具有体膨胀和剪切效应的结构陶瓷相变塑性细观本构模型:Ⅰ.非比例加载历史

本文在对结构陶瓷的四方至单斜(t→m)马氏体相变进行细观力学、热力学和微观机制分析的基础上,导出了在非比例加载条件下考虑材料的体膨胀和剪切效应的相变塑性细观本构模型。作者首次采用 Mori-Tanaka 方法以自洽的方式导出了材料构元的 Helmho-ltz 自由能及余能函数的解析表达式,它是外加宏观应力(或应变)、温度、相变夹杂体积分数以及夹杂内平均相变应变的函数,其中夹杂体积分数和平均相变应变为描述材料构元微结构变化的内变量。最后按 Hill-Rice 本构理论框架导出相变塑性屈服面方程及增量本构关系。     

A micromechanics constitutive theory for forward transformation plasticity with shear and dilatation effect: I,nonproportional loading history

A micromechanics constitutive theory which takes into account both the dilatation and shear effects of the transformation is proposed to describe the macroscopic plastic behavior of structure ceramics during forward transformation under different temperatures. Under some basic assumptions, the analytic expressions of the Helmholtz and complementary free energy of the constitutive element are derived in a self-consistent manner by using the Mori-Tanaka's method which takes into account the interaction between the transformed inclusions. In the framework of Hill-Rice's internal variable constitutive theory, the forward transformation yield function and incremental stress strain relations, in analogy to the theory of metal plasticity, for non-proportional loading histories are obtained. The project supported by National Natural Science Foundation of China     

Two-scale micromechanics-based probabilistic modeling of domain switching in ferroelectric ceramics

A two-scale micromechanics model is developed in this paper to analyze domain switching in ferroelectric ceramics, using a probabilistic domain switching criterion based on energetic analysis. The microstructure of ferroelectric ceramics at two distinct length scales, domains and grains, has been carefully analyzed. The interaction at domain level is accounted for by energy minimization theory, while the fluctuation at grain level is analyzed using ellipsoidal two-point correlation function. The model has been implemented by Monte Carlo method, and applied to simulate the electric poling and mechanical depoling of Pb(Zr_xTi_1-x)O₃ (PZT) ceramics across morphotropic phase boundary (MPB). The drastically different switching characteristics of PZT ceramics across MPB has been captured, and good agreement with experiments has been observed. The effects of the transformation strains and spontaneous polarizations are highlighted, confirming the proposition of Li et al. [2005. Domain switching in polycrystalline ferroelectric ceramics. Nature Materials 4, 776–781] that the strain compatibility plays a dominant role in domain switching in ferroelectric ceramics.     

Micromechanics of ferroelectric domain switching behavior: Part II: Constitutive relations and hysteresis

A constitutive relation is developed to describe the nonlinear behavior of ferroelectric ceramics subjected to external stress and electric field. The theoretical development considers each domain as an inclusion. The Helmholtz and Gibbs free energy of the constituent element are derived by using a micromechanics approach. They are functionals of the orientation distribution function (ODF) that represents the domain distribution patterns. By applying the internal variable theory and expanding ODF in Fourier series, the yield condition, evolution of ODF, and constitutive relation are obtained. Theoretical results agree with experiments.     

A micromechanics method to study the effect of domain switching on fracture behavior of polycrystalline ferroelectric ceramics

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On the thermodynamics of some generalized second-grade fluids

The generalized second-grade fluids, which have been used for modeling the creep of ice and the flow of coal-water and coal-oil slurries, are among the simplest non-Newtonian fluid models that can describe shear-thinning/thickening and exhibit normal stress effects. In this article, we conduct thermodynamic analysis on a class of generalized second-grade fluids, one distinguishing feature of which is the existence of a constitutive function Φ that describes frictional heating. We work within the framework of Serrin’s original formulation of neoclassical thermodynamics, where internal energy and entropy functions, if they exist for a continuous body at all, are to be derived from the classical First Law and (quantitatively reformulated) Second Law of thermodynamics for cycles. For the class of generalized second-grade fluids in question, we show from the First Law that an internal energy density u exists, and we derive the equation of energy balance; from the Second Law, we demonstrate the existence of an entropy density s and derive the Clausius–Duhem inequality that it satisfies. We obtain explicit expressions for u, s and the frictional heating Φ, and derive thermodynamic restrictions on the material functions of temperature μ, α ₁, and α ₂ that appear in the constitutive relation for the Cauchy stress. For the special case of second-grade fluids, our expressions for u and s agree with those which Dunn and Fosdick [6] derived under the theoretical framework of the rational thermodynamics of Coleman and Noll.     

