Plastic deformation of bicrystals within continuum dislocation theory

We analyze the evolution of the plastic distortion and the nucleation and accumulation of dislocations within a model bicrystal with one active slip system in each single crystal (symmetric with respect to the interface), which is subject to prescribed displacements of plane–strain shear and extension, and we present closed–form analytical solutions. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the antiplane shear deformation of finite wedges

Antiplane shear deformation of finite wedges is considered under different boundary conditions. First, the assertions and results of a recent paper, namely Chue and Liu [C.H. Chue, W.J. Liu, Comments on “Analysis of an isotropic finite wedge under antiplane deformation”, Int. J. Solids Struct. 41 (2004) 5023–5034] are invalidated. Then, closed form solutions are extracted for the stress distribution in the wedge. These closed forms have the advantages of showing the possible geometric stress singularity as well as the load singularity explicitly, in addition to the continuity or discontinuity as well as the convergence of the results in the entire region. Finally, the stress intensity factors are extracted in the special case of a circular shaft containing an edge crack under different boundary conditions.     

Weak solutions for nonlinear antiplane problems leading to hemivariational inequalities

We analyze the antiplane shear deformation of an elastic cylinder in frictional contact with a rigid foundation, for static processes, under the small deformation hypothesis. Based on the Knaster–Kuratowski–Mazurkiewicz technique in the theory of the hemivariational inequalities, we prove that the model has at least one weak solution. Moreover, we present several examples of constitutive laws and friction laws for which our theoretical results are valid. Finally, we comment on the conditions which guarantee the uniqueness of the solution.     

Application of the dislocation theory of fracture to polymeric materials

The macrocharacteristics of the mechanical strength of crystalline polymers are estimated on the basis of the dislocation theory of fracture. The values obtained for the breaking and safe stresses and the necking stress are in good agreement with the experimental results.Institute of Problems of Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 549–551, May–June, 1971.     

Boundary optimal control for nonlinear antiplane problems

We consider a nonlinear antiplane problem which models the deformation of an elastic cylindrical body in frictional contact with a rigid foundation. The contact is modelled with Tresca’s law of dry friction in which the friction bound is slip dependent.The aim of this article is to study an optimal control problem which consists of leading the stress tensor as close as possible to a given target, by acting with a control on the boundary of the body. The existence of at least one optimal control is proved. Next we introduce a regularized problem, depending on a small parameter ρ, and we study the convergence of the optimal controls when ρ tends to zero. An optimality condition is delivered for the regularized problem.     

Application of the boundary element method for analysis of the antiplane shear of anisotropic bodies containing thin-walled structures

We have ascertained the limits of reasonable application of the classical boundary element method for the solution of the antiplane problem of the theory of elasticity in the study of bodies with thin-walled elements of structure and geometry. We have proposed an approach for the regularization of singular and quasisingular integrals, which appear inevitably in analyzing thin structures. We give also numerical examples demonstrating the reliability and efficiency of the proposed approach.     

A continuum theory of superconductivity

Abstract. A continuum theory of superconductivity is formulated for a mixture consisting of three species: a superelectron fluid, a conducting fluid, and a conducting elastic solid. Each one of the three species is subject to their own electro-magnetic (E-M) fields and motions. Irreversible thermodynamics are used to obtain constitutive equations. Field equations, boundary and initial conditions are given. A special case is obtained, suitable for mathematical analysis and applications. The Pippard theory of superconductivity is shown to be a special case of the present theory.     

A nonlinear extensible 4-node shell element based on continuum theory and assumed strain interpolations

Summary A quadrilateral continuum-basedC ⁰ shell element is presented, which relies on extensible director kinematics and incorporates unmodified three-dimensional constitutive models. The shell element is developed from the nonlinear enhanced assumed strain (EAS) method advocated by Sino & Armero [1] and formulated in curvilinear coordinates. Here, the EAS-expansion of the material displacement gradient leads to the local interpretation of enhanced covariant base vectors that are superposed on the compatible covariant base vectors. Two expansions of the enhanced covariant base vectors are given: first an extension of the underlying single extensible shell kinematic and second an improvement of the membrane part of the bilinear element. Furthermore, two assumed strain modifications of the compatible covariant strains are introduced such that the element performs well even in the case of very thin shells. This paper is dedicated to the memory of Juan C. Simo In honour of Professor Juan Simo who had significant collaboration with our institute and contributed important insights to our research work. This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

