Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface

In this work, dynamic crack growth along a ductile-brittle interface under anti-plane strain conditions is studied. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behavior. Firstly, the asymptotic near-tip stress and velocity fields are derived. These fields are assumed to be variable-separable with a power singularity in the radial coordinate centered at the crack tip. The effects of crack speed, strain hardening of the ductile phase and mismatch in elastic moduli of the two phases on the singularity exponent and the angular functions are studied. Secondly, full-field finite element analyses of the problem under small-scale yielding conditions are performed. The validity of the asymptotic fields and their range of dominance are determined by comparing them with the results of the full-field finite element analyses. Finally, theoretical predictions are made of the variations of the dynamic fracture toughness with crack velocity. The influence of the bi-material parameters on the above variation is investigated.     

A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress

This is the second half of a two-part finite element investigation of quasi-static and dynamic crack growth in hardening elastic-plastic solids under mode I plane stress, steady state, and small-scale yielding conditions. The hardening materials are assumed to obey the von Mises yield criterion and the associated flow rule, and are characterized by a Ramberg-Osgood type power-law effective stress-strain curve. The asymptotic feature of the crack-tip stress and deformation fields, and the influence of hardening and crack propagation speed on these fields as well as on the size and shape of the crack-tip active plastic zone, are addressed in detail. The results of this study strongly suggest the existence of stress and strain singularities of the type [ln(R _o/r )]^s (s>0) at r=0, where r is the distance to the crack tip and R ₀ is a length scaling parameter, which is consistent with the predictions of asymptotic analyses of variable-separable type by Gao et al. [1–4]. Difficulties in estimating the values of R ₀ and s by fitting the results of the present full-field study to the type of singularities shown above are analyzed, and quantititive differences between the results of this study and those of the asymptotic analyses are discussed. As expected, findings presented here share many similarities with those reported in the first part of this study [5] for crack growth in linear hardening solids. A prominent common feature of crack growth in these two types of hardening materials is that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack surface, and kinks in the angular variations of the stress and velocity fields begin to develop near where elastic unloading is taking place.     

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid are investigated under plane strain and plane stress conditions. In each case, by expanding the plastic fields and the governing equations in the coordinate y, the problem is reduced to solving a system of nonlinear ordinary differential equations which is similar to that of mode III derived by Achenbach and Z.L.Li. An approximate solution for small values of x is obtained and matched with the elastic field of a blunt crack at the elastic-plastic boundary. The crack growth criterion of critical strain is employed to determine the value of K _II of the far-field that would be required for a steadily growing crack.     

Simulation of dynamic crack growth using the generalized interpolation material point (GIMP) method

Dynamic crack growth is simulated by implementing a cohesive zone model in the generalized interpolation material point (GIMP) method. Multiple velocity fields are used in GIMP to enable handling of discrete discontinuity on either side of the interface. Multilevel refinement is adopted in the region around the crack-tip to resolve higher strain gradients. Numerical simulations of crack growth in a homogeneous elastic solid under mode-II plane strain conditions are conducted with the crack propagating along a weak interface. A parametric study is conducted with respect to varying impact speeds ranging from 5 m/s to 60 m/s and cohesive strengths from 4 to 35 MPa. Numerical results are compared qualitatively with the dynamic fracture experiments of Rosakis et al. [(1999) Science 284:1337–1340]. The simulations are capable of handling crack growth with crack-tip velocities in both sub-Rayleigh and intersonic regimes. Crack initiation and propagation are the natural outcome of the simulations incorporating the cohesive zone model. For various impact speeds, the sustained crack-tip velocity falls either in the sub-Rayleigh regime or in the region between (c _S is the shear wave speed) and c _D (c _D is the dilatational wave speed) of the bulk material. The Burridge–Andrews mechanism for transition of the crack-tip velocity from sub-Rayleigh to intersonic speed of the bulk material is observed for impact speeds ranging from 9.5 to 60 m/s (for normal and shear cohesive strengths of 24 MPa). Within the intersonic regime, sustained crack-tip velocities between 1.66 c _S (or 0.82 c _D ) and 1.94 c _S (or 0.95 c _D ) were obtained. For the cases simulated in this work, within the stable intersonic regime, the lowest intersonic crack-tip velocity obtained was 1.66 c _S (or 0.82 c _D ).     

