The effect of residual stress on the stability of a circular cylindrical tube

Residual stresses in an unloaded configuration of an elastic material have a significant influence on the response of the material from that configuration, but the effect of residual stress on the stability of the material, whether loaded or unloaded, has only been addressed to a limited extent. In this paper we consider the level of residual stress that can be supported in a thick-walled circular cylindrical tube of non-linearly elastic material without loss of stability when subjected to fixed axial stretch and either internal or external pressure. In particular, we consider the tube to have radial and circumferential residual stresses, with a simple form of elastic constitutive law that accommodates the residual stress, and incremental deformations restricted to the cross section of the tube. Results are described for a tube subject to a level of (internal or external) pressure characterized by the internal azimuthal stretch. Subject to restrictions imposed by the strong ellipticity condition, the emergence of bifurcated solutions is detailed for their dependence on the level of residual stress and mode number.

Sharp bounds on the minimum $M$-eigenvalue and strong ellipticity condition of elasticity $Z$-tensors-tensors

<p style='text-indent:20px;'>In this paper, we establish sharp upper and lower bounds on the minimum <i>M</i>-eigenvalue via the extreme eigenvalue of the symmetric matrices extracted from elasticity <i>Z</i>-tensors without irreducible conditions. Based on the lower bound estimations for the minimum <i>M</i>-eigenvalue, we provide some checkable sufficient or necessary conditions for the strong ellipticity condition. Numerical examples are given to demonstrate the proposed results.</p>

Wrinkles and creases in the bending, unbending and eversion of soft sectors

We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoretical and numerical treatment allows us to predict the number and wavelength of the wrinkles. Guided by experimental observations, we finally look at the development of creases, which we simulate through advanced finite-element computations. In some cases, the linearized analysis allows us to predict correctly the number and the wavelength of the creases, which turn out to occur only a few per cent of strain earlier than the wrinkles.

Asymptotic Properties of Solutions of Two-Dimensional Differential-Difference Elliptic Problems

In the half-plane {−∞<x<+∞}×{0<y<+∞}, the Dirichlet problem is considered for m differential-difference equations of the kind uxx+∑mk=1akuxx(x+hk,y)+uyy=0, where the amount of nonlocal terms of the equation is arbitrary and no commensurability conditions are imposed on their coefficients a1,..., am and the parameters h1,..., hm determining the translations of the independent variable x. The only condition imposed on the coefficients and parameters of the studied equation is the nonpositivity of the real part of the symbol of the operator acting with respect to the variable x. Earlier, it was proved that the specified condition (i. e., the strong ellipticity condition for the corresponding differential-difference operator) guarantees the solvability of the considered problem in the sense of generalized functions (according to the Gel’fand-Shilov definition), a Poisson integral representation of a solution was constructed, and it was proved that the constructed solution is smooth outside the boundary line. In the present paper, the behavior of the specified solution as y → +∞ is investigated. We prove the asymptotic closedness between the investigated solution and the classical Dirichlet problem for the differential elliptic equation (with the same boundary-value function as in the original nonlocal problem) determined as follows: all parameters h1,..., hm of the original differential-difference elliptic equation are assigned to be equal to zero. As a corollary, we prove that the investigated solutions obey the classical Repnikov-Eidel’man stabilization condition: the solution stabilizes as y → +∞ if and only if the mean value of the boundary-value function over the interval (-R, +R) has a limit as R → +∞.

On Ellipticity of Hyperelastic Models Based on Experimental Data

The condition of ellipticity of the equilibrium equation plays an important role for correct description of mechanical behavior of materials and is a necessary condition for new defining relationships. Earlier, new deformation measures were proposed to vanish correlations between the terms, that dramatically simplifies restoration of defining relationships from experimental data. One of these new deformation measures is based on the QR-expansion of deformation gradient. In this paper, we study the strong ellipticity condition for hyperelastic material using the QR-expansion of deformation gradient.

THE RATE OF CONVERGENCE AND WEAKER CONVERGENT CONDITION FOR THE METHOD FOR FINDING THE LARGEST SINGULAR VALUE OF RECTANGULAR TENSORS

The applications of real rectangular tensors, among others, including the strong ellipticity condition problem within solid mechanics, and the entanglement problem within quantum physics. A method was suggested by Zhou, Caccetta and Qi in 2013, as a means of calculating the largest singular value of a nonnegative rectangular tensor. In this paper, we show that the method converges under weak irreducibility condition, and that it has a Q-linear convergence.