Multiscale modelling of damage and failure in two-dimensional metallic foams

The fracture strength of metal foams depends sensitively on the properties of the constituent material as well as the cellular architecture. A change in microscopic properties carries over to the macroscopic scale through an alteration of the mesoscopic damage and fracture mechanisms. In this paper we study these dependencies using a modelling framework that takes all these ingredients into account. We have developed a micromechanical model based on a discrete Voronoi representation of cellular metals that incorporates power-law strain hardening and damage development of the cell wall material. The influence of the relative density and material strain hardening on the cell wall damage behavior and overall fracture response is analyzed in detail. The effect of the cellular architecture is studied by varying the cell shape anisotropy and structural randomness. We also simulate the effect of post-processing heat treatments on the solid material plastic and fracture properties and how this affects the overall fracture profile and damage development. Finally, all material and architectural effects are summarized in a strength versus ductility graph, identifying trends for improved design of metallic foams.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

A Study of Large Plastic Deformations in Dual Phase Steel Using Digital Image Correlation and FE Analysis

Large plastic deformation in sheets made of dual phase steel DP800 is studied experimentally and numerically. Shear testing is applied to obtain large plastic strains in sheet metals without strain localisation. In the experiments, full-field displacement measurements are carried out by means of digital image correlation, and based on these measurements the strain field of the deformed specimen is calculated. In the numerical analyses, an elastoplastic constitutive model with isotropic hardening and the Cockcroft–Latham fracture criterion is adopted to predict the observed behaviour. The strain hardening parameters are obtained from a standard uniaxial tensile test for small and moderate strains, while the shear test is used to determine the strain hardening for large strains and to calibrate the fracture criterion. Finite Element (FE) calculations with shell and brick elements are performed using the non-linear FE code LS–DYNA. The local strains in the shear zone and the nominal shear stress-elongation characteristics obtained by experiments and FE simulations are compared, and, in general, good agreement is obtained. It is demonstrated how the strain hardening at large strains and the Cockcroft–Latham fracture criterion can be calibrated from the in-plane shear test with the aid of non-linear FE analyses. An erratum to this article can be found at     

A comparison of cohesive zone modeling and classical fracture mechanics based on near tip stress field

A mode III crack with a cohesive zone in a power-law hardening material is studied under small scale yielding conditions. The cohesive law follows a softening path with the peak traction at the start of separation process. The stress and strain fields in the plastic zone, and the cohesive traction and separation displacement in the cohesive zone are obtained. The results show that for a modest hardening material (with a hardening exponent N = 0.3), the stress distribution in a large portion of the plastic zone is significantly altered with the introduction of the cohesive zone if the peak cohesive traction is less than two times yield stress, which implies the disparity in terms of the fracture prediction between the classical approach of elastic–plastic fracture mechanics and the cohesive zone approach. The stress distributions with and without the cohesive zone converge when the peak cohesive traction becomes infinitely large. A qualitative study on the equivalency between the cohesive zone approach and the classical linear elastic fracture mechanics indicates that smaller cracks require a higher peak cohesive traction than that for longer cracks if similar fracture initiations are to be predicted by the two approaches.     

Finite element analysis of mode III steady crack growth in anisotropic hardening materials

In this paper, the steady crack growth of mode III under small scale yielding conditions is investigated for anisotropic hardening materials by the finite element method. The elastic-plastic stiffness matrix for anisotropic materials is given. The results show the significant influences of anisotropic hardening behaviour on the shape and size of plastic zone and deformation field near the crack tip. With a COD fracture criterion, the ratio of stress intensity factorsk _ss/k_c varies appreciably with the anisotropic hardening parameterM and the hardening exponentN.     

