Large strain rate-dependent response of elastomers at different strain rates: convolution integral vs. internal variable formulations

Two different viscoelastic frameworks adapted to large strain rate-dependent response of elastomers are compared; for each approach, a simple model is derived. Within the Finite Linear Viscoelasticity theory, a time convolution integral model based on an extension to solid of the K-BKZ model is proposed. Considering the multiplicative split of the deformation gradient into elastic and inelastic parts, an internal variable model based on a large strain version of the Standard Linear Solid model is considered. In both cases, the strain energy functions involved are chosen neo-Hookean, and then each model possesses three material parameters: two stiffnesses and a viscosity parameter. These parameters are set to ensure the equivalence of the model responses for uniaxial large strain quasi-static and infinitely fast loading conditions, and for uniaxial rate-dependent small strain loading conditions. Considering their responses for different Eulerian strain rates, their differences are investigated with respect to the strain rate; more specifically, both stiffness and dissipative properties are studied. The comparison reveals that these two models differ significantly for intermediate strain rates, and a closing discussion highlights some issues about their foundations and numerical considerations.     

A meso-scale damage evolution model for cyclic fatigue of viscoplastic materials

This study presents an approach to predict the degree of material degradation and the resulting changes in elastic, plastic and creep constitutive properties of viscoplastic materials, during cyclic loading in micro-scale applications. The objective of the study is to address the initiation and growth of homogeneous meso-scale damage, in the form of distributions of micro-cracks and micro-voids, due to cyclic, plastic (rate-independent inelastic) and creep (rate-dependent inelastic) deformations in viscoplastic materials and to evaluate the resulting changes in the effective meso-scale elastic, plastic and creep constitutive properties. An energy partitioning damage evolution (EPDE) model is proposed to describe the viscoplastic damage evolution. Development of the EPDE model constants is then demonstrated for a Pb-free solder, based on cyclic fatigue test data. Application of the EPDE model is demonstrated for solder joint fatigue during thermal cycling of a ball grid array (BGA) electronic assembly. A 3D viscoplastic finite element analysis is conducted, and damage evolution is modeled using a successive initiation (SI) technique reported earlier by the authors. In this approach, the local (meso-scale) material properties are progressively degraded and highly damaged sections of the macro-scale structure are ultimately eliminated, using the EPDE model. Prediction of damage initiation and propagation is presented both with and without property updating, for comparison purposes. The analysis shows that the EPDE model can realistically capture the softening observed during cyclic loading.     

Micromechanically Established Constitutive Equations for Multiphase Materials with Viscoelastic–Viscoplastic Phases

A model for viscoelastic–viscoplastic solids is incorporated in a micromechanical analysis of composites with periodic microstructures in order to establish closed-form coupled constitutive relations for viscoelastic–viscoplastic multiphase materials. This is achieved by employing the homogenization technique for the establishment of concentration tensors that relate the local elastic and inelastic fields to the externally applied loading. The resulting constitutive equations are sufficiently general such that viscoelastic, viscoplastic and perfectly elastic phases are obtained as special cases by a proper selection of the material parameters the phase. Results show that the viscoelastic and viscoplastic mechanisms have significant effect on the global stress-strain, relaxation and creep behavior of the composite, and that its response is strongly rate-dependent in the reversible and irreversible regimes.     

Material forces for inelastic models at large strains: application to fracture mechanics

Elastomeric materials show a wide range of different elastic and inelastic properties. Additionally, this class of materials is subjected to large deformations. Considering all these effects, fracture mechanical investigations are very challenging tasks and cannot be performed with standard approaches. Effects of inhomogeneities and discontinuities such as cracks can be investigated with the so-called material force approach in an efficient and elegant way. For comprehensive investigations of inelastic materials, the complete balance of the material motion problem has to be formulated. In this case, the material volume forces depend on the internal history variables which are required for the inelastic constitutive model. This paper derives a general formulation for rate-dependent and rate-independent inelastic materials based on a multiplicative split of the deformation gradient to cover viscoelastic and elastoplastic materials at finite deformations.     

