Electro-mechanical pre-fracture zones for an electrically permeable interface crack in a piezoelectric bimaterial

A plane strain problem for two piezoelectric half-spaces adhered by a very thin isotropic interlayer with a crack under the action of remote mixed mode mechanical loading and electrical flux is considered. The crack is situated either at an interface or in the interlayer. It is assumed that the substrates are much stiffer than the intermediate layer. Therefore, pre-fracture zones (plastic or damage) arise at the crack continuations. Normal and shear stresses are assumed to be constant in this zones and to satisfy some material equation, which can be taken from theory or derived experimentally. Modeling the pre-fracture zones by the crack continuations with unknown cohesive stresses on their faces reduces the problem to elastic interface crack analysis leading to a Hilbert problem. This problem is solved exactly. The pre-fracture zone lengths and stresses in these zones are found from algebraical and transcendental equations. The latter are derived from the conditions of stress finiteness at the ends of pre-fracture zones and the material equations. The electrical displacement at any point of the pre-fracture zones is found in closed form as well. Particular cases of symmetrical loading and of equivalent properties of the upper and lower bimaterial components are considered. Numerical results corresponding to certain material combinations and interlayer material equations are presented and analysed. In the suggested model, any singularities connected with the crack are eliminated, i.e., all mechanical and electrical characteristics are limited in the near-crack tip region.     

Discrete numerical investigation of the ratcheting phenomenon in granular materials

Several relevant geotechnical works, such as railway and road embankments, offshore foundations and vibrating machine foundations, are affected by the progressive accumulation of irreversible settlements. These latter represent the macroscopic evidence of the progressive rearrangement of particles under cycling loading, which is commonly referred to, in the literature, as ratcheting. This phenomenon is well known, but it is quite difficult to describe it by means of an appropriate constitutive model. As a consequence, the evaluation of durability of the aforementioned structures remains an open problem. In this article, the phenomenon will be approached by employing a Distinct Element model capable of describing the evolution of the microstructure induced by cyclic mechanical perturbations. Several analyses are performed in order to stress the influence of both the stress level and loading history on the mechanical response of a numerical model of a sand specimen. The numerical analyses are intended to provide an experimental background for conceiving a simplified macro approach based on generalised plasticity theory. In particular by means of probe test the plastic potential and the hardening parameters will be defined as a function of the current stress state and loading history.     

Un modèle viscoélastique de remodelage osseux : approche unidimensionnelleA viscoelastic model of bone remodeling: unidimensional approach

A bone remodeling model taking into account the viscoelastic properties of the material is proposed. The nonlinear equations governing the evolution of the bone apparent density are solved by a finite difference method in the unidimensional case of a n-unit elements model. The results show the effects of the viscous damping on the structure for a controlled mechanical loading. To cite this article: S. Baïotto, M. Zidi, C. R. Mecanique 332 (2004).     

Macroscopical non-linear material model for ferroelectric materials inside a hybrid finite element formulation

A new approach for modeling hysteretic non-linear ferroelectric ceramics is presented, based on a fully ferroelectric/ferroelastic coupled macroscopic material model. The material behavior is described by a set of yield functions and the history dependence is stored in internal state variables representing the remanent polarization and the remanent strain. For the solution of the electromechanical coupled boundary value problem, a hybrid finite element formulation is used. Inside this formulation the electric displacement is available as nodal quantity (i.e. degree of freedom) which is used instead of the electric field to determine the evolution of remanent polarization. This involves naturally the electromechanical coupling. A highly efficient integration technique of the constitutive equations, defining a system of ordinary differential equations, is obtained by a customized return mapping algorithm. Due to some simplifications of the algorithm, an analytical solution can be calculated. The automatic differentiation technique is used to obtain the consistent tangent operator. Altogether this has been implemented into the finite element code FEAP via a user element. Extensive verification tests are performed in this work to evaluate the behavior of the material model under pure electrical and mechanical as well as coupled and multi-axial loading conditions.     

