General plane elasticity solution for non-homogeneous materials

In this study, we consider the general plane elasticity problem for non-homogeneous materials subjected to mechanical and thermal loads. The general equations are derived in x–y coordinate system. It is shown that, if the non-homogeneity varies exponentially, then the governing equations are reduced to partial differential equations with constant coefficients. To illustrate the validity and usefulness of the formulation, problems involving both mechanical and thermal loads with the non-homogeneity varying in both directions are considered. As examples, infinite non-homogeneous beams subjected to axial load and bending moment are considered and the solutions obtained in closed form. Then an example illustrating the effect of temperature change is presented. Again a closed form solution is obtained for the stress distribution. Also for limiting values of the non-homogeneity parameter, extensive asymptotic analyses are presented. In addition two problems where the Young’s Modulus or the coefficient of thermal expansion coefficient varies in both directions are presented. From the examples solved, it is found that non-homogeneity affects the stress distribution significantly, and leads to counterintuitive results with serious implication to the study of fracture problems in functionally graded materials.     

On the solution singularity in the plane elasticity problem for a cantilever strip

The plane elasticity problem of bending of a cantilever strip whose material is assumed to be incompressible in the transverse direction is solved. It is shown that, in the classical statement of of the boundary condition for the fixed edge of the strip, the solution has a singularity at the corner points of the edge. Several cases of the strip fixation and loading characterized by the presence or absence of the solution singularity are considered. The strength of glass beams of three types, for which the theory of elasticity predicts whether the normal stress has a singularity, is studied experimentally. It is shown that the limit stresses for the beams of the types under study are practically the same, which testifies that the solution singularity does not have any physical nature.     

An approach to three-dimensional elasticity problem of cylindrical orthotropic bodies

The state equation for a static or dynamic thermal elastic problem of cylindrical orthotropic bodies is derived using the three-dimensional theory of elasticity. The governing equation for displacement components is obtained by series expansion, and for simplicity the case of an axisymmetric problem of transverse isotropic bodies is given. For illustration, an axisymmetric problem of a combined-cylinder made of two transverse isotropic materials is also presented. Some interesting and significant results are obtained.Supported by-National Natural Science Foundation and Post-Doctor's Foundation of China     

Morawetz's method for the decay of the solution of the exterior initial-boundary value problem for the linearized equation of dynamic elasticity

In this paper, the behavior of the solution of the time-dependent linearized equation of dynamic elasticity is examined.For the homogeneous problem, it is proved that in the exterior of a star-shaped body on the surface of which the displacement field is zero, the solution decays at the rate t ^-1 as the time t tends to infinity.For the non-homogeneous problem with a harmonic forcing term, it is proved that for large times, the elastic material in the exterior of the body, tends to a harmonic motion, with the period of the external force.The convergence to the steady harmonic state solution is at the rate t ^-1/2 as t tends to infinity, and is uniform on bounded sets.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

On the plane problem of orthotropic quasi-static thermoelasticity

The plane displacement boundary value problem of quasi-static linear orthotropic thermoelasticity is discussed. The thermoelastic system on a bounded simply-connected domain is decoupled. The decoupled temperature equation is investigated by using an accurate estimate and the contractive mapping principle. Representation of solution of the field equation is obtained, and some solvability results are proved. The results are of both theoretical and numerical interest.     

Hypersingular integral equation for multiple curved cracks problem in plane elasticity

The complex variable function method is used to formulate the multiple curved crack problems into hypersingular integral equations. These hypersingular integral equations are solved numerically for the unknown function, which are later used to find the stress intensity factor, SIF, for the problem considered. Numerical examples for double circular arc cracks are presented.     

On the contact problem of classical elasticity

We deal with the contact problem of homogeneous and isotropic linear elastostatics in the exterior of a bounded convex domain of ${\Bbb{R}}^{3}$ . We prove existence and uniqueness of a solution, provided the elasticity tensor is only strongly elliptic.     

On the elastic problem of the orthotropic thick plate

Summary Equilibrium equations for orthotropic media are written taking the displacement components as unknowns; these equations are integrated with operational methods by separation of the variables.The unknown quantities are six « initial funcitons » that is, displacements and their partial derivatives with respect toz, calculated on the planez=0.Following a method of structural mechanics, the cases of symmetrical and nonsymmetrical loading of plate, namely compression and flexion, are considered separately.The separation of the variables allows us to resolve in two successive stages the problem of the boundary conditions: the Cauchy conditions on the surfacesz=± h become differential equations to which we associate the condition on the cylindrical surface.The process leads to a symbolic solution of the problem from which we construct the resolvent equations in the form of power series of operators. If terms of a higher order are retained in these equations, a more accurate theory is obtained; it is shown that if only the first term is assumed, the equation for the ortho tropic plate in the Kirchhoff-Love sense is obtained.The method is applied in order to resolve a problem numerically; the results are compared with those deduced by the usual theory.

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Sommario Si scrivono le equazioni indefinite dell'equilibrio dei mezzi ortotropi assumendo come incognite le componenti di spostamento; se ne effettua l'integrazione con metodi operazionali per separazione delle variabili. Le incognite risultano esplicitate attraverso sei « funzioni iniziali », cioè spostamenti e loro derivate rispetto a z calcolate sul piano medio.In relazione ad una decomposizione dei carichi si individuano due problemi distinti, di compressione e di flessione, che vengono trattati parallelamente.La separazione delle variabili permette di risolvere in due fasi successive il problema dei valori al contorno: le condizioni di Cauchy sulle facce parallele al piano medio si traducono di fatto in equazioni differenziali cui vanno associate le condizioni sulla superficie cilindrica.Il procedimento conduce ad una soluzione simbolica del problema, a partire dalla quale si costruiscono le equazioni risolventi sotto forma di sviluppi in serie di potenze di operatori. L'ordine delle equazioni risolventi, e quindi il numero di condizioni che si possono soddisfare sulla superficie laterale, è fissato dal numero di termini che si considerano in questi sviluppi; si dimostra che il solo primo termine conduce all'equazione della piastra ortotropa ricavata sotto le ipotesi di Kirchhoff-Love.Il metodo è applicato alla soluzione di un problema concreto; i risultati sono messi a confronto con quelli dedotti dalla teoria ordinaria.

     

Fredholm integral equation for the multiple circular arc crack problem in plane elasticity

Summary An elementary solution for the multiple circular arc problem is obtained in this paper. The elementary solution is defined as a particular case of the single circular arc crack problem, in which remote stresses are equal to zero, and two pairs of concentrated forces are applied at a prescribed point of crack face. By using the principle of superposition, Fredholm integral equation for the multiple circular arc problem in plane elasticity is obtained. The suggested approach is illustrated by several numerical examples. If a smaller arc crack is surrounded by a larger arc crack, the stress intensity factors for the former become rather small. The phenomenon of shielding is illustrated by examples. Accepted for publication 17 September 1996