Modeling of ductile fracture: Significance of void coalescence

In this paper void coalescence is regarded as the result of localization of plastic flow between enlarged voids. We obtain the failure criterion for a representative material volume (RMV) in terms of the macroscopic equivalent strain (E_c) as a function of the stress triaxiality parameter (T) and the Lode angle (θ) by conducting systematic finite element analyses of the void-containing RMV subjected to different macroscopic stress states. A series of parameter studies are conducted to examine the effects of the initial shape and volume fraction of the primary void and nucleation, growth, and coalescence of secondary voids on the predicted failure surface E_c(T, θ). As an application, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a porous plasticity model is used to describe the void growth process and the expression for E_c is calibrated using experimental data. The calibrated computational model is applied to predict crack extension in fracture specimens having various initial crack configurations and the numerical predictions agree very well with experimental measurements.     

Void growth and interaction experiments: Implications to the optimal straining rate in superplastic forming

In this work, a set of parametric experiments was conducted on a superplastic material (eutectic tin–lead alloy) with one or more pre-drilled holes in each specimen. The small-sized holes were for simulating microvoids that occur and grow during superplastic forming. All holes were axially aligned with the tensile axis. The results revealed an increase in ductility with the number of holes up to 10 holes and a decrease thereafter. The ductility enhancement was explained based on the m-curve as due to a rise in the strain rate sensitivity locally around the holes. The decrease was explained due to strong void interaction that resulted in shear failure. This was further verified by a separate set of experiments of only two interacting voids with various interspacing. Finally, the void size versus applied strain was fully characterized and the results supported the ductility observations.     

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.     

Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms

This work presents a model to represent ductile failure (i.e. failure controlled by nucleation, growth and coalescence) of materials whose irreversible deformation is controlled by several plastic or viscoplastic deformation mechanisms. In addition work hardening may result from both isotropic and kinematic hardening. Damage is represented by a single variable representing void volume fraction. The model uses an additive decomposition of the plastic strain rate tensor. The model is developed based on the definition of damage dependant effective scalar stresses. The model is first developed within the generalized standard material framework and expressions for Helmholtz free energy, yield potential and dissipation potential are proposed. In absence of void nucleation, the evolution of the void volume fraction is governed by mass conservation and damage does not need to be represented by state variables. The model is extended to account for void nucleation. It is implemented in a finite element software to perform structural computations. The model is applied to three case studies: (i) failure by void growth and coalescence by internal necking (pipeline steel) where plastic flow is either governed by the Gurson–Tvergaard–Needleman model or the Thomason model, (ii) creep failure (Grade 91 creep resistant steel) where viscoplastic flow is controlled by dislocation creep or diffusional creep and (iii) ductile rupture after pre-compression (aluminum alloy) where kinematic hardening plays an important role.     

Micromechanics, macromechanics and constitutive modeling of the elasto-viscoplastic deformation of rubber-toughened glassy polymers

Under certain conditions, such as sufficiently low temperatures, high loading rates and/or highly triaxial stress states, glassy polymers display an unfavorable characteristic—brittleness. A technique used for reducing the brittleness (increasing the fracture toughness) of these materials is rubber toughening. While there is significant qualitative understanding of the mechanical behavior of rubber-toughened polymers, quantitative modeling tools for the large-strain deformation of rubber-toughened glassy polymers are largely lacking.In this paper, we develop a suite of numerical tools to investigate the mechanical behavior of rubber-toughened glassy polymers, with emphasis on rubber-toughened polycarbonate. The rubber particles are modeled as voids in view of their deformation-induced cavitation early during deformation. A three-dimensional micromechanical model of the heterogeneous microstructure is developed to study the effects of initial rubber particle (void) volume fraction on the underlying elasto-viscoplastic deformation mechanisms in the material, and how these mechanisms influence the macroscopic response of the material. A continuum-level constitutive model is developed for the large-strain elasto-viscoplastic deformation of porous glassy polymers, and it is calibrated against micromechanical modeling results for porous polycarbonate. The constitutive model can be used to study various boundary value problems involving rubber-toughened (porous) glassy polymers. As an example, the case of an axisymmetric notched bar is simulated for the case of polycarbonate with varying levels of initial porosity. The quality of the constitutive model calibration is assessed using a multi-scale modeling approach.     

