Antiplane shear deformations in a linear theory of elasticity with microstructure

In this paper, we solve fundamental boundary value problems in a theory of antiplane elasticity which includes the effects of material microstructure. Using the real boundary integral equation method, we reduce the fundamental problems to systems of singular integral equations and construct exact solutions in the form of integral potentials.Received: March 25, 2002     

Antiplane shear deformation of piezoelectric bodies in contact with a conductive support

We consider a mathematical model which describes the frictional contact between a piezoelectric body and an electrically conductive support. We model the material’s behavior with an electro-elastic constitutive law; the frictional contact is described with a boundary condition involving Clarke’s generalized gradient and the electrical condition on the contact surface is modelled using the subdifferential of a proper, convex and lower semicontinuous function. We derive a variational formulation of the model and then, using a fixed point theorem for set valued mappings, we prove the existence of at least one weak solution. Finally, the uniqueness of the solution is discussed; the investigation is based on arguments in the theory of variational-hemivariational inequalities.     

A note on non-uniqueness in linear elasticity theory

Zusammenfassung Es wird mit Hilfe von Gegenbeispielen gezeigt, dass die Annahme eines stark elliptischen Tensors der Elastizitätsmoduln für Spannungsaufgaben und gemischte Probleme der linearen Elastizität und sogar für das Verschiebungsproblem im Fall eines inhomogenen Körpers keine Eindeutigkeit garantiert. Für den Spezialfall eines isotropen Materials mit verschwindender Kompressibilität wird ein Ausdruck für die allgemeinste (nicht-eindeutige) Lösung des Spannungsproblems bei verschwindenden Oberflächenspannungen gegeben.     

On the frame invariance of linear elasticity theory

It is widely believed that classical linear elasticity theory does not conform to the Galilean frame invariance of general (non-relativistic) field theories. This view is traced here to an interpretation of the relationship between the deformation gradient and the displacement gradient which does not reflect the tensor character of the variables involved. Frame invariance is shown to follow if tensor character is imposed on this relationship at the outset.     

On the frame invariance of linear elasticity theory

It is widely believed that classical linear elasticity theory does not conform to the Galilean frame invariance of general (non-relativistic) field theories. This view is traced here to an interpretation of the relationship between the deformation gradient and the displacement gradient which does not reflect the tensor character of the variables involved. Frame invariance is shown to follow if tensor character is imposed on this relationship at the outset.     

The shell theory including transverse shear deformations and thickness change

A variational method for refining the theory of shells based on power series expansion of displacements has been described. The particular case of a cubic approximation for the tangential displacements and a quadratic approximation for the deflections is considered in detail. A constitutive system of differential equations in the canonical form for the axisymmetrical deformation of cyclindrical shells is derived. As an example, axisymmetrical deformations of a cylindrical shell made of an orthotropic composite material are discussed.Martin Luther Universität Halle-Wittenberg, Fachbereich Werkstoffwissenschaften. Germany. Kharkov State Polytechnical University, Department of Dynamics and Strength of Machines. Ukraine. Published in Mekhanika Kompozimykh Materialov, No. 6, pp. 768–780. November–December, 1997.     

Integral equations in the linear theory of elasticity in semiinfinite domains

We investigate linear integral equations on a semiaxis that appear in the course of construction of solutions of boundary-value problems in the theory of elasticity in such domains as a semiinfinite strip or a cylinder. By using the Mellin transformation and the theory of perturbations of linear operators, we establish general results concerning the solvability and asymptotic properties of solutions of the equations considered. We give examples of application of the general statements obtained to specific integral equations in the theory of elasticity. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 50, No. 5, pp. 613–622, May, 1998.     

Singularities of homogeneous deformations in finite elasticity

Is classification of the singularities of the potential, concerning the homogeneous deformations in Finite Elasticity, an important material property? The present study demonstrates that the answer to the question is positive. Since the type of singularity prescribes Maxwell's sets in the neighborhood of a singularity, the emergence of multiphase strain states depends on the performed classification. Bifurcation analysis and singularity classification for homogeneous deformations of any hyperelastic, anisotropic, homogeneous material under any type of conservative quasi-static loading is performed. Critical conditions for branching of the equilibrium paths are defined and their post-critical behavior is studied. Singularities corresponding to simple and compound branching are classified according to Arnold's classification. Unconstrained materials are only considered. Special attention is given to the compound branching $(D_k)$ cases requiring elaborate mathematical techniques. The geometry of the umbilics is described with the equilibrium paths and the cusp lines. With the help of Maxwell's sets in the neighborhood of the umbilics, the emergence of discontinuous strain fields is demonstrated. The theory is applied to orthotropic and transversely isotropic materials as well. The proposed method may directly be applied to crystals stability problems.     

On the so-called comparison theorem in the linear theory of elasticity

Summary The comparison theorems in [3] and [4] are refined to their sharpest possible common form.

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Zusammenfassung Die Vergleichstheoreme in [3] und [4] werden zu ihrer schärfsten gemeinsamen Form verfeinert.

