Dynamic stresses in a compound body with circular crack under sliding contact on an interface

We have considered the three-dimensional problem of harmonic loading of a circular crack in an elastic composite consisting of two dissimilar half-spaces under sliding contact on the surface of their bonding. The defect is situated in one of the half-spaces perpendicularly to the interface of materials. Using the representations of solutions in the form of Helmholtz potentials, we have reduced the problem to a boundary integral equation for the function of dynamic defect opening. Based on the numerical solution of this equation, we have obtained the frequency dependences of mode I stress intensity factor near the crack for different relations between the elastic moduli of components of the composite and the depths of crack location with respect to the interface.     

Optimal Design of Brittle Composite Materials: a Nonsmooth Approach

Our goal is to design brittle composite materials yielding maximal energy dissipation for a given static load case. We focus on the effect of variation of fiber shapes on resulting crack paths and thus on the fracture energy. To this end, we formulate a shape optimization problem, in which the cost function is the fracture energy and the state problem consists in the determination of the potentially discontinuous displacement field in the two-dimensional domain. Thereby, the behavior at the crack surfaces is modeled by cohesive laws. We impose a nonpenetration condition to avoid interpenetration of opposite crack sides. Accordingly, the state problem is formulated as variational inequality. This leads to potential nondifferentiability of the shape-state mapping. For the numerical solution, we derive first-order information in the form of subgradients. We conclude the article by numerical results.     

Biaxial tension of a homogeneous isotropic plate with two equal coaxial cracks with regard for plastic zones near their tips

We investigate the problem of determination of the stress-strain state of an isotropic plate with two equal cracks at a set homogeneous field of forces at infinity. It is assumed that the lips of the cracks are free of load and that, near their tips, plastic zones are formed. Using Kolosov–Muskhelishvili complex potentials, we seek a solution of the problem in the class of functions bounded at the tips of the cracks and reduce it to problems of linear conjugation. Relations for the determination of the values of plastic zones and crack tip opening displacements are obtained. We perform a numerical analysis of the problem and construct graphs of dependences of the lengths of plastic zones and crack tip opening displacements on the distance between the centers of the cracks.     

Thermal stress state of a bimaterial with a closed interfacial crack having rough surfaces

We have formulated the problem of thermoelasticity for a bimaterial whose components differ only in their shear moduli, with a closed interfacial crack having rough surfaces. The bimaterial is subjected to the action of compressive loads and heat flow normal to the interfacial surface. We have taken into account the dependence of thermal conductance of the defect on the contact pressure of its faces and heat conductivity of the medium that fills it. The problem is reduced to a Prandtl-type nonlinear singular integro-differential equation for temperature jump between the crack surfaces. An analytical solution of this problem has been constructed for the case of action of the heat flow only. We have analyzed the dependence of contact pressure of the defect faces, temperature jump between them, and the intensity factor of tangential interfacial stresses on the value of given heat flow, roughness of the surfaces, and ratio between the shear moduli of joined materials.     

粘弹体中裂纹演化的细观力学分析

根据Eshelby等效夹杂理论研究含残币形裂纹的粘弹体中裂纹张开位移随时间的缓慢增大以及含裂纹粘弹性体的等效模量随时间的变化。对于Maxwell粘弹性材料给出了模量随时间变化的显式表达式,结果表明裂纹的缓慢张开使材料模量减小更快。     

Interaction of a harmonic torsional wave with ring-shaped defects in an elastic body

We have solved the problem of determination of the stressed state in an isotropic elastic body near ring-shaped defects (a crack or a thin rigid inclusion) as a result of the action of a harmonic torsional wave. The method of solution is based on the use of discontinuous solutions of the equation of torsional vibrations and lies in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of angular displacement or tangential stress.     

The Method of Smooth Domains in the Equilibrium Problem for a Plate with a Crack

A free boundary problem is considered of the equilibrium of an elastic plate with a crack. We suppose that some boundary mutual nonpenetration conditions are given on the crack faces in the form of simultaneous equalities and inequalities. We suggest a new approach to posing the problem in a smooth domain although it was stated in a domain with cuts originally. We treat the constraints on the components of the displacement vector and stress tensor on the crack faces as interior constraints, i.e., constraints given on subsets of the smooth domain of a solution.     

