Two-dimensional radial integration displacement discontinuity method with high-order isoparametric elements

Two-dimensional displacement discontinuity method for transversely isotropic materials

ABSTRACTThis paper is concerned with stress intensity factors for cracks emanating from a triangular or square hole under biaxial loads by means of a new boundary element method. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfied and the crack‐tip displacement discontinuity elements proposed by the author. In the boundary element implementation, the left or the right crack‐tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a Hybrid Displacement Discontinuity Method (HDDM). Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors for plane elastic crack problems. In addition, the present numerical results can reveal the effect of the biaxial loads on stress intensity factors.

A boundary integral method for solving the exterior acoustic radiation problem of axisymmetric bodies with arbitrary boundary conditions is developed. The new formulation derives from the method proposed by Burton and Miller [Proc. R. Soc. London Ser. A 323, 201–210 (1971)], which uses a linear combination of the Helmholtz integral equation and its normal derivative to ensure the uniqueness of the numerical solution at all frequencies. It is shown that the use of tangential operators enables one to easily compute the hypersingular integral without much extra effort. By taking advantage of the properties of axisymmetric geometry, and using the expansion of the boundary conditions and the surface distribution functions in Fourier series with respect to the angle of revolution, the surface integral is reduced to a line integral along the generator of the body, and Fourier integrals of the Green’s function and its derivatives over the circumferential angle. The Fourier integrals with singular kernels are evaluated using recurrence formulas in terms of the complete elliptic integrals. A new approach is then proposed to evaluate the Burton and Miller formulation at a corner by using the left-hand and right-hand limits of normal derivatives. Doing so, isoparametric elements throughout the surface of the body are able to be used. In order to demonstrate the validity and accuracy of the method, numerical results with quadratic isoparametric curvilinear elements are presented for radiation problems of a pulsating sphere, an oscillating sphere, and a finite cylinder with various boundary conditions.

A numerical analysis of perpendicular cracks under general in-plane loading with a hybrid displacement discontinuity method

In this article, the interaction of arbitrary multiple cracks in an infinite plate is examined by means of a boundary element method consisting of the non-singular displacement discontinuity element presented by Crouch and Starfied and the crack tip displacement discontinuity elements proposed by the present author. In the boundary element implementation, the left or the right crack tip element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The present numerical results of stress intensity factors (SIFs) for some typical multiple crack problems show that the numerical approach is simple yet very accurate for analysing the interaction of arbitrary multiple cracks in an infinite plate.

Analysis of an interfacial crack in a piezoelectric bi-material via the extended Green’s functions and displacement discontinuity method

The orientation-preservation condition, i.e., the Jacobian determinant of the deformation gradient $\det \nabla u$ is required to be positive, is a natural physical constraint in elasticity as well as in many other fields. It is well known that the constraint can often cause serious difficulties in both theoretical analysis and numerical computation, especially when the material is subject to large deformations. In this paper, we derive a set of sufficient and necessary conditions for the quadratic iso-parametric finite element interpolation functions of cavity solutions to be orientation preserving on a class of radially symmetric large expansion accommodating triangulations. The result provides a practical quantitative guide for meshing in the neighborhood of a cavity and shows that the orientation-preservation can be achieved with a reasonable number of total degrees of freedom by the quadratic iso-parametric finite element method.

Applications of the fast multipole fully coupled poroelastic displacement discontinuity method to hydraulic fracturing problems

A new constant displacement discontinuity (CDD) element is presented for the numerical solution of Mode I, II and III crack problems, based on the strain-gradient elasticity theory in its simplest possible Grade-2 (second gradient of strain or G2 theory) variant. The accuracy of the proposed new element is demonstrated herein in a first attempt only for isolated straight cracks or for co-linear straight cracks for which closed form solutions exist. It is shown that the results based on this new element are in good agreement with the exact solutions. Moreover, the new method preserves the simplicity and hence the high speed of the CDD method originally proposed by Crouch with only one collocation point per element for plane crack problems, but it is far more efficient compared to it, especially close to the crack tips where the displacement and stress gradients are highest.

Boundary element analysis of infinite anisotropic elastic medium containing inclusions and cracks

A general and advanced boundary element transient analysis of elastoplastic plates

Revisit of the equivalence of the displacement discontinuity method and boundary element method for solving crack problems

Extended displacement discontinuity method for analysis of cracks in 2D piezoelectric semiconductors

Displacement discontinuity analysis of a nonlinear interfacial crack in three-dimensional transversely isotropic magneto-electro-elastic bi-materials

Sand production mechanisms were numerically studied by means of a displacement discontinuity boundary-element method. The method allows an explicit modelling of fracture growth in rocks in 2D. Rock fracture in hollow cylinder sand production tests was simulated. Different types of fracture growth typically observed in hollow cylinder tests were investigated, i.e. breakout formation, slit propagation and uniform collapse of the cavity. Changing the mechanical behavior of the rock, represented by the properties and constitutive behavior of the displacement discontinuity elements, from brittle to ductile resulted in a change of the failure pattern from breakouts to uniform collapse, reproducing thus the laboratory tests. A clear effect of the specimen size (external diameter) on the results was demonstrated numerically. The increase of the external diameter of the hollow cylinder results in a virtually linear increase of the hollow cylinder strength, at least in the range of diameter values investigated.