J. Eftis

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.

The inclined crack under biaxial load

Abstract The problem of a sheet with an inclined crack subject to a system of biaxial loads is examined in a manner similar to that used in our previous publications. The combined effect of load biaxiality and crack orientation on K l , K II , maximum shear, anile of initial crack extension, elastic strain energy density, as well as on the local strain energy rate is made explicit. Once again it is shown that use of the so-called “singular solution” for expression of the local crack-tip elastic stress and displacements is not sufficient to give adequate account of the biaxial load effect.

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.

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.

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.

Load biaxiality and fracture: a two-sided history of complementing errors

Abstract It is the view of many engineers associated with fracture mechanics, materials science and mechanical design that for bodies containing plane cracks, loads that are applied in a direction that is parallel to the plane of the crack, in addition to the tensile load acting in the direction perpendicular to it, can have no influence upon the fracture behavior of the body. This point of view has persisted for the past sixty years or so, and still continues to persist, despite the paucity of experimental evidence in support of it. This paper traces the two-sided historical genesis of this notion. It shows how a series of oversights or errors, both in the calculation of the elastic strain energy appearing in Griffiths global crack instability hypothesis and in the evolution of Irwins local theory of the crack-tip elastic stress intensity, have, by coincidence, had the particular effect of nullifying the presence of the loads that are applied parallel to the crack. The mishaps associated with one theory are, in a sense, complimentary to the entirely different set of errors that cause this same null effect to appear in the other theory, in that they give the appearance of consistency between the two theories in this regard. In a subsequent paper, which will serve to synthesize the results of our research work in this area, considerable experimental data will be included that lends support to our contention that loads that are applied parallel to the plane of the crack have a decided influence upon the fracture behavior of the body.

G̃c and R-curve fracture toughness values for aluminum alloys under plane stress conditions

Abstract The effect of specimen geometry and subcritical crack growth on the nonlinear energy fracture toughness, G c , has been examined for thin, center-cracked sheets of 2024-T3 and 7075-T6 aluminum alloys. The procedure followed was to independently vary the specimen length, L, width, w, andd crack length-to-specimen width ratio and to determine the toughness both at the onset of subcritical crack growth and at the initiation of unstable fracture. Comparisons were also made with the R-curve toughness, GR, evaluated at unstable fracture from which it was found that both G c and GR displayed the same trend of change with geometrical variables, with G c consistently higher than GR. When the nonlinear energy fracture toughness was evaluated at the onset of subcritical crack growth, it was found that the geometry dependence essentially disappeared. Scanning electron microscopic examination of some typical fracture surfaces showed that stable crack growth was accompanied by a gradual change of fracture mode from plane strain to plane stress. An analysis of possible errors in the experimental procedure showed that the scatter observed in G c values was not due to experimental errors, but apparently due to inhomogeneities in the materials. Several techniques were also introduced for the purpose of more directly incorporating crack growth into the G c determination, but it was found that they did not cause significant variation in the toughness values.