Plastic potential and yield function of porous materials with aligned and randomly oriented spheroidal voids

Based on an energy approach, the plastic potential and yield function of a porous material containing either aligned or randomly oriented spheroidal voids are developed at a given porosity and pore shape. The theory is applicable to both elastically compressible and incompressible matrix and, it is proved that, in the incompressible case, the theory with spherical and aligned spheroidal voids also coincides with Ponte Castaneda's bounds of the Hashin-Shtrikman and Willis types, respectively. Comparison is also made between the present theory and those of Gurson and Tvergaard, with a result giving strong overall support of this new development. For the influence of pore shape, the yield function and therefore the stress-strain curve of the isotropic porous material are found to be stiffest when the voids are spherical, and those associated with other pore shapes all fall below these values, the weakest one being caused by the disc-shaped voids. The transversely isotropic nature of the yield function and stress-strain curves of a porous material containing aligned pores are also demonstrated as a function of porosity and pore shape, and it is further substantiated with a comparison with an exact, local analysis when the void shape becomes cylindrical.     

The coupled effects of plastic strain gradient and thermal softening on the dynamic growth of voids

This paper examines the combined effects of temperature, strain gradient and inertia on the growth of voids in ductile fracture. A dislocation-based gradient plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99] is applied, and temperature effects are incorporated. Since a strong size-dependence is introduced into the dynamic growth of voids through gradient plasticity, a cut-off size is then set by the stress level of the applied loading. Only those voids that are initially larger than the cut-off size can grow rapidly. At the early stages of void growth, the effects of strain gradients greatly increase the stress level. Therefore, thermal softening has a strong effect in lowering the threshold stress for the unstable growth of voids. Once the voids start rapid growth, however, the influence of strain gradients will decrease, and the rate of dynamic void growth predicted by strain gradient plasticity approaches that predicted by classical plasticity theories.     

The evolution of voids in the plasticity strain hardening gradient materials

The main aim of this paper is to opens out the meso-mechanism of void growth and coalescence in the matrix materials with graded strain-hardening exponent distribution. For this end, detailed finite element computations of a representative cylindrical cell containing a spherical void have been carried out. According to the FE analyses, significant effects of the strain-hardening exponent gradient (SEG) in the matrix on the void growth and coalescence are revealed: (1) In the homogeneous materials, the void growth and coalescence are slightly dependent on the strain-hardening exponent, however, the SEG distribution in the matrix can increase remarkably the void growth rate and decrease seriously the void coalescence strain. (2) The critical void shapes in the homogeneous materials are mainly governed by the macroscopic stress triaxiality, but due to earlier plastic flow localization in the softer matrix layer, the SEG distribution in the matrix has very significant effects on the deformed void shapes, especially when the stress triaxiality is lower. (3) When the triaxial stress levels are lower, in the homogeneous materials, the shape change mode of the void evolution is dominate so the void growth rate is very low; however, the SEG distribution in the matrix can bring the volume change mode out, as a result of increasing the void growth rate. (4) Comparisons of the numerical results with the existing damage model indicate that the classic damage model cannot give satisfying prediction to the void growth in both the homogeneous strain-hardening matrix and the SEG materials. On the basis of large numbers of numerical computations, a new damage model, which can uniformly describe the void growing in the homogeneous and plasticity gradient materials, is suggested. A mass of element computations have validated that the new damage model can give satisfying agreement with the FE results of cell model.     

Porous materials with two populations of voids under internal pressure: II. Growth and coalescence of voids

This study is devoted to the mechanical behaviour of polycrystalline materials with two populations of voids, small spherical voids located inside the grains and larger spheroidal voids located at the grain boundaries. In part I of the work, instantaneous effective stress–strain relations were derived for fixed microstructure. In this second part, the evolution of the microstructure is addressed. Differential equations governing the evolution of the microstructural parameters in terms of the applied loading are derived and their integration in time is discussed. Void growth results in a global softening of the stress–strain response of the material. A simple model for the prediction of void coalescence is proposed which can serve to predict the overall ductility of polycrystalline porous materials under the combined action of thermal dilatation and internal pressure in the voids.     

The effects of thermal softening and heat conduction on the dynamic growth of voids

This paper seeks to examine the dynamic growth of a single void in an elastic–plastic medium through analytical and numerical approaches. Particular attention is paid to the instability of void growth, and to the effects of inertia, thermal softening and heat conduction. A critical stress is known to exist for the unstable growth of voids. The dependence of this critical stress on material properties is examined, and this critical stress is demonstrated to correspond to the lower limit for the ductile spall strength in many materials. The effects of heat conduction on the dynamic growth of voids strongly depend on the time and length scales in the early stages of the dynamic void growth.     

