损伤拉索的等效弹性模量及其参数分析

损伤拉索会出现线形松弛、应力水平降低的情况,必然会影响拉索的等效弹性模量。本文首先引入损伤程度、位置及范围3个参数,用以描述拉索损伤形态的特征,建立损伤拉索索力和线形计算公式,采用数值方法计算了损伤拉索弦向等效弹性模量精确数值,并和经典的等效弹性模量公式的计算结果进行了比较分析,分析了考虑损伤时两种不同计算方法结果的误差。计算表明,对于500m弦向长度以内的损伤拉索,拉索的弦向长度Lc越大,倾角越小,等效弹性模量的损失越大,并且应用割线模量公式计算的误差也越大,当Lc=500m时,损伤拉索相对误差值在2.5%~4.5%之间。弦向应变越小,等效弹性模量损失越大,弦向应变在[0.001,0.004]内,应用割线模量公式计算的相对误差小于3.5%。损伤程度及损伤范围对引用等效弹性模量公式的误差影响较大,倾角对等效弹模公式相对误差的影响也不容忽视。弦向长度、弦向应变、倾角和损伤程度参数都是通过改变拉索的松弛程度进而影响等效弹性模量的数值以及公式的误差。     

冲击载荷下混凝土材料的动态本构关系

利用改装的杆径为 74mm的直锥变截面式大尺寸Hopkinson压杆对混凝土材料进行冲击压缩实验 ,系统研究了混凝土的应变率硬化效应 ,采用一种新的方法损伤冻结法对混凝土材料在冲击载荷下的损伤软化效应进行了系统研究 ,给出了冲击载荷下混凝土的损伤演化方程 ;在对数据进行合理分析的基础上 ,结合粘弹性本构理论 ,得到混凝土材料的损伤型线性粘弹性本构关系。     

Damage estimation on beam-like structures using the multi-resolution analysis

This work proposes a vibration-based damage evaluation method that can detect, locate, and size damage utilizing only a few of the lower mode shapes. The proposed method is particularly advantageous for beam-like structures with uncertain applied axial load, mass density, and foundation stiffness. Based on a small damage assumption, a linear relationship between damaged and undamaged curvatures is revealed in the context of elasticity. It turns out that the resulting damage index equation inherently suffers from singularities near inflection nodes. The transformation of the problem into the multi-resolution wavelet domain provides a set of coupled linear equations. With the aid of the singular value decomposition technique, the solution to the damage index equation is achieved in the wavelet space. Next, the desired physical solution to the damage index equation is reconstructed from the one in the wavelet space. The performance of the proposed method is compared with two existing damage detection methods using a set of numerical simulations. The proposed method attempts to resolve the mode selection problem, the singularity problem, the axial force problem, and the absolute severity estimation problem, all of which remained unsolved by earlier attempts.     

Analysis of the localization of deformation and the complete stress–strain relation for mesoscopic heterogeneous brittle rock under dynamic uniaxial tensile loading

Stress redistribution induced by excavation results in the tensile zone in parts of the surrounding rock mass. It is significant to analyze the localization of deformation and damage, and to study the complete stress–strain relation for mesoscopic heterogeneous rock under dynamic uniaxial tensile loading. On the basis of micromechanics, the complete stress–strain relation including linear elasticity, nonlinear hardening, rapid stress drop and strain softening is obtained. The behaviors of rapid stress drop and strain softening are due to localization of deformation and damage. The constitutive model, which analyze localization of deformation and damage, is distinct from the conventional model. Theoretical predictions have shown to consistent with the experimental results.     

Analysis of the localization of damage and the complete stress-strain relation for mesoscopic heterogeneous brittle rock subjected to compressive loads

A micromechanics-based model is established. The model takes the interaction among sliding cracks into account, and it is able to quantify the effect of various parameters on the localization condition of damage and deformation for brittle rock subjected to compressive loads. The closed-form explicit expression for the complete stress-strain relation of rock containing microcracks subjected to compressive loads was obtained. It is showed that the complete stress-strain relation includes linear elasticity, nonlinear hardening, rapid stress drop and strain softening. The behavior of rapid stress drop and strain softening is due to localization of deformation and damage. Theoretical predictions have shown to be consistent with the exoerimental results.     

A first-order strain gradient damage model for simulating quasi-brittle failure in porous elastic solids

In order to simulate quasi-brittle failure in porous elastic solids, a continuum damage model has been developed within the framework of strain gradient elasticity. An essential ingredient of the continuum damage model is the local strain energy density for pure elastic response as a function of the void volume fraction, the local strains and the strain gradients, respectively. The model adopts Griffith’s approach, widely used in linear elastic fracture mechanics, for predicting the onset and the evolution of damage due to evolving micro-cracks. The effect of those micro-cracks on the local material stiffness is taken into account by defining an effective void volume fraction. Thermodynamic considerations are used to specify the evolution of the latter. The principal features of the model are demonstrated by means of a one-dimensional example. Key aspects are discussed using analytical results and numerical simulations. Contrary to other continuum damage models with similar objectives, the model proposed here includes the effect of the internal length parameter on the onset of damage evolution. Furthermore, it is able to account for boundary layer effects.     

