Dynamic void growth in rate-sensitive plastic solids

A dynamic void growth model in rate-sensitive plastic materials is derived. The constitutive relation of the matrix is in the overstress form proposed by Perzyna. When the rate of deformation sensitive parameter tends to zero, the Gurson model is retrieved. When the porous material element is under triaxial tension, the Carroll-Holt and Johnson models are retrieved. The normality condition of the plastic rate of deformation to the dynamic loading surface at constant equivalent rates of deformation (with the volumetric part also taken into account) is discussed, and it is shown that the normality rule no longer exists in general. Finally, an approximate expression of the dynamic loading surface that may be convenient for engineering applications is suggested.     

Two mechanisms of ductile fracture: void by void growth versus multiple void interaction

Two distinct mechanisms of crack initiation and advance by void growth have been identified in the literature on the mechanics of ductile fracture. One is the interaction a single void with the crack tip characterizing initiation and the subsequent void by void advance of the tip. This mechanism is represented by the early model of Rice and Johnson and the subsequent more detailed numerical computations of McMeeking and coworkers on a single void interacting with a crack tip. The second mechanism involves the simultaneous interaction of multiple voids on the plane ahead of the crack tip both during initiation and in subsequent crack growth. This mechanism is revealed by models with an embedded fracture process zone, such as those developed by Tvergaard and Hutchinson. While both mechanisms are based on void nucleation, growth and coalescence, the inferences from them with regard to crack growth initiation and growth are quantitatively different. The present paper provides a formulation and numerical analysis of a two-dimensional plane strain model with multiple discrete voids located ahead of a pre-existing crack tip. At initial void volume fractions that are sufficiently low, initiation and growth is approximately represented by the void by void mechanism. At somewhat higher initial void volume fractions, a transition in behavior occurs whereby many voids ahead of the tip grow at comparable rates and their interaction determines initiation toughness and crack growth resistance. The study demonstrates that improvements to be expected in fracture toughness by reducing the population of second phase particles responsible for nucleating voids cannot be understood in terms of trends of one mechanism alone. The transition from one mechanism to the other must be taken into account.     

Effects of void band orientation and crystallographic anisotropy on void growth and coalescence

The effects of void band orientation and crystallographic anisotropy on void growth and linkage have been investigated. 2D model materials were fabricated by laser drilling a band of holes into the gage section of sheet tensile samples using various orientation angles with respect to the tensile axis normal. Both copper and magnesium sheets have been studied in order to examine the role of crystallographic anisotropy on the void growth and linkage processes. The samples were pulled in uniaxial tension inside the chamber of an SEM, enabling a quantitative assessment of the growth and linkage processes. The void band orientation angle has a significant impact on the growth and linkage of the holes in copper. As the void band orientation angle is increased from 0° to 45°, the processes of coalescence and linkage are delayed to higher strain values. Furthermore, the mechanism of linkage changes from internal necking to one dominated by shear localization. In contrast, the void band orientation does not have a significant impact on the void growth and linkage processes in magnesium. Void growth in these materials occurs non-uniformly due to interactions between the holes and the microstructure. The heterogeneous nature of deformation in magnesium makes it difficult to apply a coalescence criterion based on the void dimensions. Furthermore, the strain at failure does not show a relationship with the void band orientation angle. Failure associated with twin and grain boundaries interrupts the plastic growth of the holes and causes rapid fracture. Therefore, the impact of the local microstructure outweighs the effects of the void band orientation angle in this material.     

Influences of microscope size effect of matrix on plastic potential and void growth of porous materials

Based on approximate theoretical analyses on a typical spherical cell containing a spherical microvoid, the influences of matrix materials' microscopic scale on the macroscopic constitutives potential theory of porous material and microvoid growth have been investigated in detail. By assuming that the plastic deformation behavior of matrix materials follows the strain gradient (SG) plastic theory involving the stretch and rotation gradients, the ratio (λ=l/a) of the matrix materials' intrinsic characteristic lengthl to the microvoid radiusa is introduced into the plastic constitutives potential and the void growth law. The present results indicate that, when the radiusa of microvoids is comparable with the intrinsic characteristic lengthl of the matrix materials, the influence of microscopic size effect on neither the constitutive potential nor the microvoid evolution predicted can be ignored. And when the void radiusa is much lager than the intrinsic characteristic lengthl of the matrix materials, the present model can retrogress automatically to the improved Gurson model that takes into account the strain hardening effect of matrix materials. Project supported by the National Natural Science Foundation of China (No. A10102006).     

