Journal of Elasticity最新文献

This paper is concerned with a linear theory of thermoelasticity without energy dissipation, where the second gradient of displacement and the second gradient of the thermal displacement are included in the set of independent constitutive variables. In particular, in the case of rigid heat conductors the present theory leads to a fourth order equation for temperature. First, the basic equations of the second gradient theory of thermoelasticity are presented. The boundary conditions for thermal displacement are derived. The field equations for homogeneous and isotropic solids are established. Then, a uniqueness result for the basic boundary-initial-value problems is presented. An existence theorem is established for the first boundary value problem. The problem of a concentrated heat source is investigated using a solution of Cauchy-Kowalewski-Somigliana type.

The paper presents a rigorous analysis of the singularities of elastic fields near a dislocation loop in a body of arbitrary material symmetry that extends over the entire three-space. Explicit asymptotic formulas are given for the stress, strain and the incompatible distortion near the curved dislocation. These formulas are used to analyze the main object of the paper, the renormalized energy. The core-cutoff method is used to introduce that notion: first, a core in the form of a curved tube along the dislocation loop is removed; then, the energy of the complement is determined (= the core-cutoff energy). As in the case of a straight dislocation, the core-cutoff energy has a singularity that is proportional to the logarithm of the core radius. The renormalized energy is the limit, as the radius tends to 0, of the core-cutoff energy minus the singular logarithmic part. The main result of the paper are novel formulas for the coefficient of logarithmic singularity (the ‘prelogarithmic energy factor’) and for the renormalized energy.

We propose paradigmatic examples to show how material damage phenomena can be efficiently described as a solid-solid phase transition. Starting from the pioneering work of J.L. Ericksen (J. Elast. 5(3):191–201, 1975) and the extensions of R.L. Fosdick and other authors to three-dimensional non linear elasticity, we describe the insurgence of damage as a hard → soft transition between two material states (damage and undamaged) characterized by two different energy wells. We consider the two separate constitutive assumptions of a simple Neo-Hookean type damageable material and a more complex microstructure inspired damageable Gent type material with variable limit threshold of the first invariant. In both cases we study two different deformation shear classes, one homogeneous and the other one inhomogeneous and obtain fully analytic description of the system damage response under cyclic loading. The considered constitutive assumptions and deformation classes are aimed at attaining fully analytic descriptions. On the other hand, we remark that the proposed, Griffith type, variational approach of damage, based on two different energy density functions for the damaged and undamaged material phases, and a resulting non (rank-one) convex energy, can be extended to systems with more complex energy functions, possibly with a larger number of wells representing an increasing degree of damage.

Moiré patterns result from setting a 2D material such as graphene on another 2D material with a small twist angle or from the lattice mismatch of 2D heterostructures. We present a continuum model for the elastic energy of these bilayer moiré structures that includes an intralayer elastic energy and an interlayer misfit energy that is minimized at two stackings (disregistries). We show by theory and computation that the displacement field that minimizes the global elastic energy subject to a global boundary constraint gives large alternating regions of one of the two energy-minimizing stackings separated by domain walls.

We derive a model for the domain wall structure from the continuum bilayer energy and give a rigorous asymptotic estimate for the structure. We also give an improved estimate for the (L^{2})-norm of the gradient on the moiré unit cell for twisted bilayers that scales at most inversely linearly with the twist angle, a result which is consistent with the formation of one-dimensional domain walls with a fixed width around triangular domains at very small twist angles.

Solenoids, ubiquitous in electrical engineering applications, are devices formed from a coil of wire that use electric current to produce a magnetic field. In contrast to typical electrical engineering applications that pertain to their magnetic field, of interest here is their use as actuators by studying their mechanical deformation. An analytically tractable model of parallel, coaxial circular rings is used to find the solenoid’s axial deformation when subjected to a combined electrical (current) and mechanical (axial force) loading. Both finite and infinite solenoids are considered and their equilibrium configurations as well as their stability are investigated as functions of their geometry and applied current intensity.

We achieve neutrality of a double coated circular inhomogeneity embedded in an infinite matrix subjected to uniform in-plane stresses at infinity. The introduction of the double coated circular inhomogeneity does not disturb the original uniform in-plane stress distribution in the matrix. Consequently, we obtain exact representations of the effective transverse shear modulus and effective transverse Poisson’s ratio of double coated disk assemblages of various sizes completely replacing the matrix.

We classify all exactly stress-free solutions to the cubic-to-trigonal phase transformation within the geometrically linearized theory of elasticity, showing that only simple laminates and crossing-twin structures can occur. In particular, we prove that although this transformation is closely related to the cubic-to-orthorhombic phase transformation, all its solutions are rigid. The argument relies on a combination of the Saint-Venant compatibility conditions together with the underlying nonlinear relations and non-convexity conditions satisfied by the strain components.

We propose a general and efficient method to derive various minimal representations of material tensors or pseudotensors for crystals. By a minimal representation we mean one that pertains to a specific Cartesian coordinate system under which the number of independent components in the representation is the smallest possible. The proposed method is based on the harmonic and Cartan decompositions and, in particular, on the introduction of orthonormal irreducible basis tensors in the chosen harmonic decomposition. For crystals with non-trivial point group symmetry, we demonstrate by examples how deriving restrictions imposed by symmetry groups (e.g., (C_{2}), (C_{s}), (C_{3}), etc.) whose symmetry elements do not completely specify a coordinate system could possibly miss the minimal representations, and how the Cartan decomposition of SO(3)-invariant irreducible tensor spaces could lead to coordinate systems under which the representations are minimal. For triclinic materials, and for material tensors and pseudotensors which observe a sufficient condition given herein, we describe a procedure to obtain a coordinate system under which the explicit minimal representation has its number of independent components reduced by three as compared with the representation with respect to an arbitrary coordinate system.