Showing papers in "International Journal of Fracture in 1974"

Strain-energy-density factor applied to mixed mode crack problems

Abstract: This paper deals with the general problem of crack extension in a combined stress field where a crack can grow in any arbitrary direction with reference to its original position. In a situation, when both of the stress-intensity factors,k 1,k 2 are present along the crack front, the crack may spread in any direction in a plane normal to the crack edge depending on the loading conditions. Preliminary results indicate that the direction of crack growth and fracture toughness for the mixed problem of Mode I and Mode II are governed by the critical value of the strain-energy-density factor,S cr. The basic assumption is that crack initiation occurs when the interior minimum ofS reaches a critical value designatedS cr. The strain-energy-density factorS represents the strength of the elastic energy field in the vicinity of the crack tip which is singular of the order of 1/r where the radial distancer is measured from the crack front. In the special case of Mode I crack extensionS cr is related tok 1c alone asS cr = (κ − 1)k 1 2 /8μ. In general,S takes the quadratic forma 1 1 k 1 + 2a 1 2 k 1 k 2 +a 2 2 k 2 whose critical value is assumed to be a material constant. The analytical predictions are in good agreement with experimental data on the problem of an inclined crack in plexiglass and aluminum alloy specimens. The result of this investigation provides a convenient procedure for determining the critical crack size that a structure will tolerate under mixed mode conditions for a given applied stress.

A stiffness derivative finite element technique for determination of crack tip stress intensity factors

Abstract: A finite element technique for determination of elastic crack tip stress intensity factors is presented. The method, based on the energy release rate, requires no special crack tip elements. Further, the solution for only a single crack length is required, and the crack is 'advanced' by moving nodal points rather than by removing nodal tractions at the crack tip and performing a second analysis. The promising straightforward extension of the method to general three-dimensional crack configurations is presented and contrasted with the practical impossibility of conventional energy methods.

Brittle fracture of solids with arbitrary cracks

Abstract: One of the problems of fracture mechanics is the prediction of the propagation of cracks in solids. The present paper deals mainly with linear fracture mechanics which owes its origin to the works of A. A. Griffith [1, 2] and studies the development of cracks under sufficiently low loads when the behaviour of the material within a region sufficiently remote from the edges of cracks may be regarded as linearly elastic, At present, linear fracture mechanics [3] is restricted mainly to special kinds of loading geometry, with the crack extending rectilinearly (in a plane case) or in its plane (in a three-dimensional case). The main problem here is to establish a relationship between the dimensions of cracks and the loads applied. Within the framework of linear fracture mechanics the fracture itself and other non-linear phenomena that precede it are assumed to take place only within local regions which are small compared to the dimensions of cracks. The possibility that such a situation exists is associated with the fact that when the crack dimensions are sufficiently large the characteristic dimension of the end region is fully determined by a certain intrinsic dimension of the material structure. Therefore, if the material does not exhibit time dependency, the state of the end region at the moment of rupture becomes fully independent of the loads applied and the geometry of the solid, i.e. autonomous. The notion of autonomy [4] leads to the formulation of this theory as one of limit equilibrium. If the conditions of rectilinear extension of the crack (or those of the crack extension in its plane) are disturbed, there arises a problem of determining not only the dimensions of the crack, but also the path of the crack extension under such conditions of loading that a slow, quasi-static crack development is possible. This problem can be actually subdivided into two: (1) Criteria for the determination of the dimensions and paths of the crack extension, and (2) Expressions for the characteristics of the stress-strain state which are constituents of these criteria through the geometry of solid with cracks and the loads applied. As regards (1), there have been many assertions, and the connections between them are not quite clear at present. The first of the suggested criteria, namely that of local symmetry for the plane problem formulated by Barenblatt and Cherepanov [5, 6] and by Erdogan and Sih [7] can be within certain limits substantiated and generalized for the three-dimensional case. The guiding principle here is the treatment of the theory of cracks from the standpoint of the method of inner and outer expansions or that of singular perturbations [8]. The concept of the stress intensity factor which is basic in linear fracture mechanics is decisive in matching inner and outer expansions to find the main term of the asymptotic solution of the complete problem. Actually the construction of the theory of equilibrium cracks [4] implicitly employs this technique for a certain specific model. More explicit indications are given in Ref. 9. In the treatment of the problem of plastic zones in the vicinity of notches, the idea of the boundary layer is employed in Ref. 10. The problem of fracture of a solid is analysed from this standpoint in Ref. 11. As regards (2), progress has been hampered by the lack of efficient techniques for fording the stress-strain state of a solid having non-rectilinear cuts. A number of investigations have been carried out for cuts of a particular kind an arc of a circumference [12, 13], an arc of a parabola [l4], and a three-link broken line which is close to a straight line to such an extent that the boundary conditions are assumed referable to the direction of the middle portion [15, 16]. The problem of a semi-infinite curvilinear cut slightly deviating from a rectilinear one by expanding complex elastic potentials in the magnitude of deviation of the cut from the rectilinear axis tangent to the line of cut at its end is considered in Ref. 17. An exact solution of the problem of a semi-infinite cut having the form of a two-link broken line is given in Ref. 18. The present paper is devoted to the investigation of the development of cracks under arbitrary loading conditions. In Section 1 the criterion of local symmetry is substantiated and generalized for the three-dimensional case. In Section 2 an effective procedure of finding stress intensity factors for the plane case is given, in terms of which the criterion is formulated. Closed first approximation formulas for these magnitudes are presented in the case of a slightly curved crack, numerical calculations showing the applicability of the latter with an error not exceeding 10 to 15 with the angles of deviation of the crack from the straight line coming to 20°. In Section 3 equations of extension of curvilinear cracks are derived on the basis of the first approximation formulas and criterion of local symmetry. In Section 4 some examples are considered.

