Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In the present study, mode I crack subjected to cyclic loading has been investigated for plastically compressible hardening and hardening–softening–hardening solids using the crack tip blunting model where we assume that the crack tip blunts during the maximum load and re-sharpening of the crack tip takes place under minimum load. Plane strain and small scale yielding conditions have been assumed for analysis. The influence of cyclic stress intensity factor range (\(\Delta \hbox {K})\), load ratio (R), number of cycles (N), plastic compressibility (\({\upalpha })\) and material softening on near tip deformation, stress–strain fields were studied. The present numerical calculations show that the crack tip opening displacement (CTOD), convergence of the cyclic trajectories of CTOD to stable self-similar loops, plastic crack growth, plastic zone shape and size, contours of accumulated plastic strain and hydrostatic stress distribution near the crack tip depend significantly on \(\Delta \hbox {K}\), R, N, \({\upalpha }\) and material softening. For both hardening and hardening–softening–hardening materials, yielding occurs during both loading and unloading phases, and resharpening of the crack tip during the unloading phase of the loading cycle is very significant. The similarities are revealed between computed near tip stress–strain variables and the experimental trends of the fatigue crack growth rate. There was no crack closure during unloading for any of the load cycles considered in the present study.     

Prediction of fatigue crack growth under constant amplitude loading and a single overload based on elasto-plastic crack tip stresses and strains

It is generally accepted that the fatigue crack growth (FCG) depends mainly on the stress intensity factor range (ΔK) and the maximum stress intensity factor (K_max). The two parameters are usually combined into one expression called often as the driving force and many various driving forces have been proposed up to date. The driving force can be successful as long as the stress intensity factors are appropriately correlated with the actual elasto-plastic crack tip stress-strain field. However, the correlation between the stress intensity factors and the crack tip stress-strain field is often influenced by residual stresses induced in due course.A two-parameter (ΔK_tot, K_max,tot) driving force based on the elasto-plastic crack tip stress-strain history has been proposed. The applied stress intensity factors (ΔK_appl, K_max,appl) were modified to the total stress intensity factors (ΔK_tot, K_max,tot) in order to account for the effect of the local crack tip stresses and strains on fatigue crack growth. The FCG was predicted by simulating the stress-strain response in the material volume adjacent to the crack tip and estimating the accumulated fatigue damage. The fatigue crack growth was regarded as a process of successive crack re-initiations in the crack tip region. The model was developed to predict the effect of the mean and residual stresses induced by the cyclic loading. The effect of variable amplitude loadings on FCG can be also quantified on the basis of the proposed model. A two-parameter driving force in the form of: was derived based on the local stresses and strains at the crack tip and the Smith-Watson-Topper (SWT) fatigue damage parameter: D = σ_maxΔε/2. The effect of the internal (residual) stress induced by the reversed cyclic plasticity manifested itself in the change of the resultant (total) stress intensity factors controlling the fatigue crack growth.The model was verified using experimental fatigue crack growth data for aluminum alloy 7075-T6 obtained under constant amplitude loading and a single overload.     

Analysis of beam-type fracture specimens with crack-tip deformation

A novel bi-layer beam model is developed to account for local effects at the crack tip of a bimaterial interface by modeling a bi-layer composite beam as two separate shear deformable beams. The effect of interface stresses on the deformations of sub-layers, which is referred to as the elastic foundation effect in the literature, is considered in this model by introducing two interface compliance coefficients; thus a flexible joint condition at the crack tip is considered in contrast to the rigid joint condition used in the conventional bi-layer model. An elastic crack tip deformable model is presented, and the closed-form solutions of local deformation at the crack tip are then obtained. By applying this novel crack tip deformation model, the new terms due to the local deformations at the crack tip, which are missing in the conventional composite beam solutions of compliance and energy release rate (ERR) of beam-type fracture specimens, are recovered. Several commonly used beam-type fracture specimens are examined under the new light of the present model, and the improved solutions for ERR and mode mixity are thus obtained. A remarkable agreement achieved between the present and available solutions illustrates the validity of the present study. The significance of local deformation at the crack tip is demonstrated, and the improved solutions developed in this study provide highly accurate predictions of fracture properties which can actually substitute the full continuum elasticity analysis such as the finite element analysis. The new and improved formulas derived for several specimens provide better prediction of ERR and mode mixity of beam-type fracture experiments.*Author for correspondence (E-mail address: qiao@uakron.edu)     