基于结构表面声压的自适应有源声吸收控制研究

研究了实际系统中有源声吸收自适应控制方案。首先提出三种控制准则：反射声功率最小、弹性板表面声压平方和最小及有限点声压平方和最小;然后根据求声功率的近场方法、求耦合声波／振动方程的声弹性理论及无约束最优化技术,推导了三种准则下声吸收的计算公式;最后通过仿真实例,研究了采用不同准则时,次级力源与监测传感器个数和位置对吸声效果的影响     

A rate-dependent thermo-electro-mechanical free energy model for perovskite type single crystals

The three-dimensional electro-mechanical free energy potential developed by Kim and Seelecke [S.J. Kim and S. Seelecke, A rate-dependent three-dimensional free energy model for ferroelectric single crystals, Int. J. Solids Struct. 44 (2007) 1196-1209] is generalized to model various thermal aspects of perovskite type single crystals. A total of seven energy potentials are described in the 10-dimensional space of electric displacement vector, strain tensor and temperature, the first six of them representing the six distinct types of ferroelectric tetragonal variants and the seventh the paraelectric cubic phase of the materials. Energy barrier expressions given as functions of thermodynamic driving forces are combined with evolution equations to determine the phase fractions based on the theory of thermally activated processes, thus allowing for a natural treatment of rate-dependent effects. The thermodynamic Clausius-Clapeyron relation is derived from the energy potential and the double polarization hysteresis loops near Curie temperature observed by Merz [W.J. Merz, Double hysteresis loop of BaTiO₃ at the Curie point, Phys. Rev. 91 (1953) 513-517] are predicted and compared. Besides, various nonlinear coupling behavior, such as variation of spontaneous polarization over temperature, mechanical depolarization, and rate-dependent hysteresis loops, are calculated and discussed.     

A simple kinetic theory for granular flow of binary mixtures of smooth,inelastic, spherical particles

Summary Following the approach of the kinetic theory for mixtures of dense gases, the general conservation equations for the rapid flow of a binary mixture of smooth, inelastic, spherical granular particles are derived. Explicit constitutive relations for stress and rate of energy dissipation are obtained by making simple approximations for the particle velocity distribution functions. These approximations are appropriate for cases where collisional interactions are the dominant mechanism for momentum and energy exchange in the system. The theory is applied to the case of simple shear flow. In general, the theory predicts that stresses decrease with increasing concentration of the small particles and decreasing diameter ratio of small to large particles. Theoretical predictions of stresses are compared with experimental results and reasonable agreement is found.With 9 Figures     

Thermo-mechanical fatigue modeling of advanced metal matrix composites in the presence of microstructural details

An analytical model developed for predicting the inelastic response of metal matrix composites subjected to axisymmetric loading is employed to investigate the behavior of SiC---Ti composites under thermo-mechanical fatigue loading. The model is based on the concentric cylinder assemblage consisting of arbitrary numbers of elastic or inelastic sublayers with isotropic, transversely isotropic, or orthotropic, temperature-dependent properties. In the present work, the inelastic response of the titanium matrix is modeled by the Bodner-Partom unified viscoplastic theory. These features of the model allow the investigation of microstructural effects (such as the layered morphology of the SCS-6 fiber, including the weak carbon coating, and matrix microstructure) and rate-dependent response of the matrix on the fatigue behavior.

In this paper, we employ the predictions of the multiple concentric cylinder model to study the effects of the layered morphology of the SCS-6 SiC fiber and two-phase microstructure of the Ti-15-3 matrix on the response of a SiC---Ti composite under thermo-mechanical fatigue loading. It is shown that ignoring the microstructure can lead to significant errors in the predictions of the internal stress and inelastic strain distributions.     

Ab initio calculation of the structural and dynamical properties of layered semiconductors

We present a theoretical first-principles investigation of the structure and lattice dynamics of several layered semiconductors. The equilibrium structure as obtained by minimization of the total energy of the bulk materials is in good agreement with experiment. Furthermore, we have investigated the surface of these materials in order to obtain information on the van der Waals epitaxial growth. We found that the relaxed atomic positions at the surface deviate from the ideal ones in the bulk by less than 1%, which is obviously a consequence of the weak interlayer forces. Additionally, bulk phonon-dispersion curves have been calculated along several high symmetry directions within the density-functional perturbation theory (DFPT). The weak interlayer interaction makes the vibrational properties of the bulk very similar to those of the surface. In fact, our ab initio calculations for the bulk reproduce well both the experimental bulk phonon frequencies obtained by inelastic neutron scattering and the experimental surface phonon dispersion measured with inelastic He-atom scattering (HAS).     

