Ⅰ阶梯度损伤理论

从热力学基本定律出发,将应变张量、标量损伤变量、损伤梯度作为Helmholtz自由能函数的状态变量,利用本构泛函展开法在自然状态附近作自由能函数的Taylor展开,未引入附加假设,推导出Ⅰ阶梯度损伤本构方程的一般形式．该形式在损伤为0时可退化为线弹性应力-应变本构方程,在损伤梯度为0时可退化为基于应变等效假设给出的线弹性局部损伤本构方程．一维解析解表明,随着应力增大,损伤场逐步由空间非周期解变为关于空间的类周期解,类周期解的峰值区域形成局部化带．局部化带内的损伤变量将不同于局部化带外的损伤变量,由此可以反映出介质的局部化特征．损伤局部化并不是与损伤同时发生,而是在损伤发生后逐渐显现出来,模型的局部化机制开始启动;损伤局部化的宽度同内部特征长度成正比．     

弹性波在应变梯度固体界面上的反射与透射

研究了弹性波在应变梯度固体界面上的反射和透射.首先,通过应变能密度推导出应变梯度固体中的运动方程和界面条件.然后,根据非传统界面条件推导出弹性波反射和透射的振幅比.最后,用法向能量守恒验证了数值计算结果,根据数值计算结果,讨论了微结构参数对弹性波反射和透射的影响.结果发现,应变梯度固体中不仅存在体波而且还存在表面波,尤其是入射波波长越接近材料微结构的特征长度,微结构效应就会越显著.     

用张量导数法求横观各向同性弹性材料的弹性张量的一般形式

用对张量函数求导的方法导出了横观各向同性材料和各向同性材料的弹性张量的一般形式与应力-应变关系式.从推导过程可更清楚地看出为什么横观各向同性材料和各向同性材料分别有五个和两个独立的弹性常数,即材料有几个独立的弹性常数是由其应变能函数的形式所决定的.     

单轴拉伸条件下细观非均匀性岩石损伤局部化和应力应变关系分析

岩石在拉应力状态下的力学特性不同于压应力状态下的力学特性．利用细观力学理论研究了细观非均匀性岩石拉伸应力应变关系包括:线弹性阶段、非线性强化阶段、应力降阶段、应变软化阶段.模型考虑了微裂纹方位角为Weibull分布和微裂纹长度的分布密度函数为Rayleigh函数时对损伤局部化和应力应变关系的影响,分析了产生应力降和应变软化的主要原因是损伤和变形局部化.通过和实验成果对比分析验证了模型的正确性和有效性．     

微结构固体中孤立波的演变及非光滑孤立波

给出了包含宏观应变和微形变的全部二次项以及宏观应变三次项的一种新的自由能函数.利用新自由能函数并根据Mindlin微结构理论，建立了描述微结构固体中纵波传播的一种新模型.利用近来发展的奇行波系统的动力系统理论，分析了系统的所有相图分支，并给出了周期波解、孤立波解、准孤立尖波解、孤立尖波解以及紧孤立波解.孤立尖波解和紧孤立波解的得到，有效地证明了在一定条件下，微结构固体中可以形成和存在孤立尖波和紧孤立波等非光滑孤立波.此结果进一步推广了微结构固体中只存在光滑孤立波的已有结论.     

基于简化的应变梯度理论下Kirchhoff板模型边值问题的提法及其应用北大核心CSCD

考虑应变梯度和速度梯度的影响,建立薄板控制微分方程及给出其边值问题的提法,修正了前人给出的薄板角点条件.采用Levy法,给出受分布力作用下简支板的挠度及自由振动频率的解析解.通过与文献中分子动力学数据对比,验证了该文模型的有效性并提出校核材料参数的一种方法.研究结果表明,增大弹性地基和应变梯度参数可以有效提高板的等效刚度,而速度梯度参数则相反.该文提出的板的边值问题为研究薄板在复杂支撑边界及外荷载等条件提供了理论依据.同时,有望为其有限元法、有限差分法和基于能量原理的Galerkin法等数值方法提供理论依据.     

