H. Liebowitz

Crack border stress and displacement equations revisited

Abstract It is more or less accepted in fracture mechanics that the elastic stress and displacements very near to the tip of a plane line crack can be approximated with sufficient accuracy, for all geometries and outer boundary loading conditions, by. a one-parameter representation, i.e. strictly in terms of the stress intensity factors K I and/or K II . It is shown here that this presumption which appears to be reasonable on face value, quantitatively speaking, is nevertheless unacceptable as a general proposition. The reason lies with the quite arbitrary practice of omitting the second term of the series representation for the stresses, a contribution which is independent of distance from the crack tip. It is not difficult to show by way of specific examples how such omission can lead to error of serious qualitative nature in the prediction of stress and displacement related quantities of interest.

Biaxial load effects in fracture mechanics

Abstract Our investigation into the effects of load biaxiality thus far, has produced several findings which, in our opinion, are deemed to be important. 1. (a) The standard expressions for elastic stress and displacement in the crack-tip region, i.e. the so-called “singular-solution’, cannot be considered to be approximations that are acceptable in a completely general sense. 2. (b) This conclusion is best illustrated in the instance of a biaxially loaded infinite sheet with a flat (horizontal) central crack, wherein the effect of the load applied parallel to the plane of the crack appears entirely in the second terms of the series representations for local stress and displacement. Omission of these contributions, which is the usual practice, is tantamount therefore to denial of the physical presence of the horizontal load. Thus, in calculations of stress, displacement and related quantities of interest in the crack border region by means of the standard expressions, no biaxial load effects will appear, leading thereby to the erroneous impression that load applied parallel to the plane of the crack can have no influence with regard to the fracture problem. 3. (c) For the infinite sheet problem with a horizontal central crack, our analytical analysis shows significant biaxial load effect on crack border region and crack edge displacement, on local maximum shear stress, on the pattern of maximum shear isostats, on the angle of initial crack extension, and on local elastic strain energy density and strain energy rate. On the other hand, both the elastic stress intensity factor (as to be expected) and the J -integral show no sensitivity whatsoever to the presence of the horizontal load. 4. (d) The analytical results referred to above for the infinite sheet are also seen in the results obtained for a finite sheet using finite element numerical analysis. 5. (e) A nonlinear finite element analysis of the same biaxially loaded finite specimen geometry, designed to simulate elastic-plastic material behavior under conditions of no unloading, shows that the global energy rate, the J -integral, the plastic stress and strain intensity factors (in the sense of Hilton and Hutchinson), and the size of the crack border region plastic yield, all have pronounced biaxial load dependency.

Biaxial load effects on the crack border elastic strain energy and strain energy rate

Abstract In a previous paper it was shown that the singular expressions for the elastic stress and displacements in the border region of a line crack are inadequate, generally speaking. This is nowhere more clearly demonstrated than for the case of the infinite sheet with a flat central crack, biaxially loaded along its outer boundaries. For this particular problem, the entire effect of the load applied parallel to the plane of the crack shows up in the generally discarded second (non-singular) terms of the series representations for the stresses and displacements. Omission of these contributions is, in effect, equivalent to denying the presence of the boundary load applied parallel to the crack and, consequently, leads to prediction of results at variance with experimental data. In this paper we continue with further discussion of the same problem, focusing attention on the fact that the local elastic strain energy density and strain energy rate depend significantly on the biaxiality of the applied load.

Finite Element Methods in Fracture Mechanics

Finite element methodology specific to the analysis of fracture mechanics problems is reviewed. Primary emphasis is on the important algorithmic developments which have enhanced the numerical modelling of fracture processes. Methodologies to address elasto-static problems in two- and three-dimensions, elasto-dynamic problems, elasto-plastic problems, special considerations for three-dimensional non-linear problems and the modelling of stable crack growth are reviewed. In addition, the future needs of the fracture community are discussed and open questions are identified. The need for theoretical advancements in continuum mechanics and constitutive fracture laws coupled with improved numerical algorithms is emphasized. Extensive reference is made to the open literature base for further details.