低频声有源吸收理论研究

为了吸收低频入射声 ,作者提出了一种有源吸声结构 ,本文从理论上研究了该结构的吸声性能。文章首先建立理论模型 ,然后通过近场方法和求解弹性板 -声腔耦合系统结构响应推导了有源控制前后反射声功率的计算公式 ,然后按声反射功率最小化准则求得最佳有源吸声效果。最后通过一系列算例证明了该吸声结构吸收低频声波的有效性 ,并研究了吸声效果与次级力源位置的关系     

Constitutive modeling of creep-fatigue interaction for normal strength concrete under compression

Conventional approaches to model fatigue failure are based on a characterization of the lifetime as a function of the loading amplitude. The Wöhler diagram in combination with a linear damage accumulation assumption predicts the lifetime for different loading regimes. Using this phenomenological approach, the evolution of damage and inelastic strains and a redistribution of stresses cannot be modeled. The gradual degration of the material is assumed to not alter the stress state. Using the Palmgren–Miner rule for damage accumulation, order effects resulting from the non-linear response are generally neglected.In this work, a constitutive model for concrete using continuum damage mechanics is developed. The model includes rate-dependent effects and realistically reproduces gradual performance degradation of normal strength concrete under compressive static, creep and cyclic loading in a unified framework. The damage evolution is driven by inelastic deformations and captures strain rate effects observed experimentally. Implementation details are discussed. Finally, the model is validated by comparing simulation and experimental data for creep, fatigue and triaxial compression.     

An anisotropic discrete fiber model with dissipation for soft biological tissues

Nonlinear three-dimensional constitutive equations are developed for analyzing inelastic effects that cause dissipation in biological tissues. The model combines a structural icosahedral model of six discrete fiber bundles with a phenomenological model of the inelastic distortional deformations of the matrix containing the fibers. The inelastic response of the matrix is characterized by only three material parameters, which can be used to model both rate-independent and rate-dependent response with a smooth elastic-inelastic transition. Also, a robust, strongly objective scheme is discussed, which allows the model to be easily implemented into finite element computer codes. Examples show that the model predictions compare well with experimental data for the nonlinear, anisotropic, inelastic response of a number of tissues. Specifically, the model simulated the biaxial stretching of rabbit skin with an error of 15.7%, stress relaxation of rabbit skin with an error of 17.2%, simple shear of rat septal myocardium with an error of 21.6%, and uniaxial stress in compression of monkey liver with an error of 8.3%.     

Formulation and implementation of strain rate-dependent fracture toughness in context of the phase-field method

The phase-field approach is a promising technique for the realistic simulation of brittle fracture processes, both in quasi-static and transient analysis. Considering fast loading, experimental evidence indicates a strong relationship between the rate of strain and the material's resistance against fracture, which can be considered by a dynamic increase factor for the strength of the material. The paper at hand presents a novel approach within the framework of phase-field models for brittle fracture. A rate-dependent fracture toughness is formulated as a function of the rate of crack driving strain components, which results in higher strength for faster loading. Beside the increased amount of energy necessary to evolve a crack at a high strain rate loading situation, the model incorporates quasi-viscous stress-type quantities that are not directly related to the formation of the crack and exist only in the phase-field transition zone between broken and sound material. The governing strong form equations for a transient simulation are derived and the relevant information for an implementation of the model into a finite element code is outlined in detail. The performance of the model is demonstrated for static and dynamic benchmark simulations and for a comparison to experimental findings.     

An inelastic micro/macro theory for hybrid smart/metal composites

Motivated by the emerging concept of including metal phases within piezoelectric and other smart structures and materials, this paper presents a micro/macro theory for determining the coupled thermo-electro-magneto-elasto-plastic behavior of arbitrary composite laminates. The approach involves two models capable of analyzing geometries that include inelastic materials. The first is the electro-magnetic generalized method of cells (EMGMC) (Micromechanical Prediction of the Effective Behavior of Fully Coupled Electro-Magneto-Thermo-Elastic Multiphase Composites, 2000. [1]) micromechanics model. EMGMC has been reformulated to improve its computational efficiency and has been extended to admit arbitrary anisotropic local material behavior (in terms of the thermal response, mechanical response, electric response, magnetic response, as well as the coupling behavior) and inelasticity. The second model is classical lamination theory, which has also been extended for arbitrary anisotropic material behavior and electro-magnetic and inelastic effects. The end result is a coupled theory that employs EMGMC to provide the homogenized behavior of the composite plies that constitute the thermo-electro-magneto-elasto-plastic laminate. Sample results, which address the inelastic response of a hybrid smart/metal matrix composite laminate, are presented.     

An elastic-viscoplastic constitutive model for the hot-forming of aluminum alloys

Under hot-forming conditions characterized by high homologous temperatures and strain-rates, metals usually exhibit rate-dependent inelastic behavior. An elastic-viscoplastic constitutive model is presented here to describe metal behavior during hot-forming. The model uses an isotropic internal variable to represent the resistance offered to plastic deformation by the microstructure. Evolution equations are developed for the inelastic strain and the deformation resistance based on experimental results. A methodology is presented for extracting model parameters from constant true strain-rate compression tests performed at different temperatures. Model parameters are determined for an Al-1Mn alloy and an Al-Mg-Si alloy, and the predictions of the model are shown to be in good agreement with the experimental data.     

