Journal of Elasticity最新文献

This work deals with the well-posedness and asymptotic behavior of a Shear beam model subject to internal dissipation of the fractional derivative-type. The energy functional is presented, and the dissipative property of the system is stablished. We use the semigroup theory in order to deal with the well-posedness and we prove the strong stability of the (C_{0})-semigroup using the Arendt-Batty and Lyubich-Vũ’s general criterion and also we prove the polynomial stability result applying Borichev and Tomilov’s theorem.

We address a nonlinear boundary homogenization problem associated to the deformations of a block of an elastic material with small reaction regions periodically distributed along a plane. We assume a nonlinear Winkler-Robin law which implies that a strong reaction takes place in these reaction regions. Outside, on the plane, the surface is traction-free while the rest of the surface is clamped to an absolutely rigid profile. When dealing with critical sizes of the reaction regions, we show that, asymptotically, they behave as stuck regions, the homogenized boundary condition being a linear one with a new reaction term which contains a capacity matrix depending on the macroscopic variable. This matrix is defined through the solution of a parametric family of microscopic problems, the macroscopic variable being its parameter. Among others, to show the convergence of the solutions, we develop techniques that extend those both for nonlinear scalar problems and linear vector problems in the literature. We also address the extreme cases.

We study large deformations of hyperelastic membranes using a purely two-dimensional formulation derived from basic balance principles within a modern geometric setting, ensuring a framework that is independent of an underlying three-dimensional formulation. To assess the predictive capabilities of membrane theory, we compare numerical solutions to experimental data from axisymmetric deformations of a silicone rubber film. Five hyperelastic models—Neo-Hookean, Mooney-Rivlin, Gent, Yeoh, and Ogden—are evaluated by fitting their material parameters to our experimental data using TensorFlow. Our results provide a systematic comparison of these models based on their accuracy in capturing observed deformations, establishing a framework for integrating theory, experiment, and data-based parameter identification.

This article derives homogenized bending shell theories starting from three-dimensional nonlinear elasticity. The original three-dimensional model contains three small parameters: the two homogenization scales (varepsilon ) and (varepsilon ^{2}) of the material properties and the thickness (h) of the shell. We obtain different limiting behaviors depending on the limit of various ratios of these three parameters.

We solve the inverse problem in three-dimensional elasticity associated with the design of a harmonic ellipsoidal isotropic elastic solid or compressible liquid inclusion that does not disturb the first invariant of the stress tensor in the surrounding isotropic elastic matrix subjected to uniform remote normal stresses. In order to achieve the harmonic condition, the two ratios of the remote normal stresses are uniquely determined for given geometric and material parameters.

The material tailoring problem for a hollow circular cylinder composed of an isotropic, incompressible, and linearly elastic functionally graded material has been analytically analyzed. The cylinder is deformed by torques and axial loads on the end faces, and pressures on its inner and outer surfaces. The cylinder material has one elastic parameter, the shear modulus (mu left ( r right ) ). For the direct problem it is a known positive and continuously varying function in the radial direction, (r). For the inverse problem (mu left ( r right )) is a design variable and is found to provide the desired radial variation of either the strain energy density, (W^{def} (r)), or the von Mises stress, (sigma ^{VM} left ( r right )), for the given loads and the cylinder geometry. If the three loads are simultaneously varied by a factor (gamma ) then (W^{def} left ( r right ) ) and (sigma ^{VM} left ( r right ) ), respectively, change by (gamma ^{2} ) and (gamma ) for fixed (mu left ( r right ) ) in the direct problem and (mu (r)) by (gamma ^{2} ) and (gamma ) in the inverse problem for preassigned (W^{def} (r) = W_{cr} left ( r right ) ) and (sigma ^{VM} left ( r right ) = sigma _{cr}^{VM} left ( r right )). The (W_{cr} left ( r right ) ) and (sigma _{cr}^{VM} left ( r right )) are, respectively, values at failure of the strain energy density and the von Mises stress. For the cylinder material composed of two constituents having positive shear moduli as is often the case in experiments we use a homogenization technique to find the radial variations of their volume fractions and ensure (mu (r)) is positive. We review three manufacturing techniques and propose an experimental program to find (W_{cr} left ( r right )) and (sigma _{cr}^{VM} left ( r right )). The expression for (mu (r)) is derived from the solution of the direct problem that has a unique solution. It provides reference solutions for similar nonlinear problems and verification of numerical algorithms. It supports the optimal design of cylinders.

