Macroscopic Strain Rate Research Articles

Strain rate dependent compressive properties of glass microballoon epoxy syntactic foams

Lightweight glass microballoon epoxy syntactic foams have a high strength-to-weight ratio, making them attractive for transport applications. A better understanding of the compressive properties of such foams is required to improve predictive modelling tools and develop novel formulations. In this study, the response of a foam to compressive loading was experimentally investigated over strain rates from 0.001 to 4000 s −1. The stress–strain response, deformation/damage history and volume change were examined quantitatively and/or qualitatively; all of these parameters exhibit strain rate sensitivity. Combined finite element stress analysis and microscopic observations reveal that heterogeneous (localised) damage arises in the foam due to the coexistence of two failure modes: (i) crushing of glass microballoons dominating in the central part and (ii) shear cracking of the epoxy matrix that forms and propagates from the corners. As the strength of the epoxy matrix increases with increasing strain rate, cracking of glass microballoons begins to dominate over the matrix/microballoon debonding, resulting in macroscopic strain rate dependency of compressive properties.

Dynamic structures in shock-loaded copper

Shock loading of polycrystalline copper under uniaxial strain conditions at $6.2\textdiv{}7.3\text{ }\text{GPa}$ results in nucleating the dissipative structures of $5\textdiv{}25\text{ }\ensuremath{\mu}\text{m}$ in diameter. Interior of each structure is a network of microshear bands of $100\textdiv{}300\text{ }\text{nm}$ spacing. Nucleation of structures occurs at the impact velocity where particle velocity dispersion begins to grow faster than mean particle velocity or when the local strain rate at the mesoscale becomes higher than the macroscopic strain rate. Simultaneously, defect of particle velocity at the plateau of compressive pulse, macrohardness, and spall strength grow in the same manner. Physically, nucleation of dissipative structures is estimated to be initiated under resonance condition between space periods of polarized dislocation structure and driving plastic front.

Strain Rate Sensitivities of Face-Centred-Cubic Metals Using Molecular Dynamics Simulation

We use dislocation theory and molecular dynamics (MD) simulations to investigate the effect of atom properties on the macroscopic strain rate sensitivity of fcc metals. A method to analyse such effect is proposed. The stress dependence of dislocation velocity is identified as the key of such study and is obtained via 2-D MD simulations on the motion of an individual dislocation in an fcc metal. Combining the simulation results with Orowan's relationship, it is concluded that strain rate sensitivities of fcc metals are mainly dependent on their atomic mass rather than the interatomic potential. The order of strain rate sensitivities of five fcc metals obtained by analysing is consistent with the experimental results available.

The influence of a threshold stress for grain boundary sliding on constitutive response of polycrystalline Al during high temperature deformation

A continuum polycrystal plasticity model was used to estimate the influence of a threshold stress for grain boundary sliding on the relationship between macroscopic flow stress and strain rate for the aluminum alloy AA5083 when subjected to plane strain uniaxial tension at 450°C. Under these conditions, AA5083 deforms by dislocation glide at strain rates exceeding 0.001s−1, and by grain boundary sliding at lower strain rates. The stress–strain rate response can be approximated by ε˙=Aσn, where A and n depend on grain size and strain rate. We find that a threshold stress less or equal to 4MPa has only a small influence on flow stress and stress exponent n in the dislocation creep regime (a threshold stress of 2MPa increases n from 4.2 to 4.5), but substantially increases both flow stress and stress exponent in the grain boundary sliding regime (a threshold stress of 2MPa increases n from 1.5 to 2.7). In addition, when the threshold stress is included, our model predicts stress versus strain rate behavior that is in good agreement with experimental measurements reported by Kulas et al. [M.A. Kulas, W.P. Green, E.M. Taleff, P.E. Krajewski, T.R. McNelley, Metall. Mater. Trans. A 36 (2005) 1249].

Multiscale simulations for suspensions of rod-like molecules

We study the Doi model for suspensions of rod-like molecules. The Doi model couples a microscopic Fokker–Planck type equation (Smoluchowski equation) to the macroscopic Stokes equation. The Smoluchowski equation describes the evolution of the distribution of the rod orientations; it comes as a drift–diffusion equation on the sphere at every point in physical space.For sufficiently high macroscopic shear rates (high Deborah numbers), the solution of the coupled system develops internal layers in the macroscopic strain rate (the spurt phenomenon).In the high Deborah numbers regime, the drift term in the Smoluchowski equation is dominant. We thus introduce a finite-volume type discretization of the microscopic Smoluchowski equation which is motivated by transport-dominated PDEs.We carry out direct numerical simulations of the spurt phenomenon both in the dilute and concentrated regimes. Below the isotropic–nematic transition, the solution structure is identical to the one described by purely macroscopic models (JSO model). For higher concentrations, we observe the formation of microstructure coming from a position-dependent tumbling rate. We also investigate the 2-d stability of the spurted solution.

