Micro-potential model for stress-strain hysteresis of micro-cracked materials

While developing models for nonlinear mechanical and acoustical behavior of micro-cracked materials, it is common to start from a purely phenomenological approach. Most approaches essentially assume the material to have certain given “mathematical” properties, that lead to an acceptable equation of state (stress-strain relation) containing nonlinearity and hysteresis. In this paper, we formulate a deeper physical insight on the subject of mechanical hysteresis based on physical and measurable material properties. The theory developed in this paper interprets real images of crack networks in micro-inhomogeneous materials, obtained via electron microscopy, and uses this interpretation to build up a micro-potential model for a medium containing elementary cracks with known properties. It is found that the hysteretic contribution of each crack strongly depends on its average rest opening and its asperity. As a result, a distribution of cracks with different properties yields the physical basis for a slightly more complex version of the commonly used Preisach-Mayergoyz space in rock mechanics. The effect of a uniform distribution of the crack properties on the stress-strain relation is shown as an example.     

Analytical model of thermal striping for a micro-cracked solid

Using the formal asymptotic approximation of the Mode I stress intensity factor for an edge crack in a thermoelastic half plane containing several small voids obtained in [Nieves, M.J., Movchan, A.B., and Jones, I.S., 2011. Asymptotic study of a thermoelastic problem in a semi-infinite body containing a surface-breaking crack and small perforations. QJMAM 64 (3), 349–369] we investigate the effect of micro-cracks on this stress intensity factor. In numerical examples, we show how the behaviour of the stress intensity factor as a function of crack depth is affected by micro-cracks of different orientations occurring in the half space.     

A unified constitutive model for superalloys

A physically based unified constitutive model is presented for an aircraft engine nickelbase superalloy. The model accounts for deformation modes that can be activated under different stress, time, and temperature combinations. Two internal state variables and a flow function have been utilized to prdict strain rate sensitivity, stress hold creep, strain hold relaxation, monotonic loading, cyclic loading, and thermal mechanical cycling. In the model flow function, creep deformation and plasticity deformation modes have been incorporated over a wide range of temperatures (0.4 < T/T_melt < 0.75). The model is checked with independent isothermal and thermal mechanical experiments. Different temperature ranges are explored to assess model capabilities.     

A quasi-flow constitutive model with strain-rate dependence

In this paper, the proposed is a quasi-flow constitutive model with strain-rate sensitivity for elastic plastic large deformation. The model is based on the Quasi-flow Corner theory, and is suitable for the sheet metal forming process simulation with a variable punch machine velocity. Uniaxial tensile tests and deep-drawing tests of a circular blank with square punch are carried out and numerically simulated. The consistency between the experimental and the numerically simulated results shows the validity of the present new constitutive model. The project supported by the Scientific Foundation of National Outstanding Youth of China (10125208), the National Natural Science Foundation of China (19832020), and the National Education Committee of China     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.     

A nonlinear constitutive model for magnetostrictive materials

A general nonlinear constitutive model is proposed for magnetostrictive materials, based on the important physical fact that a nonlinear part of the elastic strain produced by a pre-stress is related to the magnetic domain rotation or movement and is responsible for the change of the maximum magnetostrictive strain with the pre-stress. To avoid the complicity of determining the tensor function describing the nonlinear elastic strain part, this paper proposes a simplified model by means of linearizing the nonlinear function. For the convenience of engineering applications, the expressions of the 3-D (bulk), 2-D (film) and 1-D (rod) models are, respectively, given for an isotropic material and their applicable ranges are also discussed. By comparison with the experimental data of a Terfenol-D rod, it is found that the proposed model can accurately predict the magnetostrictive strain curves in low, moderate and high magnetic field regions for various compressive pre-stress levels. The numerical simulation further illustrates that, for either magnetostrictive rods or thin films, the proposed model can effectively describe the effects of the pre-stress or residual stress on the magnetization and magnetostrictive strain curves, while none of the known models can capture all of them. Therefore, the proposed model enjoys higher precision and wider applicability than the previous models, especially in the region of the high field.The project supported by the National Natural Science Foundation of China (10132010 and 90405005)     

