Full-field Finite Element Research Articles

Microstructurally-based modelling of intergranular creep fracture using grain elements

Abstract A new numerical method is proposed to simulate intergranular creep fracture in large polycrystalline aggregates. The method utilizes so-called grain elements to represent the polycrystal. These grain elements take care of the average elastic and creep deformation of individual grains. Grain boundary processes, like cavitation and sliding, are accounted for by grain boundary elements connecting the grains. Results are compared with full-field finite element calculations. The method is demonstrated to capture the essential features of creep fracture, like creep constrained cavitation and the interlinkage of microcracks. Also the performance in polycrystals with random variations in microstructure, in terms of grain shape, is shown to be reasonably well. For the size of the unit-cell considered, a factor of around 600 is gained in computer time as compared with the full-field calculations.

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.

Effect of reversible phase transformations on crack growth

Abstract In this work, the influence of partial or full reversibility of a stress-induced phase transformation is investigated, both in terms of the size and shape of the transformation zone around the crack tip, and the toughness development during crack growth. The constitutive equations adopted in this study are due to Sun et al. (1991, J. Mech. Phys. Solids 39 , 507) and account for transformation-induced dilatant and shear strains. These constitutive equations were designed to model the phase transformations occurring in zirconia ceramics or in shape memory alloys (SMA). The results are obtained by a full field finite element analysis of crack propagation under small scale transformation conditions. The phase transformations considered here tend to give rise to a substantial toughening of the material if the transformation were irreversible. It is shown here that in some cases, the reversibility of the transformation can significantly reduce the toughness increase. A parameter study establishes the sensitivity of this deterioration to characteristics of the constitutive response.

Finite deformation effects in homogeneous and interfacial fracture

This paper summarizes recent results of asymptotic, numerical and experimental investigations of some nonlinear effects on the mechanics of fracture in homogeneous and bimaterial sheets of a particular class of hyperelastic incompressible materials. The problem is analysed within the framework of the finite strain theory of plane stress. Material induced nonlinearities are included through the use of the generalized neo-Hookean model which is characterized by three parameters which determine the small strain, “yielding” and “hardening” responses of the component(s). The structure of the near-tip stress and deformation fields is described and compared to a full-field finite element investigation. The consequences of the local results on the propagation behavior of a crack under general in-plane loading are outlined in the special case of a homogeneous sheet. The analytical results are corroborated by experimental observations obtained on natural rubber sheets.

Butt joint tensile strength: interface corner stress intensity factor prediction

The asymptotic form of the interface corner stress field in a butt joint is discussed, and a failure analysis based on the stress intensity factor defining the magnitude of this asymptotic stress field is validated. A stress singularity of type Krδ(δ < 0) exists at an interface corner in a butt joint (i.e. where an interface intersects a stress-free edge). A simple relation defines the stress intensity factor K for an idealized butt joint composed of a thin elastic adhesive layer bonded between rigid adherends and subjected to transverse tension and uniform adhesive shrinkage. This stress intensity factor, referred to here as the free-edge stress intensity factor Kf, is applicable to both plane strain and axisymmetric geometries. The way that uniform adhesive shrinkage (thermal contraction) during cure alters interface corner stress fields is also discussed. When adhesive shrinkage is present, both constant and singular terms must be included in the asymptotic solution to attain good agreement with full field finite element results over a reasonably large interface corner region. Experiments have been carried out to investigate the applicability of a Kf-based failure criterion to butt joints. Butt joints were fabricated by bonding two stainless steel rods together with an epoxy adhesive (Epon 828/T-403). The measured joint strength increased by a factor of 2 as the bond thickness was reduced from 2.0 to 0.25 mm. The observed bond thickness effect is accurately predicted when failure is presumed to occur at a critical Kf value. This fracture criterion suggests that the butt joint tensile strength varies roughly as the reciprocal of the cube root of bond thickness when the adhesive's Poisson's ratio is between 0.3 and 0.4, residual stress levels at the interface corner are negligible, the adherends are essentially rigid relative to the adhesive, and small-scale yielding conditions hold at the interface corner.

Indentation of a power law creeping solid

The aim of this paper is to establish a rigorous theoretical basis for interpreting the results of hardness tests on creeping specimens. We investigate the deformation of a creeping half-space with uniaxial stress-strain behaviour ⋵ = ⋵ 0 (σ/σ 0 ) m , which is indented by a rigid punch. Both axisymmetric and plane indenters are considered. The shape of the punch is described by a general expression which includes most indenter profiles of practical importance. Two methods are used to solve the problem. The main results are found using a transformation method suggested by R. Hill. It is shown that the creep indentation problem may be reduced to a form which is independent of the geometry of the punch, and depends only on the material properties through m . The reduced problem consists of a nonlinear elastic half-space, which is indented to a unit depth by a rigid flat punch of unit radius (in the axisymmetric case), or unit semi-width (in the plane case). Exact solutions are given for m = 1 and m = ∞. For m between these two limits, the reduced problem has been solved using the finite element method. The results enable the load on the indenter and the contact radius to be calculated in terms of the indentation depth and rate of penetration. The stress, strain and displacement fields in the half-space may also be deduced. The accuracy of the solution is demonstrated by comparing the results with full-field finite element calculations. The predictions of the theory are shown to be consistent with experimental observations of hardness tests on creeping materials reported in the literature.

