Continuum damage mechanics modelling incorporating stress triaxiality effect on ductile damage initiation

Micromechanical modelling of void nucleation in ductile metals indicates that strain required for damage initiation reduces exponentially with increasing stress triaxiality. This feature has been incorporated in a continuum damage mechanics (CDM) model, providing a phenomenological relationship for the damage threshold strain dependence on the stress triaxiality. The main consequences of this model modification are that the failure locus is predicted to change as function of stress triaxiality sensitivity of the material damage threshold strain and that high triaxial fracture strain is expected to be even lower than the threshold strain at which the damage processes initiate at triaxiality as low as 1/3. The proposed damage model formulation has been used to predict ductile fracture in unnotched and notched bars in tension for two commercially pure α‐iron grades (Swedish and ARMCO iron). Finally, the model has been validated, predicting spall fracture in a plate‐impact experiment and confirming the capability to capture the effect of the stress state on material fracture ductility at very high stress triaxiality.     

Stress triaxiality effect on void nucleation in ductile metals

The stress triaxiality effect on the strain required for void nucleation by particle‐matrix debonding has been investigated by means of micromechanical modelling. A unit‐cell model considering an elastic spherical particle embedded in an elastic‐plastic matrix was developed to the purpose. Particle‐matrix decohesion was simulated through the progressive failure of a cohesive interface. It has been shown that the parameters of matrix‐particle cohesive interface are correlated with macroscopic material properties. Here, a simple relationship for the maximum cohesive opening at interface failure as a function of material fracture toughness and yield stress has been derived. Results seem to confirm that, increasing stress triaxiality, the strain at which void nucleation is predicted to occur decreases exponentially in a similar way as for fracture strain. This result has substantial implications in modelling of ductile damage because it indicates that if the stress triaxiality is high enough, ductile fracture can occur at plastic strain lower than that necessary to nucleate damage for moderate or low stress triaxiality regime.     

应力三维度在层裂中的应用初探

探讨了应力三维度用于层裂分析的必要性和可能性。结合体胞模型的尺寸无关性特征,讨论了高应变率下高应力三维度对孔洞扩展影响的数值模拟方法。在此基础上,对轴对称体胞在不同的应力三维度下孔洞扩展进行了数值模拟。结果表明：宏观定义的等效应力-等效应变关系、孔洞体积分数与球应力关系不存在单值对应关系,它与应力三维度密切相关,因而,以应变等价原理为基石构建的含损伤本构或者状态方程在推广到其它应力状态时存在一定的局限,特别是以空心球模型在球对称应力加载下获得的含损伤本构和状态方程不能用于其它非球对称应力状态下的损伤演化。     

Influence of Shock Pre‐Compression Stress and Tensile Strain Rate on the Spall Behaviour of Mild Steel

Abstract: A series of plate‐impact spall experiments were conducted to investigate the influence of shock pre‐compression stress and tensile strain rates on the dynamic tensile fracture (or spall) behaviour of shocked mild steel. The shock pre‐compression stress amplitude and tensile strain rate were controlled independently to ensure that only one single‐loading parameter varied for each experiment. A push–pull type velocity interferometer system for any reflector (VISAR) was used to measure the free surface velocity profiles of samples. It is observed from experimental results that the influence of shock pre‐compression stress amplitude on the spall strength is less significant in the range attained in these experiments, whereas with increasing tensile strain rate, an evident 65% increase of spall strength is determined in the present tensile strain rate range of 10⁴ to 10⁶ s^?1. VISAR data are compared with finite‐difference calculations employing a modified damage function model with a percolation–relaxation function, and a good agreement between the calculation and the experiments was obtained. Preliminary simulation results also revealed that a critical damage exists, which physically corresponds to the critical intervoid ligament distance for triggering the onset of void coalescence, and may be regarded as a material parameter for describing the dynamic tensile fracture and independent of the loading conditions.     

