Multiscale Nonlinear Constitutive Modeling of Carbon Nanostructures Based on Interatomic Potentials

Continuum-based modeling of nanostructures is an efficient and suitable method to study the behavior of these structures when the deformation can be considered homogeneous. This paper is concerned about multiscale nonlinear tensorial constitutive modeling of carbon nanostructures based on the interatomic potentials. The proposed constitutive model is a tensorial equation relating the second Piola-Kirchhoff stress tensor to Green-Lagrange strain tensor. For carbon nanotubes, some modifications are made on the planar representative volume element (RVE) to account for the curved atomic structure resulting a non-planar RVE. Using the proposed constitutive model, the elastic behavior of the graphene sheet and carbon nanotube are studied.     

Incremental constitutive equation for large strain elasto plasticity

Starting from the standard theory with intermediate configuration for finite deformations of an isotropic elasto-plastic material with isotropic hardening, rate type constitutive equations are obtained. The small elastic strain approximation is then discussed and it is shown that, in this approximation, these equations reduce to Hill's formalism of large strain elasto-plasticity obtained from the classical Prandtl-Reuss relations of infinitesimal plasticity by substituting for the infinitesimal strain rate, stress and stress rate respectively the rate of deformation tensor, the Cauchy stress tensor and the Jaumann stress rate tensor. The limiting case of perfect plasticity is also investigated.     

A Lagrangian model for hardening behaviour of materials at finite deformation based on the right plastic stretch tensor

In this paper a modified multiplicative decomposition of the right stretch tensor is proposed and used for finite deformation elastoplastic analysis of hardening materials. The total symmetric right stretch tensor is decomposed into a symmetric elastic stretch tensor and a non-symmetric plastic deformation tensor. The plastic deformation tensor is further decomposed into an orthogonal transformation and a symmetric plastic stretch tensor. This plastic stretch tensor and its corresponding Hencky’s plastic strain measure are then used for the evolution of the plastic internal variables. Furthermore, a new evolution equation for the back stress tensor is introduced based on the Hencky plastic strain. The proposed constitutive model is integrated on the Lagrangian axis of the plastic stretch tensor and does not make reference to any objective rate of stress. The classic problem of simple shear is solved using the proposed model. Results obtained for the problem of simple shear are identical to those of the self-consistent Eulerian rate model based on the logarithmic rate of stress. Furthermore, extension of the proposed model to the mixed nonlinear isotropic/kinematic hardening behaviour is presented. The model is used to predict the nonlinear hardening behaviour of SUS 304 stainless steel under fixed end finite torsional loading. Results obtained are in good agreement with the available experimental results reported for this material under fixed end finite torsional loading.     

An anisotropic damage model with dilatation for concrete

An anisotropic damage model for concrete is developed within the general framework of the internal variable theory of thermodynamics. The rate of change of the compliance tensor is described in terms of kinetic relations involving a damage parameter whose increment is governed by the consistency equation associated with a pressure-dependent damage surface in stress space. The use of the compliance tensor implies that damage is reflected through a fourth-order tensor. Dilatation is obtained as a consequence of damage, and permanent deformation due to damage is addressed via a simple evolution equation. The theory is capable of accommodating the anisotropy induced by microcracking and is very suitable for computer implementation.     

Damage-induced stress-softening and viscoelasticity of limited elastic materials

The rate-dependent behavior of filled natural rubber (NR) is investigated in tensile regime. In order to describe the viscosity-induced rate-dependent effects, a constitutive model of finite strain viscoelasticity is proposed on the basis of the multiplicative decomposition of the deformation gradient tensor into elastic and viscous parts. The total stress is decomposed into an equilibrium stress and a viscosity-induced overstress by following the rheological models of Poynting–Thomson and Zener types. To incorporate the Mullins stress-softening phenomenon into a viscoelastic material, an invariant-based stress-softening function is also proposed. To identify the constitutive equation for the viscosity from direct experimental observations, an analytical scheme is proposed that ascertains the fundamental relation between the viscous strain rate and the overstress tensor with limited elastic parent material model. Evaluation of the experimental results using the proposed analytical scheme confirms the necessity of considering both the current overstress and the current deformation as variables to describe the evolution of the rate-dependent phenomena. Based on this, an evolution equation is proposed to represent the effects of internal variables on viscosity phenomena. The proposed evolution equation has been incorporated into the finite-strain viscoelasticity model in a thermodynamically consistent way.     

Constitutive equation for a class of isotropic, perfectly elastic solids using a new measure of finite strain and corresponding stress

The definition of a measure of strain, referred to as the bi-configuration strain tensor, centres on the difference between the left Cauchy-Green deformation tensor and its inverse. A new measure of stress, coined the bi-configuration stress tensor, has been defined. This measure of stress refers the traction in the current configuration jointly to the referential and spatial configurations, that is, to an effective element of area identified as an element of bi-configuration area. The stress and strain tensors are assumed to be constitutively related by a finite strain form of a generalised Hooke's law. The predictions obtained from the proposed constitutive equation are compared with the observed mechanical behaviour of various test materials. Comparison with experiment centres on biaxial stress measurements in various simple modes of deformation identified by way of a generalised stress-strain relation. The predictions from the proposed constitutive theory are in good accord with the results of experiment.     