A viscoplastic constitutive model with effect of aging

It is recognized experimentally that differences in plastic flow stress due to the change in strain rate of SUS304 stainless steel are found to decrease after cyclic preloadings, but minimal change is observed in relaxation properties. Therefore, viscosity function based on tensile stress–strain properties differs from that obtained from relaxation behavior. For such case, the existing visco-plastic constitutive concept, such as the so-called overstress model where only viscosity is taken into consideration, has a poor capability in predicting the time-dependent mechanical properties systematically. A new viscoplastic constitutive concept is presented to analyze the phenomenon. In the constitutive concept, the dynamic strain aging even at room temperature, as well as viscosity, are introduced as the dominant factors of the time-dependent plastic deformation. An experimental technique is proposed and some experimental results are presented to estimate the effects of aging and viscosity separately on the time-dependency of a SCM435 low alloyed steel under tensile loading. The proposed constitutive model with aging is verified for the systematical predictions of both plastic flow properties and relaxation behavior of the SCM435 low alloyed steel.     

一种黏弹塑性统一本构模型

经过对大量有关统一本构模型的文献资料分析，指出了现有统一本构模型存在的问题，并通过对材料实验数据的分析，指出了黏弹性现象在实验中的表现，并据此将黏弹性引入到弹性黏塑性统一本构模型之中，建立了黏弹塑性统一本构模型，通过模型的数值模拟证明：模型计算结果无论在变形趋势上，还是在数值精度上都与实验数据符合得很好，克服了此前统一本构模型存在的问题。黏弹塑性统一本构模型的产生将统一本构模型的产生将统一本构理论的内涵扩大到黏弹性范围，进而构造了一个黏弹塑性理论的新框架。     

A constitutive model for magnetostrictive and piezoelectric materials

This paper is concerned with a macroscopic nonlinear constitutive law for magnetostrictive alloys and ferroelectric ceramics. It accounts for the hysteresis effects which occur in the considered class of materials. The uniaxial model is thermodynamically motivated and based on the definition of a specific free energy function and a switching criterion. Furthermore, an additive split of the strains and the magnetic or electric field strength into a reversible and an irreversible part is suggested. Analog to plasticity, the irreversible quantities serve as internal variables. A one-to-one-relation between the two internal variables provides conservation of volume for the irreversible strains. The material model is able to approximate the ferromagnetic or ferroelectric hysteresis curves and the related butterfly hysteresis curves. Furthermore, an extended approach for ferrimagnetic behavior which occurs in magnetostrictive materials is presented. A main aspect of the constitutive model is its numerical treatment. The finite element method is employed to solve the coupled field problem. Here the usage of the irreversible field strength permits the application of algorithms of computational inelasticity. The algorithmic consistent tangent moduli are developed in closed form. Hence, quadratic convergence in the iterative solution scheme of governing balance equations is obtained.     

A new microstructure-based constitutive model for human blood

A new constitutive equation for whole human blood is derived using ideas drawn from temporary polymer network theory to model the aggregation and disaggregation of erythrocytes in normal human blood at different shear rates. Each erythrocyte is represented by a dumbbell. The use of a linear spring law in the dumbbells leads to a multi-mode generalized Maxwell equation for the elastic stress and both the relaxation times and viscosities are functions of a time-dependent structure variable. An approximate constitutive equation is derived by choosing a single mode corresponding to the cell aggregate size where the largest number of cells are to be found. This size is identified in the case of steady flows. The model exhibits shear-thinning, viscoelasticity and thixotropy and these are clearly related to the microstructural properties of the fluid. Agreement with the experimental data of Bureau et al. [M. Bureau, J.C. Healy, D. Bourgoin, M. Joly, Rheological hysteresis of blood at low shear rate, Biorheology 17 (1980) 191–203] in the case of a simple triangular step shear rate flow is convincing.     

A variational multiscale constitutive model for nanocrystalline materials

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall-Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.     