A physically-based constitutive model for anisotropic damage in rubber-toughened glassy polymers during finite deformation

The present work focuses on the development of a physically-based model for large deformation stress-strain response and anisotropic damage in rubber-toughened glassy polymers. The main features leading to a microstructural evolution (regarding cavitation, void aspect ratio, matrix plastic anisotropy and rubbery phase deformation) in rubber-toughened glassy polymers are introduced in the proposed constitutive model. The constitutive response of the glassy polymer matrix is modelled using the hyperelastic-viscoplastic model of [Boyce et al., 1988] and [Boyce et al., 2000]. The deformation mechanisms of the matrix material are accounted for by two resistances: an elastic-viscoplastic isotropic intermolecular resistance acting in parallel with a visco-hyperelastic anisotropic network resistance, each resistance being modified to account for damage effects by void growth with a variation of the void aspect ratio. The effective contribution of the hyperelastic particles to the overall composite behaviour is taken into account by treating the overall system in a composite scheme framework. The capabilities of the proposed constitutive model are checked by comparing experimental data with numerical simulations. The deformation behaviour of rubber-toughened poly(methyl methacrylate) was investigated experimentally in tension at a temperature of 80 °C and for different constant true strain rates monitored by a video-controlled technique. The reinforcing phase is of the soft core-hard shell type and its diameter is of the order of one hundred nanometers. The particle volume fraction was adjusted from 15% to 45% by increments of 5%. The stress-strain response and the inelastic volumetric strain are found to depend markedly on particle volume fraction. For a wide range of rubber volume fractions, the model simulations are in good agreement with the experimental results. Finally, a parametric analysis demonstrates the importance of accounting for void shape, matrix plastic anisotropy and rubber content.     

COUPLING EFFECTS OF VOID SHAPE AND VOID SIZE ON THE GROWTH OF AN ELLIPTIC VOID IN A FIBER-REINFORCED HYPER-ELASTIC THIN PLATE

The growth of a prolate or oblate elliptic micro-void in a fiber reinforced anisotropic incompressible hyper-elastic rectangular thin plate subjected to uniaxial extensions is studied within the framework of finite elasticity. Coupling effects of void shape and void size on the growth of the void are paid special attention to. The deformation function of the plate with an isolated elliptic void is given, which is expressed by two parameters to solve the differential equation. The solution is approximately obtained from the minimum potential energy principle. Deformation curves for the void with a wide range of void aspect ratios and the stress distributions on the surface of the void have been obtained by numerical computation. The growth behavior of the void and the characteristics of stress distributions on the surface of the void are captured. The combined effects of void size and void shape on the growth of the void in the thin plate are discussed. The maximum stresses for the void with different sizes and different void aspect ratios are compared.     

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 the extension of the Gurson-type porous plasticity models for prediction of ductile fracture under shear-dominated conditions

One of the major drawbacks of the Gurson-type of porous plasticity models is the inability of these models to predict material failure under low stress triaxiality, shear dominated conditions. This study addresses this issue by combining the damage mechanics concept with the porous plasticity model that accounts for void nucleation, growth and coalescence. In particular, the widely adopted Gurson–Tvergaard–Needleman (GTN) model is extended by coupling two damage parameters, representing the volumetric damage (void volume fraction) and the shear damage, respectively, into the yield function and flow potential. The effectiveness of the new model is illustrated through a series of numerical tests comparing its performance with existing models. The current model not only is capable of predicting damage and fracture under low (even negative) triaxiality conditions but also suppresses spurious damage that has been shown to develop in earlier modifications of the GTN model for moderate to high triaxiality regimes. Finally the modified GTN model is applied to predict the ductile fracture behavior of a beta-treated Zircaloy-4 by coupling the proposed damage modeling framework with a recently developed J₂–J₃ plasticity model for the matrix material. Model parameters are calibrated using experimental data, and the calibrated model predicts failure initiation and propagation in various specimens experiencing a wide range of triaxiality and Lode parameter combinations.     

