Limit equilibrium of an orthotropic plate weakened by a periodic row of collinear cracks

A modified δ_c-model is used to study the limiting state of an orthotropic plate weakened by a periodic row of collinear cracks and satisfying a general failure criterion. The failure mechanism of the plate is analyzed.Astudy is made of the effects of the degree of orthotropy, the biaxiality of external loading, and the geometrical parameters on the fracture process zones at the crack tips and the limiting state of the plate. The safe loading of an orthotropic viscoelastic plate with a periodic row of collinear cracks is examined. The effect of the rheological parameters on the safe-load region is studied Translated from Prikladnaya Mekhanika, Vol. 44, No. 8, pp. 126–135, August 2008.     

Thermo-stress concentration of an orthotropic plate weakened by multiple elliptical holes

A laminate weakened by multiple elliptical holes of arbitrary distribution, arbitrary orientation and arbitrary dimensions, is treated as an anisotropic, infinite, multiple connected thin plate. By Faber series expansion [1–6] and a complex potential method in the plane theory of thermo-elasticity of an anisotropic body, the general step to deduce the thermostress concentration in the laminate subjected to arbitrary mechanical and thermal loads is obtained.Supported by The Chinese Science Foundation of Aeronautics     

Cracking of an orthotropic substrate reinforced by an orthotropic plate

A solution method is derived to determine the stress intensity factors for both an internal crack and an edge crack in an orthotropic substrate that is reinforced on its boundary by a finite-length orthotropic plate. The method utilizes the Green’s functions for a pair of dislocations and a concentrated force on the boundary while invoking the concept of superposition. Enforcing the traction-free boundary condition along the crack surfaces and the continuity of displacement gradients along the plate/substrate interface results in a coupled system of singular integral equations. An asymptotic analysis of the kernels in these equations for the region of the junction point between the plate corner and the substrate boundary reveals the strength of the singularity in the case of an edge crack. The numerical solution of the integral equations provides results for the stress intensity factors for both an internal crack and an edge crack perpendicular to the substrate boundary and aligned with one of the corners of the plate. The present results have been validated against previously published stress intensity factors for an internal crack and an edge crack in an isotropic substrate.     

First fundamental problems of anisotropic elastic plane weakened by periodic collinear cracks

Using the method of complex functions, we discuss the first fundamental problems of an anisotropic infinite elastic plane weakened by periodic collinear cracks and with periodic boundary loads on both sides of the cracks. This problem was considered by Cai [Engineering Fracture Mechanics 46（1）, 133-142 （1993）]. However, the previous method is imperfect. Therefore, the results are incorrect. Here, we revise the method and give a correct solution.     

Elastic equilibrium of a finite anisotropic plate weakened by an elliptical hole and containing cracks

Novosibirsk. Translated from Prikladnaya Mekhanika, Vol. 25, No. 8, pp. 94–100, August, 1989.     

Optimal design of a compound plate weakened by a periodic crack system

We use the minimax criterion to perform atheoretical analysis of determining the optimal interference for fitting elastic inclusions into holes in an isotropic elastic plate weakened by a periodic system of rectilinear cracks. We construct a closed system of algebraic equations that allows us to minimize the fracture parameters depending on the geometric and mechanical characteristics of the inclusions. The obtained fitting interference for the inclusions ensures an increase in the bearing strength of the compound plate under bending.     

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.

     

Limiting state of an elastoplastic orthotropic plate with a periodic system of collinear cracks

A modified Dugdale model is used to study the fracture of an orthotropic elastoplastic plate with a periodic system of rectilinear cracks. The material of the plate obeys a general yield criterion. The general form of solution is obtained in terms of Kolosov-Muskhelishvili potentials. The size of the plastic zone is expressed in terms of the external load and geometrical parameters. The equations for the determination of the stresses in the plastic zone and the crack opening displacement are derived. The effect of anisotropy on the formation of the plastic zones at the crack tip is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 80–88, May 2007.     

Asymptotic analysis of stresses in an isotropic linear elastic plane or half-plane weakened by a finite number of holes

The problem of an isotropic linear elastic plane or half-plane weakened by a finite number of small holes is considered. The analysis is based on the complex potential method of Muskhelishvili as well as on the theory of compound asymptotic expansions by Maz’ya. An asymptotic expansion of the solution in terms of the relative hole radii is constructed. This expansion is asymptotically valid in the whole domain, i.e. both in the vicinity of the holes and in the far-field. The approach leads to closed-form approximations of the field variables and does not require any numerical approximation. Several examples of the interaction between holes or holes and an edge are presented.     

Fundamental-solution-based finite element model for plane orthotropic elastic bodies

A new hybrid finite element formulation is presented for solving two-dimensional orthotropic elasticity problems. A linear combination of fundamental solutions is used to approximate the intra-element displacement fields and conventional shape functions are employed to construct elementary boundary fields, which are independent of the intra-element fields. To establish a linkage between the two independent fields and produce the final displacement-force equations, a hybrid variational functional containing integrals along the elemental boundary only is developed. Results are presented for four numerical examples including a cantilever plate, a square plate under uniform tension, a plate with a circular hole, and a plate with a central crack, respectively, and are assessed by comparing them with solutions from ABAQUS and other available results.