基于高阶Cauchy-Born准则的单壁碳纳米管本构模型

提出了一种基于高阶Cauchy—Born准则建立单壁碳纳米管本构模型的方法。通过引入高阶变形梯度,合理地修正了传统Cauchy—Born准则在描述纳米管变形几何关系时所存在的缺陷。利用原子间相互作用势以及能量等效原理,得到了基于广义连续介质模型的单壁碳纳米管的本构关系。由此得到的本构参数不仅与变形梯度张量F,而且与其梯度F相关,因此是一种广义连续介质模型。利用这样的本构模型,本文还对单壁碳纳米管的杨氏模量进行了预测,并与采用其他方法得到的结果进行了对比,从而证实了所提出方法的有效性。     

The non-uniqueness of the atomistic stress tensor and its relationship to the generalized Beltrami representation

The non-uniqueness of the atomistic stress tensor is a well-known issue when defining continuum fields for atomistic systems. In this paper, we study the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). We demonstrate the validity of these analogies using a numerical test. A program for performing the decomposition of the atomistic stress tensor called MDStressLab is available online at http://mdstresslab.org.     

Thermomechanical material modelling based on a hybrid free energy density depending on pressure,isochoric deformation and temperature

In order to represent temperature-dependent mechanical material properties in a thermomechanical consistent manner it is common practice to start with the definition of a model for the specific Helmholtz free energy. Its canonical independent variables are the Green strain tensor and the temperature. But to represent calorimetric material properties under isobaric conditions, for example the exothermal behaviour of a curing process or the dependence of the specific heat on the temperature history, the temperature and the pressure should be taken as independent variables. Thus, in the field of calorimetry the Gibbs free energy is usually used as thermodynamic potential whereas in continuum mechanics the Helmholtz free energy is normally applied. In order to simplify the representation of calorimetric phenomena in continuum mechanics a hybrid free energy density is introduced. Its canonical independent variables are the isochoric Green strain tensor, the pressure and the temperature. It is related to the Helmholtz free energy density by a Legendre transformation. In combination with the additive split of the stress power into the sum of isochoric and volumetric terms this approach leads to thermomechanical consistent constitutive models for large deformations. The article closes with applications of this approach to finite thermoelasticity, curing adhesives and the glass transition.     

多晶金属材料的三维组集式弹塑性本构模型

本文从分析多晶金属材料的细观组织在弹塑性变形中的贮能和耗能机制入手,提出一个三维组集式弹塑性本构模型。该模型将材料单元抽象成沿三维空间各方向均匀分布组件的集合体,方向组件反映材料的细观性态并在宏观上协调变形,所有方向组件的内力总效应构成宏观应力。文中导出了显式的弹塑性本构关系,并与Budiansky的复杂加载试验结果及其它塑性模型进行了对比。     

Enhancing the Bolzon–Schrefler–Zienkiewicz Constitutive Model for Partially Saturated Soil

This paper presents an enhanced version of the elasto-plastic model for partially saturated soil first proposed by Bolzon, Schrefler and Zienkiewicz in 1996, “BSZ” model, which uses the effective stress tensor and suction as independent stress variables. It is recalled that the effective stress tensor proposed by Lewis and Schrefler in 1982 is thermodynamically consistent and, compared with other choices of stress tensors, results particularly suitable for partially saturated soil mechanics. A hydraulic constitutive relationship and a hydraulic hysteresis are introduced in the model, to take into account the irreversible deformation during cyclic drying and wetting until structural collapse. For this reason the plastic rate of strain is split into the sum of two components: one depending on the effective stress tensor and the other one on suction. This is the new feature of the BSZ model. This enhanced model is then cast into a thermodynamical framework at macroscopic level and it is shown that it is possible to derive the constitutive law from the Helmholtz free energy and a dissipation function, both for associative and non- associative plasticity. Finally the model predictions have been compared with experimental data for Sion slime, with particular emphasis on the deviatoric part, and model predictions of hysteretic behaviour have been investigated in case of a wetting and drying cycle on compacted betonite–kaolin.     

A Representation Theorem for Material Tensors of?Weakly-Textured Polycrystals and?Its Applications in?Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.     

Couple stresses in crystalline solids: origins from plastic slip gradients, dislocation core distortions, and three-body interatomic potentials

In the presence of plastic slip gradients, compatibility requires gradients in elastic rotation and stretch tensors. In a crystal lattice the gradient in elastic rotation can be related to bond angle changes at cores of so-called geometrically necessary dislocations. The corresponding continuum strain energy density can be obtained from an interatomic potential that includes two- and three-body terms. The three-body terms induce restoring moments that lead to a couple stress tensor in the continuum limit. The resulting stress and couple stress jointly satisfy a balance law. Boundary conditions are obtained upon stress, couple stress, strain and strain gradient tensors. This higher-order continuum theory was formulated by Toupin (Arch. Ration. Mech. Anal. 11 (1962) 385). Toupin's theory has been extended in this work to incorporate constitutive relations for the stress and couple stress under multiplicative elastoplasticity. The higher-order continuum theory is exploited to solve a boundary value problem of relevance to single crystal and polycrystalline nano-devices. It is demonstrated that certain slip-dominated deformation mechanisms increase the compliance of nanostructures in bending-dominated situations. The significance of these ideas in the context of continuum plasticity models is also dwelt upon.     