Brittle fracture in compression

Abstract: The initiation and propagation of microfractures and their contribution to material failure in compression are examined. The early part of the fracture process, the lateral and the axial yield points of the stress-strain curves, are identified with the onset of microfracture, respectively at the tensile and the compressive stress concentrations of the elastic flaw boundary. Later stages, including the initiation of inclined shear fractures, the mobilization of total resistance and the reduction of strength to the residual level, are discussed in terms of a modified Coulomb model.

Slow crack growth in brittle materials under dynamic loading conditions

Abstract: The failure of materials due to slow crack growth, under dynamic loading conditions, is analyzed in terms of crack velocity, stress intensity relationships. It is shown that this type of analysis can fully describe the failure characteristics for both constant strain-rate and constant stress-rate loading. The analysis is used to predict the variations of strength and subcritical crack growth with strain-rate and stress-rate. Application of the analysis to several ceramic systems give data which are entirely consistent with available experimental data.

Proof testing of ceramic materials—an analytical basis for failure prediction

Abstract: An analysis is presented which permits the accurate prediction of component lifetimes after proof testing. The analysis applies to crack propagation controlled fracture but can be used as a conservative prediction when crack initiation is predominant. The analytical predictions are confirmed in a series of time-to-failure measurements.

A dynamic analysis of unstable crack propagation and arrest in the DCB test specimen

Abstract: A simple analytical model is developed to accompany experimental work on rapid crack propagation and arrest in the DCB test specimen. The present work extends the beam-on-elastic foundation model used previously by taking account of shear deformation and of both translational and rotary inertia. Crack speeds predicted with the model are found to be in good agreement with the constant-speed behavior observed experimentally. It is demonstrated that kinetic energy makes an important contribution to maintaining unstable crack propagation and to crack arrest.