We recently laid down the theoretical basis for the consistent formulation of the collocation boundary element method, as it should have been conceived from the beginning. We proposed a convergence theorem for two- and three-dimensional problems of elasticity and potential, which applies to generally curved elements in the frame of an isoparametric analysis. We also showed that the code implementation leads to controllable, highly precise and accurate results for arbitrarily small source-field distances of two-dimensional problems – limited only by the machine’s capacity to represent numbers. On the other hand, there still is the cost-benefit question of how to adequately describe a real problem’s geometry without increasing the number of degrees of freedom (h- and p-mesh refinement). We are proposing that the isoparametric implementation – with the introduced elegance of a convergence theorem – be replaced with a formulation that preserves the problem’s idealized geometry but is not isoparametric, in general. We also introduce a homothetic approach – for nodes and elements adaptively generated according to the same pattern along a boundary patch –, which is highly cost-effective. We present conceptual formulation, code implementation, and numerical illustrations that go from the simple case of an infinite plate with a circular hole to very challenging – physically unrealistic and only mathematically conceivable – topological applications: a multi-connected domain with generally curved boundary patches and presenting cracks, cusp and reentrant angles of virtually zero magnitude, and a strip of material of zero width. This cannot be manufactured in the real world but can be nevertheless simulated provided we have the proper mathematical tools, as presently proposed.

Formation of ice layers during winter is a common natural phenomenon in high-latitude regions. To evaluate the impact of the ice layer on the seismic response of a poroelastic medium, we develop a novel model to describe the dynamic interaction among the ice layer, water layer, and transversely isotropic poroelastic rock under vertical P-wave excitation. First, the general solutions for the poroelastic rock and overlying water and ice layers are derived by applying the Laplace transform. Then the dual-variable and position (DVP) method is employed to obtain a semi-analytical solution of the layered media in the transform domain. By applying a numerical inverse Laplace transform scheme, the time response of free-field motion in the layered rock under P-wave excitation is obtained. Numerical results show that the ice layer causes more complex waveforms and amplifies the vertical displacement in deeper locations in the poroelastic medium with low permeability. A higher anisotropic modulus ratio leads to an earlier arrival of displacement peaks and troughs, with this effect strengthening over time. Stiff interlayers amplify the displacement and advance the waveform, while soft interlayers have the opposite effect.

This paper develops an axisymmetric boundary element method (BEM) for analyzing a transversely isotropic (TI) layered half-space with an internal cavity subjected to pressure. The BEM formulation utilizes the fundamental solution of a TI layered solid of infinite extent under body forces uniformly concentrated along a circular ring. Three types of isoparametric elements are used to discretize the core region surrounding the cavity and an infinite element is utilized to discretize the external boundary away from the cavity. A novel numerical quadrature scheme is introduced to calculate the regular and singular integrals in the BEM formulation. Numerical verifications are carried out to confirm the accuracy and computational efficiency of the proposed BEM. The numerical results demonstrate the influence of the heterogeneity and anisotropy of the TI layered solid on the elastic fields in the surrounding rocks around the cavity of either sphere or ellipsoid.

In nonlinear elasticity, the square root of a tensor arises in the polar decomposition of the deformation gradient, and in many other applications in other areas as well. In this work, given a positive integer (p), we derive an explicit expression for the principal (p)-th root of a real-valued second-order tensor, which is not necessarily diagonalizable, whose eigenvalues do not lie on the closed negative real axis, but which is otherwise arbitrary, for any underlying space dimension (n). We also present a method for the explicit evaluation of the derivative of the (p)-th root of a tensor.