A model of dynamic failure in ductile materials

In this paper, a dynamic failure model in ductile materials under the action of a mean tensile stress is developed. The model proposed takes into account nucleation and growth of void as part of the failure process under dynamic loading conditions. In the evolution of porosity , work-hardening behavior and rate-dependent effects are included. Numerical simulations of aluminum, aluminum alloy and OFHC copper spallation processes are performed. The results of computation are in fair agreement with experimental results.Support of this work by the special grant No.9187004 from Natural Science Fundation of China is gratefully acknowledged     

EVOLUTION OF VOIDS IN DUCTILE POROUS MATERIALS AT HIGH STRAIN RATE

In the present work,a dynamic damage model in ductile materials underthe application of dynamic general stresses loading is presented.The evolution equationof ductile voids has the closed form,in which work-hardening,the change of surfaceenergy of voids,rate-dependent,inertial effects are taken into account.Theexpressions of critical stresses for the growth and compaction of voids are directlyobtained from the evolution equations of voids.From the expressions,the resultobtained by Carroll and Holt,as a special example,is given.Numerical analysis ofthe model indicates that the growth of voids is sensitive to the strain rates.The voidsgrow quickly as the increase of strain rates.It is also shown that the influence of theinertial effects on the void growth is great at high loading rates.It appears to resist thegrowth of voids.In addition,a dynamic collapse model of ductile voids is alsoproposed,which can be applied to study the problems of compaction in powder andother materials.     

结构冲击畸变问题的直接相似方法研究

结构受冲击时由不同材料引起比例模型与全尺寸原型的畸变, 通常通过修正比例模型的速度或密度进行补偿. 然而传统的修正方法, 存在需预先测试结构响应、依赖特殊本构方程、不能反映动态过程的缺陷, 因而限制了相似理论在比例模型试验中的直接应用. 本文提出了一种不同材料畸变问题的直接相似方法. 它通过建立应变率区间上比例模型预测的流动屈服应力与原型流动屈服应力的最佳逼近关系, 直接获得了修正速度或修正密度的比例因子, 完成了比例模型与原型的动态相似关系. 基于Norton-Hoff, Cowper-Symonds, Johnson-Cook三种经典的本构模型, 研究了材料应变率敏感性特征参数、参考应变率、屈服应力、密度在动态相似关系中的作用. 并通过受质量冲击的折板结构算例, 验证了直接相似方法的有效性. 分析表明, 本文提出的直接相似方法不需要预先测试结构的响应, 不依赖特殊的本构方程、强调动态相似特性, 具有直接、高效、通用的特点. 此外, 动态相似关系的最佳逼近效果, 受材料应变率敏感性特征参数控制, 屈服应力、密度和参考应变率影响不大; 当比例模型应变率敏感性特征参数与原型相近, 可获得最佳逼近.      

Accelerated void growth in porous ductile solids containing two populations of cavities

This paper presents an analytical and numerical study of accelerated void growth in porous ductile solids arising from the presence of two populations of cavities very different in size. It is based on the model problem of some hollow sphere made of porous plastic material and subjected to hydrostatic tension. The central hole plays the role of a typical big cavity of the first population while those dispersed in the matrix stand for the small cavities of the second one. The behavior of the matrix is supposed to obey Gurson's famous “homogenized” model for porous ductile solids (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. ASME J Engng Materials Technol 99, 2–15). The analytic solution of this model problem shows that the small voids located near the big one grow twice as fast as the latter void. This suggests that in a subsequent step, these small cavities may reach coalescence prior to the big ones, thus creating spherical shells of ruined matter around the cavities of the first population and leading to accelerated growth of the latter cavities; this scenario is in agreement with experimental evidence. Since this subsequent step is not amenable to a complete analytic solution, it is studied numerically. Finally, a simplified model reproducing the two steps of void growth (prior to coalescence of the small voids and after it has started) is developed on the basis of the analytical solution for the first step and some elements of a similar solution for the second one. The results derived from this simplified model are in good quantitative agreement with those obtained through the complete numerical simulations.     