A new model for porous nonlinear viscous solids incorporating void shape effects – I: Theory

The aim of this work is to present a new, improved model for porous ductile solids at high temperatures, for which viscous effects are important. The sound matrix is assumed to obey a simple Norton law without threshold. The voids are assumed to be and remain spheroidal so that their shape is characterized by a single parameter. The model makes an essential use of the notions of gauge surface and gauge function, which extend those of yield surface and yield function of classical plasticity to the nonlinearly viscous case. It is obtained by looking for a good heuristic expression of the gauge function, using specific models pertaining to special cases as references. These reference models include: (i) that of Gologanu, Leblond and Devaux for the case of an ideal-plastic material (i.e. with an infinite Norton exponent) containing voids of arbitrary shape; (ii) that of Leblond, Perrin and Suquet for arbitrary nonlinear viscous materials with spherical or cylindrical voids; (iii) that of Ponte Castañeda and Zaidman for viscous materials with arbitrary voids, but in the linear case only (i.e. for a Norton exponent of unity), although this model was developed for arbitrary nonlinear viscous materials. The nonlinear Hashin–Shtrikman bound is used as a further reference in the case of a zero macroscopic mean strain rate. The model also includes a suitable evolution equation for the void shape parameter, which is again inspired from those proposed in some special cases by Gologanu, Leblond and Devaux, and Ponte Castañeda and Zaidman.

Analysis of creep due to grain-boundary diffusion in hexagonal microstructures

For steady-state deformation caused by grain-boundary diffusion in hexagonal microstructures, the stress distribution on grain boundaries and the macroscopic strain rates are analysed by taking the effects of viscous grain-boundary sliding into account. The maximum normal stress and the extent of stress concentration are shown to decrease as the grain-boundary viscosity increases. For infinite viscosity and/or extremely small grain sizes, the distribution of the normal stress becomes uniform on grain boundaries. The strain rates are predicted by both the stress analysis and the energy balance method, and the two strain rates are consistent with each other. The predicted strain rates also decrease as the grain-boundary viscosity increases. The present analysis reveals that the grain-size exponent is dependent on the grain size and the grain-boundary viscosity: the exponent becomes unity for small grain sizes and/or high viscosity, while it is three for large grain sizes and/or low viscosity. Recent experimental observations that the strain rates of nano-sized grain are much lower than those predicted by grain-boundary diffusion are explained by the increasing contribution of viscous grain-boundary sliding with decreasing grain size.

Stress measurements of deforming olivine at high pressure

Rheological properties of mantle minerals are critical for understanding the dynamics of the Earth’s deep interior. Due to limitations in experimental technique, previous quantitative studies of the rheological properties of mantle minerals are limited to either low pressure or low temperature. The present understanding of mantle flow is mostly inferred from the extrapolation of relatively low-pressure data to mantle high-pressure conditions. However, the effect of pressure (represented by activation volume) on the rheological properties of olivine is still controversial. Therefore, deformation experiments, carried out at mantle pressures, are necessary to understand and model mantle flow. Here we report an experimental study of plastic deformation of San Carlos olivine (Mg, Fe) 2SiO 4 under upper mantle conditions. Macroscopic differential stress and strain rates have been measured in situ in a large-volume high-pressure apparatus using newly developed techniques. The differential stress at high temperature and high pressure that we measured is significantly lower than that estimated by many currently accepted olivine flow laws. We document the first in situ experimental differential stress results in a multi-anvil press. Our results give direct evidence for a relatively small activation volume (less than 10 −5 m 3 mol −1). This shows that the effect of pressure on dislocation creep is small.

Évaluation de l'influence des changements de géométrie sur les déformations de séchage d'un milieu poreux fissuré

The macroscopic response of a cracked solid subjected to drying is investigated within the framework of micromechanics. The originality of this contribution lies in the fact that the variations of the aspect ratios of cracks induced by the capillary pressure increase are accounted for. When the initial aspect ratio is small enough, it is shown that neglecting the geometrical changes yields an erroneous prediction of the sign of the macroscopic volume strain rate. To cite this article: X. Chateau et al., C. R. Mecanique 331 (2003).

