Die Spannungen ergeben sich aus der Losung im elastischen Fall. Diese werden in die Fliesbedingung eingesetzt und ergeben die Ubergangsflache. Die Moglichkeit, dies als Ubergangsphanomen bei endlicher Verformung zu behandeln, ist noch nicht untersucht worden. Wenn der plastische Zustand einzusetzen beginnt, andern sich die Spannungs-Dehnungsbeziehungen. Dies sollte in unseren Gleichungen zum Ausdruck kommen. Dies kann die lineare klassische Theorie nicht leisten. Unter Zugrundelegung der Theorie der endlichen Verformungen fuhren die Gleichgewichtsbedingungen auf nichtlineare Differentialgleichungen, welche die Ubergange des elastisch-plastischen Zustands beherrschen. Die asymptotischen Losungen an diesen Stellen geben dann die Verschiebungen und die Spannungen, ohne das eine halbempirische Fliesbedingung notwendig ist. Die Fliesspannung kann in Beziehung gebracht werden zu den Elastizitatskoeffizienten im Ubergangsbereich durch das Spannungs-Zug-Resultat bei endlicher Verformung. Diese Methode wird angewandt auf die unter Druck erfolgende plastische Verformung von Schalen und Zylinder.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/zamm.19630430706

B. R. Seth

Papers

Generalized strain measure with applications to physical problems

Measure-concept in mechanics

Finite strain in elastic problems

Transition Theory of Elastic-Plastic Deformation, Creep and Relaxation

Elastic plastic transition in shells and tubes under pressure