Relaxation of Multiwell Energies in Linearized Elasticity and Applications to Nematic Elastomers

The relaxation of a free-energy functional which describes the order–strain interaction in nematic elastomers is obtained explicitly. We work in the regime of small strains (linearized kinematics). Adopting the uniaxial order tensor theory (Frank model) to describe the liquid crystal order, we prove that the minima of the relaxed functional exhibit an effective biaxial nematic texture, as in the de Gennes order tensor model. In particular, this implies that, at a sufficiently macroscopic scale, the response of the material is soft even if the order of the system is assumed to be fixed at the microscopic scale. The relaxed energy density satisfies a solenoidal quasiconvexification formula.     

Directions of Coaxiality between Pure Strain and Stress in Linear Elasticity

For a linear anisotropic hyperelastic body subjected to a state of pure strain it is shown that there are at least three directions such that the stress and strain tensors are coaxial.     

Second-Order Fredholm Equations for the First Boundary-Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

A complete potential theory is constructed for the first boundary-value problem in the two-dimensional anisotropic theory of elasticity (the force vector is specified on the boundary) in a bounded domain on a plane with a Lyapunov boundary. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 85–94, March–April, 2006.     

弹性力学的混合方程和Hamilton正则方程

本文指出,在弹性力学基本方程中,按变量分类的位移方程,应力方程以外的第三种混合方程,以及按运算子分类的微分方程、变分原理以外的第三种Hamilton方程,它们正好是对应的。本文讨论了静力的和动力的情况以及它们可能的应用。     

多晶纯钛在高应变率不同温度下的拉伸力学行为实验研究

利用MTS809和自行研制的旋转盘冲击拉伸试验机,对多晶纯钛进行了应变率为0.001s-1和300s-1、温度为298K至973K的拉伸试验和应变率为300 s-1不同温度下的冲击拉伸复元试验,得到了多晶纯钛的拉伸应力应变曲线和高应变率等温应力应变曲线。试验结果表明,多晶纯钛的拉伸力学行为具有应变率和温度相关性。采用修正的Johnson-Cook模型进行数值拟合,结果表明,该本构模型能较好地表征多晶纯钛在试验应变率和温度范围内的拉伸力学行为。     

Reducing the Order of the Boundary Condition in a Theory of Plane Strain Elasticity with Boundary Reinforcement

We consider an elastic solid whose boundary is partly reinforced in such a way that the coated part of the boundary can be represented by the union of a finite number of open curves. We show how the (nonstandard) boundary condition on the reinforcement can be integrated resulting in an equivalent (lower-order) boundary condition which, unlike its higher-order counterpart, permits analysis of the corresponding boundary value problems using the boundary integral equation method. Mathematics Subject Classifications (2000) 74B05, 45E05, 35J25     

PMMA低、中应变率单向拉伸力学性能的实验研究

利用MTS810试验机和自行研制的中应变率材料试验机对有机玻璃(PMMA)准静态(应变率为2.00×10-2s-1,2.01×10-3s-1和2.38×10-4s-1)和中应变率(应变率为18.6s-1,2.81s-1,6.54×10-1s-1和2.92×10-1s-1)单向拉伸力学性能进行了实验研究。应变率为2.38×10-4s-1时,应力-应变曲线中存在软化段。结果显示,在上述应变率范围内,PMMA的力学性能具有明显的应变率相关性,表现为应变率强化、应变率硬化和高速脆性。PMMA的拉伸力学性能在应变率达到18.6s-1时出现了更大的应变率敏感性。采用包含一个非线性弹簧和六个松弛模式的粘弹性模型分析PMMA力学性能的应变率相关性,得到PMMA在低、中应变率下单向拉伸的本构方程,理论结果与实验结果符合较好。     

鼓泡法试验薄膜力学行为的研究

本文采用弹性理论,研究了鼓泡法试验过程中,外力(P)与薄膜中心挠度(w_0)之间的本构关系,计算出P与w_0的数学表达式.该分析模型考虑了薄膜内延展变形及薄膜中残余拉应力对本构关系的影响,并在较大挠曲区间范围内(挠度为10~(-2)～10~2倍薄膜厚度)分析本构关系特征.分析结果表明,外力与薄膜中心挠度之间本构关系可表示为:P∝w_0~n,n为指数.随着薄膜中心挠度的增加,薄膜的主要力学行为由弯曲过渡到延展,本构关系由线性过渡到立方关系,本构关系方程指数n由1增加到3.利用本构方程进行数值计算结果表明,薄膜挠曲所需外力随着薄膜半径增加而减小,随着残余拉应力的增加而增加.相对于以往不考虑延展变形的小挠度模型,本文所提出的力学模型更适合于大挠度条件下鼓泡法试验.     

