Mode I fracture-energy methods for concrete

Various energy methods are evaluated for concrete beams of different sizes with constant width and spandepth ratio and tested in three-point bending. Using load-deflection curves, two new methods to estimate the energy-release rate are proposed and compared with the other energy methods for mode I fracture. One of the new methods, which is based directly on the definition of energy-release rate, gives results which are size independent for precracked beams and also invariant with crack length.     

Mixed-mode crack-tip deformations studied using a modified flexural specimen and coherent gradient sensing

An optical mapping of deformation fields and evaluation of fracture parameters near mixed-mode cracks in homogeneous specimens under elastostatic conditions is undertaken. A modified edge notched flexural geometry is used in the study and its ability in providing a relatively wide range of mode mixities is demonstrated. A full-field, optical shearing interferometry called ‘coherent gradient sensing’ (CGS) is used in the study. Crack-tip parameters such as stress-intensity factors, mode mixity and energy-release rate are measured from the interference patterns. The patterns are analyzed using Williams' mixed-mode, asymptotic expansion field. An expression for energy-release rate for the specimen is also derived using beam theory. The theoretical stress-intensity factors are then obtained using a mode-partitioning method based on moment decomposition. Experimental measurements and theoretical predictions are found to be in good agreement. Limitations of the mode-partitioning method used in the investigation are also pointed out.     

The effects of shear on delamination in layered materials

The effect of transverse shear on delamination in layered, isotropic, linear-elastic materials has been determined. In contrast to the effects of an axial load or a bending moment on the energy-release rate for delamination, the effects of shear depend on the details of the deformation in the crack-tip region. It therefore does not appear to be possible to deduce rigorous expressions for the shear component of the energy-release rate based on steady-state energy arguments or on any type of modified beam theory. The expressions for the shear component of the energy-release rate presented in this work have been obtained using finite-element approaches. By combining these results with earlier expressions for the bending-moment and axial-force components of the energy-release rates, the framework for analyzing delamination in this type of geometry has been extended to the completely general case of any arbitrary loading. The relationship between the effects of shear and other fracture phenomena such as crack-tip rotations, elastic foundations and cohesive zones are discussed in the final sections of this paper.     

Numerical computation and analytical estimation of stress-intensity factors for strips with multiple edge cracks

In this paper, stress-intensity factors for a two-dimensional problem are determined. Strips with multiple symmetrical edge cracks in tension are investigated. A simple analytical estimation is compared to numerical results. The influence of penetration of the crack faces and mixed-mode loading on the numerical results is investigated. A simple method to estimate stress-intensity factors for strips with multiple edge cracks is proposed.     

A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity

We formulate a simple one-parameter macroscopic model of distributed damage and fracture of polymers that is amenable to a straightforward and efficient numerical implementation. We show that the macroscopic model can be rigorously derived, in the sense of optimal scaling, from a micromechanical model of chain elasticity and failure regularized by means of fractional strain-gradient elasticity. In particular, we derive optimal scaling laws that supply a link between the single parameter of the macroscopic model, namely, the critical energy-release rate of the material, and micromechanical parameters pertaining to the elasticity and strength of the polymer chains and to the strain-gradient elasticity regularization. We show how the critical energy-release rate of specific materials can be determined from test data. Finally, we demonstrate the scope and fidelity of the model by means of an example of application, namely, Taylor-impact experiments of polyurea 1000 rods.     

On Material Conservation Laws for a Consistent Plate Theory

In the paper, material conservation laws associated with a consistent second-order plate theory are derived, which takes shear deformations and strains in thickness direction of the plate into account. Three path-independent integrals are established. In the presence of inhomogeneities in the material (e.g., defects or cracks), energy-release rates due to the change of the configuration of such flaws can be calculated by these integrals. The resulting material forces may serve to assess the reliability of structures with cracks.     

Elastic-plastic J analysis for an inner surface flaw in a pressure vessel

Elastic-plastic finite-element (FE) analyses of a pressurized cylinder with a deep inner surface flaw were done to study the effect of different loading conditions, i.e., axial forces and crack-face loading. The local energy-release rate was calculated by the method of virtual crack extension including a correction term if pressure acts on the crack faces. The numerical results are compared with a small-scale yielding generalization of the Newman-Raju approximation. The FE analysis reveals that the highest triaxiality of the stress state does not coincide with the point of maximumJ. Thus, an explanation might be found for the experimentally observed canoe shape of stable crack propagation. Paper was presented at the 1986 SEM Fall Conference on Experimental Mechanics held in Keystone, CO on November 2–5.     