Characterization of the post-necking strain hardening behavior using the virtual fields method

The present study aims at characterizing the post-necking strain hardening behavior of three sheet metals having different hardening behavior. Standard tensile tests were performed on sheet metal specimens up to fracture and heterogeneous logarithmic strain fields were obtained from a digital image correlation technique. Then, an appropriate elasto-plastic constitutive model was chosen. Von Mises yield criterion under plane stress and isotropic hardening law were considered to retrieve the relationship between stress and strain. The virtual fields method (VFM) was adopted as an inverse method to determine the constitutive parameters by calculating the stress fields from the heterogeneous strain fields. The results show that the choice of a hardening law which can describe the hardening behavior accurately is important to derive the true stress–strain curve. Finally, post-necking hardening behavior was successfully characterized up to the initial stage of localized necking using the VFM with Swift and modified Voce laws.     

Effects of the strain-hardening exponent on two-parameter characterizations of surface-cracks under large-scale yielding

The influence of strain hardening exponent on two-parameter J-Q near tip opening stress field characterization with modified boundary layer formulation and the corresponding validity limits are explored in detail. Finite element simulations of surface cracked plates under uniaxial tension are implemented for loads exceeding net-section yield. The results from this study provide numerical methodology for limit analysis and demonstrate the strong material dependencies of fracture parameterization under large scale yielding. Sufficient strain hardening is shown to be necessary to maintain J-Q predicted fields when plastic flow progresses through the remaining ligament. Lower strain hardening amplifies constraint loss due to stress redistribution in the plastic zone and increases the ratio of tip deformation to J. The onset of plastic collapse is marked by shape change and/or rapid relaxation of tip fields compared to those predicted by MBL solutions and thus defining the limits of J-Q dominance. A radially independent Q-parameter cannot be evaluated for the low strain hardening material at larger deformations within a range where both cleavage and ductile fracture mechanisms are present. The geometric deformation limit of near tip stress field characterization is shown to be directly proportional to the level of stress the material is capable of carrying within the plastic zone. Accounting for the strain hardening of a material provides a more adjusted and less conservative limit methodology compared to those generalized by the yield strength alone. Results from this study are of relevance to establishing testing standards for surface cracked tensile geometries.     

Application of Laser Deposition to Mechanical Characterization of Advanced High Strength Steels Subject to Non-Proportional Loading

Background

Characterization of hardening and fracture limits of advanced high strength steels (AHSSs) undergoing strain path changes (SPCs) are particularly challenging for plane strain condition, which commonly occurs in sheet metal forming. There is a need for a simple, engineering-friendly method to characterize materials subjected to complex loading paths that mimic stress conditions in actual forming processes.

Objective

Experimental additive manufacturing techniques have been applied to reinforce AHSS specimens subjected to SPCs in order to broaden capabilities for characterizing hardening behavior and fracture limits.

Methods

Hardening curves subject to SPCs (e.g. uniaxial tension or equi-biaxial tension followed by plane strain) have been obtained with a programmable biaxial tensile testing system using cruciform-shaped specimens with load-bearing arms reinforced by laser deposition. A notched specimen selectively reinforced by laser deposition was newly designed to characterize fracture limits subjected to SPCs ending with plane strain condition.

Results

Complex loading histories were successfully enabled by applying laser deposition technology. Results show that both hardening behavior and fracture limits of a TRIP-assisted steel and a dual-phase steel are dependent on loading history.

Conclusions

It appears that the laser deposition technique can be used for material characterization under specific SPCs. Hardening behavior of AHSSs under SPCs ending with plane strain is quite different from traditional uniaxial tension-uniaxial compression tests. For materials sensitive to SPCs, multi-step forming can be a great option to reach the targeted forming shape.

     

Tensile failure of two-dimensional quasi-brittle foams

Stress redistribution caused by damage onset and the subsequent local softening plays an important role in determining the ultimate tensile strength of a cellular structure. The formation of damage process zones with struts dissipating a finite amount of fracture energy will require the macroscopic stress to be increased in order to continue structural damage. The goal of this paper is to investigate the influence of the fracture energy of the solid on the tensile fracture strength and the strain to fracture in quasi-brittle two-dimensional foams using a microstructural model. We analyze the mesoscopic damage and failure mechanisms in uniaxial tension. Relative density, strut cross-sectional profile, solid’s fracture strain, and fracture energy are varied systematically. The effect of the specific fracture energy on the peak behavior has been shown to be captured by the ratio of the fracture energy to the stored elastic energy. We have also explored the net section strength variation in the presence of a central crack at two different fracture energies. Comparison is made between two structurally identical quasi-brittle and ductile strain hardening foams to identify the differences in the damage mechanisms.     