Application of quadratic isoparametric finite elements in linear fracture mechanics

Abstract: Quadratic isoparametri c elements are shown to embody 1//r singularity f or c alculating s tress i ntensity f actors of elastic f racture mechanics. The singularity is obtained by placing the mid-side node on any side at the quarter p oint. Figure 1 shows the 2-dimensional, 8-noded quadrilateral (a) and 6-noded triangle (b), isoparametric e lements with the mid-side nodes near the crack tip at the quarter nodes. Figure 2 shows the S-dimensional elements w ith the mid-side node near the crack edge at the quarter p oints. The local s trains in these e lements vary as 1//r throughout the element. In the 3-D case, the strains along the crack edge are non-singular. A very important feature of these elements is that they satisfy the necessary requirements for convergence [I] in their singular form as well as in their non-singular form. They, therefore, pass the patch test [2], possess rigid body motion (R.B.M.), constant strain modes, interelement compatibility, and continuity of displacements. In contrast, other special crack tip elements [3,4], do not possess rigid body motion modes and do not pass the patch test, thus making their use in the problems cited below questionable. The existence of rigid body motion and constant strain modes in the proposed isoparametric elements allows the calculation of stress intensity factors for thermal gradients in 2- and 3-D problems and in problems where symmetry about the crack cannot be invoked (R.B.M. exists). In addition, since these elements are part of the element library of most general purpose programs, their use in linear fracture mechanics is very tractable. The element formulation in its non-singular form is well documented ([I], pp. 103-154). The element in the singular form is formulated exactly in the same manner except for a restriction on the location of the nodal points. In summary, the element is formulated by mapping its geometry from the cartesian space into a unit curvilinear space using special quadratic functions [i]. The same functions, in the curvilinear space, are used to interpolate the displacements within the element, hence the name isoparametric. In order to achieve the required singularity, the Jacobian of transformation [J], from the cartesian to the curvilinear space, is made singular by placing the mid-side nodes near the crack tip at the quarter points. The singularity occurs only at the crack tip point. It can be easily shown, for example, that for the rectangular form of the case in Figure la, the strain in the local x-direction along the line i-2, is given by

Adhesive fracture mechanics

Abstract: In studies of fracture mechanics the adhesive fracture energy is regarded as a fundamental property of the adhesive system. It is pointed out that the value of the adhesive fracture energy depends on surface preparation, curing conditions, and absorbed monolayers. A test method reported makes use of a disk whose peripheral part is bonded to a substrate material. Pressure is injected into the unbonded central part of the disk. At a certain critical pressure value adhesive failure can be observed. A numerical stress analysis involving arbitrary geometries is conducted.

Stress intensity factors of an elliptical crack or a semi-elliptical crack subject to tension

Abstract: This paper is concerned with the analysis of stress intensity factors of a semi-infinite body with an elliptical or a semi-elliptical crack subject to tension.

Delay in fatigue crack growth

Abstract: The importance of delay, or retardation in the rate of fatigue crack growth, produced by load interactions in variable amplitude loading on the accurate prediction of fatigue lives of engineering structures has been well recognized for some time. Heretofore, only a few simple loading combinations or spectra have been examined systematically. In this investigation the effects of a broad range of loading variables on delay in fatigue crack growth at room temperature are examined for a mill annealed Ti-6Al-4V alloy. The results are used to estimate crack growth behavior under programmed loads.

A computer simulation study of Hertzian cone crack growth

Abstract: A general method for simulating on a computer the growth of the cone-shaped fracture that forms under Hertzian contact loading is outlined The program involves an incrementing procedure in which both contact circle and cone crack are grown in piecewise manner, according to suitable rate equations The contact circle expands at a rate determined by the mode of indenter loading, and thereby sets up a time-varying stress field Appropriate fracture-mechanics criteria are then invoked to calculate the response of the growing crack to the contact stresses Effects of loading mode, specimen environment and temperature, size and location of the initial flaw from which the cone crack nucleates, are investigated systematically The computer predictions compare favourably with available experimental data The results are discussed in the light of previous theoretical treatments of the Hertzian fracture problem, and some new features in the crack-growth characteristics are pointed out Calculations are made specifically for normal contact loading on glass, but ready extension of the program to other loading situations and materials in envisaged

Antiplane shear crack terminating at and going through a bimaterial interface

Abstract: The antiplane shear problem of two bonded elastic half planes containing a crack perpendicular to the interface is considered. The cases of a semi-infinite crack terminating at the interface, a finite crack away from and terminating at the interface, two cracks one on each side of the interface, and a finite crack crossing the interface are separately investigated. The nature of the stress singularity for the crack terminating at and going through the interface is studied, and it is shown that at the irregular point on the interface, for the former the power of singularity is not -1/2 and for the latter the stresses are bounded. For a material pair of aluminum-epoxy some numerical results giving the stress intensity factors, the density functions, and the crack opening displacements are presented.