Modeling the dynamic response of visco-elastic open-cell foams

This article introduces a mesoscopic formulation for modeling the dynamic response of visco-elastic, open-cell solid foams. The effective material response is obtained by enforcing on a representative 3D unit cell the principle of minimum action for dissipative systems. The resulting model accounts explicitly for the foam topology, the elastic and viscous properties of the cell wall, and the inertial effects arising from non-affine motion within the cells. The microinertial effects become significant in retarding the foam collapse during exceedingly high strain-rate loading. As an application example, a heterogenous case of compressive deformation at high strain rate is simulated utilizing the present model as a constitutive update in a non-linear finite element analysis code. This FEM simulation shows the ability of the model to capture the progressive foam collapse during the dynamic compression as observed in experimental studies. Using the microscopic model, the inertial and viscous strain-rate effects are investigated through the foam density, viscosity, and relative density. Based on the physics incorporated into the local cell model, we provide insight into the physical mechanisms responsible for the experimentally observed strain-rate effects on the behavior of dynamically loaded foam materials.     

Dynamic analysis of a tethered satellite system for space debris capture

Herein we analyze the dynamic behavior of a tethered satellite system for space debris capture, considering the large deformation of a tether. The tethered satellite system is modeled as two point masses and a string, and the equations of motion of the tethered satellite system are derived by using the absolute nodal coordinate formulation. To calculate the net velocity after debris capture, equations are established describing the momentum exchange between the net and the space debris. By using this model, the dynamic responses of the tethered satellite system after debris capture are calculated for the variations of the capture angles and capture velocities of the debris. This allows analysis of the orbital response of the tethered satellite system and the large tensions arising from tether tumbling. Finally, we analyze the effects of varying system parameters of the tethered satellite system and the space debris upon the dynamic responses.     

Structural interfaces in linear elasticity. Part I: Nonlocality and gradient approximations

Many biological and optimal materials, at multiple scales, consist of what can be idealized as continuous bodies joined by structural interfaces. Mechanical characterization of the microstructure defining the interface can nowadays be accurately done; however, such interfaces are usually analyzed employing models where those properties are overly simplified. To introduce into the analysis the microstructure properties, a new model of structural interfaces is proposed and developed: a true structure is introduced in the transition zone, joining continuous bodies, with geometrical and material properties directly obtained from those of the interfacial microstructure. First, the case of an elliptical inclusion connected by a structural interface to an infinite matrix is solved analytically, showing that nonlocal effects follow directly from the introduction of the structure, related to the inclination of the connecting elements. Second, starting from a discrete structure, a continuous model of a structural interface is derived. The usual zero-thickness linear interface model is shown to be a special case of this more general continuous structural interface model. Then, a gradient approximation of the interface constitutive law is rigorously derived: it is the first example of the analytical derivation of a nonlocal interface model from the microstructure properties. The effects introduced in the mechanical behavior by both the continuous model and its gradient approximation are illustrated by solving, for the first time, the problem of a circular inclusion connected to an infinite matrix by a structural interface and subject to remote uniform stress.     

Crack tip blunting and cleavage under dynamic conditions

In structural materials with both brittle and ductile phases, cracks often initiate within the brittle phase and propagate dynamically towards the ductile phase. The macroscale, quasistatic toughness of the material thus depends on the outcome of this microscale, dynamic process. Indeed, dynamics has been hypothesized to suppress dislocation emission, which may explain the occurrence of brittle transgranular fracture in mild steels at low temperatures (Lin et al., 1987). Here, crack tip blunting and cleavage under dynamic conditions are explored using continuum mechanics and molecular dynamics simulations. The focus is on two questions: (1) whether dynamics can affect the energy barriers for dislocation emission and cleavage, and (2) what happens in the dynamic “overloaded” situation, in which both processes are energetically possible. In either case, dynamics may shift the balance between brittle cleavage and ductile blunting, thereby affecting the intrinsic ductility of the material. To explore these effects in simulation, a novel interatomic potential is used for which the intrinsic ductility is tunable, and a novel simulation technique is employed, termed as a “dynamic cleavage test”, in which cracks can be run dynamically at a prescribed energy release rate into a material. Both theory and simulation reveal, however, that the intrinsic ductility of a material is unaffected by dynamics. The energy barrier to dislocation emission appears to be identical in quasi-static and dynamic conditions, and, in the overloaded situation, ductile crack tip behavior ultimately prevails since a single emission event can blunt and arrest the crack, preventing further cleavage. Thus, dynamics cannot embrittle a ductile material, and the origin of brittle failure in certain alloys (e.g., mild steels) appears unrelated to dynamic effects at the crack tip.     