Damage and self-similarity in fracture

Consider applications of damage mechanics to material failure. The damage variable introduced in damage mechanics quantifies the deviation of a brittle solid from linear elasticity. An analogy between the metastable behavior of a stressed brittle solid and the metastable behavior of a superheated liquid is established. The nucleation of microcracks is analogous to the nucleation of bubbles in the superheated liquid. In this paper we have applied damage mechanics to four problems. The first is the instantaneous application of a constant stress to a brittle solid. The results are verified by applying them to studies of the rupture of chipboard and fiberglass panels. We then obtain a solution for the evolution of damage after the instantaneous application of a constant strain. It is shown that the subsequent stress relaxation can reproduce the modified Omori’s law for the temporal decay of aftershocks following an earthquake. Obtained also are the solutions for application of constant rates of stress and strain. A fundamental question is the cause of the time delay associated with damage and microcracks. It is argued that the microcracks themselves cause random fluctuations similar to the thermal fluctuations associated with phase changes.     

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

End effects for anti-plane shear deformations of sandwich structures

The purpose of this research is to further investigate the effects of material inhomogeneity and the combined effects of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects. Saint-Venant decay rates for self-equilibrated edge loads in symmetric sandwich structures are examined in the context of anti-plane shear for linear anisotropic elasticity. The problem is governed by a second-order, linear, elliptic, partial differential equation with discontinuous coefficients. The most general anisotropy consistent with a state of anti-plane shear is considered, as well as a variety of boundary conditions. Anti-plane or longitudinal shear deformations are one of the simplest classes of deformations in solid mechanics. The resulting deformations are completely characterized by a single out-of-plane displacement which depends only on the in-plane coordinates. They can be thought of as complementary deformations to those of plane elasticity. While these deformations have received little attention compared with the plane problems of linear elasticity, they have recently been investigated for anisotropic and inhomogeneous linear elasticity. In the context of linear elasticity, Saint-Venant's principle is used to show that self-equilibrated loads generate local stress effects that quickly decay away from the loaded end of a structure. For homogeneous isotropic linear elastic materials this is well-documented. Self-equilibrated loads are a class of load distributions that are statically equivalent to zero, i.e., have zero resultant force and moment. When Saint-Venant's principle is valid, pointwise boundary conditions can be replaced by more tractable resultant conditions. It is shown in the present study that material inhomogeneity significantly affects the practical application of Saint-Venant's principle to sandwich structures.     

Cosserat (micropolar) elasticity in Stroh form

The theory of defects in Cosserat continua is sketched out in strict analogy to the theory of line defects in anisotropic elasticity (Stroh theory). This rewrite of the second order equilibrium equations of elasticity in a 3-dimensional space as first order equations in a 6-dimensional space is analogous to replacing the Laplace equation by the Riemann–Cauchy equations. For generalized plane strain of anisotropic micropolar (Cosserat) elasticity one obtains a 14-dimensional coupled linear system of differential equations of first order and for plane strain of anisotropic micropolar (Cosserat) elasticity we obtain a 6-dimensional coupled linear system of differential equations of first order.     

弹脆性材料的损伤本构关系及应用

本文根据连续损伤力学方法,对弹脆性材料损伤的力学响应进行一般分析。理论分析中,材料与损伤都是各向异性的。还导出了计算损伤张量、有效弹性张量、真实应力张量以及损伤能耗率张量的实用表达式。     

On canonical equations of continuum thermomechanics

It is shown that the canonical balance of momentum of continuum mechanics can be formulated in a general way, but not independently of the usual balance of linear momentum, even in the absence of specified constitutive equations. A parallel construct is made of necessity for the accompanying time-like canonical energy equation. On specifying the energy, previous particular cases can be deduced including pure elasticity, inhomogeneous thermoelasticity of conductors, and the case of dissipative solid-like materials described by means of a diffusive internal variable (such as in damage or weakly non-local plasticity). A redefinition of the entropy flux is necessarily accompanied by a redefinition of the Eshelby stress tensor.     

Invariant characteristic representations for classical and micropolar anisotropic elasticity tensors

By extending and developing the characteristic notion of the classical linear elasticity initiated by Lord Kelvin, a new type of representation for classical and micropolar anisotropic elasticity tensors is introduced. The new representation provides general expressions for characteristic forms of the two kinds of elasticity tensors under the material symmetry restriction and has many properties of physical and mathematical significance. For all types of material symmetries of interest, such new representations are constructed explicitly in terms of certain invariant constants and unit vectors in directions of material symmetry axes and hence they furnish invariants which can completely characterize the classical and micropolar linear elasticities. The results given are shown to be useful. In the case of classical elasticity, the spectral properties disclosed by our results are consistent with those given by similar established results.     

Conservation laws in linear elasticity based upon divergence transformations

The existence of conservation laws in linear elasticity based upon divergence transformations of the Lagrangian density function is investigated. It is found that there exist a set of conservation laws which correspond to infinitesimal homogeneous perturbations of the strain and velocity fields. These conservation laws have a unique feature not shared by other conservation laws in linear elasticity in that they contain an arbitrary free parameter.     

