Plane crack propagation in a hyperelastic incompressible material

We propose a crack propagation criterion for hyperelastic materials (rubber type material) within the framework of plane elasticity in finite deformation. The criterion is based on the examination of the asymptotic elastic field near the crack tip prior to propagation. According to this criterion, the propagation will take place for a critical value of the strain energy density intensity factor. The kink angle, obtained by applying the criterion of maximum opening stress, will depend on the fracture tensile stress of the actual material. We propose to use a local iterative finite element method to compute the asymptotic quantities involved in the criterion at a reasonable cost. Examples of computation for some hyperelastic laws simulating the behavior of vulcanized rubber are presented.     

Simulation of brittle and ductile fracture in an impact loaded prenotched plate

We analyze the initiation and propagation of a crack from a point on the surface of a circular notch-tip in an impact loaded prenotched plate. The material of the plate is assumed to exhibit strain hardening, strain-rate hardening, and softening due to the rise in temperature and porosity. The degradation of material parameters due to the evolution of damage in the form of porosity is considered. Brittle failure is assumed to initiate when the maximum tensile principal stress at a point reaches a critical level. Ductile failure is assumed to ensue when the effective plastic strain reaches a critical value. A crack initiating from the node where a failure first occurs is taken to propagate to the adjacent node that has the highest value of the failure parameter (the maximum tensile principal stress or the effective plastic strain). The opening and propagation of a crack are modeled by the node release technique. Surface tractions and the normal component of the heat flux are taken to be null on the newly created crack surfaces. For the brittle failure, the stress field around the crack tip resembles that in mode-I deformations of a prenotched plate loaded in tension. The distribution of the effective plastic strain in a small region around the surface of the notch-tip is not affected much by the initiation of a ductile fracture there except for a shift in the location of the point where the effective plastic strain is maximum. The initiation of the ductile failure is delayed when a crack is opened at the point where the brittle failure ensues.     

Influence of isotropic strain hardening on plane strain dynamic growth in elastic-plastic materials

In this paper, dynamic crack growth in an elastic-plastic material is analysed under mode I, plane strain, small-scale yielding conditions using a finite element procedure. The material is assumed to obey J₂ incremental theory of plasticity with isotropic strain hardening which is of the power-law type under uniaxial tension. The influence of material inertia and strain hardening on the stress and deformation fields near the crack tip is investigated. The results demonstrate that strain hardening tends to oppose the role of inertia in decreasing plastic strains and stresses near the crack tip. The length scale near the crack tip over which inertia effects are dominant also diminishes with increase in strain hardening. A ductile crack growth criterion based on the attainment of a critical crack tip opening displacement is used to obtain the dependence of the theoretical dynamic fracture toughness on crack speed. It is found that the resistance offered by the elastic-plastic material to high speed crack propagation may be considerably reduced when it possesses some strain hardening.     

Discontinuous bifurcations in a nonassociated Mohr material

Discontinuous bifurcations of an elastic-plastic material obeying Mohr's yield criterion and a nonassociated flow rule and subjected to general three-dimensional loading are investigated. The first possibility for bifurcation is identified and expressed in terms of the critical hardening modulus and the corresponding orientation of the slip planes. In the case of perfect plasticity the Mohr and Roscoe solutions for the orientation of the slip planes are derived. The corresponding strain rate fields are determined and it is shown that the Roscoe strain rate field differs significantly from the other fields. Moreover, the stress rate field is continuous across the singular surface for the Roscoe solution.     

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.     

Mechanical characterization and modeling of the deformation and failure of the highly crosslinked RTM6 epoxy resin

The nonlinear deformation and fracture of RTM6 epoxy resin is characterized as a function of strain rate and temperature under various loading conditions involving uniaxial tension, notched tension, uniaxial compression, torsion, and shear. The parameters of the hardening law depend on the strain-rate and temperature. The pressure-dependency and hardening law, as well as four different phenomenological failure criteria, are identified using a subset of the experimental results. Detailed fractography analysis provides insight into the competition between shear yielding and maximum principal stress driven brittle failure. The constitutive model and a stress-triaxiality dependent effective plastic strain based failure criterion are readily introduced in the standard version of Abaqus, without the need for coding user subroutines, and can thus be directly used as an input in multi-scale modeling of fibre-reinforced composite material. The model is successfully validated against data not used for the identification and through the full simulation of the crack propagation process in the V-notched beam shear test.     