On nonlinear effects in fracture mechanics

Abstract Linear elastic treatment of fracture is considered applicable for net section stress up to about 0.8 the uniaxial tensile yield stress. Crack front plastic yield is still small enough to be viewed and treated as a small perturbation to the local crack front elastic stress field. Assuming these same circumstances and adopting the same point of view, an approach is presented for incorporating the nonlinear effects of small scale crack front plastic yield and slow crack extension in determination of the energy release rate and fracture toughness. Deviation from linearity of the load-displacement record in a fracture toughness test offers a quantifiable measure of these effects and is used to calculate the energy release rate G , which turns out to have the form G = (1 + P )G , where G is Irwins elastic energy release rate. The factor P is directly determined in each test case, and when added to unity defines a nonlinear correction C to G , which has a lower bound value of one. Fracture toughness values for one-eight inch thick 7075-T6 center cracked aluminum sheet based on this approach are compared with uncorrected values and with values obtained by the Irwin y ∗ method of plasticity correction.

On fracture toughness in the nonlinear range

Abstract A general definition of fracture toughness, designated by G c , is developed which is appropriate to situations of subcritical crack growth and/or large-scale crack border plastic yield. The theoretical basis as well as comparisons with other proposed measures of fracture toughness are also discussed. A simple method is given for evaluating G c which is based on use of the load-displacement test record.

Load biaxiality and fracture - Synthesis and summary

Abstract In both Griffiths global energy rate theory for crack instability and Irwins local crack-tip stress intensity theory for fracture toughness, only the tensile load perpendicular to the crack influences fracture behavior of the body. Thus according to these theories, outer boundary loads applied parallel to the crack have no effect on the fracture process. This viewpoint has been widely held since its inception with the work of Griffith in 1921, and has strongly influenced the development of fracture mechanics. Investigations by the authors have shown the contrary however, in that the load biaxiality strongly affects many aspects of the fracture behavior of a cracked body. Almost all of the characteristics of brittle fracture have been shown theoretically and/or experimentally to be sensitive to load biaxiality. Specifically, this work has shown that the stress and displacement fields, the elastic strain energy density, and the maximum shear stress near the crack tip are all altered by loads applied parallel to the crack, as are the angle of initial crack extension, the strain energy of the entire body, the fracture load, and the rate of fatigue crack growth. Since the results of this research have been published piecemeal over the years, the authors are presenting, herein, a synthesis and summary of this work for the purpose of demonstrating the overall presence and consistency of the biaxial effects.

Three-dimensional finite element and dynamic analysis of composite laminate subjected to impact

A three dimensional finite element and dynamic analysis has been made for a layered fiber-reinforced composite laminate subjected to a given impact loading. Central difference method is employed in this analysis. The numerical results for the transient response of the laminate are presented.

On surface energy and the continuum thermodynamics of brittle fracture

Abstract When a separating body is viewed as a non-equilibrium thermodynamic process, the full thermodynamic nature of the surface energy induced by crack propagation becomes apparent. Within such a general framework it is no longer possible to view the surface energy as a material constant. For the self-propagating crack the induced surface energy is shown to depend explicitly on the square of the crack propagation speed. It is also shown that a separating body produces entropy even though the mechanical response at the solid may be elastic. The introduction of surface quantities such as surface energy into the continuum description of the fracture process forces a major departure from the mechanics appropriate to the non-separating body. Local equations of balance are no longer obtainable as derived consequences of postulated global balance equations. They must instead be imposed at separate additional postulates.

Computational fracture mechanics: Research and application

Abstract This paper focuses on the impact of computational methodology on furthering the understanding of fundamental fracture phenomena. The current numerical approaches to the solution of fracture mechanics problems, e.g. finite element (FE) methods, finite difference methods and boundary element methods, are reviewed. The application of FE methods to the problems of linear elastic fracture problems is discussed. Particular emphases are placed on the stress intensity factors, energy release rate in mixed mode fracture and dynamic crack propagation. Numerical solutions of ductile fracture problems are surveyed. A special focus is placed on stable crack growth problems. The need for further research in this area is emphasized. The importance of large strain phenomena and accurate modeling of non-linearities is highlighted. An expanded version of fracture mechanics methodology is given by Liebowitz [Advances in Fracture Research 3. Pergamon Press, Oxford (1989)]; additional treatment is given in this paper to numerical results incorporating error estimates and algorithms for mesh design into the FE code. The adaptive method involves various stages which includes FE analysis, error estimation/indication, mesh refinement and fracture/failure analysis iteratively. Reference is made to integrate expert knowledge and a hierarchical, rule-based, decision process to fracture mechanics for the purpose of designing practical fracture-proof engineering products. Some further areas of research in adaptive finite element analysis are discussed.