7A04铝合金的本构关系和失效模型

使用万能材料试验机、扭转试验机和Taylor撞击实验研究了高强铝合金7A04在常温至250 ℃的 准静态、动态本构关系和失效模型。基于实验结果,修改了Johnson-Cook强度模型中的应变强化项以及 Johnson-Cook失效模型中的温度软化项,并结合数值模拟标定了模型参数。实验结果表明,7A04铝合金的 应变和应变率强化效应不显著,失效应变随温度的增加、应力三轴度的减小和应变率的减小而增加。     

Mode-3 spontaneous crack propagation in symmetric functionally graded materials

The effects of spatially varying the material properties on the mode-3 planar crack propagation characteristics are numerically investigated. The spectral scheme that is available for homogeneous materials is modified to account for the symmetrically varying material properties. Crack propagation in hardening, softening and unsymmetric type of functionally graded have been simulated. A parametric study was performed by systematically varying the material inhomogeneity length scale. Our study indicated that softening and unsymmetric graded materials reduce the resistance to fracture, while a hardening material offers higher fracture resistance with increase in inhomogeneity. Only the transient phase of crack propagation speed was affected by the material property variation, irrespective of whether the material was hardening, softening or an unsymmetric type. The crack always reached a quasi-steady-state velocity, which remained unaffected by the material property inhomogeneity.     

聚碳酸酯板裂纹顶端的弹塑性研究

本文研究了聚碳酸酯板的延性断裂特征;提出了修正的强化Dugdale条带屈服模型,并用实验-数值计算混合法确定了强化弹塑性材料裂纹顶端附近的应力场参数.实验测定结果与理论等色线,计算的裂纹前沿塑性区长度相符合.由此,验证了所提出的修正强化Dugdale条带屈服模型的合理性.     

加载速率对40Cr钢Ⅱ型动态断裂特性的影响

采用新型Ⅱ型动态断裂测试技术,对高强钢40Cr在高加载速率下的Ⅱ型动态断裂特性进行了测试研究。基于新设计的Ⅱ型动态断裂试样和分离式霍普金森压杆（split Hopkinson pressure bar, SHPB）技术,通过实验-数值方法确定了裂尖在加载过程中的应力强度因子曲线。采用应变片法确定了试样的起裂时间,最终得到40Cr的Ⅱ型动态断裂韧性值,并对其加载速率相关性和材料的失效机理进行了研究。结果表明,在1.08～5.53 TPa·m1/2/s的加载速率范围内,40Cr的Ⅱ型动态断裂韧性基本表现为与加载速率成正相关的变化趋势。通过对试样断口形貌的分析,确定了材料的失效模式及机理,发现随着加载速率的增加,存在拉伸型失效向绝热剪切型失效模式转变的现象。     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.     

Modelling of plastic flow localisation and damage development in friction stir welded 6005A aluminium alloy using physics based strain hardening law

Plastic flow localisation and ductile failure during tensile testing of friction stir welded aluminium specimens are investigated with a specific focus on modelling the local, finite strain, hardening response. In the experimental part, friction stir welds in a 6005A-T6 aluminium alloy were prepared and analysed using digital image correlation (DIC) during tensile testing as well as scanning electron microscopy (SEM) on polished samples and on fracture surfaces. The locations of the various regions of the weld were determined based on hardness measurements, while the flow behaviour of these zones was extracted from micro-tensile specimens cut parallel to the welding direction. The measured material properties and weld topology were introduced into a 3D finite element model, fully coupled with the damage model. A Voce law hardening model involving a constant stage IV is used within an enhanced Gurson type micro-mechanical damage model, accounting for void nucleation, growth and coalescence, as well as void shape evolution. The stage IV hardening, observed in Simar et al. (2010), was found to increase the stiffness during plastic flow localisation as well as to postpone the onset of fracture as determined by the void coalescence criterion. Furthermore, the presence of a second population of voids was concluded to strongly affect the fracture strain of the high strength regions of the welds. This modelling effort links the microstructure and process parameters to macroscopic parameters relevant to the optimisation of the welds.     