Free Vibration Analysis of Laminated Composite Plates Resting on Elastic Foundations by Using a Refined Hyperbolic Shear Deformation Theory

The free vibration of laminated composite plates on elastic foundations is examined by using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces, and likewise, the shear components do not contribute to bending moments. The most interesting feature of this theory is that it allows for parabolic distributions of transverse shear stresses across the plate thickness and satisfies the conditions of zero shear stresses at the top and bottom surfaces of the plate without using shear correction factors. The number of independent unknowns in the present theory is four, as against five in other shear deformation theories. In the analysis, the foundation is modeled as a two-parameter Pasternak-type foundation, or as a Winkler-type one if the second foundation parameter is zero. The equation of motion for simply supported thick laminated rectangular plates resting on an elastic foundation is obtained through the use of Hamilton’s principle. The numerical results found in the present analysis for free the vibration of cross-ply laminated plates on elastic foundations are presented and compared with those available in the literature. The theory proposed is not only accurate, but also efficient in predicting the natural frequencies of laminated composite plates.     

幂律非线性粘弹性材料中的裂纹扩展*

对蠕变不可压幂律非线性粘弹性材料中裂纹的蠕变扩展进行了分析,为描述银纹带中的力学行为,假设在裂纹尖端邻域中断裂过程区中分布着阻抗裂纹张开的粘聚应力б_f,.通过对均匀应力参考状态平凡解的摄动,将非线性粘弹性问题化成线性问题处理.对于幂指数.n≌1的弱非线性情况得到了应力与位移表达式.提出断裂过程区局域能量判据,导出了裂纹孕育时间t*与蠕变扩展率a的预测公式.     

Stress intensity factor for multiple cracks in bonded dissimilar materials using hypersingular integral equations

This paper deals with the multiple inclined or circular arc cracks in the upper half of bonded dissimilar materials subjected to shear stress. Using the complex variable function method, and with the help of the continuity conditions of the traction and displacement, the problem is formulated into the hypersingular integral equation (HSIE) with the crack opening displacement function as the unknown and the tractions along the crack as the right term. The obtained HSIE are solved numerically by utilising the appropriate quadrature formulas. Numerical results for multiple inclined or circular arc cracks problems in the upper half of bonded dissimilar materials are presented. It is found that the nondimensional stress intensity factors at the crack tips strongly depends on the elastic constants ratio, crack geometries, the distance between each crack and the distance between the crack and boundary.     

求解层间界面反平面剪切破坏的剪切梁模型(Ⅱ)——失稳特性

在研究(Ⅰ)的基础上,研究了剪切梁模型在裂纹萌生和失稳扩展阶段的行为特性,1)给出了软化失稳(snap-through)和回折失稳(snap-back)两种失稳行为发生的条件.2)对剪切梁在反平面剪切载荷及侧压力共同作用下的力学行为作了解析分析计算,给出了结构的位移—载荷全过程曲线.3)讨论了失稳过程中的能量释放问题,并给出了回折失稳过程中结构对外界的能量释放的计算式.     

The end zone of a shear crack propagating at an intersonic velocity

An asymptotic solution of the problem for a shear crack propagating at an intersonic velocity is given that determines the size of the end zone, the distribution of the displacement jump in it and the dependence of the propagation velocity on an effective stress intensity factor, introduced in this paper. Numerical data are presented for the case of linear softening in the end zone and a comparison is made with the results for the Leonov–Panasyuk–Dugdale model. It is established using an effective stress intensity factor that the propagation is unstable at velocities close to the velocity of transverse waves; it becomes stable when approaching the velocity of longitudinal waves.     

The dynamic stressed state of a rectangular composite plate rigidly fixed along its edges

Using the generalized dynamic theory of bending of plates, which takes account of the compliance of the material to transverse shear strains, we obtain an approximate solution of the problem of the dynamic stressed state of a composite plate with rigidly fixed edges subject to an impact load. We exhibit the contribution of the physico-mechanical and geometric parameters of the plate to the magnitude of the computed stresses and their time variation for different coefficients of internal dissipation of mechanical energy.Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 3, 1997, pp. 129–133.     

