-plane analysis for the fundamental problems of a stretched infinite plate weakened by curvilinear holes.
Save our stones -- hysteresis phenomenon in porous media
We present a mathematical description of wetting and drying stone pores, where the resulting mathematical model contains hysteresis operators. We describe these hysteresis operators and present a numerical solution for a simplified problem.
Scalar boundary value problems on junctions of thin rods and plates
We derive asymptotic formulas for the solutions of the mixed boundary value problem for the Poisson equation on the union of a thin cylindrical plate and several thin cylindrical rods. One of the ends of each rod is set into a hole in the plate and the other one is supplied with the Dirichlet condition. The Neumann conditions are imposed on the whole remaining part of the boundary. Elements of the junction are assumed to have contrasting properties so that the small parameter, i.e. the relative...
Scaling laws for non-euclidean plates and the isometric immersions of riemannian metrics
Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper...
Scaling laws for non-Euclidean plates and the W2,2 isometric immersions of Riemannian metrics
Recall that a smooth Riemannian metric on a simply connected domain can be realized as the pull-back metric of an orientation preserving deformation if and only if the associated Riemann curvature tensor vanishes identically. When this condition fails, one seeks a deformation yielding the closest metric realization. We set up a variational formulation of this problem by introducing the non-Euclidean version of the nonlinear elasticity functional, and establish its Γ-convergence under the proper scaling....
Scattering of SH-waves by a Griffith crack in a long strip at asymmetric position.
Second order linear Volterra integrodifferential equations.
Seismic inversion for a crak opening
The displacement field caused by the classic earthquake mechanism model consisting of a slip along the fault is extended to the case when besides the slip, also an opening occurs caused by tensional forces. The tensor matrix describing the moment tensor does not necessarily have a nil trace. The direct problem is solved finding the radiation pattern for and waves. A method to solve the inverse problem of the determination of the four parameters describing the source is presented and tested on...
Selfadjoint Extensions for the Elasticity System in Shape Optimization
Two approaches are proposed to modelling of topological variations in elastic solids. The first approach is based on the theory of selfadjoint extensions of differential operators. In the second approach function spaces with separated asymptotics and point asymptotic conditions are introduced, and a variational formulation is established. For both approaches, accuracy estimates are derived.
Self-similarly expanding networks to curve shortening flow
We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form degree angles, and expands homothetically under curve shortening flow. We also prove uniqueness of these networks.
Semicontinuity theorem in the micropolar elasticity
In this paper we investigate the equivalence of the sequential weak lower semicontinuity of the total energy functional and the quasiconvexity of the stored energy function of the nonlinear micropolar elasticity. Based on techniques of Acerbi and Fusco [Arch. Rational Mech. Anal.86 (1984) 125–145] we extend the result from Tambača and Velčić [ESAIM: COCV (2008) DOI: 10.1051/cocv:2008065] for energies that satisfy the growth of order p≥ 1. This result is the main step towards the general existence...
Sensitivity analysis of a nonlinear obstacle plate problem
We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9, 10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that...
Sensitivity Analysis of a Nonlinear Obstacle Plate Problem
We analyse the sensitivity of the solution of a nonlinear obstacle plate problem, with respect to small perturbations of the middle plane of the plate. This analysis, which generalizes the results of [9,10] for the linear case, is done by application of an abstract variational result [6], where the sensitivity of parameterized variational inequalities in Banach spaces, without uniqueness of solution, is quantified in terms of a generalized derivative, that is the proto-derivative. We prove that...
Shakedown theorems in poroplastic dynamics
The constitutive model assumed in this Note is poroplastic two-phase (solid-fluid) with full saturation and stable in Drucker’s sense. A solid or structure of this material is considered, subjected to dynamic external actions, in particular periodic or intermittent, in a small deformation regime. A sufficient condition and a necessary one are established, by a «static» approach, for shakedown (or adaptation), namely for boundedness in time of the cumulative dissipated energy.
Shape and topological sensitivity analysis in domains with cracks
The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural...
Shape and topology optimization of the robust compliance via the level set method
The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute it....
Shape and topology optimization of the robust compliance via the level set method
The goal of this paper is to study the so-called worst-case or robust optimal design problem for minimal compliance. In the context of linear elasticity we seek an optimal shape which minimizes the largest, or worst, compliance when the loads are subject to some unknown perturbations. We first prove that, for a fixed shape, there exists indeed a worst perturbation (possibly non unique) that we characterize as the maximizer of a nonlinear energy. We also propose a stable algorithm to compute...
Shape optimization for dynamic contact problems
The paper deals with shape optimization of dynamic contact problem with Coulomb friction for viscoelastic bodies. The mass nonpenetrability condition is formulated in velocities. The friction coefficient is assumed to be bounded. Using material derivative method as well as the results concerning the regularity of solution to dynamic variational inequality the directional derivative of the cost functional is calculated and the necessary optimality condition is formulated.
Shape optimization in contact problems
Shape optimization in contact problems. Approximation and numerical realization