Governing equations for the steady state creep in orthotropic materials

Summary A general constitutive equation for creep deformation is presented based upon the concept of tensorial internal variables. The consequences of the theory of tensor functions representation are discussed with respect to the evolution equations. In a particular case of steady evolution of internal variables the governing equation for the secondary creep rate is derived in terms of a scalar inelastic potential. The material parameters required to characterize the stationary creep behaviour of the orthotropic composite are obtained from the unidirectional tension creep tests performed on a glass woven fabric xylok composite. Further check on the theory is made for the bidirectionally loaded specimens.With 4 Figures     

蛋白质气泡有限变形研究

根据粘弹性材料有限变形的应变能密度函数、Maxwell模型的松弛函数及气泡的变形梯度张量,推导出蛋白质气泡有限变形的应力方程。并结合气泡的平衡方程,得到气泡在动态压力作用下有限变形时内径相对变化率随时间变化的表达式。运用该表达式,通过数值模拟方法,对蛋白质气泡有限变形的非线性特性、径向变形随气泡内外压力差、膜的厚度以及膜的粘性的变化规律进行了计算分析。结果表明:在不同载荷作用下,蛋白质气泡径向变形不但具有明显的非线性特性,而且气泡变形达到平衡时的变形大小和时间也不相同。增加气泡膜的厚度和膜的粘性既可以延长气泡变形达到平衡的时间,又可以大大增强气泡承受载荷的能力。     

The strain path dependence of multiaxial cyclic hardening behaviour

The small-strain, isotropic deformation theory is used in incremental form to model an additional cyclic hardening for any arbitrary loading path. The theory is of the unified type and does not employ yield or loading/ unloading criteria. The scalar-valued functions involved in the tensorial constitutive equations as well a growth law for these functions are identified based on idea of the equivalent state. Definitions of equivalent stress and equivalent strain have been developed to correlate step by step loading programmes taking the history of deformation into account. Use is made of the total work increment together with an interpolation method for tensor functions to generalize the simple state to a multiaxial behaviour in the strain space for a given strain increment. For the demonstration of model capability, the numerical simulation is undertaken on cyclic nonproportional paths in two-dimensional axial-shear strain space. The results are verified for stainless steel and brass by comparison with the material response experimentally obtained in the stress space.     

A theory of material divagation

The objective of this paper is to set forth a definition and the properties of a material damage or enhancement measure (i.e. divagation). The divagation of the mechanical response of a simple material is defined for a homogeneous deformation and a representation is obtained by assuming the existence of the Frechet differential of the material functional with respect to the deformation of history. The tensorial measure of material divagation is later extended to include the history of any set of constitutive parameters. The tensorial measure of material divagation is later extended to include the history of any set of constitutive parameters. Two representations for the material divagation tensor are developed from the definition and are used to establish other properties including: the definition of an ideal material, the structure of the coupling between different independent variables such as mechanical and thermal histories, a measure of the divagation using remote sensing devices, the relationship to thermodynamic dissipation, and a divagation calculation for classical viscoelasticity.     

Constitutive equation for rubber-like solids

Summary It has been found convenient to refer the deformation to the reference configuration of the body, the resulting constitutive equation being expressed in terms of the right Cauchy-Green deformation tensor, and a stress tensor which is also concerned solely with the reference configuration of the body. This particular measure of stress is referred to as the second referential stress tensor. The ratio, principal Cauchy stresses/principal stretches, identifies the proper numbers of the second referential stress tensor as the components of the force per unit underformed area. Using the Cayley-Hamilton theorem, the proposed constitutive equation is rearranged and expressed in terms of a quantity which is defined to be one quarter of the difference between the right Cauchy-Green deformation tensor and its inverse. In this way the (effective) strain properties of the proposed constitutive equation are formulated by way of the concept of a strain response function. The three response coefficients associated with the strain response function are assumed to be derivable from a strain intensity function. Two types of strain intensity function are considered. One type is characterised by being continuously differentiable, the other being characterised by the existence of vertices, but is piece-wise linear and continuous. The proposed constitutive equation is used to formulate an effective stress-effective strain relation. This effective stress-effective strain relation is used to correlate the individual stress-strain relations characteristic of various simple modes of deformation to give a single stress-strain relation. Correlation between the predictions of the proposed constitutive equation and the results of experiment proceeds by way of the concept of a generalised shear modulus.With 3 Figures     

复合固体推进剂黏弹性微裂纹损伤本构模型

为建立复合固体推进剂的损伤本构模型,在介观尺度上视其为微裂纹损伤,选取微裂纹密度为损伤内变量。在Abdel-Tawab本构方程的基础上,基于微裂纹均匀化理论,推导了损伤映射张量的一般形式。该张量通常具有非完全对称性,其物理意义是将真实应力空间中各向异性材料的多轴加载映射为等效应力空间中各向同性材料的更为复杂的多轴加载。其次,基于黏弹性动态裂纹扩展模型和裂纹扩展阻力曲线的概念,建立了损伤内变量的演化方程。该演化方程仅含4个物理意义明确的细观参数,并且参数的取值规律与宏观应力曲线的变化规律相一致。数值结果表明,建立的模型能够有效反映材料损伤的应变率、温度依赖性及各向异性特征,并且具有一定的蠕变损伤预测能力。     