The dynamic growth of a void in a plastic material and an application to fracture

The problem of the dynamic growth of a spherical void in applied stress and strain-rate fields is solved approximately using a variational method. The solution is applied to a model of fracture by void growth from brittle or incoherent inclusions. It is shown that the speed of fracture is limited by inertia near the crack tip. Some numerical results are obtained from data for high strength steels.     

A note on the effect of surface energy and void size to void growth

Recent studies revealed that rapid void growth is the dominant failure mechanism in an elasto-plastic solid under high mean tensile stress. This paper studies the effect of the surface energy and void size to the void growth. The models of a thick spherical shell and a thick spherical column in void growth are analyzed and numerically estimated. The main conclusion from this study is that, for typical metals, the surface energy effect is negligible for voids larger than 100 nm in size, but it may become significant when the void size is on the order of 10 nm.     

Effect of yield surface curvature and void nucleation on plastic flow localization

The effect of void nucleation is incorporated in a recently proposed material model that accounts for a combination of kinematic hardening and isotropic hardening of a porous ductile material. Since each of plastic dilatancy, void nucleation and yield surface curvature have a strong influence on predictions of plastic flow localization, the present material model can be used to study the interaction of these effects. Nucleation controlled by the plastic strain as well as nucleation controlled by the maximum normal stress on the particle-matrix interface are modelled. The predictions of the material model, for various combinations of parameters, are illustrated by analyses of shear band formation under plane strain or axisymmetric conditions, and by analyses of necking in biaxially stretched sheets.     

RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales

The present paper extends the Gurson and GLD models [Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth, Part I—yield criteria and flow rules for porous ductile media. J. Mech. Phys. Solids 99, 2–15; Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290–297] to involve the coupled effects of void size and void shape on the macroscopic yield behavior of non-linear porous materials and on the void growth. A spheroidal representative volume element (RVE) under a remote axisymmetric homogenous strain boundary condition is carefully analyzed. A wide range of void aspect ratios covering the oblate spheroidal, spherical and prolate spheroidal void are taken into account to reflect the shape effect. The size effect is captured by the Fleck–Hutchinson phenomenological strain gradient plasticity theory [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, pp. 295–361]. A new size-dependent damage model like the Gurson and GLD models is developed based on the traditional minimum plasticity potential principle. Consequently, the coupled effects of void size and void shape on yield behavior of porous materials and void growth are discussed in detail. The results indicate that the void shape effect on the yield behavior of porous materials and on the void growth can be modified dramatically by the void size effect and vice versa. The applied stress triaxiality plays an important role in these coupled effects. Moreover, there exists a cut-off void radius r_c, which depends only on the intrinsic length l₁ associated with the stretch strain gradient. Voids of effective radius smaller than the critical radius r_c are less susceptible to grow. These findings are helpful to our further understanding to some impenetrable micrographs of the ductile fracture surfaces.     

Micromechanical finite element calculations of temperature and void configuration effects on void growth and coalescence

We present micromechanical finite element results that quantify coalescence effects based upon temperature and different spatial arrangements of voids. We propose a critical intervoid ligament distance (ILD) to define void coalescence that is derived from micromechanical simulations in which void volume fraction evolves as a function of strain. Several parameters were varied using the temperature and strain rate internal variable plasticity model of Bammann–Chiesa–Johnson to determine the coalescence effects. The parameters include two types of materials with different work hardening rates (304L stainless steel and 6061T6 aluminum), three different temperatures (298, 400, and 600 K), several boundary conditions (force and displacement: uniaxial, plane strain, and biaxial), type of element used (plane strain and axisymmetric), different ILDs, and the number of voids (one and two void configurations). The present study provides a basis for macroscale modeling of coalescence which is briefly discussed.     