The modified Dugdale-Barenblatt model adapted to various fracture configurations in metals

Abstract: The Dugdale-Barenblatt model is extended to encompass the influence of strain hardening on the plastic enclaves developed at the tips of a crack in a plate subjected to tension at infinity. While in the DB-model the distribution of internal stresses along the plastic zone in the extension of the crack length was taken constant and equal to the yield stress, in the modified version of this model this distribution is taken variable with a minimum value the yield stress (σ0) and a maximum value (σmax depending on the loading step and the amount of strain-hardening of the material. Six different configurations of stress distribution in the plastic enclaves were considered with various values of the ratio σmax/σ0. For each stress configuration and for various loading steps the shape of the respective caustic corresponding to the singularity created at the plastic zone near the crack tip was computed by modifying appropriately Dugdale's theory to each of the six configurations in the plastic enclave. The caustic was formed by reflections of a parallel coherent light beam at the vicinity of the crack-tip.

An elastic-plastic finite element analysis of crack tip fields under biaxial loading conditions

Abstract: A square plate containing a central crack and subjected to biaxial stresses has been studied by a finite element analysis. An elastic analysis shows that the crack opening displacement and stress of separation ahead of the crack tip are not affected by the mode of biaxial loading and therefore the stress intensity factor adequately describes the crack tip states in an elastic continuum.

Fracture toughness tests of a rigid polyurethane foam

Abstract: The results of some exploratory tests for determining the fracture toughness of a rigid polyurethane foam are presented. The specimen geometries used included centre- and double-edge-cracked plates, the single-edge-cracked tensile specimen and the double cantilever beam specimen. The validity of applying the concepts of linear elastic fracture mechanics to the fracture of this foam is discussed. Some unique features of the fracture of foam are discussed.

Interaction of Cracks with Rigid Inclusions in Longitudinal Shear Deformation

Abstract: The interaction of a crack with rigid circular cylindrical inclusions is considered for the case of longitudinal shear deformation General representations of the solutions for a radial crack near a single and midway between two inclusions are given The particular case of uniform shearing stress applied at infinity is discussed in detail

An application of finite element techniques to post-yield analysis of proposed standard three-point bend fracture test pieces

Abstract: Making use of an Elastic-Plastic Finite Element Program, two recommended fracture testing geometries are analysed Special attention is given to the singular environment at a crack tip, characterised by Rice's contour integral, and its relationship with the COD concept suggested by Wells

Mixed mode stress field effect in adhesive fracture

Abstract: Numerical or analytical analyses were performed on seven different test specimens including blister test, 90-degree peel test, torsion test, and various cone tests. These specimens are in general subjected to complex stress fields having various amounts of Mode I, Mode II, and Mode III loads. The specimens were then constructed using polymethylmethacrylate (PMMA) for the adherends and a transparent polyurethane elastomer (Solithane® 113)1 for the adhesive. This combination permitted direct observation of the bondline as load was applied. Although initial debonds as well as bond end termination singularities were present in all specimens, in some cases the debond did not initiate at the singularity points as would normally have been expected. An explanation for this behavior is presented, as well as a comparison of loading mode effect on those specimens for which the debond did propagate from a bond terminus singular point. In addition, the effect of adhesive compressibility, adhesive thickness, and bondline geometric curvature are discussed for many of the test specimens.

Some calculations of the energy-release rate G for cracks in micropolar and couple-stress elastic media

Abstract: A few years ago Sternberg and Muki [I] considered the effect of couple stresses on the stress concentration at the tip of a crack. They treated the problem of a finite crack in an infinite medium under conditions of plane strain with a uniform tension acting at infinity. The main conclusions were that at the crack tip the stress and couple stress fields had singularities of the same order, the order of the stress singularities being the same as those of the classical elastic problem. It was found that the limit of the stress intensity factor as 1 (the couple stress parameter) tended to zero was different to the usual elastic result (1 identically equal to zero). However, their approach which involved the numerical solution of integral equations did not give a precise evaluation of the coefficients involved in the stress and couple stress intensity factors. The couple stress theory has been criticised by Eringen [2] who replaces it by the micropolar theory of elasticity (see [2] for a review). In this note we consider the problem of a semi-infinite crack in a strip using both theories. These solutions which are accomplished by the use of a path independent integral demonstrate that G, the energy release rate, does tend to the elastic result as 1 + 0 even though the stress intensity factor may not.

