Jay R. Walton

A study of dynamic crack growth in elastic materials using a cohesive zone model

Abstract The problem of a semi-infinite mode III crack dynamically propagating in a two-dimensional linear elastic infinite body is considered. The crack tip is assumed to be a cohesive zone whose (finite) size is determined so as to cancel the classical crack tip stress singularity caused by the applied loads. The cohesive zone behavior is assumed rate dependent and is characterized by a thermodynamically based constitutive equation. A new semi-analytical solution method has been formulated to solve the resulting initial value problem. The proposed solution method offers the capability to analyze the entire crack growth phenomenon (acceleration-steady state-arrest), without requiring special assumptions, neither on the crack propagation mode (e.g. steady state or assigned crack tip velocity), nor on the space-time discretization, so to obtain solutions that are not affected by grid size effects. Several solutions, corresponding to various values of the initial and boundary conditions as well as cohesive zone constitutive properties, are presented and analyzed.

The Convexity Properties of a Class of Constitutive Models for Biological Soft Issues

During the last three decades, the theory of nonlinear elasticity has been used extensively to model biological soft issues. Although this research has generated many different models to describe the constitutive response of these issues, surprisingly little work has been done to analyze the mathematical features of the various models proposed. Here we carefully examine the convexity properties, specifically the Legendre-Hadamard and the strong ellipticity conditions, for a class of soft-issue models related to the classical Fung model. For various versions of models within this class, we discover necessary and sufficient conditions that must be satisfied for a model to have one or both of these convexity properties. We interpret our results mechanically and discuss the related implications for constitutive modeling. Also, we explore some more general points on how convexity can be studied and on the relation among different convexity properties.

Sufficient conditions for strong ellipticity for a class of anisotropic materials

We provide sufficient conditions for strong ellipticity for a general class of anisotropic hyperelastic materials. This general class includes as a subclass transversely isotropic materials. Our sufficient conditions require that the first partial derivatives of the reduced-stored energy function satisfy some simple inequalities and that the second partial derivatives satisfy a convexity condition. We also characterize a restricted type of strong ellipticity for a subclass of transversely isotropic materials undergoing pure homogeneous deformations. We apply our results to a model of soft tissue from the biomechanics literature.

Steady growth of a crack with a rate and temperature sensitive cohesive zone

The steady-state dynamic propagation of a crack in a heat conducting elastic body is numerically simulated. Specifically, a mode III semi-infinite crack with a nonlinear temperature dependent cohesive zone is assumed to be moving in an unbounded homogeneous linear thermoelastic continuum. The numerical results are obtained via a semi-analytical technique based on complex variables and integral transforms. The relation between the thermo-mechanical properties of the failure zone and the resulting crack growth regime are investigated. The results show that temperature dependent solutions are substantially different from purely mechanical ones in that their existence and stability strongly depends on the cohesive zone thermal properties.

Numerical simulations of a dynamically propagating crack with a nonlinear cohesive zone

Numerical solutions of a dynamic crack propagation problem are presented. Specifically, a mode III semi-infinite crack is assumed to be moving in an unbounded homogeneous linear elastic continuum while the crack tip consists of a nonlinear cohesive (or failure) zone. The numerical results are obtained via a novel semi-analytical technique based on complex variables and integral transforms. The relation between the properties of the failure zone and the resulting crack growth regime are investigated for several rate independent as well as rate dependent cohesive zone models. Based on obtained results, an hypothesis is formulated to explain the origin of the crack tip velocity periodic fluctuations that have been detected in recent dynamic crack propagation experiments.

The energy release rate for a quasi-static mode I crack in a nonhomogeneous linearly viscoelastic body

Abstract The Energy Release Rate (ERR) for the quasi-static problem of a semi-infinite mode I crack propagating through an inhomogeneous isotropic linearly viscoelastic body is examined. The shear modulus is assumed to have a power-law dependence on depth from the plane of the crack and a very general behavior in time. A Barenblatt type failure zone is introduced in order to cancel the singular stress and a formula for the ERR is derived which explicitly displays the combined influences of material viscoelasticity and inhomogeneity. The ERR is calculated for both power-law material and the standard linear solid and the qualitative features of the ERR are presented along with numerical illustrations.

The quasi-static propagation of a plane strain crack in a power-law inhomogeneous linearly viscoelastic body

SummaryAn analysis is presented of the steady-state propagation of a semi-infinite mode I crack for an infinite inhomogeneous, linearly viscoelastic body. The shear modulus is assumed to have a power-law dependence on depth from the plane of the crack. Moreover, both a general and a power-law behavior in time for the shear modulus are considered. A simple closed form expression for the normal component of stress in front of the propagating crack is derived which exhibits explicitly the form of the stress singularity and its material dependency. The crack profile is examined and its dependence on the spatial and time behavior of the shear modulus is determined.

On a mathematical model of the productivity index of a well from reservoir engineering

Motivated by the reservoir engineering concept of the productivity index of a produc- ing oil well in an isolated reservoir, we analyze a time dependent functional, diffusive capacity, on the solutions to initial boundary value problems for a parabolic equation. Sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions. The dependence of the constant diffusive capacity on the type of the boundary condition (Dirichlet, Neumann, or third boundary condition) is investigated using a known variational principle and confirmed numer- ically for various geometrical settings. An important comparison between two principal constant values of a diffusive capacity is made, leading to the establishment of criteria when the so-called pseudo-steady-state and boundary-dominated productivity indices of a well significantly differ from each other. The third boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well. The questions of stabilization and unique- ness of the time independent values of the diffusive capacity are addressed. The derived formulas are used in numerical study of evaluating the productivity index of a well in a general three-dimensional reservoir for a variety of well configurations.

A Unified Perspective on the Nature of Bonding in Pairwise Interatomic Interactions

Different classes of ground electronic state pairwise interatomic interactions are referenced to a single canonical potential using explicit transformations. These approaches have been applied to diatomic molecules N2, CO, H2(+), H2, HF, LiH, Mg2, Ca2, O2, the argon dimer, and one-dimensional cuts through multidimensional potentials of OC-HBr, OC-HF, OC-HCCH, OC-HCN, OC-HCl, OC-HI, OC-BrCl, and OC-Cl2 using accurate semiempirically determined interatomic Rydberg-Klein-Rees (RKR) and morphed intermolecular potentials. Different bonding categories are represented in these systems, which vary from van der Waals, halogen bonding, and hydrogen bonding to strongly bound covalent molecules with binding energies covering 3 orders of magnitude from 84.5 to 89,600.6 cm(-1) in ground state dissociation energies. Such approaches were then utilized to give a unified perspective on the nature of bonding in the whole range of diatomic and intermolecular interactions investigated.

Modeling of a Curvilinear Planar Crack with a Curvature-Dependent Surface Tension

An approach to modeling fracture incorporating interfacial mechanics is applied to the example of a curvilinear plane strain crack. The classical Neumann boundary condition is augmented with curvature-dependent surface tension. It is shown that the considered model eliminates the integrable crack-tip stress and strain singularities of order