Constitutive modeling of strain-induced softening in swollen elastomers

Under cyclic loading, elastomeric material exhibits strong inelastic responses such as stress-softening due to Mullins effect, hysteresis and permanent set. The corresponding inelastic responses are observed in both dry and swollen rubbers. Moreover, it is observed that inelastic responses depend strongly on the swelling level. For engineering applications involving the interaction and contact between rubber components and solvent, the understanding and consideration of swelling are essential pre-requisites for durability analysis. In this paper, a simple phenomenological model describing Mullins effect in swollen rubbers under cyclic loading is proposed. More precisely, the proposed model adopts the concept of evolution of soft domain microstructure with deformation originally proposed by Mullins and Tobin. The swollen rubbers are obtained by immersing dry ones in solvent until desired degrees of swelling are achieved. Subsequently, their mechanical responses, in particular Mullins effect, under cyclic loading are investigated. These experimental data are used to assess the efficiency of the proposed model. Results show that the model agrees qualitatively well with experiments. Furthermore, the model captures well the fundamental features of strain-induced softening.     

A HYDRO-MECHANICAL-CHEMICAL COUPLING MODEL FOR GEOMATERIAL WITH BOTH MECHANICAL AND CHEMICAL DAMAGES CONSIDERED

A general framework of hydro-mechanical-chemical coupling model is proposed for geomaterial subjected to the dual effects of mechanical loading and chemical degradation. Mechanical damage due to microcracks in solid matrix and chemical damage induced by the increase of porosity due to dissolution of matrix minerals as well as their interactions are considered. A special model is proposed for sandstone. The reaction rate is formulated within the framework of mineral reaction kinetics and can thus take into account different dissolution mechanisms of three main mineral compositions under different pH values. The increase of porosity is physically defined by the dissolution of mineral composition and the chemical damage is related to the increase of porosity. The mechanical behavior is characterized by unified plastic damage and viscoplastic damage modeling. The effective stress is used for describing the effect of pore pressure. The elastic parameters and plastic evolution as well as viscoplastic evolution are dependent on chemical damage. The advection, which is coupled with mechanical damage and chemical damage, is considered as the dominant mechanism of mass transfer. The application of model proposed is from decoupled experiments to fully coupled experiment. The model offers a convenient approach to describing the hydro-mechanical-chemical coupled behavior of geomaterial.     

Calibration,validation, and verification including uncertainty of a physically motivated internal state variable plasticity and damage model

In this paper, we illustrate a formal calibration, validation, and verification process that includes uncertainty in an internal state variable plasticity-damage model that is implemented in a finite element code. The physically motivated continuum model characterizes damage evolution by incorporating material uncertainty due to microstructural spatial clustering. The uncertainty analysis is performed by introducing material variation through model validation and verification. The effect of variability in microstructural clustering and boundary conditions on the sensitivities and uncertainty of the plasticity-damage evolution for the 7075 aluminum alloy is characterized. The results show the potential of this methodology in the evaluation of material response uncertainty due to microstructure spatial clustering and its effect on damage evolution. For damage evolution, we have shown that the initial isotropic damage evolved into an anisotropic form as the deformation increased which is consistent with experimentally observed behavior for 7075 aluminum alloy in literature. Also, the sensitivities were found to be consistent with the physics of damage progression for this particular type of material. Through the sensitivity analysis, the initial defect size and number density of cracked particles are important at the beginning of deformation. As damage evolves, more voids are nucleated and grow and the sensitivity analysis illustrates this as well. Then, voids combine with each other and coalescence becomes the main driver, which is also confirmed by the sensitivity analysis. This work also shows that the microstructurally based damage evolution equations provide an accurate representation of the damage progression due to large intermetallic particles. Finally, we illustrate that the initial variation in the microstructure clustering can lead to about ±7.0%, ±8.1%, and ±9.75% variation in the elongation to failure strain for torsion, tensile, and compressive loading, respectively.     