基于微孔贯通细观损伤模型的金属韧性断裂分析

韧性材料断裂过程通常可看作是材料内部微孔洞的形核、扩展及相互贯通的积累。经典的Gurson- Tvergaard (GT)模型能够很好地模拟具有变形均匀、各向同性的孔洞的萌生及扩展过程;但无法模拟由孔洞贯通而引起的局部变形过程,因此需要对其修正,引入相应的孔洞贯通准则。该文采用两种贯通准则对经典GT模型进行修正,即Thomason的塑性极限载荷准则和临界等效塑性应变准则。借助用户自定义程序UMAT将采用这两种贯通准则修正的GT本构关系嵌入至商用有限元软件ABAQUS中,从而可通过对金属材料应力状态和断裂机理的分析控制孔洞的贯通。以一组含有不同缺口根半径的圆棒拉伸试验件为例,分析了该类金属构件自孔洞萌生至最终断裂的整个损伤演化过程,并与试验数据进行了对比,验证了该模型的有效性。该文还讨论了金属断裂过程中应力三轴度对微裂纹萌生与扩展的影响。     

Nanoscale void nucleation and growth and crack tip stress evolution ahead of a growing crack in a single crystal

A constrained three-dimensional atomistic model of a cracked aluminum single crystal has been employed to investigate the growth behavior of a nanoscale crack in a single crystal using molecular dynamics simulations with the EAM potential. This study is focused on the stress field around the crack tip and its evolution during fast crack growth. Simulation results of the observed nanoscale fracture behavior are presented in terms of atomistic stresses. Major findings from the simulation results are the following: (a) crack growth is in the form of void nucleation, growth and coalescence ahead of the crack tip, thus resembling that of ductile fracture at the continuum scale; (b) void nucleation occurs at a certain distance ahead of the current crack tip or the forward edge of the leading void ahead of the crack tip; (c) just before void nucleation the mean atomic stress (or equivalently its ratio to the von Mises effective stress, which is called the stress constraint or triaxiality) has a high concentration at the site of void nucleation; and (d) the stress field ahead of the current crack tip or the forward edge of the leading void is more or less self-similar (so that the forward edge of the leading void can be viewed as the effective crack tip).     

Investigation on fracture mechanisms of metals under various stress states

Fracture mechanisms for widely used metal materials are investigated under various loading conditions. Several specimens and different loading methods are deliberately designed to produce various stress states. The stress triaxiality is used to rate the level of tension and compression under various stress states. The stress triaxiality increases with adding a notch in the specimen under tension loading and decreases by changing the loading from tension to compression. Scanning electron microscopes are used to observe the microscopic features on the fracture surfaces. The fracture surfaces observed in the tests indicate that with the decreasing stress triaxiality the fracture mechanism for a given metal material includes intergranular cleavage, nucleation, growth, void coalescence, and local shear band expansion. With the fracture mechanisms changing from intergranular cleavage to nucleation, growth, and coalescence of voids, and expansion of a local shear band, four possible fracture modes can be observed, which are quasi-cleavage brittle fracture, normal fracture with void, shear fracture with void, and shear fracture without void. Quasi-cleavage brittle fracture and normal fracture with void are both normal stress-dominated fracture modes; however, their mechanisms are different. Shear fracture with and without void are both shear stress-dominated fracture, and shear fracture with void is also influenced by the normal stress. To a certain metal material, under high stress triaxiality, quasi-cleavage brittle fracture and normal fracture with void tend to occur, and under low stress triaxiality, shear fracture with and without void tend to occur. In addition, the critical positions and fracture criteria adapted to each fracture mode will also be different.     

Dynamic vs. quasi-static shear failure of high strength metallic alloys: Experimental issues

Ductile fracture of metals by void nucleation, growth and coalescence under positive stress triaxiality is well admitted. This is not the case when metals are submitted to negative stress triaxiality. The present work aims at contributing to a better understanding of the competition between micro-mechanisms at the origin of failure of metals when submitted to shear-pressure loading at low and high strain rates. With this aim in view, experiments were carried out on Ti–6Al–4V shear-compression samples involving a stress triaxiality range comprised between −0.2 and −0.5. Results show that the material failure is the consequence of a void growth induced process. At high strain rate, due to the localization of the deformation within adiabatic shear bands, the failure of the material occurs earlier, leading to maximum shear strain smaller at high strain rate than at low strain rate. Impact tests were also carried out on Kalthoff and Winkler type double notched plates. They showed that the interaction between tension and shear waves leads to a complex Mode I–Mode II crack propagation.     