Characterization of random microstructural stresses and fracture estimation

In this paper, the analytical approach to quantitative characterization of random microstructural residual stress field in brittle elastic materials is presented. The analysis consists of two parts. First, we expound the basic features of random microstructural stresses and show how the eigenstrain approach (of “classical” micromechanics) can be extended to the situations where randomness of the initial eigenstrains has to be taken into account. The second part of the paper deals with the effects of random microstresses on crack growth phenomena. The stress intensity induced by random dilatant transformation eigenstrains and by thermally-induced random microstresses are treated in detail, including numerical and graphical illustrations of specific crack problems.     

A 2D finite element based on a nonlocal constitutive model describing localization and propagation of phase transformation in shape memory alloy thin structures

Here, the effects of localization and propagation of martensitic phase transformation on the response of SMA thin structures subjected to thermo-mechanical loadings are investigated using nonlocal constitutive model in conjunction with finite element method. The governing equations are derived based on variational principle considering thermo-mechanical equilibrium and the spatial distribution of the nonlocal volume fraction of martensite during transformation. The nonlocal volume fraction of martensite is defined as a weighted average of the local volume fraction of martensite over a domain characterized by an internal length parameter. The local version of the thermo-mechanical behavior model derived from micromechanics considers the local volume fraction of martensite and the mean transformation strain. A 4-noded quadrilateral plane stress element with three degrees of freedom per node accounting for in-plane displacements and the nonlocal volume fraction of martensite is developed. Numerical simulations are conducted to bring out the influence of material and geometrical heterogeneities (perturbations/defects) on the localization and propagation of phase transformation in SMA thin structures. Also, a sensitivity analysis of the material response due to the localization and the other related model parameters is carried out. The detailed investigation done here clearly shows that the localization of phase transformation has significant effect on the response of shape memory alloys.     

Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics

Any attempt made to separate energy into electrical and mechanical parts may lead to inconsistencies as they do not necessarily decouple. This is illustrated by application of the energy density function in the linear theory of piezoelasticity. By assuming that a critical energy density function prevails at the onset of crack initiation, it is possible to establish the relative size of an inner and outer damage zone around the crack tip; they correspond to the ligaments at failure caused by pure electric field and pure mechanical load. On physical grounds, the relative size of these zones must depend on the relative magnitude of the mechanical and electrical load. Hence, they can vary in size depending on the electromechanical material and damage resistance properties. Numerical results are obtained for the PZT-4, PZT-5H, and P-7 piezoelectric ceramics. These two ligaments for the two damage zones may coincide for appropriate values of the applied electrical field and mechanical load.Explicit expression of the energy density factor S is derived showing the mixed mode electromechanical coupling effects. The factor S can increase or decrease depending on the direction of the applied electric field with reference to the poling direction. This is in contrast to the result obtained from the energy release rate quantity, which remains unchanged for electric field in the direction of poling or against it.     

Cohesive zone size of microcracks in brittle materials

Consideration of cohesive microcracks in continuum micromechanics is a challenging task since a lot of applications (such as, e.g., estimation of the stiffness of a microcracked solid) require a priori knowledge of the size of the cohesive zone. The latter, however, can be determined analytically only for the special case of Barenblatt–Dugdale cracks, i.e. for cracks with spatially constant cohesive tractions. Herein, we deal with the general case of spatially non-constant cohesive tractions: Generalizing the Barenblatt–Dugdale approach, we consider that each crack is surrounded by a plane annular cohesive zone characterized by a constitutive softening law (introduced as a power law) relating the vector of cohesive tractions to the displacement discontinuity. The size of this cohesive zone is then estimated using the theorem of minimum potential energy, based on a class of kinematically admissible displacement fields.     

