含多共線或平行裂縫的彈性介質受均勻平面應力波作用之破壞力學分析

Abstract

本文分析含共線或平行多裂縫之彈性體，受到平面應力波作用下，各個裂縫尖端之應力強度因子的時間變化。分析方法係將裂縫視為連續分佈的差排(continuous distribution of dislocations)，利用單一差排的動態基本解，建立連結差排密度及動態載重的積分方程式。求解時先將該積分方程式對時間作Laplace轉換，以Gauss-Chebyshev積分法則求解轉換域之應力強度因子，再利用數值方法將之逆轉回時間域。 本文的算例包含：單、雙與三裂縫於不同排列的應力強度因子，其中，單裂縫於等向性材料受縱波作用的應力強度因子、共線雙裂縫於等向性材料受縱波作用的應力強度因子、單裂縫於正交性材料受平面波作用的應力強度因子、平行裂縫於等向性材料受縱波作用的應力強度因子與現有的文獻比對一致，並得知對於分析裂縫問題本法擁有高精準度與便利性。 有關等向性介質，由算例結果可得以下結論：(1)單裂縫的第一型應力強度因子的峰值是發生於另一尖端的繞射表面波抵達該尖端之瞬時，但波松比(Poisson’s ratio)大於0.48時則不然；(2)對於共線等長雙裂縫，其內裂縫尖端的應力強度因子峰值會隨內尖端距離減小而增大，(3)平行等長雙裂縫的第一型應力強度因子峰值隨兩裂縫的垂直距離變小而降低。對於正交性介質之結論如下：(1)平行裂縫的 值越大，應力強度因子越早發生，(2)對於平行不等長雙裂縫，受到入射波作用的裂縫越長，未受入射波作用的裂縫尖端第一型應力強度因子會越小。

An analysis is presented for an array of collinear or non-collinear multiple cracks subject to a uniform plane stress wave in an isotropic or an orthotropic material. An integral equation for the problem is established by modeling the cracks as distributions of dislocations and using a dynamic fundamental solution of a discrete dislocation. The integral equation is solved by Gaussian-Chebyshev integration quadrature in the Laplace transform domain first and the solution is then inverted to obtain the dynamic stress intensity factors in the time domain. Numerical examples include: one, two or three collinear or non-collinear cracks for several configurations. Comparisons of the present results with the existing results, in cases when they are available, show the present method is highly accurate and useful for assessing structural integrity of elastic media under dynamic loading in the presence of cracks. Several conclusions can be drawn form the numerical results. For isotropic media, (1) for a single crack, the peak mode I stress intensity at either tip occurs at the arrival time of the Rayleigh surface emitted from the other tip unless Poisson’s ratio is greater than 0.48; (2) for two collinear cracks of equal length, the peak stress intensity factors at the inner tips increase with decreasing distance between the inner tips of the cracks.; (3) for two parallel cracks of equal length, the peak mode I stress intensity factors decrease with decreasing distance between the cracks. For orthotropic media, (1) the time at which the peak stress intensity factor occurs decreases with increasing along the crack line; (2) for two parallel cracks of unequal lengths, the peak mode I stress intensity factors decrease with increasing length of crack that is first struck by the stress wave.