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.     

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.     

Impact and fracture response of sintered 316L stainless steel subjected to high strain rate loading

This paper describes the use of a material testing system (MTS) and a compressive split-Hopkinson bar to investigate the impact behaviour of sintered 316L stainless steel at strain rates ranging from 10^− 3 s^− 1 to 7.5 × 10³ s^− 1. It is found that the flow stress–strain response of the sintered 316L stainless steel depends strongly on the applied strain rate. The rate of work hardening and the strain rate sensitivity change significantly as the strain rate increases. The flow behaviour of the sintered 316L stainless steel can be accurately predicted using a constitutive law based on Gurson's yield criterion and the flow rule of Khan, Huang and Liang (KHL). Microstructural observations reveal that the degree of localized grain deformation increases at higher strain rates. However, the pore density and the grain size vary as a reversible function of the strain rate. Impacts at strain rates higher than 5.6 × 10³ s^− 1 are found to induce adiabatic shear bands in the specimens. These specimens subsequently fail as a result of crack propagation along the dominant band. The fracture surfaces of the failed specimens are characterized by dimple-like structures, which are indicative of ductile failure. The depth and the density of these dimples are found to decrease with increasing strain rate. This observation indicates a reduction in the fracture resistance and is consistent with the observed macroscopic flow stress–strain response.     

Mixed-mode crack growth predictions

The problem of slow stable growth of an inclined crack in a plate subjected to uniaxial tension is studied by the strain energy density criterion. The stable crack growth process is simulated by predicting a series of crack growth steps corresponding to a piecewise loading increase when material elements along the direction of crack extension absorb a critical amount of elastic strain energy density. Crack instability takes place when the last ligament of crack extension takes a critical value which is a material constant. The critical stress at the onset of crack initiation and unstable crack extension is determined for various crack inclination angles. Three different loading step increments corresponding to three different loading rates are considered and their effect on stable crack growth is analysed. Furthermore, the influence of loading history on the crack growth process for three different loading types is studied. The complete crack growth patterns for all types of load are determined and analysed. It is obtained that the amount of slow crack growth can be increased by lowering the rate of loading. The effect of the loading history on the failure load and the crack paths is established.     

A non-local stress and strain energy release rate mixed mode fracture initiation and propagation criteria

A non-local stress condition for crack initiation and propagation in brittle materials is presented. This condition is expressed in terms of normal and tangential traction components acting on a physical plane segment (damage zone) of specified length. Next, a non-local strain energy release rate criterion is proposed. This condition is based on the assumption that initiation or propagation of cracking occurs when the maximal value of the function of opening and sliding energy release rates reaches a critical value. The value of energy release rates is determined for finite elementary crack growth. Mixed mode conditions are considered, for which both the critical load value and the crack orientation are predicted from the non-local stress and energy criteria, which are applicable to both regular and singular stress concentrations. The effect of non-singular second order term (T_σ-stress) on the crack propagation is discussed.     

Analysis of the stress intensity factor dependence with the crack velocity using a lattice model

In this work, the influence of crack propagation velocity in the stress intensity factor has been studied. The analysis is performed with a lattice method and a linear elastic constitutive model. Numerous researchers determined the relationship between the dynamic stress intensity factor and crack propagation velocity with experimental and analytical results. They showed that toughness increases asymptotically when the crack tip velocity is near to a critical. However, these methods are very complex and computationally expensive; furthermore, the model requires the use of several parameters that are not easily obtained. Moreover, its practical implementation is not always feasible. Hence, it is usually omitted. This paper aims to capture the physics of this complex problem with a simple fracture criterion. The selected criterion is based on the maximum principal strain implemented in a lattice model. The method used to calculate the stress intensity factor is validated with other numerical methods. The selected example is a finite 2D notched under mode I fracture and different loads rates. Results show that the proposed model captures the asymptotic behaviour of the SIF in function of crack speed, as reported in the aforementioned models.     

