Displacement Potential Approach Research Articles

Analytic Solution for Axial Point Load Strength Test on Solid Circular Cylinders

This paper presents a new analytic solution for the stresses within an elastic solid finite cylinder subjected to the axial point load strength test (PLST). The displacement potential approach is used to uncouple the equations of equilibrium; then the contact stresses on the end surfaces induced by the point load indentors are expanded in terms of a Fourier-Bessel expansion to yield the unknown constants of the appropriate form of the displacement potential. The solution shows that a zone of higher tensile stress is developed in the vicinity of the applied point loads, compared with the roughly uniform tensile stress in the central portion of the line between the two point loads. This peak tensile stress within the cylinder decreases with increasing Poisson's ratio and the size of the loading area, and it increases with increasing Young's modulus. The tensile stress distributions along the axis of symmetry in a cylinder under the axial PLST are remarkably similar to that observed in a sphere under the diametral PLST. The solution also demonstrates both size and shape effects on the point load strength index (PLSI) that was observed in these experiments. In particular, for a fixed length-to-diameter ratio, the larger the specimen, the smaller the PLSI; whereas for a fixed diameter, the longer the specimen the smaller the PLSI.

Spherically isotropic, elastic spheres subject to diametral point load strength test

This paper presents an analytic solution for the stress concentrations within a spherically isotropic, elastic sphere of radius R subject to diametral point load strength test. The method of solution uses the displacement potential approach together with the Fourier–Legendre expansion for the boundary loads. For the case of isotropic sphere, our solution reduces to the solution by Hiramatsu and Oka, 1966 and agrees well with the published experimental observations by Frocht and Guernsey (1953) . A zone of higher tensile stress concentration is developed near the point loads, and the difference between this maximum tensile stress and the uniform tensile stress in the central part of the sphere increases with E/ E′ (where E and E′ are the Youngs moduli governing axial deformations along directions parallel and normal to the planes of isotropy, respectively) , G′/ G (where G and G′ are the moduli governing shear deformations in the planes of isotropy and the planes parallel to the radial direction) , and ν̄/ ν′ (where ν̄ and ν′ are the Poissons ratios characterizing transverse reduction in the planes of isotropy under tension in the same plane and under radial tension, respectively) . This stress difference, in general, decreases with the size of loading area and the Poissons ratio.

On the elastic interaction between two fibers in a continuous fiber composite under thermal loading

The problem of fiber interaction in unidirectional fiber composites under thermal loading is considered. A pair of fibers is modeled by two inhomogeneities that sustain an eigenstrain loading, proportional to the difference of fiber/matrix thermal expansion coefficients. Utilizing the displacement potential approach, the plane strain problem is solved analytically. The effect of incoherent interfaces is evaluated, in comparison to the case of perfect bonding.

A degenerate triangular shell element with constant cross-sectional warping

The development of a degenerate curved general triangular element, based on the assumed displacement potential energy approach, is presented for the analysis of moderately thick shells. The present element is a quadratic triangle of C°-type in the curvilinear coordinate plane and assumes transverse inextensibility and the Reissner-Mindlin hypothesis of constant transverse shear deformation (crosssectional warping). Numerical results presented include the study of convergence and effect of two Gauss quadrature schemes. These results demonstrate the accuracy of the shell constitutive relations consistently derived from the three-dimensional elasticity theory and also the efficiency of the present element.

Triangular finite element for analysis of thick laminated shells

AbstractThe development of a general curved triangular element based on an assumed displacement potential energy approach is presented for the analysis of arbitrarily laminated thick shells. The associated laminated shell theory assumes transverse inextensibility and layerwise constant shear angle. The present element is a quadratic triangle of C0‐type in the curvilinear co‐ordinate plane, which is then mapped onto a curved surface. Convergence of transverse displacement, moments, stresses and the effect of two Gauss quadrature schemes also form a part of the investigation. Examples of two laminated shell problems demonstrate the accuracy and efficiency of the present element. Comparison of the present LCST (layerwise constant shear‐angle theory) based solutions, with those based on the CST (constant shear‐angle theory) clearly demonstrates the superiority of the former over the latter, especially in the prediction of the distribution of the surface‐parallel displacements and stresses through the laminate thickness.