Coupling effects of void size and void shape on the growth of prolate ellipsoidal microvoid

The combined effects of void size and void shape on the void growth are studied by using the classical spectrum method. An infinite solid containing an isolated prolate spheroidal void is considered to depict the void shape effect and the Fleck-Hutchinson phenomenological strain gradient plasticity theory is employed to capture the size effects. It is found that the combined effects of void size and void shape are mainly controlled by the remote stress triaxiality. Based on this, a new size-dependent void growth model similar to the Rice-Tracey model is proposed and an important conclusion about the size-dependent void growth is drawn: the growth rate of the void with radius smaller than a critical radius r_c may be ignored. It is interesting that r_c is a material constant independent of the initial void shape and the remote stress triaxiality.The project supported by the National Natural Science Foundation of China (A10102006) and the New Century Excellent Talents in Universities of China. The English text was polished by Keren Wang.     

Theoretical study of void closure in nonlinear plastic materials

Void closing from a spherical shape to a crack is investigated quantitatively in the present study. The constitutive relation of the Void-free matrix is assumed to obey the Norton power law. A representative volume element （RVE） which includes matrix and void is employed and a Rayleigh-Ritz procedure is developed to study the deformation-rates of a spherical void and a penny-shaped crack. Based on an approximate interpolation scheme, an analytical model for void closure in nonlinear plastic materials is established. It is found that the local plastic flows of the matrix material are the main mechanism of void deformation. It is also shown that the relative void volume during the deformation depends on the Norton exponent, on the far-field stress triaxiality, as well as on the far-field effective strain. The predictions of void closure using the present model are compared with the corresponding results in the literature, showing good agreement. The model for void closure provides a novel way for process design and optimization in terms of elimination of voids in billets because the model for void closure can easily be applied in the CAE analysis.     

The effect of straining mode on void growth

Large strain finite element method is employed to investigate the effect of straining mode on void growth. Axisymmetric cell model embedded with spherical void is controlled by constant triaxiality loading, while plane-stress model containing a circular void is loaded by constant ratio of straining. Elastic-plastic material is used for the matrix in both cases. It is concluded that, besides the known effect of triaxiality, the straining mode which intensifies the plastic concentration around the void is also a void growth stimulator. Experimental results are cited to justify the computation results. This paper is jointly supported by the National Natural Science Foundation of China (19872064), the Chinese Academy of Sciences (KJ951-1-201) and the Laboratory for Nonlinear mechanics of Continuous Media of the Institute of Mechanics     

空孔与运动裂纹相互作用的动焦散线实验研究

为研究预制裂纹不同偏移距离时运动裂纹与空孔的相互作用规律,采用动态焦散线实验系统,将预制裂纹的偏移距离设定为唯一变量,对含空孔的有机玻璃(PMMA)试件进行冲击三点弯实验。研究表明,存在两个临界距离：（6 mm (2 R )、9 mm (3 R )）,在该偏移距离下,裂纹扩展轨迹、动态断裂特性发生显著改变：(1) 预制裂纹偏移距离不大于3 mm时,裂纹贯穿空孔,发生二次起裂,且二次起裂的速度与应力强度因子显著大于一次起裂,无偏移时裂纹轨迹的分形维数为最小值;(2) 偏移距离增大至6 mm时,裂纹不再贯穿空孔,空孔对裂纹先吸引后排斥,裂纹速度与应力强度因子先减小后增大,裂纹轨迹的分形维数达到最大值;(3) 偏移距离大于6 mm时,空孔对裂纹的吸引作用逐渐减小,大于9 mm后,空孔对裂纹的吸引不再显著,裂纹起裂后即向落锤加载方向扩展直至贯穿试件。

     

On ductile fracture initiation toughness: Effects of void volume fraction,void shape and void distribution

This paper studies the effects of the initial relative void spacing, void pattern, void shape and void volume fraction on ductile fracture toughness using three-dimensional, small scale yielding models, where voids are assumed to pre-exist in the material and are explicitly modeled using refined finite elements. Results of this study can be used to explain the observed fracture toughness anisotropy in industrial alloys. Our analyses suggest that simplified models containing a single row of voids ahead of the crack tip is sufficient when the initial void volume fraction remains small. When the initial void volume fraction becomes large, these simplified models can predict the fracture initiation toughness (J_Ic) with adequate accuracy but cannot predict the correct J–R curve because they over-predict the interaction among growing voids on the plane of crack propagation. Consequently, finite element models containing multiple rows of voids should be used when the material has large initial void volume fraction.     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