Time-derivative equations for fatigue crack growth in metals

Predicting fatigue crack growth in metals remains a difficult task because available models are based on cycle-derivative equations, such as the Paris law, while service loads are often far from being cyclic. The main objective of this paper is therefore to propose a set of time-derivative equations for fatigue crack growth. The model is based on the thermodynamics of dissipative processes. For this purpose, three global state variables are introduced in order to characterize the state of the crackthe crack length a, the plastic blunting at crack tip and the intensity of crack opening C. Thermodynamics counterparts are introduced for each variable. Special attention is paid to the elastic energy stored inside the crack tip plastic zone, because, in practice, residual stresses at crack tip are known to considerably influence fatigue crack growth. The stored energy is included in the energy balance equation, and this leads to the appearance of a kinematics hardening term in the yield criterion for the cracked structure. No dissipation is associated with crack opening, but to crack growth and to crack tip blunting. Finally, the model consists in two laws: a crack propagation law, which is a relationship between d dt and da/dt and which observes the inequality stemmed from the second principle, and an elastic-plastic constitutive behaviour for the cracked structure, which provides d dt versus applied-load. The model was implemented and tested. It reproduces successfully the main features of fatigue crack growth as reported in the literature, such as the Paris law, the stress-ratio effect and the overload retardation effect.     

Surface effects on mode-I crack tip fields: A numerical study

Surface energy often significantly influences the deformation and failure behavior of materials and devices at the nanoscale. However, how it alters the local deformation around a crack tip remains unclear. In the present paper, we investigate the surface effects on the near-tip fields of a mode-I blunt crack (or notch). The theory of surface elasticity is incorporated into the finite element method. It is found that when the curvature radius of the crack root shrinks to nanometers, surface effects considerably affect the local stress distributions near the crack tip. We also calculate the J-integral, which is almost independent of surface effects except when the integral path approaches the crack tip. This demonstrates that surface effects are localized in a small zone around the crack tip, where the classical fracture mechanics solutions neglecting surface effects should be modified.     

Quasi-static extension of cohesive crack described by the energy partition technique

Nonlinear differential equation governing mode I fracture under small scale yielding condition has been derived on the basis of the energy partition concept. This technique is associated with a cohesive crack model. The nonlinear zone which precedes a propagating crack has been assumed to have a structured nature. In addition to this microstructural assumption, it has been postulated that the energy dissipated within the process zone (), embedded in a larger nonlinear zone (R), remains invariant to the extent of crack growth.Upper and lower bounds of the tearing modulus have been related to the material ductility via closed form expressions. It has been demonstrated that the energy screening, measured by the ratio of the true fracture energy () to the total work expended in the cohesive zone during the process of irreversible deformation, is a monotonic function of the crack growth increment, resembling a reciprocal of the apparent material resistance to cracking described by an R-curve.     

On elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under mixed mode loading

A generalized Irwin model is proposed to investigate elastic–plastic fracture behavior of a bi-layered composite plate with a sub-interface crack under combined tension and shear loading. The dependence of the stress intensity factors, the plastic zone size, the effective stress intensity factor and the crack tip opening displacement on the crack depth h, the Dundurs’ parameters and the phase angle θ is discussed in detail. Numerical results show that in most cases, if the crack is embedded in a stiffer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will increase. On the contrary, if the crack is embedded in a softer material, when the crack is close to the interface, the plastic zone size and the crack tip opening displacement will decrease.     