A modified Gurson model and its application to punch-out experiments

Recent experimental evidence has reiterated that ductile fracture is a strong function of stress triaxiality. Under high stress triaxiality loading, failure occurs as a result of void growth and subsequent necking of inter-void ligaments while under low stress triaxiality failure is driven by shear localization of plastic strain in these ligaments due to void rotation and distortion. The original Gurson model is unable to capture localization and fracture for low triaxiality, shear-dominated deformations unless void nucleation is invoked. A phenomenological modification to the Gurson model that incorporates damage accumulation under shearing has been proposed. Here we further extend the model and develop the corresponding numerical implementation method. Several benchmark tests are performed in order to verify the code. Finally, the model is utilized to model quasi-static punch-out experiments on DH36 steel. It is shown that the proposed modified Gurson model, in contrast to the original model, is able to capture the through-thickness development of cracks as well as the punch response. Thus, the computational fracture approaches based on the modified Gurson model may be applied to shear-dominated failures.     

Dynamic fracture of tungsten heavy alloys

Dynamic fracture of tungsten heavy alloys was induced by two different test techniques. The first was spall (e.g., 1-D strain fracture). The second was transverse impact, as occurs in a yawed penetrator. Spall failure is stress driven, and spall stress corresponds to the threshold for void formation, which is 2.6 GPa for a 91% WNiCo alloy and 2.1–2.5 GPa for a 95% WNiFe alloy. Yaw-induced fracture, on the other hand, is strain driven. Surface flaws can provide fracture sites. At the meso scale, grain cleavage is mainly responsible for transverse fracture. Grain fracture also appears to play a critical role in the initiation of spall fracture.     

Cell model simulation of void growth in nodular cast iron under cyclic loading

The failure of cast iron under high plastic cyclic strains is controlled by the mechanisms of formation, growth and coalescence of voids. A cell model approach is used to simulate nodular cast iron as a periodic array of loosely bonded spherical inclusions in the matrix material. The models are analyzed by the finite element method under cyclic loading while keeping the stress triaxiality constant. Different types of matrix hardening are used: isotropic, kinematic and combined hardening. The graphite inclusions are simulated by a rigid body. Deformation and void growth are studied in dependence on stress triaxiality and strain range. In most cases after a few cycles a non-symmetric stationary mesoscopic cyclic stress–strain curve is established. The deformation response and the development of the void volume fraction are strongly affected by the value of triaxiality. The void volume is incrementally increasing with each load cycle in a ratcheting manner. The void growth rate depends on the chosen hardening type and is smallest for kinematic hardening. The comparison with simulations in absence of graphite inclusions revealed that void evolution is favored by the inclusions.     

Analysis of fracture limit curves and void coalescence in high strength interstitial free steel sheets formed under different stress conditions

Void formation, which is a statistical event, depends on inhomogeneities present in the microstructure. The analysis on void nucleation, their growth and coalescence during the fracture of high strength interstitial free steel sheets of different thicknesses is presented in this article. The analysis shows that the criterion of void coalescence depends on the d-factor, which is the ratio of relative spacing of the ligaments (δd) present between the two consecutive voids to the radius of the voids. The computation of hydrostatic stress (σ_m), the dominant factor in depicting the evolution of void nucleation, growth and coalescence and the dimensional analysis of three different types of voids namely oblate, prolate and spherical type, have been carried out. The ratio of the length to the width (L/W) of the oblate or prolate voids at fracture is correlated with the mechanical properties, microstructure, strains at fracture, Mohr’s circle shear strains and Triaxiality factors. The Lode angle (θ) is determined and correlated with the stress triaxiality factor (T), ratio of mean stress (σ_m) to effective stress (σ_e). In addition, the Void area fraction (V _a), which is the ratio of void area to the representative area, is determined and correlated with the strain triaxiality factor (T_o).     