Constitutive equation for a class of non-simple elastic materials

Summary Initial yield is the upper limit of the purely elastic deformation behaviour of an elasticplastic solid. Thus the choice of the constitutive equation describing the purely elastic deformation behaviour determines the initial yield function. The constitutive equation of a simple elastic material is only compatible with von Mises yield criterion, a conclusion which applies also to the classical infinitesimal theory. A more general form of constitutive equation for an elastic material is formulated by way of the concept of a stress loading function, the proposed constitutive equation being quadratic in the stress. The two loading coefficients associated with the stress loading function are assumed to be deriveable from a generalised isotropic yield criterion which is now assumed to hold over the entire range of deformation, and in this context is referred to as the stress intensity function. The proposed constitutive equation has the same representation in terms of the left Cauchy-Green deformation tensor as that for a simple elastic material. Using the Cayley-Hamilton theorem, this representation is rearranged and expressed in terms of a measure of finite strain which is defined to be one quarter of the difference between the left Cauchy-Green deformation tensor and its inverse. In this way the strain properties of the proposed constitutive equation are formulated by way of the concept of a strain response function. The three response coefficients associated with the strain response function are assumed to be deriveable from a generalised, isotropic, strain intensity function. The predictions of the proposed constitutive equation are considered in the context of the combined stressing of a thin sheet of incompressible material. In this way, it is shown that the proposed constitutive equation is not limited in the same way as the constitutive equation of a simple elastic material.     

A circular inclusion in a finite domain I. The Dirichlet-Eshelby problem

Summary This is the first paper in a series concerned with the precise characterization of the elastic fields due to inclusions embedded in a finite elastic medium. A novel solution procedure has been developed to systematically solve a type of Fredholm integral equations based on symmetry, self-similarity, and invariant group arguments. In this paper, we consider a two-dimensional (2D) circular inclusion within a finite, circular representative volume element (RVE). The RVE is considered isotropic, linear elastic and is subjected to a displacement (Dirichlet) boundary condition. Starting from the 2D plane strain Navier equation and by using our new solution technique, we obtain the exact disturbance displacement and strain fields due to a prescribed constant eigenstrain field within the inclusion. The solution is characterized by the so-called Dirichlet-Eshelby tensor, which is provided in closed form for both the exterior and interior region of the inclusion. Some immediate applications of the Dirichlet-Eshelby tensor are discussed briefly.     

单个蛋白质气泡的振动特性分析

根据粘弹性材料有限变形的应变能密度函数、Maxwell模型的松弛函数及气泡的变形梯度张量,推导出蛋白质气泡有限变形的应力方程.并结合气泡的动力学方程,得到气泡在内外压力差、弹性有限变形应力及粘性耗散应力共同作用下内径的非线性振动方程.利用该方程,通过数值模拟方法,对蛋白质气泡有限变形时的振动特性进行了分析,研究了气泡内外压力差、膜的厚度、膜的粘性以及气泡大小对气泡振动特性的影响.结果表明,蛋白质气泡的振动具有非线性特性,当初始压力差不同时,气泡的振动频率、振幅、速度的变化是不同的,停止振动时的大小也不相同;增加膜的厚度和膜的粘性会抑制气泡的振动,增强气泡承受载荷的能力;对于大小不同的气泡,尺寸较小的气泡振动频率高,速度衰减慢.     

Analysis of viscoelastic behaviour of transversely isotropic materials

In this paper a constitutive equation to describe the mechanical behaviour of materials, reinforced with unidirectional fibres, is presented. The material behaviour of both matrix and fibres may be viscoelastic. The constitutive equation is a linear relation between the second Piola–Kirchhoff stress tensor and the Green–Lagrange strain tensor. The effective relaxation functions in the constitutive equation are composed of component relaxation functions employing the structural model of Hashin and Rosen. A two-dimensional membrane element incorporating this constitutive equation is implemented in a finite element program. The results of several calculations are presented in order to demonstrate the possibilities of the numerical tool. One calculation concerns a square membrane with a circular hole in its centre. The effect of fibre orientation on deformation and stresses will be displayed for this structure as well as for another membrane structure.     

Aspects of strain measures and strain rates

Summary In finite homogeneous deformation processes, the principal triad generally rotates with respect to a material element during the deformation. The material derivative of the logarithmic strain is no longer simply related to the rate of deformation tensor, and this is exemplified herein. A mathematical procedure is provided for the analysis and the derivations vations are formulated using the co-rotational rate technique in hope, that this technique may be extended to other applications in future modeling. It will be apparent in the article that the co-rotational rate formulation provides a convenient mathematical procedure for handling problems in finite deformation.