基于非局部应变梯度理论功能梯度纳米板的弯曲和屈曲研究

以纳米机器人等智能器件中的功能梯度纳米板结构为研究对象,基于非局部应变梯度理论,研究了其弯曲和屈曲问题.推导了一般情况下的功能梯度纳米板运动方程,弯曲和屈曲作为其特例可简化而成.分析了非局部尺度参数、材料特征尺度参数、梯度指数、纳米板尺寸等对弯曲挠度和临界屈曲载荷的影响.结果表明:不同高阶连续介质力学理论下的最大挠度都随梯度指数的增大而增大,正方形纳米板挠度较小,且板厚越大,弯曲挠度越小;最大挠度随非局部尺度参数的增大而增大,随材料特征尺度参数的增大而减小.临界屈曲载荷随梯度指数的增大而减小,随板厚、长宽比的增大而增大,随非局部尺度的增大而减小,随材料特征尺度的增大而增大.非局部应变梯度高阶弯曲和屈曲中存在结构软化与硬化机制,两个内特征参数之间具有耦合效应,当非局部尺度大于材料特征尺度时,非局部效应在功能梯度纳米板力学性能中占主导作用;当材料特征尺度大于非局部尺度时,应变梯度效应占主导作用.解析结果还证明了当非局部尺度等于材料特征尺度时,非局部应变梯度理论结果退化为经典结果.     

考虑纤维弯曲刚度的橡胶-帘线复合材料各向异性超弹性本构模型

针对纤维增强复合材料的有限变形,基于Spencer的连续介质力学不变量理论,提出了一种考虑纤维弯曲刚度的非线性超弹性本构模型.通过引入变形后纤维方向向量的梯度项,把单位体积的自由应变能分解为便于参数识别的体积变形、等容变形、各向异性变形和弯曲刚度4部分.理论和实验分析均表明传统的基于连续介质力学的纤维增强复合材料有限变形理论不适用于弯曲变形,必须考虑纤维弯曲刚度的影响.数值仿真结果也验证了在应变能函数中增加弯曲刚度项是必要的.     

各向同性弹性损伤本构方程的一般形式

直接从不可逆热力学基本定律出发,推导出弹性各向同性损伤材料本构方程的一般形式,克服了由应变等效假设建立的经典损伤本构方程的缺陷,并阐明了两种各向同性弹性损伤模型(单标量模型与双标量模型)之间的联系.研究表明,采用单标量描述的损伤模型,在材料损伤本构方程中含有两个“损伤效应函数”,反映损伤对于两个弹性常数的不同影响.应变等效假设给出的损伤本构方程,是该文方程的一个近似形式,常常不能满意地描述实际材料的损伤行为.     

非线性弹性理论的混合能量形式广义变分原理

本文首先对弹性材料的应变能函数∑(E_ij)和余应能函数∑C(S_ij)的部分“对应”变量作Legendre变换,引进“对应”的混合余应变能函数∑_klC和混合应变能函数∑_kl。进而,给出非线性弹性理论的各种“对应”的混合能量形式广义变分原理。线性弹性理论也有相应结果,它是本文结果的特殊情况。     

The representation of the displacement gradient of isotropic elastic body in terms of the piola stress tensor : PMM vol. 40, n 6, 1976, pp. 1070–1077

The representation of the displacement gradient of an isotropic elastic body is analyzed. It is shown on the basis of a single controlling inequality and a polar expansion of the Piola tensor that such representation has generally four branches. The mechanical meaning and the nature of that ambiguity is explained. It is established that when the angles of turn of material fibers are not excessively large, only one of the four branches is obtained. Particular cases in which the nature of ambiguity is more complex are investigated. It is noted that in many practical problems the representation of the displacement gradient by the Piola stress tensor is unambiguous.The considered problem is associated with the variational principle of complementary energy in the nonlinear theory of elasticity, where the statistically feasible fields of the asymmetric Piola stress tensor is varied [1], A method was proposed there for expressing the displacement gradient in terms of the Piola stress tensor for an isotropic elastic body. Later the concept of complementary energy and the representation of the strain gradient in terms of the Piola stress tensor were considered in [2, 3]. Examples of the use of the complementary energy concept are given in [2] and the case of an anisotropic body is considered in [3], These investigations disclosed that the considered representation of the strain tensor leads to ambiguity, but the character and nature of the ambiguity were not fully investigated.     