Constitutive relation of particulate-reinforced viscoelastic composite materials with debonded microvoids

In this paper, the constitutive relation of particulate-reinforced viscoelastic composite materials is studied by considering the evolution of microvoids. Owing to the difference in mechanical properties between matrix and second phase particles, debonding of particle-matrix interface may occur under the condition of high stress triaxiality. This kind of damage will lead to the nucleation and growth of microvoids. The reinforcing effect due to rigid particles and the weakening effect due to microvoids caused by debonding on the overall mechanical properties of particulate-reinforced composites are investigated. Using the energy criterion of interfacial debonding, an expression for the nucleation rate of microvoids is suggested. Then, the growth of these voids is studied by means of Eshelby’s equivalent inclusion method. It is shown that the strain of the microvoids depends not only on the remote strain history but also on their nucleation time. By considering the effect of nucleation time, a new definition of an average strain of microvoids is given. According to the Mori–Tanaka scheme, a macroscopic constitutive relation of the composite material is finally derived. The results show that macroscopic strain rate, particle-size dispersity, relaxation time of matrix, and interface adhesive strength all play key roles in the overall mechanical properties of particulate-reinforced composites.

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.

Micromechanical modelling of porous materials under dynamic loading

The behaviour of porous material under dynamic conditions is assessed by a micromechanical approach. By averaging, a general form for the dynamic macrostress is proposed which recovers the static definition when inertia effects are neglected. In this work, a representative volume element for the porous material is defined as a hollow sphere. Using an approximation of the velocity field and the principle of virtual work, an explicit relationship is found between the macroscopic stress and strain rate. The macrostress tensor is proved to be symmetric, in the present formulation proposed for porous materials. Illustrations are shown for hydrostatic tension or compression and also for axisymmetric loading. In the latter case, the effect of stress triaxiality is captured.

Numerical simulation of the plastic flow properties of iron single crystals

The present paper deals with the numerical simulation of the plastic flow properties of iron single crystals as well as their influence on the macroscopic elastic-plastic deformation and localization behavior affected by superimposed hydrostatic pressure. Based on experimental observations the onset of plastic yielding on the microscale is described by an extended microscopic yield condition taking into account various microscopic stress components acting on the respective slip systems. In addition, to be able to compute inelastic deformations from a plastic potential, the latter is expressed in terms of workconjugate microscopic stress and strain measures which leads to a non-associated flow rule for the macroscopic plastic strain rate. On the numerical side, generalized functions for constitutive parameters will be used to be able to simulate the single crystal's microscopic deformation behavior observed in experiments. Estimates of the current microscopic stresses and strains are obtained via an efficient and remarkably stable plastic predictor-elastic corrector technique which is incorporated into a nonlinear finite element program. Numerical simulations of uniaxial tests show quantitatively the influence of hydrostatic pressure on current material data. Further numerical studies on the additional constitutive non-Schmid terms elucidate their effect on iron single crystal's macroscopic deformation and localization behavior.

Anisotropic behaviour of porous, ductile media

An orthotropic constitutive model for porous, ductile media is developed, which is centred on the micromechanical analysis of a cylindrical representative volume element (RVE) with elliptic cross-section containing a coaxial and confocal elliptic-cylindrical cavity. The constitutive model is obtained in the case of a rigid ideally plastic behaviour of the matrix material, whose yield condition obeys J 2 flow theory of plasticity with an associated flow rule. The following condition is assumed throughout: the longitudinal axis of the hollow cylindrical RVE is a principal direction of the macroscopic strain rate and stress tensors. Cases for which the principal directions of the macroscopic strain rate tensor in the RVE cross-section plane are aligned or rotated with respect to the ellipse axes are both considered. The constitutive behaviour is characterized by the homogenized yield domain in the macroscopic stress tensor space, by associated flow rules for the plastic components of the macroscopic strain rate tensor and by the evolution laws for the internal state variables. These are the void volume fraction, already appearing in Gurson's model, the aspect ratio of the cavity and its orientation in the RVE cross-section plane, assuming that during the deformation process, the void retains an elliptic shape. The theoretical results are compared with finite element computations of the RVE strength at plastic collapse to assess the capability of the model to describe the actual micromechanical response to the applied boundary conditions.

Micromechanics of void growth at high rates

The aim of this work, based on a micromechanical approach, is to caracterise the behavior of porous material under dynamic conditions. The porous media is represented by an aggregate of hollow spheres. With use of a virtual velocity field, an analytical link between the porosity, the macroscopic strain rate and the macroscopic stress is obtained and provides the viscoplastic behavior of the porous media. The results of Carroll and Holt (1972) [1] concerning the dynamic compression of a porous aluminium are retrieved. The present model is not limited to spherical loadings. An application to an axisymmetric tensile path enlights the role of triaxiality on the evolution of voids.