Plane Stress Crack Tip Field Around a Rapidly Growing Ductile/Rigid Interface Crack

Plane stress dynamic crack growth along a ductile/rigid interface is investigated. The ductile material is taken to be ideally plastic and obey the J₂ flow theory of plasticity. Under steady-state conditions, the asymptotic structure of the crack-tip stress, velocity and strain fields has been obtained. The study reveals that two types of crack-tip sectors exist, namely uniform and nonuniform plastic sectors and that the stress, strain and velocity fields are bounded (nonsingular) in all sectors. In a uniform sector, the rectangular Cartesian components of the stress, strain and velocity fields are constant, and there is no plastic strain accumulation. In a nonuniform sector, the stress, strain and velocity components at a point depend on the angular position of the point in the crack-tip polar coordinate system and are governed by a system of simultaneous ordinary differential equations. This is a sector plastic strains can accumulate. A general crack-tip sector assembly is obtained for a practical range of crack growth speeds. Several nontrivial families of admissible solutions of the crack-tip fields based on this general assembly of uniform and nonuniform crack-tip sectors are presented and discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

A local modulus analysis of rapid crack propagation in polymers

This work is concerned with the analysis of rapid crack propagation (RCP) in Polymethylmethacrylate (PMMA), Polycarbonate (PC) and two-layer PMMA/PC systems. Remarkably constant crack speeds were observed, and higher crack speeds corresponded to the higher preloads. Uniform fracture surfaces were associated with these constant speed RCPs. An indirect method was used to characterise dynamic fracture properties of the materials. The method relies on the recorded crack length histories and boundary conditions which are incorporated in a dynamic Finite Element (FE) code to generate the crack resistance (G _ID). The numerical simulation of the constant speed RCPs generated highly scattered G _ID data. Very large variations of the computed G _ID with the crack length did not correspond to fracture surface appearances. Geometry dependent and multivalued crack resistance results with respect to the crack speed cast doubt on the uniqueness of G _ID. In this work, attempts were made to overcome these difficulties by exploring the concept that the anomalies arise from large local strains around the rapidly moving crack tip, resulting in the crack seeing a low local modulus. It is demonstrated that the critical source of error on the analysis of RCP, is the improper linear elastic representation of the material behaviour around the propagating crack tip. Since the parameters describing the behaviour of the materials near the propagating crack tip were unknown, local non-linear effects were approximated by a local low modulus strip along the prospective crack path. The choice of the local modulus was justified by measurements of the strain histories along the crack path during RCP. The local strip low modulus model generated a larger amount of the kinetic energy in the sample and the crack resistance was reduced compared to results from the single constant modulus approach. Most importantly, G _ID data were nearly independent of the crack length, crack speed and the specimen size. This local modulus concept was also successfully applied to the analysis of RCP in the duplex specimen configuration.     

Numerical simulations of dynamic crack growth along an interface

Dynamic crack growth is analyzed numerically for a plane strain bimaterial block with an initial central crack. The material on each side of the bond line is characterized by an isotropic hyperelastic constitutive relation. A cohesive surface constitutive relation is also specified that relates the tractions and displacement jumps across the bond line and that allows for the creation of new free surface. The resistance to crack initiation and the crack speed history are predicted without invoking any ad hoc failure criterion. Full finite strain transient analyses are carried out, with two types of loading considered; tensile loading on one side of the specimen and crack face loading. The crack speed history and the evolution of the crack tip stress state are investigated for parameters characterizing a PMMA/Al bimaterial. Additionally, the separate effects of elastic modulus mismatch and elastic wave speed mismatch on interface crack growth are explored for various PMMA-artificial material combinations. The mode mixity of the near tip fields is found to increase with increasing crack speed and in some cases large scale contact occurs in the vicinity of the crack tip. Crack speeds that exceed the smaller of the two Rayleigh wave speeds are also found.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

Dynamic fields generated by rapid crack growth

Stresses and strains near a rapidly propagating crack tip are affected by the mass density of the material. This paper starts with a brief summary of analytical results for near-tip dynamic fields as predicted by linear elastic fracture mechanics. Next, exact expressions are derived for dynamic crack-line strains, for mode-III crack propagation in a nonlinear elastic material and in an elastic perfectly-plastic material. These expressions are valid on the crack line from the moving crack tip to the moving boundary with the region of linearly elastic deformation. For steady-state crack growth, a critical strain criterion is used to compute the relation between external load and crack tip speed. The required external load increases with crack-tip speed.     