Intense bending waves in a bar

Summary In this paper, the work presented in [1] is extended to study higher-order approximations of nonlinear effects in a bar. It has been found that long bending waves, being the low-frequency modes involved in resonant triads, are stable against small perturbations. Consequently, a bending wave with group velocity which is less than that of longitudinal waves should behave as a linear quasi-harmonic wavetrain. On the other hand, one may expect self-modulation instability of intense bending wavetrains during the long-time evolution. This paper overcomes such a contradiction. To describe the nonlinear dynamics in detail, one should allow for higher-order approximation effects in the model. Such effects are associated with the diffusion of linear wave packets due to different group velocities, and amplitude dispersion caused by nonlinearity. Within the second-order approximation analysis, an amplitude modulation is indeed experienced for intense bending waves. As a result, envelope solitons can be formed from unstable bending wavetrains. The group matching of long longitudinal and short bending waves, being a particular case of the self-modulation, is of special interest as a limit case of the triple-wave resonant interactions. It demonstrates the relation between the first- and the second-order approximation effects. Accepted for publication 20 July 1996     

尖头弹丸撞击下金属靶板弹道极限的两种工程模型

研究了圆锥形头和卵形头刚性弹垂直撞击塑性金属靶板扩孔冲塞型和延性扩孔型穿孔模式,考虑靶板背面自由边界的影响,提出两种两阶段工程分析模型,得到最小穿透能量的解析解。由球形空腔膨胀理论和两阶段总耗能最小确定第一阶段的侵彻深度,由功能原理和圆柱形空腔膨胀理论计算第一阶段侵彻扩孔耗能,延性扩孔型第二阶段耗能近似按Taylor扩孔理论计算,扩孔冲塞型第二阶段耗能考虑了加速塞块和剪断塞块所损耗的能量。与铝合金和装甲钢靶板弹道试验数据比较表明,本文两阶段模型的计算结果与试验结果吻合较好。     

Time-temperature dependence in fracture behavior of high impact polystyrene

High Impact Polystyrene (HIPS) is one of the first toughened systems in which the brittle polystyrene becomes more ductile with the addition of an elastomer. However, it exhibits a ductile behavior only above a certain temperature and below a certain loading rate. Fracture in this material, like in most toughened systems, can become brittle when the temperature is lowered or the loading rate is increased. The correlation between temperature and loading rate seems to be controlled by the molecular relaxation according to the Arrhenius equation. The objective of this work is to foster the understanding of the effects of time and temperature on the fracture behavior of HIPS. The time and temperature dependence in fracture performance has been found to be governed by the strain energy density criterion. The theory allows prediction of fracture performance at various loading rates and temperatures. The brittle–ductile transition is controlled by an energy activation process. A peak in fracture energy always occurs at the transition region. This is attributed to the relaxation of the polymer macromolecules. The time and temperature dependence of this relaxation can be predicted by the Arrhenius equation. The rise in fracture energy at high loading speeds is not due to the higher frequency oscillations from dynamic effect but is controlled by the critical strain energy density.     