Mesoscopic natural element analysis of elastic moduli, yield stress and fracture of solids containing a number of voids

A two-dimensional mesoscale simulation method based on the natural element method, which is a kind of meshless method, is developed and applied to the analysis of overall elastic moduli, macro yield stress and void-linking fracture. The calculated results are compared with the theoretical solutions for overall elastic moduli, the experimental macro yield stress for aluminum and brass as well as the improved Gurson’s yield function, and the experimental void linking fracture to show the validity of the proposed method.     

The Piola-Kirchhoff stress may depend linearly on the deformation gradient

It is shown that, whenever the residual stress does not vanish, the response function delivering the Piola-Kirchhoff stress in terms of the deformation gradient may be genuinely linear, and yet independent of the observer; moreover, an explicit representation formula for such a function is obtained.     

The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids

A model of a mixture of bone tissue and bioresorbable material with voids was used to numerically analyze the physiological balance between the processes of bone growth and resorption and artificial material resorption in a plate-like sample. The adopted model was derived from a theory for the behavior of porous solids in which the matrix material is linearly elastic and the interstices are void of material. The specimen—constituted by a region of bone living tissue and one of bioresorbable material—was acted by different in-plane loading conditions, namely pure bending and shear. Ranges of load magnitudes were identified within which physiological states become possible. Furthermore, the consequences of applying different loading conditions are examined at the end of the remodeling process. In particular, maximum value of bone and material mass densities, and extensions of the zones where bone is reconstructed were identified and compared in the two different load conditions. From the practical view point, during surgery planning and later rehabilitation, some choice of the following parameters is given: porosity of the graft, material characteristics of the graft, and adjustment of initial mixture tissue/bioresorbable material and later, during healing and remodeling, optimal loading conditions.     

A unified criterion for the growth and coalescence of microvoids

A yield criterion is developed which unifies void growth and void coalescence theories. Standard void growth theory assumes that plastic flow is diffuse, if not prevalent everywhere within the matrix of the elementary cell considered. On the other hand, void coalescence theory assumes states of post-localized plasticity whereby plastic flow is restricted to intervoid ligaments. The new theory accommodates both scenarios through some appropriate choice of microscopic velocity fields. An important implication for actual evolution problems is a seamless transition from void growth to void coalescence. This is in contrast with previous hybrid approaches whereby abrupt transitions are associated with the presence of unavoidable corners in the effective yield surface. More generally, the new criterion is applicable to describe yielding in porous metal plasticity for both low and high void volume fractions.     

The measurement of the yield stress of liquids

An analysis has been made of different methods of measuring the yield stress of liquids. In the experimental program, a comparison is made of measurements of the yield stress using an Instron 3250 Rheometer in several geometries (cone-plate, parallel plate and eccentric disk) in shear flow and stress relaxation, a laboratory vane and a cone penetrometer. Good agreement has been obtained between the shear flow data and the laboratory vane, while stress relaxation appears to underestimate the yield stress.     

梯度润湿表面上液滴定向迁移及合并行为的格子Boltzmann模拟

采用改进的格子Boltzmann方法,对梯度润湿性表面上液滴的定向迁移及合并行为进行了数值模拟,该模型在精度和稳定性上都有很大改善,同时,研究了梯度润湿性表面上液滴定向迁移和合并的动力学特性,并对液滴尺寸及润湿梯度对液滴动力学特性的影响规律进行了分析。数值结果表明,液滴在梯度润湿性表面运动时会发生形变,且动态接触角逐渐减小。润湿梯度对液滴定向迁移行为有显著影响,润湿梯度越大,液滴左右侧接触线位移越大,润湿长度增加越快。但是液滴尺寸对接触线位移影响较小。润湿梯度对液桥宽度基本无影响,但对液滴初始合并时间有显著影响。     

Effects of voids on postbuckling delamination growth in unidirectional composites

This work examines the effects of manufacturing induced voids on the postbuckling behavior of delaminated unidirectional composites. In the finite element model developed, a through-width delamination is introduced close to one surface of a flat panel, and a void is placed in the delamination plane ahead of each delamination front. The panel is subjected to compression in the fiber direction. The postbuckling delamination growth is studied by calculating the strain energy release rate (SERR) using the virtual crack closure technique. Local stress analyses of the region near the delamination front are also performed to further investigate the void effects. It is found that although the presence of void does not significantly alter the postbuckling transverse displacement of the delaminated panel, the induced stress perturbation by the void affects the SERR. The Mode II SERR as well as the total SERR increase depending on the size of the void and its distance from the delamination front. Since the Mode I SERR shows non-monotonic behavior with the applied load, the effects of voids are studied on its maximum value.     