Effects of pressure-sensitive yielding on stress distributions in crankshaft sections under fillet rolling and bending fatigue tests

In this paper, the evolution equation for the active yield surface during the unloading/reloading process based on the Drucker–Prager yield function and a recently developed anisotropic hardening rule is first presented. A user material subroutine based on the anisotropic hardening rule and the constitutive relation was written and implemented into the commercial finite element program ABAQUS. Computations were first conducted for a simple plane strain finite element model under uniaxial monotonic and cyclic loading conditions. The results indicate that the anisotropic hardening rule with the non-associated flow rule describes well the strength-differential effect and the asymmetric closed hysteresis loops as observed in the uniaxial cyclic loading tests of cast irons. Then, a two-dimensional plane strain finite element analysis of a crankshaft section under fillet rolling and subsequent bending was conducted. For the pressure sensitivity corresponding to the cast iron crankshaft of interest, the critical locations for fatigue crack initiation according to the stress distributions for pressure-sensitive materials agree with the experimental observations in bending fatigue tests of crankshaft sections.     

Effect of material plasticity on fatigue crack propagation under complex stress state

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. However, this fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, a modification has been made to G _max criterion to implement the consideration of the plastic strain energy. This criterion is extended to study the fatigue crack growth characteristics of mixed mode cracks in steel pipes. To predict crack propagation due to fatigue loads, a new elasto-plastic energy model is presented. This new model includes the effects of material properties like strain hardening exponent n, yield strength σ_y and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.     

A finite element analysis of plane strain dynamic crack growth in materials displaying the Bauschinger effect

In this paper dynamic crack growth in an elastic-plastic material is analyzed under mode I plane strain small-scale yielding conditions using a finite element procedure. The main objective of this paper is to investigate the influence of anisotropic strain hardening on the material resistance to rapid crack growth. To this end, materials that obey an incremental plasticity theory with linear isotropic or kinematic hardening are considered. A detailed study of the near-tip stress and deformation fields is conducted for various crack speeds. The results demonstrate that kinematic hardening does not oppose the role of inertia in decreasing the plastic strains and stresses near the crack tip with increase in crack speed to the same extent as isotropic strain hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small micro-structural distance behind the tip is used to obtain the dependence of the theoretical dynamic fracture toughness with crack speed. It is found that for any given level of strain hardening, the dynamic fracture toughness displays a much more steep increase with crack speed over the quasi-static toughness for the kinematic hardening material as compared to the isotropic hardening case.     

Powder compaction interpretation using the power law

The effects of strain hardening and axial strain rate, over a wide range of rates (10^–3 to 10⁵ s^–1), on the compaction properties of a variety of pharmaceutical powders have been investigated. The powders tested are: Di Pac sugar, paracetamol d.c., Avicel and lactose. These materials have been assessed using the constants derived from the power law as a criterion to describe their behaviour. All the materials tested show, with varying degrees, a non-linear increase in the yield pressure (flow stress), the constantG, the strain rate exponentm and the strain hardening exponentn as the strain rate increases. These variations are more clear in the materials known to deform plastically, such as Avicel. This is attributable to a change either from ductile to brittle behaviour or a reduction in the amount of plastic deformation due to the time-dependent nature of the plastic flow. This, however, is explained in terms of dislocation and diffusion processes involved in the plastic deformation mechanisms during the compaction process. As the speed of compaction increases the characteristics of deformation, including the value of the strain rate exponent, the shape of the creep curve and the nature of creep rate, suggest that the creep behaviour is therefore controlled by some form of diffusion process. Meanwhile, the creep characteristics of the low and medium rate tests appear to be consistent with dislocation climb and viscous glide. For the materials tested, Avicel is found to be the most strain-rate sensitive material, while paracetamol d.c. is found to be the least strain-rate sensitive material.     