Influence of crack tip constraint on void growth in ductile FCC single crystals

Effect of constraint (stress triaxiality) on void growth near a notch tip in a FCC single crystal is investigated. Finite element simulations within the modified boundary layer framework are conducted using crystal plasticity constitutive equations and neglecting elastic anisotropy. Displacement boundary conditions based on mode I, elastic, two term K-T field are applied on the outer boundary of a large circular domain. A pre-nucleated void is considered ahead of a stationary notch tip. The interaction between the notch tip and the void is studied under different constraints (T-stress levels) and crystal orientations. It is found that negative T-stress retards the mechanisms of ductile fracture. However, the extent of retardation depends on the crystal orientation. Further, it is found that there exists a particular orientation which delays the ductile fracture processes and hence can potentially improve ductility. This optimal orientation depends on the constraint level.     

Role of nanoscale Cu/Ta interfaces on the shock compression and spall failure of nanocrystalline Cu/Ta systems at the atomic scales

Molecular dynamics (MD) simulations are used to investigate the role of size and distribution of nanoscale Cu/Ta interfaces on the nucleation and evolution of defects during shock loading and spall failure of nanocrystalline (nc) Cu/Ta alloys. Cu/Ta interfaces are introduced through the embedding of Ta clusters in nc-Cu matrix. The phase stability of the embedded Ta clusters either as FCC or BCC clusters is first investigated and reveals that the FCC Ta clusters have a lower energy for diameters less than 4 nm, whereas the BCC Ta clusters have a lower energy for the larger diameters. The shock simulations are then carried out for Ta clusters with an average diameter of 1 and 3 nm and concentrations of 3.0, 6.3 and 10.0% to investigate the role of size and distribution of Cu/Ta interfaces (due to presence of clusters) on the nucleation and evolution of dislocations as well as the spall strength of the alloy. The MD simulations indicate that the Cu/Ta interfaces reduce the capability of nc-Cu to accommodate plasticity through nucleation of dislocations and create void nucleation sites during spallation. The MD simulations further reveal that the impact strengthening effects due to the presence of nanoscale Cu/Ta interfaces are strongly dependent upon the size and distribution of Ta clusters, as well as the grain size of Cu matrix. Smaller size of interfaces (cluster size), higher concentration of Ta (smaller spacing between interfaces) and larger matrix grain size render higher spall strengths of nc-Cu/Ta microstructures.     

Effect of annealing temperature on void coalescence in 5086 aluminium alloy formed under different stress conditions

In this paper, the influence of annealing temperature on formability, fracture behavior, void nucleation, its growth, and coalescence are studied. The voids and fracture behavior are studied as a function of various void parameters, namely δd-factor (ligament thickness between consecutive voids), d factor (ratio of δd and radius of the void), void area fraction (V_a) and L/W ratio (ratio of length to width of the void). The L/W ratio of the oblate or prolate voids at fracture is correlated with the mechanical properties, microstructure, minor strain at fracture (ε₂), Mohr’s circle shear strains, stress, and strain triaxiality factors. The Lode factor (θ) is determined and correlated with the Stress triaxiality factor (T), which is the ratio of mean stress (σ_m) to effective stress (σ_e). In addition, the Void area fraction (V_a), which is the ratio of void area to the representative sample area, is determined correlated with the Strain triaxiality factor (T_o). It is found that the sheets annealed at 300°C, possesses better formability due to lower d-factor, higher and -values, greater void area fraction and lower L/W ratio of void accommodating more plastic deformation.     

Growth and coalescence of penny-shaped voids in metallic alloys

The growth and coalescence of penny-shaped voids resulting from particle fracture is a common damage process for many metallic alloys. A three steps modeling strategy has been followed to investigate this specific failure process. Finite element cell calculations involving very flat voids shielded or not by a particle have been performed in order to enlighten the specific features of a damage mechanism starting with initially flat voids with respect to more rounded voids. An extended Gurson-type constitutive model supplemented by micromechanics-based criteria for both void nucleation and void coalescence is assessed for the limit of very flat voids towards the FE calculations. The constitutive model is then used to generate a parametric study of the effects of the stress state, the microstructure and the mechanical properties on the ductility. Based on these results, a simple closed-form model for the ductility is finally proposed. The main outcomes of this study are that (i) the ductility of metal alloys involving penny-shaped voids is primarily controlled by the relative void spacing; (ii) the definition of an effective porosity in terms of an equivalent population of spherical voids is valid for low particle volume fraction; (iii) the presence of a particle shielding the void does not significantly affect the void growth rates and void aspect evolution; (iv) early fracture by void coalescence can occur under very low stress triaxiality conditions if the particle volume fraction is large enough, explaining that some alloys and composites can fail through a transgranular ductile fracture mode under uniaxial tension condition before the onset of necking; (v) the fracture mechanism moves from void growth controlled to void nucleation controlled when increasing the void nucleation stress, lowering the stress triaxiality, and increasing the initial void aspect ratio.     

