Plane Elasticity Equations Research Articles

On Optimum Truss Layout by a Homogenization Method

A new approach to the truss layout optimization problem is possible by using homogenization theory. Unlike traditional methods for optimization of truss topology, the new method does not require that the set of admissible connecting points be defined a-priori. This represents a great advantage over other strategies, which may inadvertently exclude optimum solutions by prescribing a set of connecting points that is inconsistent with the optimum connectivity. The homogenization approach, however, is based on the solution of the equations of plane elasticity over a two dimensional domain and its solutions are only truss-like structures. The relationship between these solutions and optimum truss structures is examined.

Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary

Uniform asymptotic energy estimates for solutions of the equations of dynamic plane elasticity with nonlinear dissipation at the boundary

A family of higher order mixed finite element methods for plane elasticity

The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL 2(Ω), and optimal order negative norm estimates are obtained inH s (Ω)′ for a range ofs depending on the index of the finite element space. An optimal order estimate inL ∞(Ω) for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.

Analysis of Some Mixed Finite Element Methods for Plane Elasticity Equations

Analysis of Some Mixed Finite Element Methods for Plane Elasticity Equations

Analysis of some mixed finite element methods for plane elasticity equations

We analyze some mixed finite element methods, based on rectangular elements, for solving the two-dimensional elasticity equations. We prove error estimates for a method proposed by Taylor and Zienkiewicz and for some new variants of the known equilibrium methods. A numerical example is given demonstrating the performance of the various algorithms considered.

Indentation of an Elastic Layer by an Array of Punches Moving With Steady Velocity

A solution is presented for the problem of an elastic layer that is indented by an infinite array of punches moving with a steady velocity below the Rayleigh wave speed. The layer is assumed to be loaded symmetrically about its midplane. The equations of plane elasticity are used to develop dual series equations which are reduced to a single Fredholm integral equation. An asymptotic solution to the integral equation is developed for the case when the ratio of contact length to punch spacing is small, and it is compared with a numerical solution. Numerical calculations for the effective stiffness of the layer and the stresses along the midplane are included.

Stress and deformation analysis of fibrous composite materials by point matching

AbstractThe point matching numerical method and its generalization, the method of boundary point least squares, have been successfully applied to numerous boundary value and eigenvalue problems. The present paper demonstrates the application of these techniques to problems in the micromechanics of fibrous composite materials, i.e. determination of elastic moduli and stress concentrations for parallel‐fibre materials which are loaded transversely with respect to the fibres. The solution technique utilizes exact solutions of the governing equations of plane elasticity for each component fibre and its surrounding matrix material in a typical repeating section of the composite material. The continuity conditions for stresses and displacements between fibre and matrix and the repeatability conditions at the boundary of the repeating section are satisfied approximately in a pointwise manner. Some special numerical techniques which were found to be particularly useful in applying the point matching method to these problems are delineated. The method is demonstrated for composite materials having circular, elliptical and square fibres in regular, staggered arrays. Numerical results are given which show the accuracy of the method as well as stress concentration and composite elastic moduli data.

Higher-Order Theories for Structural Analysis Using Legendre Polynomial Expansions

Governing equations of plane elasticity are examined to define suitable approximate theories. Each dependent variable in the problem is considered as a series expansion in Legendre polynomials; attention is focused on establishment of a logical approach to truncation of the series. Important variables for approximate theories of any order are established from energy considerations, and the desired approximate theories are established by direct reduction of the field equations and also from an energy viewpoint. A new “classical” beam theory is developed capable of treating displacement boundary conditions on lateral surfaces. Higher-order approximate theories are studied to make certain comparisons with exact solutions; the results of these comparisons indicate that the new method yields approximate theories which may be more accurate than previous theories with similar levels of approximation.

An estimate of the limiting speed of a propagating ductile crack

A steady state dynamic solution for a Dugdale crack propagating at a constant speed is obtained with the Sneddon-Radok formulation of the dynamic plane elasticity equations. The dynamic stress distribution and the Tresca yield condition are used to determine the limiting ductile crack propagation speed. It is shown that the limiting brittle crack speed criteria of Yoffe and of Craggs are satisfied as well. A simple relation between the flow stress and the crack speed based on observations of ductile crack propagation in steel foil is utilized in order to express the limiting speed in terms of the applied load and material properties. Calculated values of the limiting speed in steel and aluminium sheets are found to be not strongly dependent on the applied load and are about 0-1C 2 and 0-3C 2, respectively.

Higher order effects in thick rectangular elastic beams

Governing equations for the rectangular strip are investigated to obtain solutions which show the relationship between classical beam theory and higher order theories. The exact equations of plane elasticity are reduced to a coupled set of ordinary differential equations by expressing all dependent variables as series solutions containing Legendre Polynomials in the thickness coordinate. Legendre Polynomials are particularly advantageous for this analysis because their completeness, convergence and orthogonality properties are well formulated, and because the usual stress resultants of classical beam theory appear naturally as coefficients of the polynomials P0 and P1. The coupled ordinary differential equations are obtained in a form such that proper truncation of the series to obtain approximate theories is immediately apparent. The coupling effects are investigated and a possible method of obtaining an approximate solution to the fully coupled equations without truncation of the series solutions is suggested. A sample problem is worked out in detail to illustrate the application of a new approximate theory.