Experimental Research into Fracture of EN-AW 2024 and EW-AW 2007 Aluminum Alloy Specimens with Notches Subjected to Tension

The paper presents the results of experimental tests of elasto-plastic fracture of axially symmetric specimens with circumferential notches with various round notch radii at the notch’s root. The tests were conducted for two materials: EN-AW 2024 and EN-AW 2007 aluminum alloys. The specimens were subjected to monotonic uniaxial tensile test. Special attention was paid to the shape of the fracture surface, the change of critical force value and the maximum displacement, depending on the shape of the notch. Additionally, hardening curves are presented for the adopted materials and the results of Finite Element calculations of stress and strain fields in the specimens with different notch radii.     

薄壁柱壳在内部爆炸载荷下膨胀断裂的研究

以柱形爆炸装置为研究对象 ,讨论了其薄壁壳体在爆炸载荷下的动态断裂准则。给出了典型的FAE模型及在数值计算时必要的简化条件。对回收的壳壁破片进行了SEM扫描电镜观测 ,分析了金属材料在高应变率下膨胀断裂的微观机制。采用解耦的、考虑应变强化的粘塑性本构方程 ,根据损伤破坏理论推导出以临界应变为依据的改进断裂准则 ,计算表明断裂参数理论解与数值模拟结果一致 ,并得到了实验方面的例证。     

Decaying Turbulence in the Generalised Burgers Equation

This work is concerned with the derivation of optimal scaling laws, in the sense of matching lower and upper bounds on the energy, for a solid undergoing ductile fracture. The specific problem considered concerns a material sample in the form of an infinite slab of finite thickness subjected to prescribed opening displacements on its two surfaces. The solid is assumed to obey deformation-theory of plasticity and, in order to further simplify the analysis, we assume isotropic rigid-plastic deformations with zero plastic spin. When hardening exponents are given values consistent with observation, the energy is found to exhibit sublinear growth. We regularize the energy through the addition of nonlocal energy terms of the strain-gradient plasticity type. This nonlocal regularization has the effect of introducing an intrinsic length scale into the energy. Under these assumptions, ductile fracture emerges as the net result of two competing effects: whereas the sublinear growth of the local energy promotes localization of deformation to failure planes, the nonlocal regularization stabilizes this process, thus resulting in an orderly progression towards failure and a well-defined specific fracture energy. The optimal scaling laws derived here show that ductile fracture results from localization of deformations to void sheets, and that it requires a well-defined energy per unit fracture area. In particular, fractal modes of fracture are ruled out under the assumptions of the analysis. The optimal scaling laws additionally show that ductile fracture is cohesive in nature, that is, it obeys a well-defined relation between tractions and opening displacements. Finally, the scaling laws supply a link between micromechanical properties and macroscopic fracture properties. In particular, they reveal the relative roles that surface energy and microplasticity play as contributors to the specific fracture energy of the material.     

On the identification of a high-resolution multi-linear post-necking strain hardening model

The Finite Element Model Updating (FEMU) technique is an inverse method that enables to arrive at a complete solution to the problem of diffuse necking of a thick tensile specimen. Conventionally, FEMU relies on the identification of a phenomenological strain hardening law that inherently limits the accuracy of the method due to the predefined character of the adopted strain hardening law. A high-resolution multi-linear post-necking strain hardening model enables to describe more generically the actual strain hardening behaviour. A numerical concept study is used to scrutinise the identification of such a model using FEMU. It is shown that, unlike progressive identification strategies, a global identification strategy followed by a smoothing operation based on area conservation yields sufficiently accurate results. To study the experimental feasibility, the latter strategy is used to identify the post-necking strain hardening behaviour of a thick S690QL high-strength steel. To this purpose, a notched tensile specimen was loaded up to fracture, while the elongation was measured using Digital Image Correlation (DIC). It is shown that the global identification strategy suffers from experimental noise associated with DIC and the load signal.