Spallation of single crystal nickel by void nucleation at shock induced grain junctions

Molecular dynamics (MD) simulations of spallation in single crystal nickel were performed for a range of system sizes and impact velocities. The initial compressive wave leaves a rich microstructure in its wake. The subsequent tensile waves create multiple grains and grain junctions between regions of differing crystal orientation. These grain junctions serve as void nucleation sites when the reflected tensile waves interact, leading to ductile failure. In this way, the mechanism for failure in an initially single-crystalline sample is similar to that seen experimentally in high-purity, poly-crystalline metals, in which grain boundaries are sites for void nucleation.     

Effect of stress triaxiality levels in crack tip regions on the characteristics of void growth and fracture criteria

The behaviour of void growth in the crack tip regions of four specimen geometries with different stress triaxiality levels have been investigated by the FEM method and experimental observations in plane strain and plane stress cases respectively. It was found that the shape change of growing voids, the configurations of a blunting crack tip and the sizes of decreasing ligament between the void and the crack tip are strongly dependent upon the stress triaxiality levels. Under the condition of plane stress, the stress triaxiality on the ligaments of cracks are nearly the same for different specimen geometries, also the void growth, crack tip blunting asnd decreasing of ligament size are identical for various specimens with increasing load levels, which lead to the conclusion that the J_i-value is independent of specimen geometries. However, in the plain strain case, different void growth, crack tip blunting and decreasing in ligament size for various stress triaxiality levels directly caused the J_i-value to be dependent on the specimen geometries. It was found that when the void is linked to the blunting crack tip by the extrapolation to the zero ligament from FEM calculations, the J_i values, measured experimentally, are underestimated slightly.     

A void coalescence-based spall model

A void coalescence-based spall model is presented using stress relaxation equations based on the assumption that the main effect of the microcracks is to reduce the area over which the stress acts in the early stages and the stress decreases to porosity-dependent value in the void coalescence stages. The stress- (or pressure-) dependent spall porosities given by Thomason, by Tonks et al and by Cochran et al. are, respectively, combined with conservation equations, equation of state and constitutive equations for the damaged aggregate to establish a series of closed equations for all variables including the damage. The void coalescence-based spall model contains only two parameters: the spall strength and critical damage, the values of which can be initially estimated for plate-impact spall tests and be finally determined to make the computed results of spall tests under the initial and boundary conditions consistent with experimental velocity (stress) profile and the observed damage at spall plane in general. The computer simulations of spall experiments for copper, uranium and steel are performed with the one-dimensional Lagrangian finite difference method. The computed results based on the pressure-dependent spall porosities given by Tonks et al., and by Cochran et al., are consistent in general, but different from the computed results based on Thomason's 2D stress-dependent spall porosity to a considerable extent.     

High strain rate fragmentation of liquid systems at atmospheric pressure

The fragmentation characteristics of liquid systems at atmospheric pressure has been investigated experimentally and compared to hydrodynamic calculations as well as theoretical predictions. The geometry is a one-dimensional (1-D) nylon flat plate impacting a flat plate liquid system at velocities of approximately 0.3 km/s. The experiments were conducted at the Marquette University's gas gun facility. Hydrocodes calculations were used to investigate early time shock evolution, material deformation and strain rate. High-speed photography and witness cards were used to capture the impact and fragmentation event as well as drop distributions. The experimental drop distributions are compared to distributions obtained from a Grady–Kipp (GK) fragmentation model. The liquid spall GK fragmentation model predicted the correct median drop size while the flow stress spall over predicted the median drop size. The standard GK Poisson drop distribution superimposed over the median drop size produces a distribution which is wider than the data obtained.