Satya N. Atluri

Analytical solution for embedded elliptical cracks, and finite element alternating method for elliptical surface cracks, subjected to arbitrary loadings

Abstract The complete solution for an embedded elliptical crack in an infinite solid and subjected to arbitrary tractions on the crack surface is rederived from Vijayakumar and Atluris general solution procedure. The general procedure for evaluating the necessary elliptic integrals in the generalized solution for elliptical crack is also derived in this paper. The generalized solution is employed in the Schwartz alternating technique in conjunction with the finite element method. This finite element-alternating method gives an inexpensive way to evaluate accurate stress intensity factors for embedded or elliptical cracks in engineering structural components.

NEW CONCEPTS IN MESHLESS METHODS

Meshless methods have been extensively popularized in literature in recent years, due to their flexibility in solving boundary value problems. Two kinds of truly meshless methods, the meshless local boundary integral equation (MLBIE) method and the meshless local Petrov–Galerkin (MLPG) approach, are presented and discussed. Both methods use the moving least-squares approximation to interpolate the solution variables, while the MLBIE method uses a local boundary integral equation formulation, and the MLPG employs a local symmetric weak form. The two methods are truly meshless ones as both of them do not need a ‘finite element or boundary element mesh’, either for purposes of interpolation of the solution variables, or for the integration of the ‘energy’. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable. In essence, the present meshless method based on the LSWF is found to be a simple, efficient and attractive method with a great potential in engineering applications. Copyright

Path-independent integrals, energy release rates, and general solutions of near-tip fields in mixed-mode dynamic fracture mechanics

In this paper the following topics are addressed: (i) the physical meaning of pathindependent integrals for elastodynamically propagating cracks introduced earlier by Atluri, Bui and Kishimoto et al. (ii) the relation of these integrals to the energy release rates, for propagating cracks and (iii) the relation between these integrals and the time-dependent stress-intensity factors KI(t), KII(t) and KIII(t) in general mixed mode dynamic crack propagation. Finally, a new path-independent integral which has the meaning of energy-release-rate for a propagating crack, is introduced.

Path-independent integrals in finite elasticity and inelasticity, with body forces, inertia, and arbitrary crack-face conditions

Abstract In this paper, certain path-independent integrals, of relevance in the presence of cracks, in elastic and inelastic solids are considered. The hypothesized material constitutive properties include: 1. (i) finite and infinitesimal elasticity 2. (ii) rate-independent incremental flow theory of elasto-plasticity 3. (iii) rate-sensitive behaviour including elasto-viscoplasticity and creep. In each case, finite deformations are considered, along with the effects of body forces, material acceleration, and arbitrary traction/displacement conditions on the crack-face. Also the physical interpretations of each of the integrals either in terms of crack-tip energy release rates or simply energy-rate differences in two comparison cracked-bodies are explored. Several differences between the results in the present work and those currently considered well established in literature are pointed out and discussed.

ALTERNATE STRESS AND CONJUGATE STRAIN MEASURES, AND MIXED VARIATIONAL FORMULATIONS INVOLVING RIGID ROTATIONS, FOR COMPUTATIONAL ANALYSES OF FINITELY DEFORMED SOLIDS, WITH APPLICATION TO PLATES AND SHELLS-I

Abstract Attention is focused in this paper on: (i) definitions of alternate measures of “stress-resultants” and “stress-couples” in a finitely deformed shell (finite mid-plane stretches as well as finite rotations); (ii) mixed variational principles for shells, undergoing large mid-plane stretches and large rotations, in terms of a stress function vector and the rotation tensor. In doing so, both types of polar decomposition, namely rotation followed by stretch, as well as stretch followed by rotation, of the shell midsurface, are considered; (iii) two alternate bending strain measures which depend on rotation alone for a finitely deformed shell; (iv) objectivity of constitutive relations, in terms of these alternate strain/“stress-resultants”, and “stress-couple” measures, for finitely deformed shells. To motivate these topics, and for added clarity, a discussion of relevant alternate stress measures, work-conjugate strain measures, and mixed variational principles with rotations as variables, is presented first in the context of three-dimensional continuum mechanics. Comments are also made on the use of the presently developed theories in conjunction with mixed-hydrid finite element methods. Discussion of numerical schemes and results is deferred to the Part II of the paper, however.

Effects of a Piezo-Actuator on a Finitely Deformed Beam Subjected to General Loading

The deformation of a beam-column, the upper and lower surfaces of which are bonded in segments with piezo-ceramic liners, is studied for the purpose of obtaining appropriate expressions for the force transferred to the structural member by the piezo-actuator. This concept may be employed for the control of large dynamic deformations of a lattice-type flexible space-structure. The present model, which is based upon a static analysis, accounts for the effects of transverse shear and axial forces in addition to a bending moment on the beam in formulating the governing equilibrium equations. The present model provides more complete expressions for the force transmitted to the structural member than a model reported earlier in literature, in which the shear and axial forces are neglected.

Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media

A complete form of stress and electric displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic piezoelectric media, is derived by using the complex function theory. New definitions of real-valued stress and electric displacement intensity factors for the interfacial crack are proposed. These definitions are extensions of those for cracks in homogeneous piezoelectric media. Closed form solutions of the stress and electric displacement intensity factors for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media are also obtained by using the mutual integral.

An effective method for solving the hyper‐singular integral equations in 3‐D acoustics

The application of the boundary integral methods to the problem of acoustics, exterior to a three‐dimensional surface, suffers, in general, from the nonexistence of nonuniqueness of its solutions at frequencies that are characteristic of the associated interior problem. The formulation that is most suitable for numerical implementation still appears to be that proposed by Burton and Miller [Proc. R. Soc. London Ser. A 323, 201–210 (1971)]. However, the hypersingular kernels present in such a formulation render it computationally unattractive. Previous attempts to regularize such hypersingular kernels involved the use of double surface integrals, or implicit use of tangential operators, or closed‐form evaluations of hypersingular integrals. The method of double surface integrals is computationally highly inefficient, even though it allows higher‐order interpolation schemes on the surface. The other two approaches are more conducive to the assumption of a constant value for each of the variables (pressure a...

On constitutive relations at finite strain - Hypo-elasticity and elasto-plasticity with isotropic or kinematic hardening

Abstract The question of ‘generalization’ to finite strains of the constitutive relations of infinitesimal strain theory of elasticity, and classical elasto-plasticity with isotropic and kinematic hardening, is critically examined. Simple generalizations, which lead to physically plausible material behaviour, are presented. The current controversies surrounding (i) the choice of stress-rate in the above generalization and (ii) the ‘anomaly’ of oscillatory stresses in kinematic hardening plasticity, as discussed by E.H. Lee and others, are analyzed. It is found that these ‘controversies’ are easily resolvable.

Incremental path-independent integrals in inelastic and dynamic fracture mechanics

Abstract Certain incremental path-independent integrals, of relevance in the mechanics of fracture of elastic-plastic materials described by a classical flow theory of plasticity, are presented. Both quasi-static as well as dynamic fracture situations are considered. The topics discussed include: (i) incremental path-independent integrals that characterize the crack-tip fields in elastic-plastic materials; (ii) incremental integrals related to the incremental total potential energy difference; and (iii) the complementary or dual representations of these integrals. The use of these integrals is illustrated through some numerical examples. Comments are made on the utility of these integrals in postulating rational fracture criteria.