Ranbir S. Sandhu

Variational principles for boundary value and initial-boundary value problems in continuum mechanics

Abstract A procedure for setting up variational principles for a class of linear coupled field problems in continuum mechanics is presented. Some generalizations of the principle and their relationship with existing variational theorems are examined. Alternative schemes useful for direct methods of solution are discussed. Examples of typical application are included.

A variational principle for linear, coupled field problems in continuum mechanics

Abstract A variational principle applicable to linear, coupled field problems in continuum mechanics is presented. An important feature for direct methods of calculation, particularly the finite element method, is the inclusion of initial conditions on the field variables as part of the variational principle. Applications to problems in several areas of continuum mechanics are included.

Variational formulation of linear problems with nonhomogeneous boundary conditions and internal discontinuities

Abstract A variational formulation applicable to linear operators with nonhomogeneous boundary conditions and jump discontinuities is presented. For the formulation to be applicable, the boundary condition and discontinuities have to be consistent with the operator governing the field problem. The problem is set up in a space of suitable continuous bilinear mapping. Thus, operators on inner product spaces, convolution spaces and energy spaces are included as specializations. The basic construction can be used to generate dual-complementary variational principles. Implementation is illustrated by examples. The role of boundary terms in finite element discretizations based on interpolants of limited smoothness is discussed.

Reduced integration for improved accuracy of finite element approximations

Abstract For completeness the finite element bases which are used for approximate solutions of elliptic problems of order 2p by the Ritz method must include the functions corresponding to the constant value of the pth derivative. In actual usage, to ensure a positive definite system of algebraic equations, additional interpolating functions are introduced. This leads to “multiple covering” of some of the system modes and results in overestimation of stiffness. Reduced integration techniques eliminate some of this multiple covering and thereby give improved accuracy. Selective reduced integration has been found useful in the analysis of flexural problems. In this paper we suggest the use of only the minimal covering that is sufficient for convergence. A technique for solution of the discretized system is given. Numerical performance data show remarkable improvement over conventional procedures. The proposed scheme yields good approximation even for very coarse meshes. This indicates the possibility of considerable economy in the cost of obtaining finite element solutions to complex problems, e.g. coupled field problems, three-dimensional problems, stress concentration etc.

A three-field finite element procedure for analysis of elastic wave propagation through fluid-saturated soils

A three-field mixed finite element procedure for the analysis of wave propagation through fluid-saturated porous soils is presented. This approach uses the solid displacement, the fluid displacement relative to the solid and the pore-fluid pressure as the field variables. Thus, a continuous pore pressure field is ensured and pore pressure boundary conditions, e.g., free-draining surfaces can be easily handled. Application of the procedure to a one-dimensional problem shows good agreement with the analytical solution. However, the method is more expensive than the customary two-field formulation using the solid and the relative fluid displacements as the field variables.

Finite Element Analysis of Coupled Deformation and Fluid Flow in Porous Media

Application of tne finite element method to problems of coupled deformation and fluid flow in porous media is briefly reviewed. Equations governing the physical process, variational formulations appropriate for use with finite element methods, and finite element discretization as well as solution strategies are discussed. To illustrate various methods and procedures, darcy flow of an incompressible fluid in a linear elastic solid is considered in some detail. Other cases are occasionally referred to.

A higher order discrete theory for laminated plates

Abstract A higher order layer-wise laminated theory based on quadratic ‘inplane’ displacement variation and linear transverse displacement variation over the thickness of each layer is presented. A generalization of Reissners variational theory is employed to set up consistent coupled constitutive equations for force resultants in the lamina. Continuity of tractions as well as displacements across interfaces is enforced. The theory is able to represent the distribution of stress over the thickness much better than the lower order discrete laminate theories currently available. As an illustration, the theory is applied to stress analysis of free-edge delamination specimens.

Continuous stress finite element analysis of a free-edge delamination specimen

Abstract A mixed finite element formulation based on minimization of potential energy is developed to analyze the free-edge delamination problem in composite laminate coupons under uniform longitudinal strain. In this displacement-based procedure continuities of both displacement and traction, along interelement as well as interlaminar boundaries, are ensured through appropriate displacement-stress transformations.

A parametric study of the finite element eigenfunction method for the scattering of SH-waves

Abstract The performance of a coupled Finite Element—Analytic method in solving elastic wave scattering in infinite spaces is studied for plane SH-waves. The influence of several parameters on the accuracy of the results is investigated. These parameters are the number of terms in the series solution, the number of compatibility points across an imaginary circular interface (i.e. the points for which the continuity of both stresses and displacements are enforced), and the ratio of element dimension to wavelength in the direction of propagation. It was found that the method converges rapidly with the number of terms. Furthermore, both element dimension to wavelength ratio and the number of compatibility points significantly influence the accuracy of the approximate solution.

On Theories of Mixtures and Their Applications to Dynamics of Fluid Saturated Porous Media

A finite strain theory for motion and deformation of fluid-saturated solids is proposed. A convected coordinate frame of reference is preferred for the solid particles that are enclosed in a material region that moves with them. The fluid is taken to be in relative motion with respect to these solid particles. This may be regarded as an extension of Gibson’s quasistatic one-dimensional theory to dynamics of two-phase mixture in three dimensions. Reduction to small strains yields Biot’s equations of motion but without Biot’s simplifying assumptions of existence of energy functions. Numerical examples include wave propagation analysis of an elastic-perfectly-plastic solid and a fluid-saturated elastic-plastic strain-hardening soil column subjected to dynamic excitation.