Fatigue crack propagation and cyclic deformation at a crack tip

Abstract: The fatigue crack propagation relation da/dN = f(R) Delta K squared can be derived with three assumptions: small-scale yielding, material homogeneity, and that crack tip stresses and strains are not strongly affected by plate thickness. The function f(R) is a constant at a given stress ratio, R. The effects of plate thickness and stress ratio on crack tip deformation and fatigue crack growth in 2024-T351 aluminum alloy were studied. High Delta K level in a thin specimen causes crack tip necking. Necking is more pronounced at high stress ratio. Necking causes high maximum strain near a crack tip and fast crack growth rate.

The fracture of spherical shells under pressure and circular tubes with angled cracks in torsion

Abstract: Experiments are reported on spherical PMMA domes with thickness variations. The results were correlated with the parameter a2/Rt where I is a mean thickness based on equivalent energy. The curvature effect could be adequately represented by a published formulae. Crack angles and failure stresses for tubes broken in torsion were compared with the maximum hoop stress and minimum strain energy theories of Sih.

On the effect of orthotropy in a cracked cylindrical shell

Abstract: A pressurized cylindrical shell containing a longitudinal crack is considered. The shear modulus of the sheet may be evaluated from the measured Young's moduli and the Poisson's ratios rather than being an independent material constant. Two examples, one for a mildly orthotropic (titanium) and the other for a strongly orthotropic (graphite) material approximately satisfying the condition of special orthotropy are given. The results show that the stress intensity factors are rather strongly dependent on the degree of orthotropy.

Thermal fracture of bonded dissimilar media containing a penny-shaped crack

Abstract: The problem of uniform heat flow disturbed by an insulated penny-shaped crack along the common plane between two semi-infinite elastic media with different thermo-mechanical properties is formulated in terms of two potential functions in a half-space which in turn is reduced to a plane problem solvable by Muskhelishvili's method in complex function theory. Explicit expressions for the stress-intensity factors and the local stress field are derived and used in conjunction with Griffith's energy criterion to obtain the critical temperature gradient which motivates and produces initial crack extension along the bonding surface. The stress analysis involved is also applicable to a penny-shaped crack between two dissimilar solids under shear loadings.

Some applications of elastic-plastic analysis to fracture mechanics

Abstract: The purpose of analysis in fracture mechanics is to determine characterising parameters which reflect the influence of loading and geometry on the crack tip environment of flawed bodies. Here practical methods are developed which permit the determination of such parameters in general situations. Extensive use of finite element methods has been made to provide relevant field values which are then manipulated to determine the required parameters. However the choice of method to determine field values is arbitrary and is dictated by the ease with which such field values may be found. It is in the manipulation of these values that the fracture mechanics philosophy is introduced. Contributions are made in three areas. First economic methods for the determination of the linear fracture mechanics parameter in general stiuations are developed which are of direct relevance to design procedures. Detailed discussion of the Dugdale model of fracture behaviour is then given and a general method for determining Dugdale model solutions is provided. This method is used to provide solutions for standard specimen geometries and it is suggested that such solutions will enable a rational evaluation of the general applicability of the model. However the method is such that, should sufficient confidence in the model be established, design calculations on its premises may be performed. Finally, it is demonstrated that materials which allow extensive plastic flow at a flaw tip prior to fracture may be analysed using the basic ideas of fracture mechanics. It is shown that a line integral provides a flaw tip environment parameter for materials deforming according to the physically appropriate Prandtl-Reuss laws of plasticity. It is hoped that these results will indicate a rational approach to correlating fracture behaviour in such situations.