Motion of dislocations and interfaces during deformation of martensitic Cu–Al–Ni crystals

Investigations of the ultrasonic strain amplitude-dependent internal friction (ADIF) and of the influence of ultrasonic vibrations on the macroplastic strain (the acoustoplastic effect) have been performed in situ during deformation of quenched Cu–Al–Ni single crystals in the β′ 1 martensitic phase. The effect of the macroscopic plastic strain rate on the ADIF and acoustoplastic effect as well as the kinetics of the acoustoplastic effect has been studied. The results of in situ ADIF measurements are compared with data on the ADIF temperature dependence. Observed regularities are attributed to differences in the mechanisms of the macroplastic deformation of the martensitic phase, related to the motion of intervariant interfaces, and of the reversible anelastic strain, which, at ultrasonic frequencies and moderate strain amplitudes, is largely due to the oscillatory motion of the partial dislocations. The conclusion has been drawn that the dynamics of partial dislocations at temperatures of 210–300 K is to a great extent determined by their interaction with atmospheres of mobile pinners with saturated density. Simultaneous measurements of the ADIF and acoustoplastic effect allows the conclusion that, for high oscillatory strain amplitudes, the breakthrough of partial dislocations beyond the mobile atmospheres of pinners initiates the step-like accumulation of the macroplastic strain due to the motion of intervariant boundaries.

The Kinetics of Irreversible Structural Relaxation and Homogeneous Plastic Flow of Metallic Glasses

The paper is devoted to a comprehensive review of the Directional Structural Relaxation model developed recently. The model treats homogeneous plastic flow of metallic glasses as a result of structural relaxation oriented by the external stress. The analytical expressions for the macroscopic plastic strain rate due to irreversible structural relaxation of metallic glasses are derived. A comparison of the predictions of the model with the experimental data is given. It is concluded that the model provides a good basis for the understanding of homogeneous plastic flow of metallic glasses under various experimental conditions.

Yield Functions and Flow Rules for Porous Pressure-Dependent Strain-Hardening Polymeric Materials

To characterize the response of progressively damaged glassy polymers due to the presence and evolution of voids, yield functions and flow rules were developed systematically for a pressure-dependent matrix following the modified von Mises criterion. A rigid-perfectly plastic material was first assumed. The upper bound method was used with a velocity field which has volume preserving and shape changing portions. Macroscopic yield criterion in analytical closed form was first obtained for spherical voids which is valid for all possible macroscopic strain rate fields. Macroscopic yield criteria in analytical closed form were then obtained for cylindrical voids for the special cases of axisymmetric and plane-strain modes of deformation. The upper-bound solutions were subsequently improved to better match analytical solutions for pure hydrostatic loading. Characteristics of the yield function as a function of pressure dependency and void fraction were studied in detail. Generalization of the model for spherical voids to include elasticity as well as strain hardening of the matrix was then obtained. An example for the uniaxial response of a progressively damaged material was then used to illustrate one possible application of the full set of constitutive equations. [S0021-8936(00)02902-0]

Finite elastic–plastic deformation behaviour of crystalline solids based on a non-associated macroscopic flow rule 1Paper presented at the Sixth International Symposium on Plasticity and Its Current Applications, Juneau, Alaska (U.S.A.), July 14–18, 1997. 1

The present paper deals with a nonlinear finite element analysis of the macroscopic elastic–plastic deformation and localization behaviour of crystalline solids. The description includes elastic strains arising from the distortion of the lattice as well as crystallographic deformations due to irreversible microscopic slip along preferred lattice planes and in corresponding lattice directions. In addition, the effect of plastic volume changes on the microstructure is taken into account. Macro- and microscopic stress measures are related to Green’s macroscopic strains via a hyperelastic constitutive law based on a free energy potential function, and the onset of plastic yielding on the microscale is described by a modified yield condition which includes appropriate microscopic stress components. To be able to compute inelastic deformations from plastic potentials, the latter are expressed in terms of work-conjugate microscopic stress and strain measures thus leading to a non-associated flow rule for the macroscopic plastic strain rate. The numerical integration of the rate formulation of the constitutive equations is performed using a plastic predictor-elastic corrector technique. Its implementation into a nonlinear finite element program is discussed, and numerical solutions of finite strain elastic-plastic boundary value problems involving highly localized deformations are presented. They demonstrate both the efficiency of the algorithm and the influence of various model parameters on the deformation and load-carrying behaviour of crystalline solids.

On the sintering with rigid inclusions

AbstractThe sintering of porous matrices containing rigid, i.e. non‐densifying, inclusions is discussed on the basis of the composite sphere model of Scherer. Recognizing that internal stresses are generated by the presence of the non‐densifying inclusions, a new equation for the densification rate of the composite is proposed. It takes into account three contributions: (i) the free densification strain rate, which is independent of the stresses present, (ii) the strain rate due to the microscopic effect of the stresses on the sintering mechanism and (iii) the macroscopic deformation strain rate induced by the stresses. This result indicates that the presence of rigid inclusions may generate higher internal stresses in the densify ing compact than those calculated by the original Scherer's approach.