Characterization of the nonlinear fracture resistance parameters for an aviation GTE turbine disc

The effects of crack growth rate model formulation, based on the elastic‐plastic and undamaged/damaged creep crack tip fields on the behaviour of low‐cycle fatigue and creep fracture resistance parameter behaviour, are represented by numerical calculations. The crack growth rate models include the fracture process zone size and damage parameters. An aviation gas turbine engine (GTE) rotating turbine disc is the focus of this innovative application of basic analytical and numerical solutions. For the GTE turbine disc, the constraint parameters, local fracture process zone sizes, and nonlinear plastic (K_p) and creep (K_cr) stress intensity factors are calculated by finite element analysis to characterize the fracture resistance along the semielliptical crack front as a function of the flaw aspect ratio, operation temperature, and disc rotation speed. Predictions of the creep‐fatigue crack growth rate and residual lifetime are given for different combinations of operation loading conditions and damage of the GTE turbine disc.     

Intersonic shear crack growth along weak planes

Classical dynamic fracture theories predict the Rayleigh surface wave speed (c _R ) to be the limiting speed of propagation for mode-I cracks in constitutively homogeneous, isotropic, linear elastic materials subjected to remote loading. For mode-II cracks, propagating along prescribed straight line paths, the same theories, while excluding the possibility of crack growth in the speed regime between c _R and the shear wave speed, c _s , do not exclude intersonic (c _s <υ<c _l ) crack tip speeds. In the present study, we provide the first experimental evidence of intersonic crack growth in such constitutively homogeneous and isotropic solids, ever recorded in a laboratory setting. Intersonic shear dominated crack growth, featuring shear shock waves, was observed along weak planes in a brittle polyester resin under far-field asymmetric loading. The shear cracks initially propagate at speeds just above c _s and subsequently accelerate rapidly to the longitudinal wave speed (c _l ) of the solid. At longer times, when steady state conditions are attained, they propagate at speeds slightly higher than √2–c _s . The experimental results compare well with existing asymptotic theories of intersonic crack growth, and the significance of the preferred speed of √2–c _s is discussed. Received: 13 September 1999 / Reviewed and occerted: 19 November 1999     

Analysis of dynamic interaction between an inclusion and a nearby moving crack by BEM

In this paper, the dynamic interaction between an inclusion and a nearby moving crack embedded in an elastic medium is studied by the boundary element method (BEM). To deal with this problem, the multi-region technique and two kinds of time-domain boundary integral equations (BIEs) are introduced. The system is divided into two parts along the interface between the inclusion and the matrix medium. Each part is linear, elastic, homogeneous and isotropic. The non-hypersingular traction boundary integral equation is applied on the crack surfaces; while the traditional displacement boundary integral equation is used on the interface and external boundaries. In the numerical solution procedure, square root shape functions are adopted as to describe the proper asymptotic behavior in the vicinity of the crack-tips. The crack growth is modeled by adding new elements of constant length to the moving crack tip, which is controlled by the fracture criterion based on the maximum circumferential stress. In each time step, the direction and the speed of the crack advance are evaluated. The numerical results of the crack growth path, speed, dynamic stress intensity factors (DSIFs) and dynamic interface tractions for various material combinations and geometries are presented. The effect of the inclusion on the moving crack is discussed.     

A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress

The asymptotic structures of crack-tip stress and deformation fields are investigated numerically for quasi-static and dynamic crack growth in isotropic linear hardening elastic-plastic solids under mode I, plane stress, and small-scale yielding conditions. An Eulerian type finite element scheme is employed. The materials are assumed to obey the von Mises yield criterion and the associated flow rule. The ratio of the crack-tip plastic zone size to that of the element nearest to the crack tip is of the order of 1.6 × 10⁴. The results of this study strongly suggest the existence of crack-tip stress and strain singularities of the type r ^s (s < 0) at r=0, where r is the distance to the crack tip, which confirms the asymptotic solutions of variable-separable type by Amazigo and Hutchinson [1] and Ponte Castañeda [2] for quasi-static crack growth, and by Achenbach, Kanninen and Popelar [3] for dynamic crack propagation. Both the values of the parameter s and the angular stress and velocity field variations from the present full-field finite element analysis agree very well with those from the analytical solutions. It is found that the dominance zone of the r ^s-singularity is quite large compared to the size of the crack-tip active plastic zone. The effects of hardening and inertia on the crack-tip fields as well as on the shape and size of the crack-tip active plastic zone are also studied in detail. It is discovered that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack flank, and kinks in stress and velocity angular variations begin to develop. This dynamic phenomenon observable only for rapid crack growth and for low hardening materials may explain the numerical difficulties, in obtaining solutions for such cases, encountered by Achenbach et al. who, in their asymptotic analysis, neglected the existence of reverse yielding zones along the crack surfaces.     

