Mazen R. Tabbara

Element-free galerkin methods for static and dynamic fracture

Abstract Element-free Galerkin (EFG) methods are presented and applied to static and dynamic fracture problems. EFG methods, which are based on moving least-square (MLS) interpolants, require only nodal data; no element connectivity is needed. The description of the geometry and numerical model of the problem consists only of a set of nodes and a description of exterior boundaries and interior boundaries from any cracks. This makes the method particularly attractive for growing crack problems, since only minimal remeshing is needed to follow crack growth. In moving least-square interpolants, the dependent variable at any point is obtained by minimizing a function in terms of the nodal values of the dependent variable in the domain of influence of the point. Numerical examples involving fatigue crack growth and dynamic crack propagation are presented to illustrate the performance and potential of this method.

DYNAMIC FRACTURE USING ELEMENT‐FREE GALERKIN METHODS

The element-free Galerkin method for dynamic crack propagation is described and applied to several problems. This method is a gridless method, which facilitates the modelling of growing crack problems because it does not require remeshing; the growth of the crack is modelled by extending its surfaces. The essential feature of the method is the use of moving least-squares interpolants for the trial-and-test functions. In these interpolants, the dependent variable is obtained at any point by minimizing a weighted quadratic form involving the nodal variables within a small domain surrounding the point. The discrete equations are obtained by a Galerkin method. The procedures for modelling dynamic crack propagation based on dynamic stress intensity factors are also described.

Element-free Galerkin method for wave propagation and dynamic fracture

Abstract Element-free Galerkin method (EFG) is extended to dynamic problems. EFG method, which is based on moving least square interpolants (MLS), requires only nodal data; no element connectivity is needed. This makes the method particularly attractive for moving dynamic crack problems, since remeshing can be avoided. In contrast to the earlier formulation for static problems by authors, the weak form of kinematic boundary conditions for dynamic problems is introduced in the implementation to enforce the kinematic boundary conditions. With this formulation, the stiffness matrix is symmetric and positive semi-definite, and hence the consistency, conergence and stability analyses of time integration remain the same as those in finite element method. Numerical examples are presented to illustrate the performance of this method. The relationship between the element-free Galerkin method and the smooth particle hydrodynamics (SPH) method is also discussed in this paper. Results are presented for some one-dimensional problems and two-dimensional problems with static and moving cracks.

Finite element derivative recovery by moving least square interpolants

Abstract A simple, accurate technique for recovery of displcements and derivatives, such as strains is presented. The technique is based on local interpolation of nodal displacements using a moving least square method. The strains are then recovered by taking appropriate derivatives of this interpolant. Numerical experiments in linear elasticity and heat conduction on the convergence and accuracy of the recovered derivatives show very good results and superconvergence for strains in many cases; the technique is also effective for displacement interpolation for projection methods.

New method of analysis for slender columns

The paper presents a simple new method to calculate column-interaction diagrams, which takes into account slenderness effects. The method consists of a simple incremental loading algorithm that traces the load-deflection curve at constant eccentricity of the axial load. The column failure is defined for design purposes as the peak of the diagram of axial load versus midlength bending moment at constant load eccentricity. The tangent modulus load is found to be apporximately equal to the peak load of column with load eccentricity 0.01 of the cross-sectional thickness and represents a lower bound for the maximum loads at still smaller eccentricities. Strain irreversibility at unloading can be taken into account but its effect is very small. The method is compared with the ACI moment magnification method and with the CEB Model column method based on moment-curvature relations. The agreement with the CEB method is very close, but with respect to the ACI method there are large discrepancies.

Simulations of the Penetration of 6061-T6511 Aluminum Targets by Spherical-Nosed VAR 4340 Steel Projectiles

In certain penetration events it is proposed that the primary mode of deformation of the target can be approximated by known analytical expressions. In the context of an analysis code, this approximation eliminates the need for discretizing the target as well as the need for a contact algorithm. Thus, this method substantially reduces the computer time and memory requirements. In this paper a forcing function which is derived from a spherical-cavity expansion (SCE) analysis has been implemented in a transient dynamic finite element code. This irnplementation is capable of computing the structural and component responses of a projectile due to a three dimensional penetration event. Simulations are presented for 7.1 l-mm-diameter, 74.7-mm-long, spherical-nose, vacuum- arc-remelted (VAR) 4340 steel projectiles that penetrate 6061-T6511 aluminum targets. Final projectile configurations obtained from the simulations are compared with post-test radiographs obtained from the corresponding experiments. It is shown that the simulations accurately predict the permanent projectile deformation for three dimensional loadings due to incident pitch and yaw over a wide range of striking velocities.

Bifurcation and stability of structures with interacting propagating cracks

A general method to calculate the tangential stiffness matrix of a structure with a system of interacting propagating cracks is presented. With the help of this matrix, the conditions of bifurcation, stability of state and stability of post-bifurcation path are formulated and the need to distinguish between stability of state and stability path is emphasized. The formulation is applied to symmetric bodies with interacting cracks and to a halfspace with parallel equidistant cooling cracks or shrinkage cracks. As examples, specimens with two interacting crack tips are solved numerically. It is found that in all the specimens that exhibit a softening load-displacement diagram and have a constant fracture toughness, the response path corresponding to symmetric propagation of both cracks is unstable and the propagation tends to localize into a single crack tip. This is also true for hardening response if the fracture toughness increases as described by an R-curve. For hardening response and constant fracture toughness, on the other hand, the response path with both cracks propagating symmetrically is stable up to a certain critical crack length, after which snapback occurs. A system of parallel cooling cracks in a halfspace is found to exhibit a bifurcation similar to that in plastic column buckling.

Stable path of interacting crack systems and micromechanics of damage

ABSTRACT Micromechanics of damage in brittle heterogeneous materials and composites requires analysis of a system of interacting cracks. The response path of a crack system typically exhibits bifurcations such that the states on each post-bifurcation branch can be stable yet only one branch can be reached in a stable manner. Recent results on thermodynamic criteria for stable states and stable response paths of inelastic structures are reviewed and formulated in terms of the incremental internal entropy of the system. The incremental entropy, which can be expressed in terms of second order work, is then calculated for various points on the response paths of some typical symmetric crack systems. It is shown that while the symmetric states may be stable, the path which leads to them is unstable and cannot occur in reality. Generally, nonsymmetry develops at the beginning of softening. The results show that it is insufficient to model distributed cracking only by means of crack systems and linear elastic fracture mechanics. Further aspects, such as material heterogeneity, residual stresses, and cohesive fracture zones for the microcracks, might have to be taken into account.