Triangular finite element for analysis of thick laminated plates

AbstractThe development of a general triangular C0 element, based on an assumed quadratic displacement potential energy approach, is presented for the analysis of arbitrarily laminated thick plates. The element formulation assumes transverse inextensibility and layerwise constant shear‐angle. Convergence of transverse displacement, moments and stresses, the effects of two different Gauss quadrature schemes and comparison of the present solutions with the available analytical/finite‐element results also form a part of the investigation. Furthermore, numerical results indicate close agreement between the LCST (layerwise constant shear‐angle theory) and the three‐dimensional elasticity theory with the length (or width) to thickness ratio as low as 4. Detailed comparison of the LCST‐based finite‐element solutions with those based on the CST (constant shear‐angle theory) and the CLT (classical lamination theory) clearly demonstrates the superiority of the former over the latter two, especially in the prediction of the distribution of the in‐plane displacements and stresses through the laminate thickness. This paper also introduces a new non‐dimensionalized parameter, Δθ*, which is shown to be a very useful measure for classification of the laminated plates and the suitability of different plate theories over various ranges of length‐to‐thickness ratio.

Stress concentration around a part-through hole weakening a laminated plate

A quasi-three-dimensional-elasticity type theory, based on the assumptions of transverse inextensibility and layerwise constant shear-angle, is presented, in the context of an assumed quadratic displacement potential energy approach, for analyzing an edge-loaded laminated plate weakened by the presence of a part-through hole. Numerical results, obtained using the C°-type triangular finite element, indicate the existence of severe cross-sectional warping in the vicinity of the hole and plate boundaries. Furthermore, the three-dimensional nature of the stress concentration factor in the neighborhood of the hole boundary is clearly exhibited. Besides, very high stress concentration factors are found in the layer weakened by the part-through hole. The numerical results presented should serve as bench-mark solutions for future comparisons.

A simple and efficient shear-flexible plate bending element

A shear-flexible triangular element formulation, which utilizes an assumed quadratic displacement potential energy approach and is numerically integrated using Gauss quadrature, is presented. The Reissner/Mindlin hypothesis of constant cross-sectional warping is directly applied to the three-dimensional elasticity theory to obtain a moderately thick-plate theory or constant shear-angle theory (CST), wherein the middle surface is no longer considered to be the reference surface and the two rotations are replaced by the two in-plane displacements as nodal variables. The resulting finite-element possesses 18 degrees of freedom (DOF). Numerical results are obtained for two different numerical integration schemes and a wide range of meshes and span-to-thickness ratios. These, when compared with available exact, series or finite-element solutions, demonstrate accuracy and rapid convergence characteristics of the present element. This is especially true in the case of thin to very thin plates, when the present element, used in conjunction with the reduced integration scheme, outperforms its counterpart, based on discrete Kirchhoff constraint theory (DKT).

Thermal transient stresses due to rapid cooling in a thermally and elastically orthotropic medium

This paper reports the residual thermal transient stresses that develop within a thermally and elastically orthotropic medium with a rectangular boundary. The material is rapidly cooled from the steady-state thermal environment under prescribed surface temperatures down to room temperature. Based upon the solution of transient temperature field, the thermal stress analysis is performed by using an elastic displacement potential approach. The two-dimensional temperature and stress fields have been obtained in series expansion forms. Numerical calculations are performed for a typical unidirectional fiber composite. The results demonstrate that the residual thermal stresses exhaust a significant portion of the tensile strength of the composite material.

Triangular element for analysis of perforated plates under inplane and transverse loads

A C 0-type triangular element formulation in orthogonal curvilinear co-ordinates has been developed, based on assumptions of transverse inextensibility and constant shear angle through thickness for analysis of perforated plates subjected to inplane and transverse loads. The assumed quadratic displacement potential energy approach is utilized in obtaining an element stiffness matrix and consistent load vector, which are numerically integrated. Numerical results have been obtained using a straight-sided triangular version, which behaves like a subparametric element, for stretching and bending analyses of perforated plates.

Triangular element for analysis of a stretched plate weakened by a part-through hole

A C 0-type triangular element formulation in orthogonal curvilinear coordinates has been developed, based on assumptions of transverse inextensibility and layerwise constant shear angle for analysis of a stretched homogeneous plate weakened by a part-through hole. The element stiffness matrix and consistent load vector have been derived using assumed quadratic displacement (in curvilinear coordinate plane) potential energy approach. Numerical results obtained using the straight-sided triangular element version indicate the presence of transverse shear deformation in a stretched homogeneous plate weakened by a symmetrically located concentric hole

Stress Concentrations in Three-Dimensional Electrostriction

Using the techniques employed in developing a Papkovich-Neuber representation for the displacement vector in classical elasticity, a particular integral of the kinematical equations of equilibrium for the uncoupled theory of electrostriction is developed. The particular integral is utilized in conjunction with the displacement potential function approach to problems of the theory of elasticity to obtain closed-form solutions of several stress concentration problems for elastic dielectrics. Under a prescribed uniform electric field at infinity, the problems of an infinite elastic dielectric having first a spherical cavity and then a rigid spherical inclusion are solved. The rigid spheroidal inclusion problem and the penny-shaped crack problem are also solved for the case where the prescribed field is parallel to their axes of revolution.