哈密顿体系与弹性楔体问题

将哈密体系引入到级坐标下的弹性力学楔体问题,利用该体系辛空间的性质,将问题化为本征值和本征向量求解上,得到了完备的解空间,从而改变了弹性力学传统的拉格朗日体系以应力函数为特征的半逆法的讨论去解决该类问题的思路,给出了一条求解该类问题的直接法。     

On the Size of RVE in Finite Elasticity of Random Composites

This paper presents a quantitative study of the size of representative volume element (RVE) of random matrix-inclusion composites based on a scale-dependent homogenization method. In particular, mesoscale bounds defined under essential or natural boundary conditions are computed for several nonlinear elastic, planar composites, in which the matrix and inclusions differ not only in their material parameters but also in their strain energy function representations. Various combinations of matrix and inclusion phases described by either neo-Hookean or Ogden function are examined, and these are compared to those of linear elastic types.     

结合内时理论研究岩石衰减与应变振幅的关系

本文将解决复杂弹塑性问题的有效工具内时理论,引入到岩石衰减的研究中来。根据内时理论,简要推导了在中等应变条件下,岩石衰减与应变振幅的用非弹性参数α表示的具体关系式。在MTS压机上,对砂岩、花岗岩等多种岩石样品进行了一系列的循环加卸载实验。利用应力应变滞回曲线的面积,根据衰减的能量损耗的定义,计算了各种实验条件下的岩石样品的衰减值。实验数据表明,在各次循环平均应力相等的条件下,岩石样品的衰减随应变振幅的增加而增加,与内时理论的推导结果相一致。最后,对表示岩石非弹性的参数α进行了简要的讨论。砂岩的α值较为稳定,在0.5左右;而花岗岩的α值变化较大,但二者都随着应力的增加而增加。该实验结果对地震波在地球介质中的衰减研究有一定的参考意义。     

弹性体表面分层压缩失稳问题的数学弹性力学解

采用数学弹性力学的稳定平衡方程并结合富氏积分变换的方法研究了含表面平行裂纹的弹性体在压缩载荷下的表面分层失稳问题。导出了一级显式的精确齐次奇异积分方程组,然后.通过Gauss-Chebyshev积分公式,得到一组齐次代数方程组,从而求出临界压缩载荷。并将结果与经典的材料力学梁板稳定的研究方法所得结果进行了比较,指出经典方法误差太大而不适于求解此问题。最后,利用数学弹性力学解求出的等效弹性支承常数给出一个简单精确的临界压缩载荷计算公式。     

On the Interior Transmission Problem in Nondissipative, Inhomogeneous, Anisotropic Elasticity

In this paper, the interior transmission problem for the non absorbing, anisotropic and inhomogeneous elasticity is investigated. The direct scattering problem for the penetrable inhomogeneous, anisotropic and nondissipative scatterer is first studied and the existence and uniqueness of its solution are established. In the sequel, the interior transmission problem in its classical and weak form is presented and suitable variational formulations of it are settled. Finally, it is proved that the interior transmission eigenvalues constitute a discrete set. This revised version was published online in June 2006 with corrections to the Cover Date.     

