Mathematics and Mechanics of Solids最新文献

We use Muskhelishvili's complex variable formulation to solve the plane-strain problem of a compressible liquid inclusion of arbitrary shape embedded within an infinite isotropic elastic matrix under uniform remote in-plane stresses. First, the exterior of the domain occupied by the liquid inclusion is mapped onto the exterior of a unit circle in the image plane using a mapping function that contains an arbitrary number of terms. With the aid of a modified form of analytic continuation, a set of linear algebraic equations with relatively simple structure is obtained. Once this set of linear algebraic equations is solved, the internal uniform hydrostatic stress field within the liquid inclusion and the elastic field in the matrix (characterized by a pair of analytic functions) are fully determined. We illustrate our theory by deriving a closed-form solution for a hypotrochoidal liquid inclusion and comparing our results with those available in the existing literature. In addition, numerical results for the internal uniform hydrostatic stress within liquid inclusions with an n-fold axis of symmetry are presented graphically to examine the influence of the number of terms used in the mapping function. Finally, we determine the internal hydrostatic stress for the case of a rectangular liquid inclusion with various aspect ratios.

We recall in this note that the induced tangent stiffness tensor $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ appearing in a hypoelastic formulation based on the Zaremba-Jaumann corotational derivative and the rate constitutive equation for the Kirchhoff stress tensor τ is minor and major symmetric if the Kirchhoff stress τ is derived from an elastic potential $W\left(F\right)$ . This result is vaguely known in the literature. Here, we expose two different notational approaches which highlight the full symmetry of the tangent stiffness tensor $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ . The first approach is based on the direct use of the definition of each symmetry (minor and major), i.e., via contractions of the tensor with the deformation rate tensor D. The second approach aims at finding an absolute expression of the tensor $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ , by means of special tensor products and their symmetrisations. In some past works, the major symmetry of $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ has been missed because not all necessary symmetrisations were applied. The analogous tangent stiffness tensor $H^{\mathrm{ZJ}}\left(\sigma \right)$ , relating the Cauchy stress tensor σ to the Zaremba-Jaumann corotational derivative is also obtained, with both methods used for $H_{\tau }^{\mathrm{ZJ}}\left(\tau \right)$ . The approach is exemplified for the isotropic Hencky energy. Corresponding stability checks of software packages are shortly discussed.

We study the plane strain problem of a circular elastic inhomogeneity partially debonded from an infinite elastic matrix subjected to an edge dislocation at an arbitrary position. The debonded portion of the circular interface is occupied by an incompressible liquid slit inclusion. The original boundary value problem is reduced to a standard Riemann-Hilbert problem with discontinuous coefficients which can be solved analytically. The two unknown constants appearing in the analytical solution are determined by imposing the incompressibility condition of the liquid inclusion. Closed-form expressions for the internal uniform hydrostatic stress field within the liquid slit inclusion, the average mean stress within the circular elastic inhomogeneity, the rigid body rotation at the center of the circular inhomogeneity, and the two complex stress intensity factors at the two tips of the debonded portion induced by the edge dislocation are obtained.

In the definition of Noll, a body is uniform if all points are made of the same material. As shown by Noll himself and by Epstein and Maugin, uniformity makes the Helmholtz free energy depend on the material point exclusively through a tensor field, called uniformity tensor or implant tensor or material isomorphism. Uniformity is therefore a particular case of inhomogeneity. In turn, uniformity includes homogeneity as a particular case: indeed, homogeneity is attained when the uniformity tensor happens to be integrable. This work focuses on the non-linear large-deformation behaviour of uniform dielectric elastomers. Building on the foundational works of Toupin, Eringen and others, this work integrates continuum mechanics with electrostatics to develop a finite element framework for analysing uniform dielectric elastomers. This framework allows for considering the inherent inhomogeneity in materials exhibiting non-linear electromechanical coupling such as electro-active polymers. The inhomogeneity is assumed to be self-driven, i.e., not implied by the second law of thermodynamics: rather, it depends on the torsion of the connection (covariant derivative) induced by the uniformity tensor. A MATLAB^®-based finite element solver is developed and applied to the simulation of an electromechanical beam-type actuator. The solver is robust and capable of addressing various simulation scenarios. Numerical simulations demonstrate the significant impact of material uniformity on actuator performance. This research provides a tool for future applications in dielectric elastomers, particularly in sensors, actuators and bio-inspired robotics.

We study the steady-state response of a three-phase composite composed of an internal hypotrochoidal compressible liquid inclusion, an intermediate isotropic elastic coating and an outer isotropic elastic matrix with simultaneous interface slip and diffusion occurring on the solid-solid interface when the matrix is subjected to a uniform hydrostatic stress field. We design a neutral coated hypotrochoidal liquid inclusion that does not disturb the prescribed uniform hydrostatic stress field in the surrounding matrix. The neutrality is achieved when the plane-strain bulk modulus or the compressibility of the elastic matrix is determined by solving a system of simultaneous linear algebraic equations for given geometric and material parameters of the coated liquid inclusion.