Hiromichi Itou

This paper is concerned with the reconstruction issue of an inverse crack problem in a two-dimensional bounded domain, which may have a possible application to the nondestructive evaluation of materials. It is assumed that the domain consists of two elastic plates welded together and contains unknown cracks on the junction line. The governing equation for the in-plane displacement is the Navier equation. The problem is to extract information about the location of cracks from observation data consisting of a single set of applied surface traction and the resulting in-plane displacement field measured on the boundary of the domain. It is shown that the enclosure method combined with the Kelvin transform yields explicit formulae for extracting such information from the observation data.

The quasi-static problem describes a nonlinear porous body with a non-penetrating Barenblatt’s crack driven by the fracturing fluid, and its propagation is under investigation. By this, a bulk modulus of the porous body depends linearly on the density, the fracture faces allow contact with cohesion, and leak-off of the fluid into reservoir is accounted by the model. The mathematical problem consists in finding time-continuous functions of a displacement and a mean fluid pressure in the fracture, which satisfy the coupled system of the variational inequality and the fluid mass balance, which is controlled by the volume of fracking fluid pumped into the fracture. Well-posedness of the governing relations is proved rigorously by applying the method of Lagrange multipliers and using optimality conditions for the constrained minimization problem. As anillustrative example, a numerical benchmark problem of the fluid-driven fracture is presented in one dimension and computed by a Newton-type algorithm.

This study examines a class of time-dependent constitutive equations used to describe viscoelastic materials under creep in solid mechanics. In nonlinear elasticity, the strain response to the applied stress is expressed via an implicit graph allowing multi-valued functions. For coercive and maximal monotone graphs, the existence of a solution to the quasi-static viscoelastic problem is proven by applying the Browder–Minty fixed point theorem. Moreover, for quasi-linear viscoelastic problems, the solution is constructed as a semi-analytic formula. The inverse viscoelastic problem is represented by identification of a design variable from non-smooth measurements. A non-empty set of optimal variables is obtained based on the compactness argument by applying Tikhonov regularization in the space of bounded measures and deformations. Furthermore, an illustrative example is given for the inverse problem of isotropic kernel identification. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

The response of a body described by a quasi-linear viscoelastic constitutive relation, whose material moduli depend on the mechanical pressure (that is one-third the trace of stress) is studied. The constitutive relation stems from a class of implicit relations between the histories of the stress and the relative deformation gradient. A-priori thresholding is enforced through the pressure that ensures that the displacement gradient remains small. The resulting mixed variational problem consists of an evolutionary equation with the Volterra convolution operator; this equation is studied for well-posedness within the theory of maximal monotone graphs. For isotropic extension or compression, a semi-analytic solution of the quasi-linear viscoelastic problem is constructed under stress control. The equations are studied numerically with respect to monotone loading both with and without thresholding. In the example, the thresholding procedure ensures that the solution does not blow-up in finite time.

A generalization of the Kelvin–Voigt model that can represent viscoelastic materials whose moduli depend on the mechanical pressure is derived from an implicit constitutive relation in which both the Cauchy stress and the linearized strain appear linearly. For consistency with the assumption of small deformation, a thresholding approach is applied. The proposed mixed variational problem is investigated for its well‐posedness within the context of maximal monotone and coercive graphs. For isotropic extension or compression, a semi‐analytic solution of the generalization of the Kelvin–Voigt problem under stress control is presented. The corresponding numerical simulation for monotone and cyclic pressure loading is carried out, and the results then compared against the linearized model.

Mathematical methods based on the variational approach are successfully used in a broad range of applications, especially those fields that are oriented on partial differential equations. Our problem area addresses a wide class of nonlinear variational problems described by all kinds of static and evolution equations, inverse and ill-posed problems, non-smooth and non-convex optimization, and optimal control including shape and topology optimization. Within these directions, we focus but are not limited to singular and unilaterally constrained problems arising in mechanics and physics, which are governed by complex systems of generalized variational equations and inequalities. Whereas classical mathematical tools are not applicable here, we aim at a non-standard well-posedness analysis, numerical methods, asymptotic and approximation techniques including homogenization, which are successful within the primal as well as the dual variational formalism. In a broad scope, the theme issue objectives are directed toward advances that are attained in the mathematical theory of non-smooth variational problems, its physical consistency, numerical simulation and application to engineering sciences. This article is part of the theme issue ‘Non-smooth variational problems and applications’.

We consider dynamic motion of a linearized elastic body with a crack subject to a modified contact law, which we call the Signorini contact condition of dynamic type , and to the Tresca friction condition. Whereas the modified contact law involves both displacement and velocity, it formally includes the usual non-penetration condition as a special case. We prove that there exists a unique strong solution to this model. It is remarkable that not only existence but also uniqueness is obtained and that no viscosity term that serves as a parabolic regularization is added in our model. This article is part of the theme issue ‘Non-smooth variational problems and applications’.