Simulation of size effects by a phase field model for fracture

In phase field fracture models the value of the order parameter distinguishes between broken and undamaged material. At crack faces the order parameter interpolates smoothly between these two states of the material, which can be regarded as phases. The crack evolution follows implicitly from the time integration of an evolution equation of the order parameter, which is coupled to the mechanical field equations. Among other phenomena phase field fracture models are able to reproduce crack nucleation in initially sound materials. For a 1D setting it has been shown that crack nucleation is triggered by the loss of stability of the unfractured, spatially homogeneous solution, and that the stability point depends on the size of the considered structure. This work numerically investigates to which extend size effects are reproduced by the 2D phase field model. Exemplarily, a finite element study of the hole size effect is performed and the simulation results are compared to experimental data.     

An endochronic constitutive model for rate-dependent and nonproportional cyclic plasticity

The total stress response of material is decomposed into a sum of an equilibrium stress response and a non-equilibrium overstress response. Correspondingly, the rate-independent intrinsic time and the rate-dependent intrinsic time are defined respectively. Additional hardening functions for describing the isotropic and anisotropic nonproportional effects are assumed to be related to the accumulation of plastic strain component along the normal of equilibrium stress trajectory, in which the effects of geometry of the loading path are included. An endochronic constitutive model for rate-dependent, nonproportional cyclic plasticity is formulated and applied to simulate the stress responses of stainless steel XCrNi18.9 for some typical loading programs at different loading rates. A comparison between predicted results and experimental ones by Haupt and Lion shows that the former are in agrreement with the latter.     

Evolution of motion of a double planet in the gravitational field of a massive viscoelastic body

The motion of a double planet in the gravitational force field of a massive planet modeled by a viscoelastic body is studied. The double planet is modeled by a viscoelastic body and a material point. These viscoelastic bodies are homogeneous, isotropic and, in their natural undeformed state, occupy spherical regions in the three-dimensional Euclidean space. The problem is solved in the framework of a linear model in the theory of viscoelasticity. The motion separation method and the averaging method are applied to derive an approximate system of equations describing the evolution of translationalrotational motion for the mechanical system under study. The Earth-Moon system is considered in the gravitational field of the Sun as an example of a double planet.     

Irreversible deformation with subsequent unloading of a spherical viscoelastoplastic layer

Analytic solutions are obtained for a sequence of one-dimensional quasistatic problems describing viscoelastic deformation processes in the material of a hollow ball and the plastic flow nucleation and evolution processes occurring in the ball as the pressure on the outer boundary increases. The unloading process under slow removal of the loading pressure is considered as well. The stress fields and the elastic and plastic strain fields in the spherical layer material, the law of motion of the elastoplastic boundary, and the residual stress level and distribution are computed. It is assumed that at the stage preceding the plastic flow the material obeys the viscoelastic Voigt model and the loading surface is determined by the von Mises plastic flow condition.     

A constitutive model dealing with damage due to cavity growth and the Mullins effect in rubber-like materials under triaxial loading

In this work, we attempted to describe the evolution of damage in rubber-like materials due to the Mullins effect and the cavity growth process. To this end we introduced two distinct internal variables into the constitutive laws; the first one essentially describes the Mullins damage and the second describes the cavity growth. The Mullins effect was considered as a continuous type of damage that can be modelled within the continuum damage theory. The cavity growth, being discontinuous at the microscopic scale, was also modelled by a continuous variable after a homogenization procedure. These analyses allow the establishment of a compressible constitutive law describing the strain-softening phenomena for rubber-like materials. In order to identify the material parameters and to verify the efficiency of the model, we carried out experimental studies involving uniaxial, biaxial, and hydrostatic tensions under monotonic and cyclic loading. Comparison between the model-predicted results and the experimental data shows that the present model can efficiently describe both the Mullins damage and the porosity evolution of rubber-like materials under triaxial monotonic or cyclic loading with a satisfactory accuracy. The proposed concept is simple and easy to apply to engineering calculations.     

Time domain modeling of damping using anelastic displacement fields and fractional calculus

A fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and elastic part is developed. The evolution equation for the anelastic part is then a differential equation of fractional order in time. By using a fractional order evolution equation for the anelastic strain the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, the model parameters are fit to available frequency domain data for a high damping polymer. By studying the relaxation modulus and the relaxation spectrum the material parameters of the present viscoelastic model are given physical meaning. The use of this viscoelastic model in structural modeling is discussed and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag–Leffler function type. Finally a time step algorithm for solving the finite element equations are developed and some numerical examples are presented.     