The influence of matrix viscoelasticity on the rheology of polymer blends

We examine the effects of matrix phase viscoelasticity on the rheological modeling of polymer blends with a droplet morphology. Two contravariant, second-rank tensor variables are adopted along with the translational momentum density of the fluid to account for viscoelasticity of the matrix phase and the ellipsoidal droplet shapes. The first microstructural variable is a conformation tensor describing the average extension and orientation of the molecules in the matrix phase. The other microstructural variable is a configuration tensor to account for the average shape and orientation of constant-volume droplets. A Hamiltonian framework of non-equilibrium thermodynamics is then adopted to derive a set of continuum equations for the system variables. This set of equations accounts for local conformational changes of the matrix molecules due to droplet deformation and vice versa. The model is intended for dilute blends of both oblate and prolate droplets, and droplet breakup and coalescence are not taken into account. Only the matrix phase is considered as viscoelastic; i.e., the droplets are assumed to be Newtonian. The model equations are solved for various types of homogeneous deformations, and microstructure/rheology relationships are discussed for transient and steady-state conditions. A comparison with other constrained-volume rheological models and experimental data is made as well.     

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.     

Experimental study of microhole growth and coalescence in ductile materials

The growth and coalescence of two microholes in copper foil were studied experimentally byin situ tensile tests under a scanning electronic microscope. Two microholes of 15–35 μm in diameter were arranged in different distances and orientations. It was found that the mechanisms of microhole evolution were represented by slipping band creation, and then crack initiation and propagation along the slipping bands in ligament. The process of microhole growth and coalescence was influenced by the inter-center distance and orientation of microholes. The critical surface of microholes at coalescence is about 2–2.5 times that of the initial one. The variation of both the inter-center distance and orientation depends on the initial angle.     

Atomistic insights into dislocation-based mechanisms of void growth and coalescence

One of the low-temperature failure mechanisms in ductile metallic alloys is the growth of voids and their coalescence. In the present work we attempt to obtain atomistic insights into the mechanisms underpinning cavitation in a representative metal, namely Aluminum. Often the pre-existing voids in metallic alloys such as Al have complex shapes (e.g. corrosion pits) and the defromation/damage mechanisms exhibit a rich size-dependent behavior across various material length scales. We focus on these two issues in this paper through large-scale calculations on specimens of sizes ranging from 18 thousand to 1.08 million atoms. In addition to the elucidation of the dislocation propagation based void growth mechanism we highlight the observed length scale effect reflected in the effective stress-strain response, stress triaxiality and void fraction evolution. Furthermore, as expected, the conventionally used Gurson's model fails to capture the observed size-effects calling for a mechanistic modification that incorporates the mechanisms observed in our (and other researchers') simulation. Finally, in our multi-void simulations, we find that, the splitting of a big void into a distribution of small ones increases the load-carrying capacity of specimens. However, no obvious dependence of the void fraction evolution on void coalescence is observed.     

Lattice orientation effects on void growth and coalescence in fcc single crystals

Void growth and coalescence in fcc single crystals were studied using crystal plasticity under uniaxial and biaxial loading conditions and various orientations of the crystalline lattice. A 2D plane strain unit cell with one and two cylindrical voids was employed using three-dimensional 12 potentially active slip systems. The results were compared to five representative orientations of the tensile axis on the stereographic triangle. For uniaxial tension conditions, the void volume fraction increase under the applied load is strongly dependent on the crystallographic orientation with respect to the tensile axis. For some orientations of the tensile axis, such as [1 0 0] or [1 1 0], the voids exhibited a growth rate twice as fast compared with other orientations ([1 0 0], [2 1 1]). Void growth and coalescence simulations under uniaxial loading indicated that during deformation along some orientations with asymmetry of the slip systems, the voids experienced rotation and shape distortion, due mainly to lattice reorientation. Coalescence effects are shown to diminish the influence of lattice orientation on the void volume fraction increase, but noteworthy differences are still present. Under biaxial loading conditions, practically all differences in the void volume fraction for different orientations of the tensile axes during void growth vanish. These results lead to the conclusion that at microstructural length scales in regions under intense biaxiality/triaxiality conditions, such as crack tip or notched regions, the plastic anisotropy due to the initial lattice orientation has only a minor role in influencing the void growth rate. In such situations, void growth and coalescence are mainly determined by the stress triaxiality, the magnitude of accumulated strain, and the spatial localization of such plastic strains.