A. C. Pipkin

Small deformations superposed on large deformations in materials with fading memory

The non-linear stress-deformation relation for materials with memory has been discussed by Green & Rivlin [1] and Green,Rivlin & Spencer [2]. In the present paper we derive the restricted form applicable to small dynamic deformations superposed on large static deformations. Although this can be done by making an appropriate linearization of the final result of the non-linear theory, we avoid some of the complexities of that theory by introducing the linearization at an earlier stage. The results obtained will apply in particular to problems of wave propagation in finitely deformed viscoelastic bodies.

Equilibrium of Tchebychev Nets

Cloth is deformed mainly by changing the angle between the threads of the warp and woof. The additional deformation due to fiber-stretching is ordinarily negligible in comparison to the finite distortion that can be produced without stretching. In a lecture published only in summary, Tchebychev [1] suggested a continuum model for cloth in which the fibers are treated as continuously distributed and inextensible. Another summary of Tchebychev’S lecture appears in his collected works [2], but the manuscript was not published until 1951 [3]. Independently, Voss [4, 5] investigated the geometrical properties of the networks that are formed on curved surfaces by initially plane, orthogonal networks of inextensible fibers. Probably because of the influence of Darboux [6], such networks came to be known as Tchebychev nets (see also Bianchi [7, 8] for example).

Convexity conditions for strain-dependent energy functions for membranes

The forms that the convexity, polyconvexity, and rank-one convexity inequalities take when the strain energy is required to be a function of the strain G are studied. It is shown in particular that W(G) must be an increasing function of G, in the sense that W(G′)≧W(G) if G′ − G is non-negative definite. Relatively simple sufficient conditions in terms of G alone are given. Necessary and sufficient conditions in terms of G alone are found to be rather complex.

Some Examples of Crinkles

Many physical problems can be phrased as the problem of minimizing some energy functional E[f] over a given class of admissible functions f. It can happen that there is a minimizing sequence fn that approaches a limit f but f does not minimize E, either because f is not in the admissible class or because E is not lower semicontinuous. In the examples that I discuss here, this happens because the derivatives # are highly discontinuous and do not approach f’ in the limit. I call such sequences crinkles, and call the limiting function f the carrier of the crinkle. Young [1] has written a book on the subject; he calls such sequences generalized curves. In control theory the same sort of thing is also called a chattering state.