On the evolution of the linear material properties of PZT during loading history—an experimental study

Ferroelectrics exhibit material behavior which is strongly affected by its loading history. Among other phenomena, the coefficients describing the linear material behavior are known to change when the state of polarization is altered. There are several approaches to modeling ferroelectric/ferroelastic behavior. However, with all models, assumptions have to be made on how the linear coefficients depend on the state of polarization. Often the elastic and dielectric coefficients are defined to be constant for the sake of simplicity. Alternatively, their evolution and that of the piezoelectric constants are described rather intuitively, while systematic experimental data are sparse. The present study explores the impact of large signal mechanical and electrical loading on the low frequency linear response of a soft PZT ceramic. This is accomplished via cyclic tests with progressively increasing maximum electrical or mechanical load. Upon load reversal, the quasi-linear response is measured. Remanent polarization and remanent strain are used as internal variables to describe the material behavior as a function of loading history. While the dielectric permittivity κ₃₃ is shown to exhibit only minor variation, Young’s modulus and the piezoelectric coefficient d₃₃ change significantly in the course of loading.     

A crack-mechanics based model for damage and plasticity of brittle materials under dynamic loading

A rate-dependent model for damage and plastic deformation of brittle materials under dynamic loading is presented. The model improves upon a recently developed micromechanical damage model (Zuo et al., 2006) by incorporating plastic deformation of the material. The distribution of the microcracks in the material is assumed to remain isotropic, and the damage evolution is through the growth of the average crack size. Plasticity is considered through an additive decomposition of the total strain rate, and a rate-independent, von Mises model is used. The model was applied to simulate the response of a model material (SiC) under uniaxial strain loading. To further examine the behavior of the model, cyclic loading and large-strain compressive loading were considered. Numerical results of the model predictions are presented, and comparisons with those from a previous model are provided.     

Modeling the mechanical response of rolled Zircaloy-2

An elastoplastic self-consistent model has been implemented to perform a systematic study of the response of rolled Zircaloy-2 subjected to mechanical loading. The intergranular stresses induced by cooling the material from 898 K to room temperature are calculated, accounting for the experimental texture, and compared with experimental data. The elastoplastic response in tension and compression along the rolling and the transverse directions of the sheet is predicted and compared against the results of uniaxial tensile and compressive tests performed in the same material. The role of the internal stresses on the yield stress and the elastoplastic transition is analyzed, and information about the active deformation systems in the individual grains is inferred from the comparison. Indirect inference of the parameters describing the deformation mechanisms is the only available means, because it is not possible to grow single crystals of these alloys. The results of this study demonstrate the adequacy of self-consistent schemes for predicting intergranular stresses and the significance of the latter on the mechanical behavior of the material.     

Phase field model for coupled displacive and diffusive microstructural processes under thermal loading

A non-isothermal phase field model that captures both displacive and diffusive phase transformations in a unified framework is presented. The model is developed in a formal thermodynamic setting, which provides guidance on admissible constitutive relationships and on the coupling of the numerous physical processes that are active. Phase changes are driven by temperature-dependent free-energy functions that become non-convex below a transition temperature. Higher-order spatial gradients are present in the model to account for phase boundary energy, and these terms necessitate the introduction of non-standard terms in the energy balance equation in order to satisfy the classical entropy inequality point-wise. To solve the resulting balance equations, a Galerkin finite element scheme is elaborated. To deal rigorously with the presence of high-order spatial derivatives associated with surface energies at phase boundaries in both the momentum and mass balance equations, some novel numerical approaches are used. Numerical examples are presented that consider boundary cooling of a domain at different rates, and these results demonstrate that the model can qualitatively reproduce the evolution of microstructural features that are observed in some alloys, especially steels. The proposed model opens a number of interesting possibilities for simulating and controlling microstructure pattern development under combinations of thermal and mechanical loading.