The minimal error method for the Cauchy problem in linear elasticity. Numerical implementation for two-dimensional homogeneous isotropic linear elasticity

In this paper, yet another iterative procedure, namely the minimal error method (MEM), for solving stably the Cauchy problem in linear elasticity is introduced and investigated. Furthermore, this method is compared with another two iterative algorithms, i.e. the conjugate gradient (CGM) and Landweber–Fridman methods (LFM), previously proposed by Marin et al. [Marin, L., Háo, D.N., Lesnic, D., 2002b. Conjugate gradient-boundary element method for the Cauchy problem in elasticity. Quarterly Journal of Mechanics and Applied Mathematics 55, 227–247] and Marin and Lesnic [Marin, L., Lesnic, D., 2005. Boundary element-Landweber method for the Cauchy problem in linear elasticity. IMA Journal of Applied Mathematics 18, 817–825], respectively, in the case of two-dimensional homogeneous isotropic linear elasticity. The inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions. The numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method (BEM) for two-dimensional homogeneous isotropic linear elastic materials.     

Waves in Constrained Linear Elastic Materials

This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by Hoger and Johnson in [12, 13]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant. This revised version was published online in July 2006 with corrections to the Cover Date.     

On Isotropic Invariants of the Elasticity Tensor

Isotropic invariants of the elasticity tensor always yield the same values no matter what coordinate system is concerned and therefore they characterize the linear elasticity of a solid material intrinsically. There exists a finite set of invariants of the elasticity tensor such that each invariant of the elasticity tensor can be expressed as a single-valued function of this set. Such a set, called a basis of invariants of the elasticity tensor, can be used to realize a parametrization of the manifold of orbits of elastic moduli, i.e. to distinguish different kinds of linear elastic materials. Seeking such a basis is an old problem in theory of invariants and seems to have been unsuccessful until now. In this paper, by means of the unique spectral decomposition of the elasticity tensor every invariant of the elasticity tensor is shown to be a joint invariant of the eigenprojections of the elasticity tensor, and then by utilizing some properties of the eigenprojections a basis for each case concerning the multiplicity of the eigenvalues of the elasticity tensor is presented in terms of joint invariants of the eigenprojections. In addition to the foregoing properties, the presented invariants may also be used to form invariant criteria for identification of elastic symmetry axes.     

基底弹性对蒸发超薄液膜去润湿过程的影响

研究了基底的弹性变形对蒸发超薄膜的稳定性和去润湿动力学过程的影响. 基于长波近似, 得到了关于液体薄膜厚度的演化方程. 运用线性稳定性理论和数值模拟两种方法, 研究了基底弹性、范德华力以及液体蒸发等因素对液体薄膜的稳定性和去润湿过程的影响. 研究结果表明增大基底的弹性系数或者减小液体的表面张力, 都能加速液膜的破碎, 并且能够影响气液界面波的波长; 液体蒸发能促进气液界面扰动的增长, 有助于液膜的破裂.      

二维Nonlocal线弹性理论的变分原理与对偶方程

基于Eringen提出的Nonlocal线弹性理论的微分形式本构关系,导出了相应的能量密度表达式,进而得到二维Nonlocal线弹性理论的变分原理.利用变分原理导出了对偶平衡方程和相应的边界条件.进而给出了非局部动力问题的Lagrange函数,并引入对偶变量和Hamilton函数,得到了对偶体系下的变分方程.在Hamilton体系下,通过变分得到了二维Nonlocal线弹性理论的对偶平衡方程和相应的边界条件.     

Uniqueness for plane crack problems in dipolar gradient elasticity and in couple-stress elasticity

The present work deals with the uniqueness theorem for plane crack problems in solids characterized by dipolar gradient elasticity. The theory of gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory employed here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. These cases are also treated in the present study. We consider an anisotropic material response of the cracked plane body, within the linear version of gradient elasticity, and conditions of plane-strain or anti-plane strain. It is emphasized that, for crack problems in general, a uniqueness theorem more extended than the standard Kirchhoff theorem is needed because of the singular behavior of the solutions at the crack tips. Such a theorem will necessarily impose certain restrictions on the behavior of the fields in the vicinity of crack tips. In standard elasticity, a theorem was indeed established by Knowles and Pucik [Knowles, J.K., Pucik, T.A., 1973. Uniqueness for plane crack problems in linear elastostatics. J. Elast. 3, 155–160], who showed that the necessary conditions for solution uniqueness are a bounded displacement field and a bounded body-force field. In our study, we show that the additional (to the two previous conditions) requirement of a bounded displacement-gradient field in the vicinity of the crack tips guarantees uniqueness within the general form of the theory of dipolar gradient elasticity. In the specific cases of couple-stress elasticity and pure strain-gradient elasticity, the additional requirement is less stringent. This only involves a bounded rotation field for the first case and a bounded strain field for the second case.