Breakdown of elasticity theory for jammed hard-particle packings: conical nonlinear constitutive theory

Hard-particle packings have provided a rich source of outstanding theoretical problems and served as useful starting points to model the structure of granular media, liquids, living cells, glasses, and random media. The nature of “jammed” hard-particle packings is a current subject of keen interest. We demonstrate that the response of jammed hard-particle packings to global deformations cannot be described by linear elasticity (even for small particle displacements) but involves a “conical” nonlinear constitutive theory. It is the singular nature of the hard-particle potential that leads to the breakdown of linear elasticity. Interestingly, a nonlinear theory arises because the feasible particle displacements (leading to unjamming) depend critically on the local spatial arrangement of the particles, implying a directionality in the feasible strains that is absent in particle systems with soft potentials. Mathematically, the set of feasible strains has a conical structure, i.e., components of the imposed strain tensor generally obey linear inequalities. The nature of the nonlinear behavior is illustrated by analyzing several specific packings. Finally, we examine the conditions under which a packing can be considered to “incompressible” in the traditional sense.     

On the inversion of the lagrange-dirichlet theorem in the case of an analytic potential

We examine the following conjecture concerning equilibrium instability for con-servative systems: if the potential is analytic and has no minimum at the origin, the vanishing solution is unstable. It is well-known that for a system with two degrees of freedom, this conjecture is true if the Hessian matrix of the potential is not zero. We prove that the conjecture is still true if the Hessian matrix is zero and the third-order terms of the potential do not constitute a perfect cube. Then, for an n-dimensional system, we extend a result of Koiter to the case of a potential for which the Hessian matrix has (n?2) strictly positive eigenvalues and two zero eigenvalues.     

Converse problem of elasticity theory for an anisotropic medium

To prevent stress concentration it is of considerable interest that the body contour be found whose portions show no preference to brittle failure or plastic deformation. Such contours are called by us “equally rigid.” Two-dimensional problems are considered for finding the “equally rigid” form of the hole in an anisotropic medium. The lack of stress concentration on the coutour of the hole is the criterion which decides whether or not the hole is “equally rigid.” For an isotropic medium this converse problem of elasticity theory was solved in [1].     

Use of inversion transformation for modeling in certain problems of the theory of elasticity and plasticity

Certain possibilities of using the inversion transformation for modeling problems of the theory of elasticity and plasticity are considered, in particular, for cases where this allows us to make the experimental investigation on the model considerably easier to carry out.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 134–136, March–April, 1972.     

On the well-posedness of the equilibrium problem for linear elasticity in unbounded regions

In this paper we establish some continuous dependence and uniqueness theorems for equilibrium solutions of the equations of general anisotropic linear elasticity in exterior domains. The method we employ is that of the weight function which we introduced in previous papers. However, this is the first example where the method is applied to a static problem. The above theorems are obtained by allowing the strain to be unbounded at large spatial distances. In some cases, no growth condition is assumed. Moreover, the displacement and the elasticities are also possibly allowed to grow.     

On various variational principles in nonlinear theory of elasticity

In the present paper functionals for the various possible main variational principles in the nonlinear theory of e-lasticity are derived from the "full energy principle" and several of them are not found yet in the literatures available. Through the derivation of this paper we suggest a conjecture on the nonexistence of the eleventh and the sixth classes for the variational principles in Table 6.1 of H.C. Hu’s monograph [1].     

On uniqueness for the displacement problem in the dynamical theory of elasticity

By means of a weighted form of Garding's inequality, we prove a uniqueness theorem for the initial boundary value problem of finite elastodynamics in unbounded domains.

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Sommario Mediante un'opportuna forma pesata della disuguaglianza di Garding, che ricaviamo preliminarmente, dimostriamo un teorema di unicità per il problema degli spostamenti in elastodinamica non lineare relativo a corpi occupanti regioni non limitate dello spazio.

     

On the theory of the fractal scaling-law elasticity

Meccanica - In the article we invoke a new treatment of the non-traditional elasticity with use of the fractal scaling-law vector calculus. The fundamental theorems for the fractal scaling-law...