Variant of the geometrically nonlinear theory of anisotropic shells with allowance for transverse shear

A very simple variant of the geometrically nonlinear theory of anisotropic shells with allowance for the high compliance of the material in transverse shear is proposed. From this theory there follow, as a special case, the equations for an isotropic shell; these differ from the relations of [2] with respect to terms of the order of the ratio of the thickness of the shell to the radii of curvature small as compared with unity. The equations obtained are used to solve the problem of the stability of orthotropic shells of revolution relative to the starting axisymmetric state of stress.Translated from Mekhanika Polimerov, No. 5, pp. 863–871, September–October, 1969.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

A continuum theory of dense suspensions

A continuum theory is introduced for viscous fluids carrying dense suspensions (such as blood) or emulsions of arbitrary shape and inertia. Suspended particles possess microinertia that make the mixture an anisotropic fluid whose viscosity changes with motion and orientation of suspensions. The microinertia balance law coupled with the equations of motion of an anisotropic fluid govern the ultimate outcome. By means of the second law of thermodynamics, constitutive equations are obtained in terms of the frame-independent tensors. In a special case, a theory of bar-like suspensions is obtained. The field equations, boundary and initial conditions are given for both the arbitrarily-shaped suspensions and the bar-like suspensions. The theory is demonstrated with the solution of the channel flow problem. The mean viscosity of the fluid with suspensions is determined. The motions of suspensions down flow are demonstrated.     

Application of foliation theory and approximative groups to controllability analysis of nonlinear systems

Translated from Optimal'nost Upravlyaemykh Dinamicheskikh Sistem, No. 19, pp. 4–12, Vsesoyuznyi Nauchno-Issledovatel'skii Institut Sistemnykh Issledovanii, Moscow (1988).     

Contribution to the nonlinear theory of viscoelasticity

The author presents a generalization of certain results obtained in [3] relating to the region of rigid behavior of plastics for which the nonlinearity of the viscoelastic properties is important even in the area of small strains. Certain questions connected with the derivation of the basic equations of viscoelasticity are considered for small strains. Formulas are obtained for the resolvents of kernels of arbitrary order. The general equations of the "principal quadratic theory of viscoelasticity" are derived.Mekhanika Polimerov, Vol. 2, No. 4, pp. 498–507, 1966     

Imperfect interface effect for nano-composites accounting for fiber section shape under antiplane shear

This paper investigates the elastic responses of fibrous nano-composites with imperfectly bonded interface under longitudinal shear. The proposed imperfect interface model is the shear lag (or the spring layer) model; the presented nano interfacial stress model is the Gurtin–Murdoch surface/interface model; and the three-phase confocal elliptical cylinder model is the geometry model accounting for the fiber section shape. By virtue of the complex variable method, a generalized self-consistent method is employed to derive the closed from solution of the effective antiplane shear modulus of the fibrous nano-composites with imperfect interface. Five existing solutions can be regarded as the limit form the present analytic expression. The influences of the interface elastic constant, the interfacial imperfection parameter, the size of the elliptic section fiber, the fiber section aspect ratio, the fiber volume fraction and the fiber elastic property on the effective antiplane shear modulus of the nano-composites are discussed. Particularly, numerical results demonstrate that the interfacial elastic imperfection will always cause a significant reduction in the effective antiplane shear modulus; and the fiber interface stress effect on the effective modulus of the fibrous nano-composites will weaken with the interfacial imperfection increases.     

Orientation-preserving condition and polyconvexity on a surface: Application to nonlinear shell theory

We propose in this paper a definition of a “polyconvex function on a surface”, inspired by the definitions set forth in other contexts by J. Ball (1977) [3] and by J. Ball, J.C. Currie, and P.J. Olver (1981) [5]. When the surface is thought of as the middle surface of a nonlinearly elastic shell and the function as its stored energy function, we show that it is possible to assume in addition that this function is coercive for appropriate Sobolev norms and that it satisfies specific growth conditions that prevent the vectors of the covariant bases along the deformed middle surface to become linearly dependent, a condition that is the “surface analogue” of the orientation-preserving condition of J. Ball. We then show that a functional with such a polyconvex integrand is weakly lower semi-continuous, a property which eventually allows to establish the existence of minimizers. We also indicate how this new approach compares with the classical nonlinear shell theories, such as those of W.T. Koiter and P.M. Naghdi.