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The dynamic stress intensity factors (DSIFs) of two 3D rectangular cracks in a transversely isotropic elastic material under an incident harmonic stress wave are investigated by generalized Almansi’s theorem and the Schmidt method in the present paper. Using 2D Fourier transform and defining the jumps of displacement components across the crack surface as the unknown functions, three pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement components across the crack surfaces are expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the geometric shape of the rectangular crack, the characteristics of the harmonic wave and the distance between two rectangular cracks on the DSIFs of the transversely isotropic elastic material. [ABSTRACT FROM AUTHOR]

In this work, a unified phase field theory for two coupled fracture types is proposed. Compared with the existing unified phase field theory for single fracture type, it considers the coupling behavior between different fracture types and is consistent with the single fracture type theory when one type of fracture is suppressed. Based on a new form of energy decomposition, that is, the strain energy is decomposed into deviatoric, tensile volumetric and compressive volumetric parts, and specifying the driving energy for each fracture type, a double phase-field model for coupled spall and adiabatic shear banding, which are two kinds of typical dynamic ductile fracture, is realized with the proposed unified phase field theory for two coupled fracture types. This coupled model can be used to predict the potential spall and adiabatic shear banding failures in ductile metals under dynamic loading, revealing the corresponding fracture types under different loadings without any additional criteria. The entire damage and fracture evolution (that is, from damage evolution, through crack expansion, and to fragmentation) are captured with this unified phase-field modeling framework. In addition, as an advantage of the double phase-field model, the complex multicrack distributions of spall and adiabatic shear bands in ductile metals can be presented with the proposed model. Based on these features, the expanding characteristics of complex cracks in metallic expanding shells are numerically studied. It is found that, if multilayer spall occurs, the specimen tends to fracture into three layers, that is, a complete spall layer, fragmentations in multilayer areas, and the main body, which is consistent with the experimental observations under certain conditions and is not the regular multiple layers predicted by the multilayer spall theory with ideal material assumption. For multiple adiabatic shear bands (ASBs) in the shell, the spiral patterns of ASBs' distributions, the initial positions, and the propagating path are clarified for the collapsed and expanding shells. The collapsed shell tends to form a single-direction spiral pattern whereas the expanding shell tends to form a double-direction spiral pattern. In addition to the inner surface, the intersection points of two ASBs can initiate cracks, and, as the intersection points have the chance of being distributed on the outer surface, the expanding shell has a greater potential to initiate a crack on the outer surface than the collapsed shell. The propagating path will be deflected after interactions of two ASBs, and the quantitative relationship between the deflection angle and the stress state before the interactions is given, which explains why the deflection angle is significantly larger in the collapsed shell compared with that in the expanding shell. Based on the above results and analysis, four typical stages of fractures in the expanding shell under internal explosive loading are identified. [ABSTRACT FROM AUTHOR]