Shape control of thin rigid inclusions and cracks in elastic bodies

The paper is concerned with a control of thin rigid inclusion and crack shapes in elastic bodies. It is assumed that rigid inclusions are delaminated; thus, cracks are located on the boundary of inclusions as well as outside of inclusions. We provide the problem formulations and analyze the shape sensitivity with respect to geometrical perturbations in the frame of free boundary models. Inequality type boundary conditions are considered at the crack faces to guarantee a mutual non-penetration between crack faces. Inclusion and crack shapes are considered as control functions. The cost functional, which is based on the Griffith rupture criterion, characterizes the energy release rate and provides the shape sensitivity with respect to a change of the geometry of the structure. We prove an existence of optimal shapes in the problems considered.     

THE INTERACTION PROBLEM BETWEEN THE ELASTIC LINE INCLUSIONS

ＩｎｔｒｏｄｕｃｔｉｏｎＯｆａｌｌｔｈｅｆｉｂｅｒ＿ｒｅｉｎｆｏｒｃｅｄｃｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌｓ,ｔｈｅｓｈｏｒｔ＿ｆｉｂｅｒｃｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌｎｏｔｏｎｌｙｓｔｒｅｎｇｔｈｅｎｓｔｈｅｍａｔｒｉｘｂｕｔａｖｏｉｄｓｄｅｆｅｃｔｉｏｎｓｏｆｔｈｅｌｏｎｇ＿ｆｉｂｅｒｃｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌａｓｗｅｌｌ.Ｔｈｅｍｉｃｒｏ＿ｍｅｃｈａｎｉｃｓａｂｏｕｔｉｔｓｕｃｈａｓｆｒａｃｔｕｒｅ ,ｆａｔｉｇｕｅａｎｄｄａｍａｇｅｉｓｖｅｒｙｃｏｍｐｌｅｘ .Ｉｎｔｈｅｐｒｅｖｉ…     

Local-buckling-induced elastic interaction between inclusions in a free-standing film

The local-buckling-induced elastic interaction between two circular inclusions in a free-standing film is reported using numerical simulation. The simulation relies on a continuum model based on the modified Föppl-von Kármán plate theory for a film with arbitrarily distributed eigenstrain and eigencurvature. It is shown that due to the overlapping of the nonlinear local buckling the elastic interaction between the two inclusions with the same eigencurvature is repulsive, while the interaction between them with the opposite eigencurvature is attractive. The interaction strength in both cases decays with their mutual distance. In addition, the inclusion with positive/negative eigenstrain above critical values can trigger an axisymmetric/non-axisymmetric buckling, respectively, and the buckling induced elastic interaction between the two inclusions with eigenstrain shows a nonmonotonic behavior.     

Problem of the periodic welding of anisotropic elastic plane with different materials

ＩｎｔｒｏｄｕｃｔｉｏｎＵｐｔｏｎｏｗ,ｔｈｅｒｅｈａｖｅｂｅｎｍａｎｙｒｅｓｅａｒｃｈｅｓｏｎｔｈｅｐｌａｎｅｗｅｌｄｉｎｇｐｒｏｂｌｅｍｏｆｉｓｏｔｒｏｐｉｃｍａｔｅｒｉａｌｓ．ｅ．ｇ．［１］ａｎｄ［２］ｅｔｃ．Ｈｏｗｅｖｅｒ,ｆｏｒｔｈｅｐｌａ...     

周期裂纹和刚性线尖端场干涉效应研究

研究双周期裂纹和刚性线夹杂非均匀材料的反平面剪切问题。基于保角变换技术和椭圆函数理论,获得了问题应力场的全场精确解,给出了裂纹和刚性线尖端应力强度因子的封闭形式解答,讨论了裂纹和刚性线尖端场的干涉效应。数值结果表明：改变水平和垂直分布周期对裂纹和刚性线尖端场影响明显不同;裂纹长度2a逐渐增大时(0≤a/ω1≤0.5),裂纹尖端应力强度因子从1逐渐增大到无限大,而刚性线的尖端场变化不大;刚性线长度2d逐渐增大时(0≤d/ω2≤1),刚性线尖端应力强度因子逐渐减小,而裂纹的尖端场仅略微增大。     

Some three-dimensional problems for an elastic medium with isolated rigid inclusions

Problems of determining stresses in isolated ellipsoidal rigid inclusions contained in an isotropic elastic space subjected to uniformly distributed loading at infinity are considered. A solution in a closed form is constructed for inclusions shaped as ellipsoids of revolution.     