A multi-axial constitutive model for metal matrix composites

Metal matrix composites (MMCs) generally do not follow the classical plasticity theory, even though the matrix metals do deform plastically. A tension-compression yield asymmetry is typically observed in MMCs. For particulate-reinforced MMCs, this non-classical response is mainly due to the variation of damage evolution with loading modes. In this paper, a viscoplastic multi-axial constitutive model for plastic deformation of MMCs is constructed using the Mises-Schleicher yield criterion. The subsequent plastic flow is characterized by an associated and decomposed flow rule considering effects from both deviatoric and hydrostatic stresses. This model is capable of describing the multi-axial yield and flow behavior of MMCs by using simulated or measured asymmetric tensile and compressive stress-strain responses as input. As an example, the influence of damage evolution in terms of interfacial debonding in MMCs (obtained from FEM simulations) is incorporated through the different tensile and compressive stress-strain behaviors. Applying this model to predict the torsion and the pressure-dependant tensile responses of some commonly used MMCs provides good agreement with experimental data.     

A three-dimensional constitutive model for shape memory alloys

Shape memory alloys (SMAs) are materials that, among other characteristics, have the ability to present high deformation levels when subjected to mechanical loading, returning to their original form after a temperature change. Literature presents numerous constitutive models that describe the phenomenological features of the thermomechanical behavior of SMAs. The present paper introduces a novel three-dimensional constitutive model that describes the martensitic phase transformations within the scope of standard generalized materials. The model is capable of describing the main features of the thermomechanical behavior of SMAs by considering four macroscopic phases associated with austenitic phase and three variants of martensite. A numerical procedure is proposed to deal with the nonlinearities of the model. Numerical simulations are carried out dealing with uniaxial and multiaxial single-point tests showing the capability of the introduced model to describe the general behavior of SMAs. Specifically, uniaxial tests show pseudoelasticity, shape memory effect, phase transformation due to temperature change and internal subloops due to incomplete phase transformations. Concerning multiaxial tests, the pure shear stress and hydrostatic tests are discussed showing qualitatively coherent results. Moreover, other tensile–shear tests are conducted modeling the general three-dimensional behavior of SMAs. It is shown that the multiaxial results are qualitative coherent with the related data presented in the literature.     

A dislocation-mechanics-based constitutive model for dynamic strain aging

Dynamic strain aging (DSA) is an important phenomenon in solute hardened metals and seriously affects the mechanical properties of metals. DSA is generally induced by the interaction between the moving dislocations and the mobile solute atoms. In this paper, only the interaction between moving dislocations and mobile solute atoms in a dislocation core area (core atmosphere) will be taken into account. To establish the constitutive model which can describe the DSA phenomenon, we improved the Zerilli-Armstrong dislocation-mechanics-based thermal viscoplastic constitutive relation, and added the effect of the interaction between the moving dislocations and core atmosphere. Because the constitutive relation established is based on the Zerilli-Armstrong relation, it can describe not only the DSA phenomenon, but also the mechanical behavior of metals over a broad range of temperatures (77K∼1000K) and strain rate (10⁻⁴∼10⁴ s⁻¹). The model prediction for tantalum fits well with the experimental data. Projected supported by the Chinese Academy of Sciences and the High Technical Project.     

A unified model for piezocomposites with non-piezoelectric matrix and piezoelectric ellipsoidal inclusions

In this paper, the closed-form solutions of the electroelastic Eshelbys tensors of a piezoelectric ellipsoidal inclusion in an infinite non-piezoelectric matrix are obtained via the Greens function technique. Based on the generalized Budianskys energy-equivalence framework and the closed-form solutions of the electroelastic Eshelbys tensors, a unified model for multiphase piezocomposites with the non-piezoelectric matrix and piezoelectric inclusions is set up. The closed-form solutions of the effective electroelastic moduli of piezocomposites are also obtained. The unified model has a rigorous but simple form, which can describe the multiphase piezocomposites with different connectivities, such as 0–3, 1–3, 2–2, 2–3, 3–3 connectivities, etc. It can also describe the effects of non-interaction and interaction among the inclusions. As examples, the closed-form solutions of the effective electroelastic moduli are given by means of the dilute solution for the 0–3 piezocomposite with transversely isotropic piezoelectric spherical inclusions and by means of the dilute solution and the Mori–Tanakas method for the 1–3 piezocomposite with two kinds of transversely isotropic piezoelectric cylindrical inclusions. The predicted results are compared with experimental data, which shows that the theoretical curves calculated by means of the Mori–Tanakas method agree quite well with the experimental values, but the theoretical curves obtained by the dilute solution agree well with the experimental values only when the volume fraction of the ceramic inclusion is less than 0.3. The results in this paper can be used to analyze and design the multiphase piezocomposites.