The energy-release rate and “self-force” of dynamically expanding spherical and plane inclusion boundaries with dilatational eigenstrain

In the context of the linear theory of elasticity with eigenstrains, the radiated field including inertia effects of a spherical inclusion with dilatational eigenstrain radially expanding is obtained on the basis of the dynamic Green's function, and one of the half-space inclusion boundary (with dilatational eigenstrain) moving from rest in general subsonic motion is obtained by a limiting process from the spherically expanding inclusion as the radius tends to infinity while the eigenstrain remains constrained, and this is the minimum energy solution. The global energy-release rate required to move the plane inclusion boundary and to create an incremental region of eigenstrain is defined analogously to the one for moving cracks and dislocations and represents the mechanical rate of work needed to be provide for the expansion of the inclusion. The calculated value, which is the “self-force” of the expanding inclusion, has a static component plus a dynamic one depending only on the current value of the velocity, while in the case of the spherical boundary, there is an additional contribution accounting for the jump in the strain at the farthest part at the back of the inclusion having the time to reach the front boundary, thus making the dynamic “self-force” history dependent.     

Experimental and theoretical study on numerical density evolution of short fatigue cracks

Fatigue testing was performed using a kind of triangular shaped specimen to obtain the characteristics of numerical density evolution for short cracks at the primary stage of fatigue damage. The material concerned is a structural alloy steel. The experimental results show that the numerical density of short cracks reaches the maximum value when crack length is slightly less than the average grain diameter, indicating grain boundary is the main barrier for short crack extension. Based on the experimental observations and related theory, the expressions for growth velocity and nucleation rate of short cracks have been proposed. With the solution to phase space conservation equation, the theoretical results of numerical density evolution for short cracks were obtained, which were in agreement with our experimental measurements. The project supported by the National Natural Science Foundation of China and the Chinese Academy of Sciences.     

SOLUTION OF MULTIPLE CRACKS IN A FINITE PLATE OF AN ELASTIC ISOTROPIC MATERIAL WITH THE DISTRIBUTED DISLOCATION METHOD

This paper presents a numerical solution to model multiple cracks in a finite plate of an elastic isotropic material. Both the boundaries and the cracks are modeled by distributed dislocations. This method results in a system of singular integral equations with Cauchy kernels which can be solved by Gauss-Chebyshev quadrature method. Four examples are provided to assess the capability of this method.     

THE CONSTITUTIVE RELATION OF CRACK-WEAKENED ROCK MASSES UNDER AXIAL-DIMENSIONAL UNLOADING

An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.     

The thermoelastic material-momentum equation

The equation of material momentum, or pseudomomentum, is obtained for thermoelastic materials. This is done in the classic theory, based on the heat conduction hypothesis, and also in the framework of a thermoelasticity approach involving no dissipation of energy, as recently proposed by Green and Naghdi. The results are applied to the thermoelastic fracture problem. When the pseudomomentum equation is written in global form, for a fractured body, it provides path-domain invariant expressions for the thermoelastic energy-release rate.Dedicated to the memory of Paul M. Naghdi.     

INTERACTION OF THREE PARALLEL CRACKS IN RECTANGULAR PLATE UNDER CYCLIC LOADS

This paper presents a numerical approach for modeling the interaction between multiple cracks in a rectangular plate under cyclic loads. It involves the formulation of fatigue growth of multiple crack tips under ruixed-mode loading and an extension of a hybrid displacement discontinuity method （a boundary element method） to fatigue crack growth analyses. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, remeshing of existing boundaries is not necessary for each increment of crack extension. Crack extension is conveniently modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. As an example, the numerical approach is used to analyze the fatigue growth of three parallel cracks in a rectangular plate. The numerical results illustrate the validation of the numerical approach and can reveal the effect of the geometry of the cracked plate on the fatigue growth.     

Analysis of three-dimensional edge cracks under tensile loading

Three-dimensional edge cracks are analyzed using the Self-Similar Crack Expansion (SSCE) method with a boundary integral equation technique. The boundary integral equations for surface cracks in a half space are presented based on a half space Green's function (Mindlin, 1936). By using the SSCE method, the stress intensity factors are determined by the crack-opening displacement over the crack surface. In discrete boundary integral equations, the regular and singular integrals on the crack surface elements are evaluated by an analytical method, and the closed form expressions of the integrals are given for subsurface cracks and edge crakcs. This globally numerical and locally analytical method improves the solution accuracy and computational effort. Numerical results for edge cracks under tensile loading with various geometries, such as rectangular cracks, elliptical cracks, and semi-circular cracks, are presented using the SSCE method. Results for stress intensity factors of those surface breaking cracks are in good agreement with other numerical and analytical solutions.     