LONGITUDINAL SHEAR PROBLEMS OF COLLINEAR RIGID LINE INCLUSIONS IN ANISOTROPIC MATERIALS

ＬＯＮＧＩＴＵＤＩＮＡＬＳＨＥＡＲＰＲＯＢＬＥＭＳＯＦＣＯＬＬＩＮＥＡＲＲＩＧＩＤＬＩＮＥＩＮＣＬＵＳＩＯＮＳＩＮＡＮＩＳＯＴＲＯＰＩＣＭＡＴＥＲＩＡＬＳＪｉａｎｇｃｎｉ－ｐｉｎｇ（蒋持平）（ＤｅｐａｒｉｔｍｅｎｔｏｆＦｌｉｇｈｔＶｅｈｉｃｌｅＤｅｓ...     

Hydroelastic interaction between water waves and thin elastic plate floating on three-layer fluid

The wave-induced hydroelastic responses of a thin elastic plate floating on a three-layer fluid, under the assumption of linear potential flow, are investigated for two-dimensional cases. The effect of the lateral stretching or compressive stress is taken into account for plates of either semi-infinite or finite length. An explicit expression for the dispersion relation of the flexural-gravity wave in a three-layer fluid is analytically deduced. The equations for the velocity potential and the wave elevations are solved with the method of matched eigenfunction expansions. To simplify the calculation on the unknown expansion coefficients, a new inner product with orthogonality is proposed for the three-layer fluid, in which the vertical eigenfunctions in the open-water region are involved. The accuracy of the numerical results is checked with an energy conservation equation, representing the energy flux relation among three incident wave modes and the elastic plate. The effects of the lateral stresses on the hydroelastic responses are discussed in detail.     

Stress-strain state of an anisotropic plate with curvilinear cracks and thin rigid inclusions

A mixed problem of linear elasticity for an infinite anisotropic plate with cuts and thin undeformable inclusions located along arbitrary open smooth curves is solved with the use of complex potentials. Special representations of the solutions are constructed and a governing system of singular integral equations is obtained. A numerical algorithm for determining the stress-strain state of the plate, including the stress-intensity factors at the tips of cuts and rigid inclusions, is proposed. Calculation results are given. Novosibirsk State Technical University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 3, pp. 213–219, May–June, 2000.     

Stress-strain state of an anisotropic plate with curved cracks and thin rigid inclusions

We solve the bending problem for an anisotropic plate with flaws like smooth curved nonoverlapping through cracks and rigid inclusions. The problem is solved by the method of Lekhnitskii complex potentials specified as Cauchy type integrals over the flaw contours with an unknown integrand density function. We use the Sokhotskii—Plemelj formulas to reduce the boundary-value problem to a system of singular integral equations with the additional conditions that the displacements in the plate are single-valued when going around the cut contours and the equilibrium conditions for stress-free rigid inclusions. After the singular integrals are approximated by the Gauss-Chebyshev quadrature formulas, the problem is reduced to solving a system of linear algebraic equations. We study the local stress distribution near flaw tips. We analyze the mutual influence of flaws on the stress distribution character near their vertices and compare the well-known solutions for isotropic plates with the solutions obtained by passing to the limit in the anisotropy parameters (“weakly anisotropic material”) and by using the method proposed here.     

Interaction of a crack with a pair of thin, elastic inclusions

The influence of a pair of thin, elastic inclusions on the stress intensity at a crack tip is considered. The interaction problem is formulated as a set of coupled singular integral equations and the solution is obtained by the Gauss-Chebyshev numerical technique. The stress intensity is computed as a function of certain geometric and elastic parameters of the inclusions, and the toughening effect of these parameters is evaluated.     

Axisymmetric scattering of plane stationary waves on absolutely rigid inclusions in an elastic medium

International Applied Mechanics -     

螺型位错偶极子与圆形界面刚性线的干涉效应研究

本文重点研究螺型住错偶极子和圆形夹杂界面刚性线的弹性干涉效应。利用复变函数方法,得到了该问题的一般解;此外还求出了只含一条界面刚性线时的封闭解答,得到了刚性线尖端的应力强度因子以及作用在螺型位错偶板子中心的像力和像力偶矩。研究结果表明：位错偶板子对应力强度因子具有很强的屏蔽或反屏蔽效应;软夹杂吸引位错偶极子,而刚性线排斥位错偶板子,在一定条件下,位错偶极子在刚性线附近出现一个平衡位置;当刚性线的长度争材料剪切模量比达到临界值时,可以改变偶极子和界面之间的干涉机理;刚性线长度对位错偶极子中心像力偶矩也有很大的影响。