Crack growth by grain boundary cavitation in the transient and extensive creep regimes

Crack growth due to cavity growth and coalescence along grain boundaries is analyzed under transient and extensive creep conditions in a compact tension specimen. Account is taken of the finite geometry changes accompanying crack tip blunting. The material is characterized as an elastic-power law creeping solid with an additional contribution to the creep rate arising from a given density of cavitating grain boundary facets. All voids are assumed present from the outset and distributed on a given density of cavitating grain boundary facets. The evolution of the stress fields with crack growth under three load histories is described in some detail for a relatively ductile material. The full-field plane strain finite element calculations show the competing effects of stress relaxation due to constrained creep, diffusion and crack tip blunting, and of stress increase due to the instantaneous elastic response to crack growth. At very high crack growth rates the Hui-Riedel fields dominate the crack tip region. However, the high growth rates are not sustained for any length of time in the compact tension geometry analyzed. The region of dominance of the Hui-Riedel field shrinks rapidly so that the near-tip fields are controlled by the HRR-type field shortly after the onset of crack growth. Crack growth rates under various conditions of loading and spanning the range of times from small scale creep to extensive creep are obtained. We show that there is a strong similarity between crack growth history and the behaviour of the C(t) and C _t parameters, so that crack growth rates correlate rather well with C(t) and C _t .A relatively brittle material is also considered that has a very different near-tip stress field and crack growth history.Visiting Professor, Brown University, August 1988 through December 1989.     

Dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

A complete asymptotic solution is given for the fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic-perfectly-plastic solid.For Mode I crack growth in the plane strain condition, the following noteworthy results are revealed: (1) The entire crack tip in steady crack growth is surrounded by a plastic region, and no elastic unloading is predicted by the complete dynamic asymptotic solution. Thus, the elastic unloading region predicted by the result of neglecting the important influence of the inertia terms in the equations of motion. (2) Unlike the quasi-static solution, the dynamic solution yields strain fields with a logarithmic singularity everywhere near the crack tip. (3) The stress field varies throughout the entire crack tip neighborhood, but does display behavior which can be approximated by a constant field followed by an essentially centered-fan field and then by another constant field, especially for small crack growth speeds. Indeed, the stress field reduces to that for the stationary crack, as the crack tip velocity - measured by the Mach number, M - reduces to zero; the strain field, however, does not reduce to that for the static solution, as M vanishes. (4) There are two shock fronts emanating from the crack tip across which certain stress and strain components undergo jump discontinuities. The location of the shock fronts and the magnitude of the jumps depend on the crack growth speed. The stress jump vanishes while the strain jump becomes unbounded, as the crack tip speed goes to zero.Finally, the Mode III steady-state crack growth is reviewed and, on the basis of Mode I and Mode III results, it is concluded that ductile fracture criteria for nonstationary cracks must be based on solutions which include the inertia effects, and that for this purpose, quasi-static solutions may be inadequate. Then, a possible ductile fracture criterion is suggested and discussed.One interesting feature of the complete dynamic asymptotic solution is that, unlike the quasi-static solution, it yields the same strain singularity for all three fracture modes.     

Measuring overload effects during fatigue crack growth in bainitic steel by synchrotron X-ray diffraction

In this work we present the results of in situ synchrotron X-ray diffraction measurements of fatigue crack-tip strain fields following a 100% overload (OL) under plane strain conditions. The study is made on a bainitic steel with a high toughness and fine microstructure. This allowed a very high (60 μm) spatial resolution to be achieved so that fine-scale changes occurring around the crack-tip were captured along the crack plane at the mid-thickness of the specimen. We have followed the crack as it grew through the plastic/residually stressed zone associated with the OL crack location. We observed two effects; one when the enhanced plastic zone is ahead of the crack and one after it has been passed. Regarding the former it was found that the compressive stress at the crack-tip initially falls sharply, presumably due to the increased plastic stretch caused by the OL. This is associated with a concomitant fall in peak tensile stress at K_max, the elastic excursion between K_min and K_max remaining essentially unchanged from before OL. Subsequently discontinuous closure as seen previously for plane stress caused by crack face contact at the OL location limits the elastic strain range experienced by the crack tip and thereby retards crack growth.     