Energy-dissipation mechanisms associated with rapid fracture of concrete

A hybrid experimental-numerical procedure, involving moiré interferometry and dynamic finite-element analysis, was used to analyze rapid crack growth in an impact loaded three-point-bend concrete specimen with an offset straight precrack. The dissipated energy rates in the fracture process zone (FPZ), which trails the rapidly extending crack, and in the frontal FPZ ahead of the crack tip, the kinetic-energy rate and energy-release rate were computed. The results showed that while the trailing FPZ was the dominant energy dissipation mechanism, much of the released energy was converted to kinetic energy in the fracturing concrete specimen. Paper was presented at the 1993 SEM Spring Conference on Experimental Mechanics held in Dearborn, MI on June 11.     

On energy changes due to the formation of a circular hole in an elastic plate

The appearance of holes in crystals, developed in stressed biological structures, has motivated us to study the energy release of an initially undisturbed plate subjected to a biaxial stress state which is disturbed by a growing hole. An energy balance allows for a kinetic equation of the hole radius a. The main emphasis is placed on the calculation of the total mechanical energy-release rate by different independent concepts. Specifically, path-independent integrals represent the most convenient approach.     

Strip-coalesced interior zone model for two unequal collinear cracks weakening piezoelectric media

In this paper, a mathematical strip-saturation model is proposed for a poled transversely isotropic piezoelectric plate weakened by two impermeable unequal-collinear hairline straight cracks. Remotely applied in-plane unidirectional electromechanical loads open the cracks in mode-I such that the saturation zone developed at the interior tips of cracks gets coalesced. The developed saturation zones are arrested by distributing over their rims in-plane normal cohesive electrical displacement. The problem is solved using the Stroh formalism and the complex variable technique. The expressions are derived for the stress intensity factors （SIFs）, the lengths of the saturation zones developed, the crack opening displacement （COD）, and the energy release rate. An illustrative numerical case study is presented for the poled PZT-5H ceramic to investigate the effect of prescribed electromechanical loads on parameters affecting crack arrest. Also, the effect of different lengths of cracks on the SIFs and the local energy release rate （LERR） has been studied. The results obtained are graphically presented and analyzed.     

Maximum-energy-release-rate criterion applied to a tension-compression specimen with crack

The title problem is studied by using the explicit asymptetic analysis presented in the accompanying paper. The asymptotic analysis indicates that the very basic problem is a semi-infinite L-shaped crack governed by a single integral equation. This equation is discretized to a system of complex algebraic equations and solved by a standard HARWELL subroutine. It is found that the maximum-energy-release-rate criterion has two branches, one for tensile loads and one for compressive loads. Our numerical results indicate that the maximum energy-release rate is always associated with maximum K ₁ and K ₂=0, where K ₁ and K ₂ are the stress-intensity factors at the fractured tip. Thus, the well-known K-G relation valid for crack-parallel propagation also holds for non-crack-parallel propagations. This conclusion is, however, purely numerical.Supported by U.S. Army Research Office-Durham under Grant DAAG-29-76-G-0272.     

Longitudinal and Bending Stresses in a Beam with Saw Cut Edge Cracks∗

Abstract

The Griffith-Irwin theory of brittle fracture of elastic solids predicts the propagation of cracks on the basis of the energy release rate. This depends upon the stress intensity factors for a given crack configuration. The present paper provides these informations for the problem of an infinite number of periodic, non-coplanar, parallel edge cracks in a strip. Two types of crack configurations, namely, periodic cracks of equal length starting from one edge and a set of two coplanar symmetrical edge cracks of equal length are solved for constant and linearly varying pressure distributions. These problems arise naturally in structural mechanics while investigating stresses in extension and bending of cracked strips. Final results are obtained from the numerical solution of certain Fredholm integral equations of the second kind derived from a dual series of Papkovich-Fadle eigenfunctions     

Numerical and experimental determination of three-dimensional multiple crack growth in fatigue

This work is concerned with the assessment of propagation of multiple fatigue cracks in three-dimensions. Computational modelling of fatigue crack propagation is made together with detection and monitoring of the crack shape development. The boundary element method (BEM) is used for automating the modelling of crack propagation in linear elastic as well as elastic–plastic regimes. Strain at several positions on the specimen surface near the crack mouth is measured to monitor crack initiation, shape development and closure levels. Examples are provided to validate the model by comparing the experimental results with those obtained by numerical predictions.