Stochastic damage model for concrete based on energy equivalent strain

Starting with the framework of conventional elastoplastic damage mechanics, a class of stochastic damage constitutive model is derived based on the concept of energy equivalent strain. The stochastic damage model derived from the parallel element model is adopted to develop the uniaxial damage evolution function. Based on the expressions of damage energy release rates (DERRs) conjugated to the damage variables thermodynamically, the concept and its tensor formulations of energy equivalent strain is proposed to bridge the gap between the uniaxial and the multiaxial constitutive models. Furthermore, a simplified coupling model is proposed to consider the evolution of plastic strain. And the analytical expressions of the constitutive model in 2-D are established from the abstract tensor expression. Several numerical simulations are presented against the biaxial loading test results of concrete, demonstrating that the proposed models can reflect the salient features for concrete under uniaxial and biaxial loading conditions.     

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。     

Construction of eigentensor columns in the linear micropolar theory of elasticity

The internal structure of the block tensor matrix of the elastic modulus tensors is studied for the case of micropolar theory. In particular, the problem of finding the eigenvalues and eigentensor columns of block tensor matrices is considered. The complete orthonormal system of eigentensor columns for a block tensor matrix is constructed. A number of definitions and theorems are formulated. Several newly introduced terms are used to propose various representations of the specific strain energy and the corresponding constitutive relations.     

Mathematical modeling of volumetric material growth

Growth (resp. atrophy) describes the physical processes by which a material of solid body increases (resp. decreases) its size by addition (resp. removal) of mass. In the present contribution, we propose a sound mathematical analysis of growth, relying on the decomposition of the geometric deformation tensor into the product of a growth tensor describing the local addition of material and an elastic tensor, which is characterizing the reorganization of the body. The Blatz-Co hyperelastic constitutive model is adopted for an isotropic body, satisfying convexity conditions (resp. concavity conditions) with respect to the transformation gradient (resp. temperature). The evolution law for the transplant is obtained from the natural assumption that the evolution of the material is independent of the reference frame. It involves a modified Eshelby tensor based on the specific free energy density. The heat flux is dependent upon the transplant. The model consists of the constitutive equation, the energy balance, and the evolution law for the transplant. It is completed by suitable boundary conditions for the displacement, temperature and transplant tensor. The existence of locally unique solutions is obtained, for sufficiently smooth data close to the stable equilibrium. The question of the global existence is examined in the simplified situation of quasistatic isothermal equations of linear elasticity under the assumption of isotropic growth.     

The Payne effect in finite viscoelasticity: constitutive modelling based on fractional derivatives and intrinsic time scales

As we know from experimental testing, the stiffness behaviour of carbon black-filled elastomers under dynamic deformations is weakly dependent on the frequency of deformation but strongly dependent on the amplitude. Increasing strain amplitudes lead to a decrease in the dynamic stiffness, which is known as the Payne effect. In this essay, we develop a constitutive approach of finite viscoelasticity to represent the Payne effect in the context of continuum mechanics. The starting point for the constitutive model resulting from this development is the theory of finite linear viscoelasticity for incompressible materials, where the free energy is assumed to be a linear functional of the relative Piola strain tensor. Motivated by the weak frequency dependence of the dynamic stiffness of reinforced rubber, the memory kernel of the free energy functional is of the Mittag Leffler type. We demonstrate that the model is compatible with the Second Law of Thermodynamics and equal to a fractional differential equation between the overstress of the Second Piola Kirchhoff type and the Piola strain tensor. In order to represent the dependence of the dynamic stiffness on the amplitude of strain, we replace the physical time by an intrinsic time variable. The temporal evolution of the intrinsic time is driven by an internal variable, which is a measure for the current state of the material's microstructure. The material constants of the model are estimated using a stochastic identification algorithm of the Monte Carlo type. We demonstrate that the constitutive approach pursued here represents the combined frequency and amplitude dependence of filler-reinforced rubber. In comparison with the micromechanical Kraus model developed for sinusoidal strains, the theory set out in this essay allows the representation of the stress response under arbitrary loading histories.     

Anisotropic effective higher-order response of heterogeneous Cauchy elastic materials

The homogenization results obtained by Bacca et al. (2013a), to identify the effective second-gradient elastic materials from heterogeneous Cauchy elastic solids, are extended here to the case of phases having non-isotropic tensors of inertia. It is shown that the nonlocal constitutive tensor for the homogenized material depends on both the inertia properties of the RVE and the difference between the effective and the matrix local elastic tensors. Results show that: (i) orthotropic nonlocal effects follow from homogenization of a dilute distribution of aligned elliptical holes and, in the limit case, of cracks; (ii) even under the dilute assumption and isotropic local effective behaviour, homogenization may lead to effective nonlocal orthotropic properties.     

关于返回映射算法中应力的四阶张量值函数

针对各向同性材料,基于一组相互正交的基张量,建立了一套有效的相关运算方法.基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴.根据张量函数表示定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组基所表示.推演结果表明:这些张量之间的运算,表现为对应系数矩阵之间的简单关系.其中,四阶张量求逆归结为对应的3×3系数矩阵求逆,它对二阶张量的变换则表现为该矩阵对3×1列阵的变换.最后,对这些变换关系应用于返回映射算法的迭代格式进行了相关讨论.     

n the fourth order tensor valued function of the stress in return map algorithm

针对各向同性材料,基于一组相互正交的基张量,建立了一套有 效的相关运算方法. 基张量中的两个分别是归一化的二阶单位张量和偏应力张量,另一个则 使用应力的各向同性二阶张量值函数经过归一化构造所得,三者共主轴. 根据张量函数表示 定理,本构方程和返回映射算法中所涉及到的应力的二阶、四阶张量值函数及其逆都由这组 基所表示. 推演结果表明：这些张量之间的运算,表现为对应系数矩阵之间的简单 关系. 其中,四阶张量求逆归结为对应的3\times3系数矩阵求逆,它对二阶张量的变换 则表现为该矩阵对3times 1列阵的变换. 最后,对这些变换关系应用于返回映 射算法的迭代格式进行了相关讨论.