Time dependence of effective properties of polycrystalline ferroelectric ceramics

In this paper, based on Merz[7] experimental results and classical nucleation theory, a micromechanics statistical model is proposed to describe the relation between the special microstructure-level evolution phenomena-domain switching and macro-response. The polycrystalline ferroelectric ceramics treated as a composition of switched domain and unswitched domain, the approaches of Eshelby's equivalent inclusion and Mori-Tanaka's mean field theory are used to analyze and predict its effective electroelastic properties. The model can incorporate the effects of time dependence of domain switching and shape of individual crystalline. To the BaTiO₃ polycrystalline ceramics, the analytical results are in good agreement with the experimental results.     

Granular micromechanics based micromorphic model predicts frequency band gaps

Granular materials are typically characterized by complex structure and composition. Continuum modeling, therefore, remains the mainstay for describing properties of these material systems. In this paper, we extend the granular micromechanics approach by considering enhanced kinematic analysis. In this analysis, a decomposition of the relative movements of interacting grain pairs into parts arising from macro-scale strain as well as micro-scale strain measures is introduced. The decomposition is then used to formulate grain-scale deformation energy functions and derive inter-granular constitutive laws. The macro-scale deformation energy density is defined as a summation of micro-scale deformation energy defined for each interacting grain pair. As a result, a micromorphic continuum model for elasticity of granular media is derived and applied to investigate the wave propagation behavior. Dispersion graphs for different cases and different ratios between the microscopic stiffness parameters have been presented. It is seen that the model has the capability to present band gaps over a large range of wave numbers.     

Soret and Dufour effect on double diffusion mixed convection from a vertical surface in a porous medium saturated with a non-Newtonian fluid

A non-similar boundary layer analysis is presented to study the flow, heat and mass transfer characteristics of non-Darcian mixed convection of a non-Newtonian fluid from a vertical isothermal plate embedded in a homogeneous porous medium with the effect of Soret and Dufour and in the presence of either surface injection or suction. The value of the mixed-convection parameter lies between 0 and 1. In addition, the power-law model is used for non-Newtonian fluids with exponent n < 1 for pseudoplastics n = 1 for Newtonian fluids and n > 1 for dilatant fluids. Furthermore, the coordinates and dependent variables are transformed to yield computationally efficient numerical solutions that are valid over the entire range of mixed convection, from the pure forced-convection limit to the pure free-convection limit, and the whole domain of non-Newtonian fluids, from pseudoplastics to dilatant fluids. The numerical solution of the problem is derived using a Runge–Kutta integration scheme with Newton–Raphson shooting technique. Distributions for velocity, temperature and concentration, as well as for the rate of wall heat and mass transfer, have been obtained and discussed for various physical parametric values.     

Analysis on small scale bridging in fibrous monolithic ceramics

Based on shear lag model of interface between fiber and matrix, a new formula that relates the crack opening displacement and bridging force in fibrous monolithic ceramics was constructed under the framework of small scale bridging. This formula was applied to predict the fracture resistance orR-curve response of a three-point bending specimen made of fibrous monolithic ceramics. A parametric investigation on the influences of fiber volume fraction, fiber radius, characteristics of constituents, BN's fracture toughness and specimen's geometry on the bridging forces and fracture resistance in Si₃N₄/BN composite was carried out. The upper and lower limits of theR-curve of Si₃N₄/BN in small scale bridging were derived. This research revealed the role played by the above parameters in the fracture toughness of fibrous monolithic ceramics. Supported by National Natural Science Foundation of China (59632090).     