Scale-dependent plasticity potential of porous materials and void growth

In the present paper, a boundary value problem about the macroscopic response and its microscopic mechanism of a representative spherical cell with a spherical microvoid under axisymmetric triaxial tension has been theoretically investigated. To capture the size effects of local plastic deformation in the matrix, the strain gradient constitutive theory including the rotation and the stretch gradients developed by Fleck and Hutchinson [Strain gradient plasticity, in: J.W. Hutchinson, T.Y. Wu (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, 1997, p. 295] is adopted. By means of the principle of minimum plasticity potential and the Lagrange multipliers method, the self-contained displacement field within the matrix has been computationally determined. Based on these, a size-dependent constitutive potential theory for porous material has been developed. The results indicate clearly that the microvoid evolution predicted by the present constitutive model displays very significant dependences on the void size especially when the radius a of microvoids is comparable with the intrinsic characteristic length l of the matrix. And when the void radius a is much lager than l, the present model can retrogress automatically to the Gurson model improved by Wang and Qin [Acta Mech. Solid. Sin. 10 (1989) 127, in Chinese].     

Cylindrical void in a rigid-ideally plastic single crystal II: Experiments and simulations

Experimental results and finite element simulations of plastic deformation around a cylindrical void in single crystals are presented to compare with the analytical solutions in a companion paper: Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals [Kysar, J.W., Gan, Y.X., Mendez-Arzuza, G., 2005. Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals, International Journal of Plasticity, 21, 1481–1520]. In the first part of the present paper, the theoretical predictions of the stress and deformation field around a cylindrical void in face-centered cubic (FCC) single crystals are briefly reviewed. Secondly, electron backscatter diffraction results are presented to show the lattice rotation discontinuities at boundaries between regions of single slip around the void as predicted in the companion paper. In the third part of the paper, the finite element method has been employed to simulate the anisotropic plastic deformation behavior of FCC single crystals which contain cylindrical voids under plane strain condition. The results of the simulation are in good agreement with the prediction by the anisotropic slip line theory.     

Constitutive potential for void-containing nonlinear materials and void growth

Based on an analysis of the deformation of an isolated void in a finite nonlinear viscous material, we establish the constitutive potentials for voided nonlinearly viscous materials, from which the related curves of the macroscopic stress, the average flow stress of the matrix material and the void volume fraction f are derived. However, the theory applies equally well to small strain, rate-independent J₂ deformation theory solid. By considering the effects of the strain-hardening directly, a modifies Gurson equation are developed. Finally, we calculate the void relative growth-rates for the nonlinear materials, and in good agreement with existed numerical results.     

On plastic localization and failure in plane strain and round void containing tensile bars

Gurson's constitutive model describing effects of void nucleation and growth in ductile metals is adopted for numerical analysis of plastic localization and failure modes in axisymmetric and plane strain tensile tests. A simple numerically oriented treatment of void coalescence is utilized. The analysis is performed in the framework of the bifurcation approach and no initial imperfections are needed to trigger localizations. The unsymmetric eigenvalue problem is solved by using a perturbation scheme which essentially reduces the problem to a symmetric eigenvalue formulation for a recently proposed “symmetrized comparison solid” of nonassociated plasticity plus the easily computable perturbation correction. The results obtained are shown to qualitatively reproduce the essential features observed experimentally in uniaxial tests.     

FCC晶体中不均匀变形对空洞长大的影响

通过编制率相关有限元用户子程序,采用一个单胞模型研究了FCC晶体中孔洞在单晶及晶界的长大行为,分析了由于晶体取向及变形失配对孔洞长大和聚合的影响。研究结果表明：孔洞的形状和长大方向与晶体取向密切相关;晶界上孔洞的长大速度大于单晶中孔洞的长大速度;晶粒间的变形失配加速了晶界上孔洞的长大趋势,因而使材料易发生沿晶断裂,随着晶粒间取向因子差异的增加,孔洞越易沿着晶界长大。