Effect of loading and geometry on the subsonic/intersonic transition of a bimaterial interface crack

An experimental investigation was conducted to study the nature of intersonic crack propagation along a bimaterial interface. A single edge notch/crack oriented along a polymer/metal interface was loaded predominantly in shear by impacting the specimen with a high velocity projectile fired from a gas gun. The stress field information around the propagating crack tip was recorded in real time by two different optical techniques--photoelasticity and coherent gradient sensing, in conjunction with high speed photography. Intersonic cracks on polymer/metal interfaces were found to propagate at speeds between the shear wave speed (c_s) and of the polymer. The nature of the crack tip fields during subsonic/intersonic transition and the conditions governing this transition were examined. Experimental observations showed the formation of a crack face contact zone as the interfacial crack speed exceeds the Rayleigh wave speed of the polymer. Subsequently, the contact zone was observed to expand in size, shrink and eventually collapse onto the intersonic crack tip. The recorded isochromatic fringe patterns showed multiple Mach wave formation associated with such a scenario. It is found that the nature of contact zone formation as well as its size and evolution differ substantially depending on the sign of the opening component of loading.     

Modeling of Bonding at an Interface Crack

A mechanical and mathematical model is suggested for an interface crack with bonding in its end zones. Normal and shear bond tractions occurring under the action of the external loads are searched for by solving a system of two singular integrodifferential equations. The stress intensity factors at the crack tip are calculated taking the compensating action of the bonds into account. Energetic characteristics of the interface crack (the deformation energy release rate and the rate of the energy absorption by the bonds) are analyzed. A sensitivity analysis is performed of the force and energetic characteristics of the interface crack to the end zone size, bond compliance and limit stretching. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stress wave emission and plastic work of notched specimens

The stress wave energy released from notched specimens of structural steel was measured in order to compare it with the recently proposedJ-integral which takes account of the effect of large plastic deformation around the crack tip in ductile materials. Very close agreement was observed between theJ-integral and the differential stress wave energy released. This suggests that the increment of the stress wave energy released is proportional to the decrement of the work done on the specimen during tensile testing under the plane stress condition. This result, combined with information obtained from linear elastic fracture mechanics, leads to a relationship between the differential stress wave energy released and the stress intensity factorK, [Δ(SWER)/Δa] ∝K ². It was also found that in the region before general yielding, the stress wave energy release was proportional to the development of plastic zone size. A larger portion of the accumulated stress wave energy released was generated after general yielding due to void formation and coalescence. The accumulated stress wave energy released at the catastrophic crack growth point reached virtually the same value for each specimen, independent of the initial crack length. This implies that void formation and coalescence are not influenced by the initial crack length, but by the geometry of the crack tip.     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

On contact zone models for an electrically limited permeable interface crack in a piezoelectric bimaterial

An electrically limited permeable crack with a frictionless contact zone at the right crack tip between two semi-infinite piezoelectric spaces under the action of a remote electromechanical loading is considered. Attention is focused on the influence induced from the permittivity of the medium inside the crack gap on the contact zone length and the fracture mechanical parameters. Assuming the electric displacement constant inside the open region of the crack, the problem is reduced to a combined Dirichlet-Riemann and Hilbert boundary value problems which have been solved exactly. Stress and electric displacement intensity factors as well as the crack tip energy release rate are found in a clear analytical form. Furthermore, transcendental equations for the determination of the real contact zone length have been obtained for a general case and for a small contact zone length in an especially simple form. The dependencies of the mentioned values on the intensities of the electromechanical loading are presented in tables and associated diagrams.     

Analysis of fatigue crack tip plasticity in Fe-2.6Si

Saurface roughness analysis, etch-pit, and recrystallization techniques were employed to characterize the plastic deformation zones around fatigue cracks in Fe-2.6wt% Si. Both surface and sub-surface cross-sections are evaluated. The plastic zone size changes significantly in the surface layers. It is correlated with energy release rate, J, because of the elastoplastic nature of crack propagation and a linear correlation is found. The plastic deformation experienced inside the plastic zone is analysed and is reported in terms of equivalent tensile strains. The material in the immediate vicinity of the crack tip experiences large plastic strains under a steep gradient. This region, however, occupies a very small fraction of the plastic zone, the rest of which is deformed to less than 4% plastic strain. In the light of the experimental data, an attempt is made to identify potential sources for the variation in measured plastic zone parameters reported in the literature.     

