Boundary Element Analysis For Couple Stress Elastic Ststic Problems

It is well known that classical continuum theories fail to adequately describe the size-dependence of the material microstructures due to the lack of intrinsic length-scale parameters in their constitutive equations.Consequently,a number of generalized continuum theories have been proposed one after another.The couple stress theories are especially characterized by their concise forms and explicit physical meanings.So they have attracted much attention.Among these theories,the HD consistent couple stress theory in which the mean curvature tensor and the couple-stress tensor was assumed to be skew-symmetric,avoid the inconsistency in the traditional couple stress theory and explains many dimensional effect phenomena.It is no doubt that the gradient elasticity problems with complex geometry and load conditions should resort to numerical methods.Meanwhile,the outstanding advantages of the boundary element method(BEM),such as the reduction of the problem dimensionality by one and the high accuracy due to their semi-analytical nature by using fundamental solutions or Green’s functions and no special requirements for element continuity etc.,render the BEM particularly attractive.Based on the HD consistent couple stress elasticity theory,a two-dimensional BEM is developed to solve the boundary value problems in the couple stress elastic materials.The main contents of this dissertation are:(1)The displacement boundary integral equations(DBIEs)are applied to present the generalized strains and stresses boundary integral equations at internal points and the explicit expressions for the high-order fundamental solutions.The collocation method is adopted for the spatial discretization of the DBIEs.A semi-analytical techniques are provided for the evaluation of weakly singular integrals and the method of rigid-body motion is applied to the evaluation of strongly singular integrals.The sinh transformation is used to deal with the nearly singular integrals for near-boundary internal points or boundary nodes in thin structures.The non-singular integrals can be computed numerically by the standard Gauss-Legendre quadrature.A DBEM program using FORTRAN language is developed for the present consistent couple stress theory.This method is used to simulate a square plate under uniaxial tension,simple shear and cylindrical bending and their related nearly singular problems.The correctness and accuracy of the method is verified by comparison with the analytical solution.This method is also used to further analyze the force problems of cantilever beams,cylinders,and plate with elliptical hole.The stress concentration problem of the plate with holes and the problem of the plane ring under displacement load are further studied.(2)The traction boundary integral equations(TBIEs)and the explicit expressions for the high-order fundamental solutions are further deduced.The discontinuous quadratic boundary elements is adopted for discretize the boundary.Some semianalytical techniques are provided for the evaluation of weakly and strongly singular integrals.The method of rigid-body motion is applied to the evaluation of hypersingular integrals.Developing a corresponding TBEM procedure to study a square plate under uniaxial tension,cylindrical bending and stress concentration problem of the plate with a hole.The accuracy of the method is verified by comparison with the analytical and DBEM solution.For crack problems,the displacement extrapolation formulas are derived for computing the stress and couple-stress intensity factors and the symmetry is used to analyze the flat plate with central crack.(3)For the elastic body with cracks,there will be two overlapping crack surfaces geometrically,which will lead to the degradation of DBIEs into ill-posed and cannot be solved for the whole.Special cases can be analyzed using symmetry,and more often it is necessary to artificially introduce new boundaries into the elastic body.The original problem is divided into two or more sub-regions without cracks and solved with the coupling conditions on the new boundary.Then the center crack plate and the edge crack plate are studied.