The molecular theory of viscoelasticity for thermoplastic elastomer SBS (SIS) at large deformations

A microstructure model for SBS and SIS triblock copolymers with hard domains as multifunctional reinforcing fillers is proposed. Based on this model and proposed mechanism of large deformations, the probability distribution function of the end-to-end vector for each constituent chain and the free energy of deformation for the total networks was calculated by the combination of statistical thermodynamics and kinetics. A new molecular theory of non-linear visco-elasticity for SBS and SIS at large deformations is presented. It is successful in relating the viscoelastic state to molecular constitution by three important parameters (C ₁₀₀,C ₀₂₀, andC ₂₀₀) of the networks. The relations of stress to strain for four types of deformation, the elastic modulus and the constitutive equation for the stress relaxation were derived from this theory. It provides a theoretical foundation for studying the relationships of multiphase network structures and mechanical properties at large deformations. An excellent agreement between the theoretical relationships and experimental data from the experiments and the reference was obtained.Project supported by the National Natural Foundation of China     

Stiffness and strength of phase transformation ceramics containing misoriented microcracks

Microcracks exert a large influence on the behavior of phase transformation ceramics. In this paper, the cause of microcracking is interpreted and the interaction between transformation particles and microcracks is described to estimate the stiffness and strength through the modified equivalent inclusion method by the authors. Three point bend experiments on Al₂O₃/ZrO₂ ceramics are performed and a comparison between the theoretical and measured results confirms the rationality of the analytical model proposed in this paper and the important role of microcracks.     

Piezomagnetic and piezoelectric poling effects on mode I and II crack initiation behavior of magnetoelectroelastic materials

The ferrite and ferroelectric phase of magnetoelectroelastic (MEE) material can be selected and processed to control the macroscopic behavior of electron devices using continuum mechanics models. Once macro- and/or microdefects appear, the highly intensified magnetic and electric energy localization could alter the response significantly to change the design performance. Alignment of poling directions of piezomagnetic and piezoelectric materials can add to the complexity of the MEE material behavior to which this study will be concerned with.Appropriate balance of distortional and dilatational energy density is no longer obvious when a material possesses anisotropy and/or nonhomogeneity. An excess of the former could result in unwanted geometric change while the latter may lead to unexpected fracture initiation. Such information can be evaluated quantitatively from the stationary values of the energy density function dW/dV. The maxima and minima have been known to coincide, respectively, with possible locations of permanent shape change and crack initiation regardless of material and loading type. The direction of poling with respect to a line crack and the material microstructure described by the constitutive coefficients will be specified explicitly with reference to the applied magnetic field, electric field and mechanical stress, both normal and shear. The crack initiation load and direction could be predicted by finding the direction for which the volume change is the largest. In contrast to intuition, change in poling directions can influence the cracking behavior of MEE dramatically. This will be demonstrated by the numerical results for the BaTiO₃–CoFe₂O₄ composite having different volume fractions where BaTiO₃ and CoFe₂O₄ are, respectively, the inclusion and matrix.To be emphasized is that mode I and II crack behavior will not have the same definition as that in classical fracture mechanics where load and crack extension symmetry would coincide. A striking result is found for a mode II crack. By keeping the magnetic poling fixed, a reversal of electric poling changed the crack initiation angle from θ₀=+80° to θ₀=−80° using the line extending ahead of the crack as the reference. This effect is also sensitive to the distance from the crack tip. Displayed and discussed are results for r/a=10⁻⁴ and 10⁻¹. Because the theory of magnetoelectroelasticity used in the analysis is based on the assumption of equilibrium where the influence of material microstructure is homogenized, the local space and temporal effects must be interpreted accordingly. Among them are the maximum values of (dW/dV)_max and (dW/dV)_min which refer to as possible sites of yielding and fracture. Since time and size are homogenized, it is implicitly understood that there is more time for yielding as compared to fracture being a more sudden process. This renders a higher dW/dV in contrast to that for fracture. Put it differently, a lower dW/dV with a shorter time for release could be more detrimental.     

Microhardness indentation application and limitation in fracture-toughness evaluation of ceramics

The use of a microhardness indentation as a controlled flaw in a flexural specimen for measuring fracture toughness in advanced ceramics is critically evaluated. From a review of fracture-toughness testing methodologies for ceramics it is shown that currently there are not any standardized or widely agreed uponK _Ic test procedures. The problems associated with the controlled flaw test are discussed and analyzed. An experimental program is used to investigate the effects of residual stress on the measurement of fracture toughness in a wide variety of sintered and hot-pressed advanced ceramics including TiB₂, Al₂O₃, SiC, ZrO₂, Si₃N₄. The validity of the test procedure for evaluatingK _Ic of advanced ceramics in general is put forth.