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Dedicated to Professor Henry Görtler on the occasion of his forthcoming 70th birthday     

On the displacement boundary-value problem of static linear elasticity theory

Zusammenfassung Unter der Voraussetzung, dass die Verschiebungsgleichungen der linearen Elastostatik für isotrope homogene Materialien stark elliptisch sind, haben wir einige globale Extremalprinzipien, ein Reziprozitätstheorem, einen Satz betreffend die dazu verwandte Greensche Abbildung und zwei Mengen von oberen und unteren Grenzen für die erste Randwertaufgabe hergeleitet.     

Singularly perturbed Dirichlet boundary-value problem for a stationary system in the linear elasticity theory

We consider a singularly perturbed Dirichlet boundary-value problem for an elliptic operator of the linear elasticity theory in a bounded domain with a small cavity. The main result is the proof of the theorem about the convergence of eigenelements of the perturbed boundary-value problem to eigenelements of the corresponding limiting boundary-value problem, when the parameter ? which defines the diameter of the small cavity tends to zero.     

Optimal adaptive preconditioners in static problems of the linear theory of elasticity

For a static problem of the linear theory of elasticity in dual statements, we construct and justify optimal adaptive two- and three-layer iterative methods with sharp estimates for the convergence rate.     

Antiplane shear of a strip containing a staggered array of rigid line inclusions

Motivated by the increased use of fibre-reinforced materials, we illustrate how the effective elastic modulus of an Isotropic and homogeneous material can be increased by the insertion of rigid inclusions. Specifically, we consider the two-dimensional antiplane shear problem for a strip of material. The strip is reinforced by introducing two sets of ribbon-like, rigid inclusions perpendicular to the faces of the strip. The strip is then subjected to a prescribed uniform displacement difference between its faces, see Figure 1. It should be noted that the problem posed is equivalent to that of the uniform antiplane shear problem for an infinite two-dimensional material containing a staggered array of rigid inclusions (see [1] for a review of antiplane problems in the literature). The problem is reduced in standard fashion [2–6] to a mixed boundary value problem in a rectangular domain, whose closed form solution given in terms of integrals of Weierstrassian Elliptic functions, is obtained via triple sine series techniques. The effective shear modulus of the reinforced strip can now be calculated and compared with the shear modulus of a strip without inclusions. Also obtained are the stress singularity factors at the end tips of the inclusions. Numerical results are presented for several different reinforcement geometries.     

Adjoint-coordinated approximations and economical discrete implementations in a dynamic problem of the linear theory of elasticity

For dynamic three-dimensional problems of the elasticity theory, we construct a new class of economical implicit difference schemes with high degree of parallelism. They include difference schemes whose parallelism degree is the same as for ordinary explicit schemes. So far, even the very existence of implicit schemes with the same parallelism degree has been strongly doubted.     

Computation of large shear deformations of a thermoplastic material

A specialized finite difference method with grid refinement and variable time steps is created to approximate the deformation velocity and the temperature in a simple model of the shearing of a thermoplastic material. A specific problem where the solution exhibits “blowup” in the adiabatic case is considered. The numerical method retains this property and is used to study the shape of the “blowup” function. The code is then used to investigate the solution in the closely related case where thermal conduction is included with a small conductivity coefficient. The computations indicate that the solution does not “blowup” in the nonadiabatic case. © 1996 John Wiley & Sons, Inc.     

Dual variational problems for boundary functionals of the linear theory of elasticity

Dual variational problem for use with the problem of minimization of the boundary functionals of three-dimensional theory of elasticity, is formulated using the method of orthogonal expansions at the boundary of the region constructed in /1/. Solutions of the initial and the dual problem obtained yield the estimates for the error of the approximate solutions of the boundary value problems of the theory of elasticity.     

On parallel solution of linear elasticity problems: Part I: theory

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg-method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M-matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg-method is analysed for different preconditionings based on the sdc-part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.     

Partial solutions of finite elasticity — Plane deformations

For plane deformations of isotropic incompressible elastic materials new reduced forms of the equilibrium equations are given. From these equations new exact solutions are deduced for the Mooney material. These solutions are shown to be possible approximate solutions to a number of mixed boundary value problems of finite elasticity.

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Zusammenfassung Neue reduzierte Formen von Gleichgewichtsgleichungen für ebene Deformationen von isotropischen, unzusammendrück baren, elastischen Materialien werden gezeigt. Von diesen Gleichungen werden neue, exakte Lösungen für das Mooney-Material abgeleitet. Diese Lösungen erweisen sich als mögliche ungefähre Lösungen für eine Anzahl von gemischten Grenzwertproblemen von begrenzter Elastizität.

     

Error estimates for certain approximate solutions of problems in the linear theory of elasticity

Zusammenfassung Eine Näherungslösung eines linearen Elastizitätsproblems kann manchmal erreicht werden mittels einer (kleinen) Änderung des Ausdrucks für die elastische Energie. Obere Schranken für die Fehler einer solchen Näherungslösung werden abgeleitet. Das Ergebnis hat auch Bedeutung vom grundsätzlichen Gesichtspunkt, weil die «exakte» Form des Energieausdrucks für wirkliche elastische Systeme niemals genau bekannt ist.

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Dedicated to Professor Dr. Hans Ziegler on the occasion of his 60th birthday