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.     

Void fraction for random loose packing of the cylindrical particles considering filling rate,material and shape

Particle packing is widely applied in organic pollutant adsorption, catalytic reaction, biomass combustion, nuclear cooling, and other scenarios. Due to the complexity of the shape, the studies on the void fraction of the cylindrical particles are not as thorough as the spherical particles. This study investigated the influence of the filling rate, material properties and sphericity on the void fraction of cylinders through experiments and simulation. DEM (discrete element method) was validated by the internal structures of the packing obtained by CT (computed tomography). Based on the logarithmic correlation between the void fraction and filling rate, an ingenious framework for predicting the void fraction of cylindrical particles was presented with two intermediate coefficients. By correlating the coefficients with the material property and sphericity, a novel void-fraction prediction model was established with R-squared of 0.996. The mechanism of void fraction under random loose packing for cylinders was eventually found in this study.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

Damage in edge cracking of rolled metal slabs

Materials get damaged under shear deformations. Edge cracking is one of the most serious damage to the metal rolling industry, which is caused by the shear damage process and the evolution of anisotropy. To investigate the physics of the edge cracking process, simulations of a shear deformation for an orthotropic plastic material are performed. To perform the simulation, this paper proposes an elasto-aniso-plastic constitutive model that takes into account the evolution of the orthotropic axes by using a bases rotation formula, which is based upon the slip process in the plastic deformation. It is found through the shear simulation that the void can grow in shear deformations due to the evolution of anisotropy and that stress triaxiality in shear deformations of (induced) anisotropic metals can develop as high as in the uniaxial tension deformation of isotropic materials, which increases void volume. This echoes the same physics found through a crystal plasticity based damage model that porosity evolves due to the grain-to-grain interaction. The evolution of stress components, stress triaxiality and the direction of the orthotropic axes in shear deformations are discussed.     

含空洞非线性材料的本构势和空洞扩展率

本文基于体胞模型的解析分析,分析了含空洞非线性材料的宏观本构势,得到了各种幂硬化指数下宏观应力和基体平均流变应力之间的相关曲线.当基体是遵循经典塑性全量理论时,这些曲线方程就是一簇依赖于空洞体积比和硬化指数的屈服面方程.当基体是粘性体时,这些方程就是粘性约束方程.通过曲线拟合的方法,本文发展了修正的Gurson方程,使之适合于不同硬化指数的情况.最后本文计算了粘性体中空洞的相对扩展率,结果与已有体胞模型的数值模拟计算结果相当一致.     

Bubbly-to-cap bubbly flow transition in a long-26 m vertical large diameter pipe at low liquid flow rate

The concurrent upward two-phase flow of air and water in a long vertical large diameter pipe with an inner diameter (D) of 200 mm and a height (z) of 26 m (z/D = 130) was investigated experimentally at low superficial liquid velocities from 0.05009 to 0.3121 m/s and the superficial gas velocities from 0.01779 to 0.5069 m/s. The resultant void fractions range from 0.03579 to 0.4059. According to the observations using a high speed video camera, the flow regimes of bubbly, developing cap bubbly and fully-developed cap bubbly flows prevailed in the flows. The developing cap bubbly flow appeared as a flow regime transition from bubbly to fully-developed cap bubble flow in the vertical large diameter pipe. The developing cap bubbly flow changes gradually and lasts for a long time period and a wide axial region in the flow direction, in contrast to a sudden transition from bubbly to slug flows in a small diameter pipe. The analysis in this study showed that the flow regime transition depends not only on the void fraction but also on the axial distance in the flow and the pipe diameter. The axial flow development brings about the transition to happen in a lower void fraction flow and the increase of pipe diameter causes the transition to happen in a higher void fraction flow. The measured void fraction showed an N-shaped axial changing manner that the void fraction increases monotonously with axial position in the bubbly flow, decreases non-monotonously with axial position in the developing cap bubbly flow, and increases monotonously again with axial position in the fully-developed cap bubbly flow. The temporary void fraction decrease phenomenon in the transition region from bubbly to cap bubbly flow can be attributed to the formation of medium to large cap bubbles and their gradual growth into the maximum size of cap bubble and/or cluster of large cap bubbles in the developing cap bubbly flow. In order to predict the N-shaped axial void fraction changing behaviors in the flow regime transition from bubbly to cap bubbly flow, the existing 12 drift flux correlation sets for large diameter pipes are reviewed and their predictabilities are studied against the present experimental data. Although some drift flux correlation sets, such as those of Clark and Flemmer (1986) and Hibiki and Ishii (2003), can predict the present experimental data with reasonable average relative deviations, no drift flux correlation set for distribution parameter and drift velocity can give a reliable prediction for the observed N-shaped axial void fraction changing behaviors in the region from bubbly to cap bubbly flow in a vertical large diameter pipe.     