DYNAMIC TENSILE TESTING OF SHEET MATERIAL USING THE SPLIT-HOPKINSON BAR TECHNIQUE

We have described an experimental method based on the SHPB technique which allows dynamic tensile testing of 1.0-mm-thick sheet material. The test method employs a framing camera to make strain measurements based on direct observation of the deforming test sample. Stress in the sample is determined using traditional SHPB wave analysis. This information is used to construct engineering stress-strain behavior. The stress-strain behavior determined using the test method is shown to be reasonably accurate for annealed tantalum and less accurate for annealed copper. We believe that this phenomenon is related to the constitutive behavior of the test material. Although the stress-strain behavior extracted from the test is in some cases suspect, the test technique is shown to provide highly reproducible results and thus can be employed to examine effects of microstructural features, alloying elements, etc., on dynamic deformation and fracture of ductile materials.     

A dynamic meso-scale model of the pullout behavior for non-straight fibers and corresponding DEM3 simulation

Lots of work had been reported in the experimental investigation and modeling of the debonding and pullout behavior for straight crack-bridging fibers in brittle matrix composites, but few of them involving in the dynamic and/or non-straight fiber case. In this paper, a dynamic meso-scale model for non-straight fiber was proposed on the work of Chanvillard. The model was time-dependent and related to the impact loading through the interfacial shearing stress along the embedded part of the steel fiber, and properly setting the thresholds, the dynamic meso-damage model of the fiber's debonding and pullout process was eventually constructed. The model prediction fitted the experimental data of Chanvillard for non-straight steel fiber under static condition quite well and could demonstrate the rate sensitivity of the reinforcing effects of fibers. Then, a 3-D discrete meso-element method (DEM3) was used to simulate the dynamic pullout behavior of such fiber configurations as wiredrawn fibers from cementitious matrix with pores. Comparing to afro-mentioned model, DEM3 was more effective in simulating the complete procedures of matrix dynamic failure and fiber pullout and would be more helpful in the analyses of complicated configurations of fiber.     

延性动态损伤的细观描述

对动载荷下材料的延性动态损伤的细观力学研究现状进行了评述，详细讨论了现有的几类典型的动态延性损伤的微孔生长模型。针对目前该领域工作中存在的问题，指出了今后应开展的工作方向。     

Higher-Order Terms in the Capillary Limit Approximation of Viscous-Capillary Flow

We propose a new approximation to the solution of the steady state equations describing two-phase immiscible flow in a porous medium. It is demonstrated that the general procedure contains the capillary equilibrium approximation as a special case. The solution is approximated by a perturbation series in a parameter related to the capillary number. The expansion of the solution results in a sequence of decoupled linear elliptic boundary value problems. This sequence is solved numerically by a Finite Element method, and the accuracy of the approximations is evaluated.     

A Reissner-type plate theory for monoclinic material derived by extending the uniform-approximation technique by orthogonal tensor decompositions of <Emphasis Type="Italic">n</Emphasis>th-order gradients

The uniform-approximation approach is an a-priori assumption free structured approach for the derivation of hierarchies of lower-dimensional theories for thin structures with increasing approximation accuracy. In this publication, we derive a second-order consistent plate theory for monoclinic material and investigate several theories that arise from the original theory by a pseudo-reduction approach which aims to reduce the number of PDEs that are to solve. A one-variable model that governs only the interior solution is presented and, in addition, an extended two-variable model that also covers edge effects. Since the second introduced variable is a rotation of a vector field, we have to uniquely identify the rotation dependent parts in general gradients of the vector field, which is resolved by the introduction of an orthogonal decomposition. The final two-variable model is equivalent to the Reissner–Mindlin theory for the special case of isotropic material, whereas the one-variable model is equivalent to the first Reissner PDE. In contrast to this special case, the two-variable model is a coupled system of two PDEs for general monoclinic material.     