TC21钛合金中应变速率拉伸力学行为实验研究

对网篮组织TC21钛合金进行了0.001 s-1～50 s-1的中应变速率室温拉伸试验。试验结果表明,TC21拉伸力学行为在试验应变速率范围内具有明显的应变速率强化效应、应变硬化效应和随应变速率升高而逐渐增大的温升软化效应;屈服应力的应变速率相关性在6 s-1时发生转折;随应变率的升高,应变硬化效应减小,断裂应变和失稳应变增大;试验应变速率范围内TC21的变形机制为位错的热激活机制。SEM和金相观察结果表明,TC21的断裂方式均为韧性断裂,断裂机理为微孔洞的聚集和长大。     

Elastic-plastic analysis of an arbitrarily-oriented mode III crack touching an interface

In this paper we investigate a semi-infinite crack terminating at an arbitrarily oriented interface between two elastic-plastic materials under an anti-plane shear loading. An analytical solution is first developed for general power-law hardening materials under a mode III loading. If both materials have the same hardening exponent, the formulation results in a nonlinear eigenequation which can be solved numerically. The stress singularities are determined as a function of two material constants: the hardening exponent n and parameter G which represents the relative resistance of the two materials. In addition to the power of the singularity, the stress, strain and displacement asymptotic fields are also determined. If the hardening exponents are not the same, the leading order terms of an expansion model ensure the stress continuity across the interface. The results show that the stress singularity mainly depends upon the material having the larger hardening exponent, with the highest stresses in the material having the smaller hardening exponent. By taking the hardening exponent n , the perfectly plastic bimaterial problem is studied. It has been found that if the crack lies in the less stiff material, the entirely plastic asymptotic fields around the crack tip can be determined. On the other hand, if the crack lies in the stiffer material, the crack-tip fields are partially elastic and partially plastic. For both cases, unique asymptotic fields can be determined explicitly. For those cases when the materials present a strain hardening property, different mathematical models are established.     

Loading‐rate dependence of mode I crack growth in concrete

Mode I crack propagation process of concrete under relatively low loading rates which cover four orders of magnitude (0.2 μm/s to 2.0 mm/s) is investigated with three‐point bending (TPB) beams. All measured material properties exhibit rate sensitivity and follow a log‐linear relationship with the loading rate. A rate‐sensitive softening curve is established. The complete load‐crack mouth opening displacement (P‐CMOD) curve, crack propagation length, and fracture process zone (FPZ) length are simulated based on crack growth criterion with the fitted material parameters under those loading rates. Results show that the simulated P‐CMOD curves agree well with those of experimental measurements. It is clear that the peak load increases with the loading rate and so is the critical crack mouth opening displacement. Moreover, under the same load level, the length of the FPZ and the cohesive stress at the initial crack tip also increase with the increasing loading rate.     

Dynamic behavior of a woven glass-fiber-reinforced polymer composite at high strain rates and its dynamic constitutive relationship

The dynamic behavior of a woven glass-fiber-reinforced polymer composite at different strain rates was investigated. Regarded as a transversely isotropic material, dynamic compression tests of the composite in the normal and tangent directions were conducted by Split-Hopkinson pressure-bar devices. The experimental results show: (1) This composite is a viscoelastic material, and the dynamic compression-failure stress of the material indicates a distinct strain-rate hardening effect. The compression-failure stress in the normal direction increases by almost 30% when the strain rate changes from 900/s to 2000/s. The failure stress in the tangent direction is enhanced by 34% when the strain rate increases from 700/s to 2000/s. The failure stresses of the composite depend linearly on the strain rates, and the corresponding strain-rate hardening factors are 0.14 and 0.046 for the normal and tangent directions, respectively. (2) The dynamic compression-failure stresses and failure strains in the normal direction are significantly larger than the corresponding values in the tangent direction at different strain rates. Based on these results, the modified ZWT viscoelastic model was adopted, and the constitutive relationships of the composite in two directions are presented for characterizing their dynamic compression response.