An equilibrium penny-shaped crack in an inhomogeneous elastic medium

The problem of a penny-shaped tensile crack in a continuously-inhomogeneous space is considered. The problem reduces to a dual integral equation for which an approximate analytical solution is constructed. It is proved that the approximate solution of the integral equation is asymptotically exact for both small and large values of the dimensionless geometric parameter of the problem. The accuracy of the solution obtained is investigated. Expressions are presented for the stress intensity factor, the energy of the opening of the crack, the displacements of its sides and the normal components of the stress tensor in the neighbourhood of its contour. In the numerical analysis of the solution of the problem, special attention is paid to analysing of the problem when the first derivative of the change in the elastic properties of the material changes sign.     

A general method for constructing Timoshenko-type theories

The equations of the plane theory of for the elasticity bending of a long strip are reduced by the method of simple iterations to the solution of a system of two equations for the displacement of the axis of the strip and the shear stress. If the transverse load varies slowly along the strip, the resolvent equations reduce to a single equation that is identical to the classical equation for the bend of a beam. When a local load is applied, the resolvent equation acquires an additional singular term that is the solution of the equation for the shear stresses under the assumption that the displacement (deflection) is a function of small variability. The convergence of the solution in an asymptotic sense is demonstrated. The application of the method of simple iterations to the dynamic equations for the bending of a strip also leads to a system of two resolvent equations in the displacement of the axis of the strip and the shear stress. These equations reduce to a single equation that is identical with the well-known Timoshenko equation. Hence, the procedure for using the method of simple iterations that has been developed can be classified as a general method for obtaining Timoshenko-type theories. An equation is derived for the bending of a strip on an elastic base with an isolated functional singular part with two bed coefficients, corresponding to the transverse and longitudinal springiness of the base.     

Dynamic crack propagation in a 2D elastic body. The out-of plane case

We discuss the propagation of a running crack under shear waves in a rigorous mathematical way for a simplified model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u (y, t) and the one-dimensional crack tip trajectory h = h (t). We handle both equations separately, assuming at first that the crack position is known. Existence and uniqueness of strong solutions of the wave equation are studied and the crack-tip singularities are derived under the assumption that the crack is straight and moves tangentially. Using an energy balance law and the crack tip behaviour of the displacement fields we finally arrive at an ordinary differential equation for h (t), called equation of motion for the crack tip. We demonstrate the crack-tip motion with corresponding nonuniformly crack speed by numerical simulations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The equilibrium problem for a Timoshenko-type shallow shell containing a through crack

Under study is the nonlinear equilibrium problem for an elastic Timoshenko-type shallow shell containing a through crack. Some boundary conditions in the form of inequalities are imposed on the curve defining the crack. We establish the unique solvability of the variational statement of the nonlinear problem of the equilibrium of a shell. We prove that, for sufficient smoothness of the solution, the initial variational statement is equivalent to the differential formulation of the problem. We deduce the boundary conditions on the inner boundary that describes the crack. In the case of the zero opening of the crack, we prove the local infinite differentiability of the solution function with additional assumptions on the functions defining the curvatures of the shell and the external loads.     

剪切载荷作用下含损伤胶接材料界面动应力强度因子的研究

主要针对剪切载荷作用下,胶接材料接合区域界面裂纹尖端动态应力强度因子进行了分析,其中考虑了裂尖区域的损伤．通过积分变换,引入位错密度函数,奇异积分方程被简化为代数方程,并采用配点法求解;最后经过Laplace逆变换,得到动态应力强度因子的时间响应．Ⅱ型动应力强度因子随着黏弹性胶层的剪切松弛参量、弹性基底的剪切模量和Poisson比的增加而增大;随膨胀松弛参量的增加而减小．损伤屏蔽发生在裂纹扩展的起始阶段．裂纹尖端的奇异性指数(-0.5)是与材料参数、损伤程度和时间无关的,而振荡指数由黏弹性材料参数控制．     

The displacement initial-boundary value problem in bending of thermoelastic plates weakened by cracks

The displacement initial-boundary value problem for bending of a thermoelastic plate with transverse shear deformation weakened by a crack is studied, and its unique solvability in spaces of distributions is proved by means of a combination of the Laplace transformation and variational methods.