层裂损伤的细观数值分析

采用体胞模型的分析方法推导了材料在弹塑性变形阶段的孔洞增长方程。假设所有孔洞的内外半径之比相同,用数值方法定性分析了材料在层裂损伤过程中孔洞数密度分布的变化。通过分析孔洞数密度分布、孔洞体积累积百分比、不同大小孔洞所占体积份额的计算结果,指出初始损伤对损伤演化有直接影响。     

Experimental analysis of local void fractions measurements for boiling hydrocarbons in complex geometry

The present paper is part of a research program on two-phase flows and heat transfer studies in tube bundles. An experimental study was carried out to analyse the void fraction for vertical two-phase flows. Boiling across a horizontal tube bundle for three hydrocarbons (n-pentane, propane and iso-butane) under saturated conditions is investigated. The experiments were performed on a tube bundle with 45 plain copper tubes of 19.05 mm outside diameter in a staggered configuration with a pitch to diameter ratio of 1.33. An optical probe has been developed to determine the local void fraction at the minimum cross section between the tubes.     

Interfacial measurements in stratified types of flow. Part II: Measurements for R-22 and R-410A

A new optical void fraction measurement system has been coupled to a flow boiling test facility to obtain dynamic and time-averaged void fractions in a horizontal tube. A series of evaporation tests have been run for two refrigerants. R-22 was tested under mass velocity conditions of 70, 100, 150 and 200 kg/m² s and R-410A for 70, 150, 200 and 300 kg/m² s in a 13.6 mm diameter glass tube. Using our newly developed image processing system, about 227 000 images have been analyzed in this study to provide the same number of dynamic void fraction measurements. From these images, 238 time-averaged void fraction values have been obtained for vapor qualities from 0.01 to 0.95. These experimental points show very good agreement with the horizontal version of the Rouhani–Axelsson drift flux void fraction model.     

Void growth in power-law creeping solids: Effect of surface diffusion and surface energy

This paper addresses the growth of a void in a nonlinearly creeping material in the presence of the void-surface energy effect and void-surface diffusion driven by surface curvature gradients. Large strain finite element analysis of the coupled problem indicates that microstructural variables (porosity and void aspect ratio), as well as macroscopic deformation rates are strongly affected by the relative strength of the void-surface energy effect and the void-surface diffusion process vis-a-vis the rate of creep deformation in the bulk of the solid. The phenomenon is characterized by two-dimensionless groups, one measuring the strength of the surface diffusion process with respect to the nonlinear creep deformation in the interior of the solid, and the other the magnitude of the surface energy of the void in relation to the applied load and the size of the void. The computations reveal a rich variety of solutions that reflect a wide range of external load, material, and geometric parameters. Classical void growth studies that ignore both surface diffusion and surface energy effects are shown to recover only one case of this family of solutions. The computations also serve to quantitatively evaluate recent constitutive theories for porous nonlinear materials that account for continuously evolving microstructure, but do not include surface diffusion or surface energy effects.