Complex Function Method To Analyze Periodic Crack Tip Field Problems

Fiber composites are an emerging class of structural materials widely usedin many critical applications such as national defense and space technology.Because low-density, high strength, high specific stiffness as well as thecharacteristics of controlling bonding the fiber and matrix and processingtechnology to design material properties, the composite materials in engineeringis very promising. Defects in composite materials are the main cause of its lowerintensity, because these defects can cause singular stress and crack. Periodiccrack is the important mechanical model to study the interaction of multiplecracks.Complex function method is widely used in the periodic crack problems,which cracks is seen as a boundary condition, gives the boundary conditionsoutside the boundary forces, and solving boundary value problems which meetsthe basic differential equation, and then studying the crack tip stress field, strainand displacement fields. There is some systematic theoretical results areobtained in the studies of periodic cracks fracture problems in isotropicmaterials under far-field uniform load. But the research of periodic cracksfracture problems in anisotropic materials under concreter load is seldom seen.In this paper, by using complex function method, the periodic cracksfracture problems of orthotropic composite materials are researched mainlythrough introduction of the Westergaard stress function and constructing theconformal mapping of Jacobi elliptical function, which makes the rectangulargeometric crack shapes of physical plane be mapped into the Half-plane on thecomplex plane ξ=ε+iη. And then with the introduction of the Westergaardstress function and application of complex function theory and undeterminedcoefficients method, mechanical problem is turned into partial differentialboundary value problem. The undetermined coefficients and the stress functionare obtained with the help of boundary conditions. Stress intensity factor on thecrack tip is defined in the first place. The analytical form of open-type crackstress fields is then expressed by stress intensity factor. The result analysis of stress interaction between the periodic cracks and scale effect of stress field nearthe crack tip can provide a meaningful reference to the study of the interactionbetween multiple cracks and the strength of the material design. In this paper theanalytical solutions of the stress intensity factor, stress field and displacementfield near the crack tip for Mode I, Mode II and Mode III in composite materialsare obtained. The theoretical results have important significance in fracturetheory research and engineering application.