双金属层状板中软界面层对裂纹扩展的影响

采用带软界面层的双金属层状板，分析了裂纹垂直于界面扩展过程中的力学行为。带单边裂纹的钢／铝层板的弯曲疲劳实验表明，裂尖接近软铝界面层时裂纹扩展速率da／dN变化很大；裂纹扩展到达界面层时发生偏折。利用有限元软件分析了裂纹尖端前沿应力和应变的变化规律，计算结果表明：软铝界面层出现较大塑性应变，增加了裂纹扩展阻力，并使裂纹扩展力△K减小，导致了da／dN降低．而软界面层中的T应力增高，造成裂纹偏折现象。     

中国电测理论和应用技术发展十年

电测是实验力学的一个大的分支.十年来,在理论和应用技术研究方面都得到了很大的发展,这篇综述包括了四个方面:(1)电测理论研究的进展;(2)电测元件及应用技术的发展;(3)当前电测工程应用有发展前途的几个方面;(4)电测未来发展的展望.文中力图给出中国有关上述标题内容研究的概貌.     

Transport of Multi-Electrolytes in Charged Hydrated Biological Soft Tissues

A mechano-electrochemical theory for charged hydrated soft tissues with multi-electrolytes was developed based on the continuum mixture theory. The momentum equations for water and ions were derived in terms of a mechanochemical force (gradient of water chemical potential), electrochemical forces (gradient of Nernst potentials) and an electrical force (gradient of electrical potential). The theory was shown to be consistent with all existing specialized theories. Using this theory, some mechano-electrokinetic properties of charged isotropic tissues were studied. The well-known Hodgkin–Huxley equation for resting cell membrane potential was derived and the phenomenon of electro-osmotic flow in charged hydrated soft tissues was investigated. Analyses show that the tissue fixed charge density plays an important role in controlling the transport of water and ions in charged hydrated soft tissues.     

Swelling Induced Finite Strain Flexure in a Rectangular Block of an Isotropic Elastic Material

The deformation of a rectangular block into an annular wedge is studied with respect to the state of swelling interior to the block. Nonuniform swelling fields are shown to generate these flexure deformations in the absence of resultant forces and bending moments. Analytical expressions for the deformation fields demonstrate these effects for both incompressible and compressible generalizations of conventional hyperelastic materials. Existing results in the absence of a swelling agent are recovered as special cases.     

基于对数应力率大应变本构关系的参数标定研究

在所有率型弹塑性本构模型中,只有对数应力率对应的本构模型能够满足自适应准则.基于对数应力率,采用实心圆轴扭转实验,对大应变弹塑性本构模型中的参数标定问题进行了讨论.推导出了考虑Swift效应时端部自由实心圆轴扭转变形的变形率、对数旋率、Kirchhoff应力及Kirchhoff应力的对数应力率.对于等向强化大应变弹塑性本构关系,给出了由实心圆轴扭转实验标定的、基于Kirchhhoff应力对数应力率的本构关系中塑性刚度函数的表达式.分析了扭转圆轴的Swift效应对塑性刚度函数的影响.结果表明,实心圆轴扭转的轴向伸长变形和径向变形对基于对数应力率大应变本构关系中的塑性刚度函数都有影响.当不考虑Swift效应时,所得塑性刚度函数表达式与不考虑Swift效应时基于Jaumann应力率的塑性刚度函数表达式相同.     

基于统一强度理论的岩石弹塑性本构模型及其数值实现

基于弹塑性力学理论,以统一强度准则为屈服准则,建立了考虑硬化/软化行为和应变率效应的岩石弹塑性本构模型;采用Fortran语言通过LS-DYNA的用户自定义材料接口(Umat)对该弹塑性本构模型进行编程,并把该程序生成求解器以达到对该模型进行应用的目的;通过岩石的单轴压缩实验和SHPB实验对所建的弹塑性本构模型进行验证,结果表明,该弹塑性本构模型能够反映岩石在准静态和动态下的力学行为。     

Strain analysis of nonlocal viscoelastic Kelvin bar in tension

Based on viscoelastic Kelvin model and nonlocal relationship of strain and stress, a nonlocal constitutive relationship of viscoelasticity is obtained and the strain response of a bar in tension is studied. By transforming governing equation of the strain analysis into Volterra integration form and by choosing a symmetric exponential form of kernel function and adapting Neumann series, the closed-form solution of strain field of the bar is obtained. The creep process of the bar is presented. When time approaches infinite, the strain of bar is equal to the one of nonlocal elasticity.