Discontinuous Mappings and the Limit Load in Boundary Value Problems of Nonlinear Elasticity

Abstract

The paper considers a boundary-value problem of nonlinear elasticity for a mapping (deformation) in two weak formulations: in the form of a variational equilibrium equation and in the form of minimization of a multidimensional integral energy functional. From a mathematical point of view, both formulations refer to problems of functional analysis, in terms of which their mathematical correctness is discussed. Using the methods of variational calculus, with the example of two simple problems, it is proved that, for some nonlinear elastic models, in the corresponding boundary-value problems, there can be mappings with slip-type discontinuities and there can be a limit load, i.e., a value of external forces above which the boundary-value problem has no solution. Among such models there are elastic potentials with linear growth with respect to the mapping’s gradient, e.g., the well-known statistical Bartenev–Khazanovich model and the phenomenological Treloar model. The relation between these effects is discussed. It is also noted that the results obtained should be taken into account in the practical use of linear-growth elastic potentials. With the example of the problem of the axisymmetric torsion or tension of a circular cylinder, lower estimates for the limit load are constructed analytically using the methods of variational calculus and optimization theory. An analysis of the relations obtained shows that a polynomial growth of order p is characterized by polynomial hardening of order p – 1. With a linear increase in the specific strain energy with the strain gradient, a saturation effect is observed, which corresponds to the limit load. This behavior is typical of boundary-value problems of deformation plasticity, where a limit load at zero hardening also exists, i.e., of ideal elastoplasticity.

APPENDIX

Elastic potentials are usually built phenomenologically after processing experimental data [1–4]. But some models are based on the molecular-statistical analysis of elastic materials. These models include the well-known Bartenev–Khazanovich model, which describes the behavior of incompressible cross-linked elastic polymers operating in water or oil [8].

We consider the multiplicative expansion of the mapping’s gradient F = Q · Λ, where Q is the orthogonal second-rank tensor (rotation tensor) and Λ is a symmetric second-rank tensor of the extension ratio [1–4].

The Bartenev–Khazanovich elastic potential has the following form:

$$W = \mu ({{\lambda }_{1}} + {{\lambda }_{2}} + {{\lambda }_{3}} - 3),\quad {{\lambda }_{1}}{{\lambda }_{2}}{{\lambda }_{3}} = 1,$$

where λ_i > 0 are the eigenvalues of the tensor Λ (the moduli of the eigenvalues of the tensor F or the principal relative elongations) and μ is the shear modulus at small strains.

The potential was obtained on the basis of molecular considerations. In the theory of Khazanovich networks, it is assumed that the action of external forces is transmitted through the direct interaction of chains rather than nodes, as in the classical theory. It is convincingly shown that μ = $3kT{{\bar {t}}_{0}}{\text{/}}(2m)$, where k ≈ 1.38 × 10^–21 (N m/K) is the Boltzmann constant, T is the absolute temperature, ${{\bar {t}}_{0}}$ is the average relative stress of molecular chains in the undeformed state, and m is the volume of the freely connected segment of the molecular chain of the elastomer.

It is easy to show that the Bartenev–Khazanovich potential has linear growth with respect to |F|. Indeed, the main properties of the orthogonal tensor Q are as follows: Q^–1 = Q^T and |Q|² = 3 [3]. Therefore, the relations F = Q · Λ and Λ = (F^T · F)^1/2 imply two main estimates: |Λ| ≤ |F| ≤ $\sqrt 3$|Λ|.

On the principal axes, Λ = diag(λ₁, λ₂, λ₃) and, since all λ_i > 0, the simplest inequalities |Λ| ≤ λ₁ + λ₂ + λ₃ ≤ $\sqrt 3$|Λ| hold. As a result, the Bartenev–Khazanovich elastic potential satisfies the following estimates:

$$\frac{\mu }{{\sqrt 3 }}\left| {\mathbf{F}} \right| - 3\mu \leqslant W \leqslant \sqrt 3 \mu \left| {\mathbf{F}} \right| - 3\mu ,$$

which corresponds to linear growth of the function W with respect to |F|.