The forms of the relation between the stress and strain tensors in a non-linearly elastic material

New relations for the stress and strain tensors, which comprise energy pairs, are obtained for a non-linearly elastic material using a similar method to that employed by Novozhilov, based on a trigonometric representation of the tensors. Shear strain and stress tensors, not used previously, are introduced in a natural way. It is established that the unit tensor, the deviator and the shear tensor form an orthogonal tensor basis. The stress tensor can be expanded in a strain-tensor basis and vice versa. By using this expansion, the non-linear law of elasticity can be written in a compact and physically clear form. It is shown that in the frame of the principal axes the stresses are expressed in terms of the strains and vice versa using linear relations, while the non-linearity is contained in the coefficients, which are functions of mixed invariants of the tensors, introduced by Novozhilov, the generalized moduli of bulk compression and shear and the phase of similitude of the deviators. Relations for different energy pairs of tensors are considered, including for tensors of the true stresses and strains, where the generalized moduli of elasticity have a physical meaning for large strains.     

剪切变形下非晶态高聚物的力学行为

基于非平衡态热力学理论，提出了一个适用于不可压材料的新的热粘弹性本构模型．该模型将橡胶弹性理论中的非高斯分子网络模型推广到计及粘性和热效应的情形．通过引入一组二阶张量形式的内变量，建议了一个新的Helmholtz自由能表达式，从而可以用来合理描述内变量的演化规律．根据以上模型，重点研究了热粘弹性材料在简单剪切变形下的力学行为，考察了由于分子链取向分布的变化而产生的“粘性耗散诱导”各向异性，讨论了应变率效应和由于粘性耗散而导致的热软化效应对剪应力的影响．理论预测结果与G’Sell等人的实验数据的定性比较表明了新的本构模型的有效性。     

Homogenisation of random composites via the multiscale finite‐element method

The transition between the chosen microstructure and microvariables and the material properties on the macrolevel is always a sensitive point in the theory of homogenisation. In this talk we will observe the transfer of data between the scales based on the multiscale finite element method where in each Gauss point of the macromesh a micromesh is attached. For a given deformation gradient provided from the macroscale one calculates microfluctuations satisfying periodic boundary conditions and from those the effective first Piola‐Kirchhoff stress tensor for each Gauss point. The latter provides a possibility to calculate the elasticity tensor on the macrolevel. We study a microstructure containing elliptical cracks of random aspect ratio and orientation. The results based on such procedure show the dependence of the macrovariables on the crack ellipticity. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Second strain gradient theory in orthogonal curvilinear coordinates: Prediction of the relaxation of a solid nanosphere and embedded spherical nanocavity

In this paper, Mindlin’s second strain gradient theory is formulated and presented in an arbitrary orthogonal curvilinear coordinate system. Equilibrium equations, generalized stress-strain constitutive relations, components of the strain tensor and their first and second gradients, and the expressions for three different types of traction boundary conditions are derived in any orthogonal curvilinear coordinate system. Subsequently, for demonstration, Mindlin’s second strain gradient theory is represented in the spherical coordinate system as a highly-practical coordinate system in nanomechanics. Second strain gradient elasticity have been developed mainly for its ability to capture the surface effects in the presence of micro-/nano- structures. As a numeric illustration of the theory, the surface relaxation of spherical domains in Mindlin’s second strain gradient theory is considered and compared with that in the framework of Gurtin–Murdoch surface elasticity. It is observed that Mindlin’s second strain gradient theory predicts much larger value for the radial displacement just near the surface in comparison to Gurtin–Murdoch surface elasticity.     

The equations of elastostatics in a Riemannian manifold

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify).     

Stability of strain gradient elastic bars in tension

Equilibrium of a bar under uniaxial tension is considered as optimization problem of the total potential energy. Uniaxial deformations are considered for a material with linear constitutive law of strain second gradient elasticity. Applying tension on an elastic bar, necking is shown up in high strains. That means the axial strain forms two homogeneously deformed sections in the ends of the bars and a section in the middle with high variable strain. The interactions of the intrinsic (material) lengths with the non linear strain displacement relations develop critical states of bifurcation with continuous Fourier’s spectrum. Critical conditions and post-critical deformations are defined with the help of multiple scales perturbation method. An erratum to this article can be found at