Asymptotic interface corner solutions for butt tensile joints

In previous work, the intensity of the stress singularity of type rδ (δ < 0) found at the interface corner between a thin elastic adhesive layer and one of a pair of rigid adherends was fully determined for a butt tensile joint. This stress intensity factor, referred to here as the free-edge stress intensity factor Kf, can be applied to both plane strain and axisymmetric geometries. This study investigates the potential application of a Kf-based failure criterion to butt tensile joints. Detailed elastic-plastic finite element calculations for adhesive properties representative of a high strength epoxy indicate that when residual cure stress can be neglected (1) the region dominated by the interface corner singularity is reasonably large relative to adhesive layer thickness, (2) the plastic yield zone is contained within the singular field at nominal failure loads, and (3) the plastic zone size is characterized by kf and displays the expected load level and layer thickness dependence. The way uniform adhesive shrinkage (thermal contraction) during cure alters interface corner stress fields is also discussed. When adhesive shrinkage is present, both constant and singular terms must be included in the asymptotic solution to get good agreement with full field finite element results. In general, there is no unique relation between the size of the interface corner yield zone and Kf, although for a prescribed shrinkage strain, Kf does characterize the extent of plastic yielding. Calculated results suggest that the presence of residual stress can have a considerable effect on the relation between bond thickness and joint strength.

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 rs (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 Castaneda [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 rs-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.

Elastic-plastic analysis of combined mode I, II and III crack-tip fields under small-scale yielding conditions

Within the context of the small-strain approach, combined mode I, II and III near-tip fields of stationary cracks in power-law hardening materials are investigated. We use a finite element technique to obtain asymptotic angular stress solutions for combined mode I and II perturbed from mode III. These perturbation solutions with the same stress singularities as those of pure mode III are presented for different hardening materials. The perturbation results further suggest that the order of crack-tip stress singularities varies smoothly with changing mode mixity. We also employ full-field finite element compulations to study the small-scale yielding near-tip fields for several combinations of prescribed remote mode I. II and III elastic K fields. These solutions verify an interesting pattern which agrees with the previous solutions for combined mode I and III loading as well as those for combined mode II and III loading: well within the plastic zone, under near mode I mixed-mode loadings, the in-plane stresses are slightly more singular than r−t/(n + 1) while the out-of-plane shear stresses are slightly less singular than r−t/(n + 1), where r is the radial distance to the lip and n is the strain hardening exponent of the material. To explain the complex behavior of the near-tip stresses, we introduce an effective in-plane shear stress and an effective out-of-plane shear stress to quantify the in-plane and out-of-plane plastic shear at different orientations in a consistent manner. The full-field solutions also corroborate the observation that the singularities of the in-plane slresses and the out-of-plane shear stresses vary smoothly with mode mixity.

Mixed mode crack-tip fields in monolithic ceramics

Asymptotic and full field finite element solutions are given for a semi-infinite planar crack in a monolithic ceramic subjected to remote mixed mode loading. The material is assumed to undergo damage in the form of elastic degradation as a result of stable microcracking. Microcracks are assumed to be preferentially oriented normal to the direction of maximum tension. An outcome of the analysis is that microcracks shield the crack tip less effectively under mode II than under mode I conditions. Other issues addressed concern the relation between the mixities of the applied loads and the near-tip fields, the path-independence of the J integral, the validity of deformation theories of damage under proportional loading, and the conditions for dominance of the singular near-tip fields

Three-Dimensional Stress Field Near the Crack Front of a Thin Elastic Plate

Based on very detailed full-field finite element analysis of the near tip region of a thin isotropic elastic plate, the three-dimensional stress state in the vicinity of a through-crack front is characterized. The computed stress field reveals strong three-dimensional effects within a radial distance of about one-half thickness from the crack-tip. Further away from the tip, through-thickess variation of field quantities decreases and, at the radial distance of approximately 1.5 times the thickness, in-plane stresses merge with the dominant two-dimensional plane stress solutions. These “two-dimensional-three-dimensional” transition distances are essentially independent of the material Poisson’s ratio, yet the amplitude of variation is greatly affected by its value. The influence of Poisson’s ratio is clearly illustrated by local J along the crack front, which shows much higher variation through the thickness for nearly incompressible solids. At points very close to the crack front, relative magnitudes of out-of-plane strain components become very small, and asymptotic plane strain conditions prevail locally. On the mid-plane of the plate, the crack tip field converges to that given by the local plane strain stress intensity factor solution within a radial distance from the tip of less than 0.5 percent of thickness. In addition, it is found that the field near the intersection of crack front and free surface may be characterized by the corner singularity of a quarter infinite crack in a half space. The size of this domain is inferred from the gradient of local stress intensity factor with respect to distance from the free surface, and it appears that the corner singularity region extends up to a spherical radius of about 3 percent of plate thickness away from the intersection. Also the amplitude of the corner singularity field is described by a corner stress intensity factor, and its magnitude is determined for thin plates of various Poisson’s ratios.

J-dominance under plane strain fully plastic conditions: the edge crack panel subject to combined tension and bending

Full field plane-strain solutions are obtained for a semi-infinite crack aligned perpendicularly to the free-edge of a semi-infinite half space where the ligament is subject to arbitrary combinations of tension and bending. The material behavior is described by the incompressible fully plastic pure-power-law deformation theory relation. The question of J-dominance of the crack-tip region is addressed by comparing the full field finite element solutions with the Hutchinson-Rice-Rosengren singularity fields. These comparisons show that the region of agreement is strongly dependent on the relative amount of bending to tension. In the vicinity of pure bending, the zone of J-dominance is a significant fraction of the ligament, and the size of the zone does not depend on hardening behavior. In the vicinity of pure stretch, the zone of J-dominance is substantially smaller and depends strongly on hardening characteristics. Size requirements which are necessary to ensure J characterization of the near-tip fields of cracked bodies subject to combined tension and bending are discussed.