Linear elasticity

About: Linear elasticity is a research topic. Over the lifetime, 9080 publications have been published within this topic receiving 258684 citations.

Unrestricted lower bounds for eigenvalues for classes of elliptic equations and systems of equations with applications to problems in elasticity

Abstract: Lower bounds for the eigenvalues of some elliptic equations and elliptic systems over bounded regions are obtained. The bounds are universal in that they depend only upon the volume of the region. Specific applications include the clamped plate, the buckling problem for the clamped plate and the equations of linear elasticity. Our results are consequences of extensions of the methods of Li and Yau (Comm. Mat. Phys. 88 (1983) 309–318) who obtained such results for the eigenvalues of the fixed membrane problem.

Structural interfaces in linear elasticity. Part I: Nonlocality and gradient approximations

Abstract: Many biological and optimal materials, at multiple scales, consist of what can be idealized as continuous bodies joined by structural interfaces. Mechanical characterization of the microstructure defining the interface can nowadays be accurately done; however, such interfaces are usually analyzed employing models where those properties are overly simplified. To introduce into the analysis the microstructure properties, a new model of structural interfaces is proposed and developed: a true structure is introduced in the transition zone, joining continuous bodies, with geometrical and material properties directly obtained from those of the interfacial microstructure. First, the case of an elliptical inclusion connected by a structural interface to an infinite matrix is solved analytically, showing that nonlocal effects follow directly from the introduction of the structure, related to the inclination of the connecting elements. Second, starting from a discrete structure, a continuous model of a structural interface is derived. The usual zero-thickness linear interface model is shown to be a special case of this more general continuous structural interface model. Then, a gradient approximation of the interface constitutive law is rigorously derived: it is the first example of the analytical derivation of a nonlocal interface model from the microstructure properties. The effects introduced in the mechanical behavior by both the continuous model and its gradient approximation are illustrated by solving, for the first time, the problem of a circular inclusion connected to an infinite matrix by a structural interface and subject to remote uniform stress.

Lower Order Rectangular Nonconforming Mixed Finite Elements for Plane Elasticity

Abstract: In this paper, we present two stable rectangular nonconforming mixed finite element methods for the equations of linear elasticity in two space dimensions which produce direct approximations for the stress and displacement. In the first method, the normal stress space of the matrix-valued stress space is taken as the second order rotated Brezzi-Douglas-Fortin-Marini element space [F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York, 1991], the enriched nonconforming rotated $Q_1$ element [Q. Lin, L. Tobiska, and A. H. Zhou, IMA J. Numer. Anal., 25 (2005), pp. 160-181] is taken for the shear stress, and the lowest order Raviart-Thomas element space [P. A. Raviart and J. M. Thomas, in Mathematical Aspects of the Finite Element Method, Lecture Notes in Math. 606, Springer-Verlag, New York, 1977, pp. 292-315] is employed to approximate the vector displacement field. The second method is obtained from the first one through dropping the interior degrees of the normal stress on each element. A first order convergence rate is obtained for both the stress and the displacement for these methods based on the superconvergence of the enriched nonconforming rotated $Q_1$ element.

Matched asymptotic expansions in nonlinear fracture mechanics—III. In-plane loading of an elastic perfectly-plastic symmetric specimen

Abstract: T he matched expansion method, introduced by the authors in two earlier papers (1976) devoted to mode III loading, is applied to the practically important case of mode I loading of a symmetric specimen. The method allows the linear elastic far-field to be considered separately from the elasto-plastic near-tip field, except for coupling through a set of parameters that are determined explicitly in the matching. The effects of the plasticity are thus found, once and for all, from the solution of a set of standard elasto-plastic problems for a semi-infinite crack in an infinite body, whose properties may be tabulated. The solution for any particular specimen geometry and loading then follows from a small set of linear elastic solutions for the specimen, which define, through coefficients γij appearing in their near-tip expansions, all the parameters in the “inner” and “outer” solutions. The effects of plasticity appear in these parameters only through a set of constants Cti that define the far-field expansions of the “inner” (near-tip) solutions: they are material constants, depending upon the constitutive relation for the material, but not upon specimen geometry and loading. The J-integral, being obtainable from the far-field, is expressed as an explicit asymptotic series in the loading parameter e, whose coefficients are given as functions of the “elastic” parameters γij and the material constants Ci. It is demonstrated that a plastic-zone correction term, ry, can be chosen to yield a two-term asymptotic expansion for J; the value of ry depends upon the yielding model only through the constant C1. The Dugdale (1960) model of yielding is treated, as a simple example for which all calculations can be performed analytically, and for which exact solutions are available for comparison. Finally, the near-tip solutions are constructed for a material obeying the Mises yield criterion and associated flow-rule, using a specially developed finite element program. The first eight of the constants Ci are tabulated, which suffice to define the J-integral up to terms of order e6 (where e is a loading parameter) and some representative near-tip features are displayed graphically. The computed value of C1 shows that the conventionally adopted value for the plastic-zone correction ry is too large by a factor of roughly 2.8, if it is to yield a genuine asymptotic estimate for J. As an example, the “elastic” parameters γij are found, from a boundary collocation program, for a centre-cracked square plate subjected to tensile loading; and a plot of J versus load, and the plastic-zone shape at a particular load level, are displayed.