Analysis of bent crack in unidirectional fibre reinforced composites

This paper presents a fracture analysis for a bent crack in an infinite orthotropic plate subjected to a far-field uniform tensile stress. To determine parameters relevant to the mixed-mode fracture conditions at the tip of the bent crack, the problem is formulated in terms of singular integral equations with generalized Cauchy kernels. The resulting system of equations is then solved numberically employing a Gaussian quadrature and the collocation method. Stress intensity factors, k1 and k2, and the strain energy release rates, GI and GII at the tip of the bent crack are obtained for various values of fibres direction and L2/L1 ratios. Extensive results for a graphite-epoxy unidirectional composite laminate are presented.     

Modeling of fatigue crack growth threshold development for a microstructurally small crack

A method for predicting the fatigue crack growth threshold using finite element analysis is investigated. The proposed method consists of monitoring the plastic strain hysteresis energy dissipation in the crack tip plastic zone, with the threshold being defined in terms of a critical value of this dissipated energy. Two-dimensional plane-strain elastic-plastic finite element analyses are conducted to model fatigue crack growth in a middle-crack tension M(T) specimen. A single-crystal constitutive relationship is employed to simulate the anisotropic plastic deformation near the tip of a microstructurally small crack without grain boundary interactions. Variable amplitude loading with a continual load reduction is used to generate the load history associated with fatigue crack growth threshold measurement. Load reductions with both constant load ratio R and constant maximum stress intensity K_max are simulated. In comparison with a fixed K_max load reduction, a fixed R load reduction is predicted to generate a 35% to 110% larger fatigue crack growth threshold value.     

On the fracture of constrained layers

The problem of a crack embedded in a layer sandwiched between two elastic adherends is analysed accounting for the influence material property mismatch on the crack tip plastic deformation, which is contained in the layer. The cohesive crack model developed by Dugdale and Barrenblatt is adopted to model the strip yielding behaviour in a constrained layer. It is found that, due to the constraint imparted by elastic adherends with higher moduli, the near tip plastic deformation exhibits a sharp transition (plastic zone grows faster than the square of stress intensity factor) from small scale to large scale yielding. Because the region of singularity dominance for a crack embedded in a layer is generally much smaller than the layer thickness when the layer has a modulus much lower than the adherends, the prevailing failure mode of most bonded joints should be under large scale yielding conditions. A model based on energy balance is proposed to determine the fracture energy of bonded joints under such condition, taking into account of the plastic dissipation in the constrained layer. Comparison with experimental results demonstrates that the theory correctly predicts the dependence of fracture toughness on layer thickness as observed in experiments. This revised version was published online in July 2006 with corrections to the Cover Date.     

Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

Crack-tip fields in anisotropic shells

Asymptotic crack-tip fields including the effect of transverse shear deformation in an anisotropic shell are presented. The material anisotropy is defined here as a monoclinic material with a plane symmetry at x ₃=0. In general, the shell geometry near the local crack tip region can be considered as a shallow shell. Based on Reissner shallow shell theory, an asymptotic analysis is conducted in this local area. It can be verified that, up to the second order of the crack tip fields in anisotropic shells, the governing equations for bending, transverse shear and membrane deformation are mutually uncoupled. The forms of the solution for the first two terms are identical to those given by respectively the plane stress deformation and the antiplane deformation of anisotropic elasticity. Thus Stroh formalism can be used to characterize the crack tip fields in shells up to the second term and the energy release rate can be expressed in a very compact form in terms of stress intensity factors and Barnett–Lothe tensor L. The first two order terms of the crack-tip stress and displacement fields are derived. Several methods are proposed to determine the stress intensity factors and `T-stresses'. Three numerical examples of two circular cylindrical panels and a circular cylinder under symmetrical loading have demonstrated the validity of the approach.