A viscoelastic constitutive model of rubber under small oscillatory load superimposed on large static deformation

Summary  A viscoelastic constitutive equation of rubber that is under small oscillatory load superimposed on large static deformation is proposed. The model is derived through linearization of Simo's nonlinear viscoelastic constitutive model and reference configuration transformation. Most importantly, in this model, static deformation correction factor is introduced to consider the influence of pre-strain on the relaxation function. Natural statically pre-deformed state is served as reference configuration. The proposed constitutive equation is extended to a generalized viscoelastic constitutive equation that includes widely used Morman's model as a special case using objective stress increment. The proposed constitutive model is tested for dynamic behavior of rubber specimens with different carbon black content. It is concluded from the test that the assumption that the effects of static deformation can be separated from time effects, which is the basis of Morman's model, is only applicable to unfilled rubber. The viscoelastic constitutive equation for filled rubber must include, therefore, the influence of the static deformation on the time effects. The suggested constitutive equation with static deformation correction factor shows good agreement with test values. Received 4 January 2001; accepted for publication 13 June 2001     

Determination of Early Flow Stress for Ductile Specimens at High Strain Rates by Using a SHPB

In a dynamic experiment to obtain the high-rate stress–strain response of a ductile specimen, it takes a finite amount of time for the strain rate in the specimen to increase from zero to a desired level. The strain in the specimen accumulates during this strain-rate ramping time. If the desired strain rate is high, the specimen may yield before the desired rate is attained. In this case, the strain rates at yielding and early plastic flow are lower than the desired value, leading to inaccurate determination of the yield strength. Through experimentation and analysis, we examined the validity and accuracy of the flow stresses for ductile materials in a split Hopkinson pressure (SHPB) bar experiment. The upper strain-rate limit for determining the dynamic yield strength of ductile materials with a SHPB is identified.     

A unified approach for pressure and temperature effects in dynamic failure criteria

In most of the existing dynamic failure criteria, effects of temperature and pressure are ignored. In a few recent contributions, attempts to include initial temperature effects have been proposed. However, the combination of both temperature and pressure effects is still lacking in all existing criteria. The aim of the present work is to model the effects of both pressure and temperature on dynamic failure. We propose a unified approach that can be implemented within several dynamic failure criteria to simulate spallation at different initial temperatures and in a wide range of pressure. This approach is based on the scaling of the threshold failure stress similar to the scaling of the flow stress used in pressure dependent plasticity. This scaling uses the shear modulus for which we propose a thermodynamically based evolution law as function of temperature and pressure. For strain controlled failure, the threshold strain is also modified to account for triaxiality. We summarize existing dynamic failure criteria and we present our proposed approach for temperature and pressure effects as it applies to each of these criteria. Results from this approach are discussed and compared with experimental observations. As an illustration, we selected one of these failure criteria and applied it to simulate spallation in planar impact test. Predicted results for this test are also discussed in comparison with experimental observations.     

Homogenization-based constitutive models for magnetorheological elastomers at finite strain

This paper proposes a new homogenization framework for magnetoelastic composites accounting for the effect of magnetic dipole interactions, as well as finite strains. In addition, it provides an application for magnetorheological elastomers via a “partial decoupling” approximation splitting the magnetoelastic energy into a purely mechanical component, together with a magnetostatic component evaluated in the deformed configuration of the composite, as estimated by means of the purely mechanical solution of the problem. It is argued that the resulting constitutive model for the material, which can account for the initial volume fraction, average shape, orientation and distribution of the magnetically anisotropic, non-spherical particles, should be quite accurate at least for perfectly aligned magnetic and mechanical loadings. The theory predicts the existence of certain “extra” stresses—arising in the composite beyond the purely mechanical and magnetic (Maxwell) stresses—which can be directly linked to deformation-induced changes in the microstructure. For the special case of isotropic distributions of magnetically isotropic, spherical particles, the extra stresses are due to changes in the particle two-point distribution function with the deformation, and are of order volume fraction squared, while the corresponding extra stresses for the case of aligned, ellipsoidal particles can be of order volume fraction, when changes are induced by the deformation in the orientation of the particles. The theory is capable of handling the strongly nonlinear effects associated with finite strains and magnetic saturation of the particles at sufficiently high deformations and magnetic fields, respectively.