Continuous near-tip fields for a dynamic crack propagating in a power-law elastic-plastic material

By use of the J ₂ flow theory and the rectangular components of field quantities, the near-tip asymptotic fields are studied for a dynamic mode-I crack propagating in an incompressible power-law elastic-plastic material under the plan strain conditions. Through assuming that the stress and strain components near a dynamic crack tip are of the same singularity, the present paper constructs with success the fully continuous dominant stress and strain fields. The angular variations of these fields are identical with those corresponding to the dynamic crack propagation in an elastic-perfectly plastic material (Leighton et al., 1987). The dynamic asymptotic field does not reduce to the quasi-static asymptotic field in the limit as the crack speed goes to zero. This revised version was published online in July 2006 with corrections to the Cover Date.     

Variation of elastic T-stresses along slightly wavy 3D crack fronts

The non-singular terms in the series expansion of the elastic crack-tip stress field, commonly referred to as the elastic T-stresses, play an important role in fracture mechanics in areas such as the stability of a crack path and the two-parameter characterization of elastic-plastic crack-tip deformation. In this paper, a first order perturbation analysis is performed to study some basic properties of the T-stress variation along a slightly wavy 3D crack front. The analysis employs important properties of Bueckner-Rice 3D weight function fields. Using the Boussinesq-Papkovitch potential representation for the mode I weight function field, it is shown that, for coplanar cracks in an infinite isotropic and homogeneous linear elastic body, the mean T-stress along an arbitrary crack front is independent of the shape and size of the crack. Further, a universal relation is discovered between the mean T-stress and the stress field at the same crack front location under the same loading but in the absence of a crack. First-order-accurate solutions are given for the T-stress variation along a slightly wavy crack front with nearly circular or straight confifurations. Specifically, cosine wave functions are adopted to describe smooth polygonal and slightly undulating planar crack shapes. The results indicate that T ₁₁, the 2D T-stress component acting normal to the crack front, increases with the curvature of the crack front as it bows out but T ₃₃, acting parallel to the crack front, decreases with the crack front curvature.     

High speed double torsion tests on tough polymers. I: Linear elastic steady state and dynamic analysis

The High Speed Double Torsion test has been used to generate steady rapid crack propagation in tough pipe-grade polyethylenes, at speeds of up to 350 ms^-1. Dynamic plane-strain fracture resistance GD data, computed from the measured displacement and crack length using a linear elastic steady-state analysis, were systematically scattered. The computed fracture loads exceeded measured values by up to 50 percent. Two possible reasons for these discrepancies are the neglect of unsteady deformation, and the use of small-strain dynamic elastic modulus data to represent the material. Since the torsional wave speed calculated from this modulus provided a good estimate for the limiting crack speed, this paper pursues the first possibility. High speed photography was used to study the deformation field, which proved to be less steady than assumed. The observations were used to support development of a fully dynamic, linear elastic torsion-beam-on-elastic-foundation model for computing the transient deformation field from boundary data. The foundation stiffness, computed by matching predicted and observed deformations in the crack tip vicinity, was consistent with that estimated from earlier quasi-static tests. As judged by the continuity of the computed energy release rate G ^dyn (and hence, equivalently, of GD), numerical integration along characteristics is more suitable than a conventional explicit finite difference scheme for solving this one-dimensional problem. The GD solution for intermediate crack lengths is also insensitive to assumed initial conditions, which are therefore chosen to minimise the settling time. The curved double-torsion crack front shape, predicted using an earlier quasi-static criterion, agrees closely with that observed from dynamic arrest lines on the fracture surface, but simply assuming the crack front to be straight and normal to the specimen plane has little effect on computed GD data. The dynamic model, used to compute GD as a function of crack speed for several pipe-grade polyethylenes, reduces but does not eliminate scatter; nevertheless, in provides a more reliable and versatile tool for reconsidering the question of material representation in Part II of the paper.     

On the influence of a bimaterial interface on dynamic stress intensity factors

Summary The method of dynamic Green's function and the integral transforms are applied to investigate the elasto-dynamic stress intensity factor of a crack straddling an interface of a bimaterial composite. The crack which extends to infinity on one side is assumed to extend an arbitrary distancea on the other side of the interface. Anti-plane line loads are suddenly applied at timet=0 on either side of the crack surface at arbitrary distancesl ₁ andl ₂ from the interface. The effect of the interface on the dynamic stress field near the crack tip is studied. It is found that the transmitted wave through the interface and reflected wave from the interface serve to increase or decrease the stress field in the vicinity of the crack tip depending on the elastic properties of the two materials.