A plane stress softening plasticity model for orthotropic materials

A plane stress model has been developed for quasi-brittle orthotropic materials. The theory of plasticity, which is adopted to describe the inelastic behaviour, utilizes modern algorithmic concepts, including an implicit Euler backward return mapping scheme, a local Newton–Raphson method and a consistent tangential stiffness matrix. The model is capable of predicting independent responses along the material axes. It features a tensile fracture energy and a compressive fracture energy, which are different for each material axis. A comparison between calculated and experimental results in masonry shear walls shows that a successful implementation has been achieved. © 1997 John Wiley & Sons, Ltd.     

Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach

Basalt columns, septarias, and mud cracks possess beautiful and intriguing crack patterns that are hard to predict because of the presence of cracks intersections and branches. The variational approach to brittle fracture provides a mathematically sound model based on minimization of the sum of bulk and fracture energies. It does not require any a priori assumption on fracture patterns and can therefore deal naturally with complex geometries. Here, we consider shrinkage cracks obtained during unidirectional drying of a colloidal suspension confined in a capillary tube. We focus on a portion of the tube where the cross-sectional shape cracks does not change as they propagate. We apply the variational approach to fracture to a tube cross-section and look for two-dimensional crack configurations minimizing the energy for a given loading level. We achieve qualitative and quantitative agreement between experiments and numerical simulations using a regularized energy (without any assumption on the cracks shape) or solutions obtained with traditional techniques (fixing the overall crack shape a priori). The results prove the efficiency of the variational approach when dealing with crack intersections and its ability to predict complex crack morphologies without any a priori assumption on their shape.     

A multi-objective GRASP procedure for reactive power compensation planning

A multi-objective approach based on the GRASP (Greedy Randomized Adaptive Search Procedure) meta-heuristic is proposed to provide decision support in the problem of locating and sizing capacitors for reactive power compensation in electrical radial distribution networks. The installation of capacitors (local sources of reactive power) in the network is aimed at correcting the power factor to improve the quality of service, particularly the network voltage profile, and reduce energy losses and power peak. The mathematical model explicitly considers two conflicting objective functions: the minimization of the network active losses and the minimization of the capacitor installation cost. An algorithmic approach based on GRASP is presented for the characterization of the non-dominated solution set.     

On the elastoviscoplasticity of ferromagnetic crystals

Based on a semi-phenomenological approach, a theory of finitely deformable elastoviscoplastic ferromagnetic crystals is constructed with the ultimate purpose to study the influence of dislocations on the dynamical properties of ferromagnets. The initial reference configuration of the body is a so-called one-domain ferromagnetic fundamental phase. Thermomagnetoelastic and viscoplastic parts of the finite strain are obtained by means of a multiplicative decomposition and the introduction of a “natural” local configuration (which corresponds to a relaxation of both elastic and ferromagnetic processes). The viscoplastic processes are accounted for through evolution equations (for internal variables) and are related to the microdynamics of dislocations. It is thus shown that a generalized notion of resolved shear stress can be introduced which involves both the Cauchy stress and ferromagnetic exchange effects. The evolution equation of dislocations densities in different glide systems is established on account of the activation criterion for dislocations. The model thus obtained is complete once the energy densities and the initial material symmetry of the material are specified.     

A multiplicative finite strain finite element framework for the modelling of semicrystalline polymers and polycarbonates

This paper presents a phenomenological model for the simulation and analysis of stress‐induced orientational hardening in semicrystalline polymers and polycarbonates at finite strains. The notion of intermediate (local) stress‐free configuration is used to develop a set of constitutive equations, and its relation to the multiple natural (stress‐free) configurations in the class of materials being considered here is discussed. A hyperelastic stored energy function, written with respect to the intermediate stress‐free configuration is presented to model the finite elastic response. It is then combined with the J₂‐flow theory to model the finite inelastic response. The isochoric constraint during inelastic deformation is treated via an exact multiplicative decomposition of the deformation gradient into volume‐preserving and spherical parts. The numerical solution algorithm is based on the use of operator splitting technique that results in a product formula algorithm with elastic‐predictor/inelastic‐corrector components. Numerical results are presented to show the behaviour of the model. Copyright © 2000 John Wiley & Sons, Ltd.