On a geometrically nonlinear damage model based on a multiplicative decomposition of the deformation gradient and the propagation of microcracks

We aim to derive a damage model for materials damaged by microcracks. The evolution of the cracks shall be governed by the maximum energy release rate, which was recently shown to be a direct consequence of the variational principle of a body with a crack (Arch. Appl. Mech. 69 (5) (1999) 337). From this, we get the path of the growing crack by introducing a series of thermodynamically equivalent straight cracks. The equivalence of the energy dissipated by microcrack growth and the damage dissipation leads to our damage evolution law. This evolution law will be embedded in a finite deformation framework based on a multiplicative decomposition into elastic and damage parts. As a consequence of this, we can present the anisotropic damaged elasticity tensor with the help of push and pull operations. The connection of this approach to other well known damage theories will be shown and the advantages of a finite element framework will be worked out. Numerical examples show the possibilities of the proposed model.     

Self-Contact for Rods on Cylinders

We study self-contact phenomena in elastic rods that are constrained to lie on a cylinder. By choosing a particular set of variables to describe the rod centerline the variational setting is made particularly simple: the strain energy is a second-order functional of a single scalar variable, and the self-contact constraint is written as an integral inequality. Using techniques from ordinary differential equation theory (comparison principles) and variational calculus (cut-and-paste arguments) we fully characterize the structure of constrained minimizers. An important auxiliary result states that the set of self-contact points is continuous, a result that contrasts with known examples from contact problems in free rods.     

弹性力学Hamilton方法广义解的适定性

首先引入了Hamilton体系中平面应力弹性力学问题正则方程的Galerkin变分方程,证明了Galerkin变分方程和目前文献中所用的Ritz变分方程的等价性,以及相应广义解的适定性,从而为目前的数值方法提供了理论基础。从证明过程中可以看到广义解实际上是Ritz变分泛函的一个鞍点。     

物理变分原理及其在电磁介质中的应用

和数学变分原理的意义不同,物理变分原理是物理界的客观规律,是基本规律.热力学定律是能量守恒定律,指任一自然过程的能量总是守恒的;但同时又是物理变分原理,指从一种状态变化到另一无限接近的状态,在所有可能的稳定过程中,真实过程的能量取极小值,因而又是动量定律.特别是对于存在迁移变分的过程和偏离平衡态不大的不可逆过程,物理变分原理特别有效,可以用来推求连续介质的控制方程,且尚未完全研究透彻.本文对这一原理及其在电磁介质中的某些应用进行了一些研究.     

On a variational principle in thermodynamics

The dynamic principle is proposed that a thermodynamic system evolves in time so that the total energy balance including the energy drawn from the environment becomes stationary for all admissible variations of thermodynamic states. It is shown that this principle allows to obtain variational characterizations of contact Hamiltonian equations (even in presence of ports), reaction equations and doubly nonlinear reaction–diffusion equations. Further, examples are discussed which support this principle.     

The evolution of laminates in finite crystal plasticity: a variational approach

The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material’s macroscopic, mechanical behavior. In particular, the arrangement of dislocations and other lattice defects to particular structures and patterns on the microscale as well as the resultant inhomogeneous distribution of localized strain results in a highly altered stress–strain response. Energetic models predicting the mechanical properties are commonly based on thermodynamic variational principles. Modeling the material response in finite strain crystal plasticity very often results in a non-convex variational problem so that the minimizing deformation fields are no longer continuous but exhibit small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. This results in fine structures that can be interpreted as the observed microstructures. In this paper, we first review the underlying variational principles for inelastic materials. We then propose an analytical partial relaxation of a Neo-Hookean energy formulation, based on the assumption of a first-order laminate microstructure, thus approximating the relaxed energy by an upper bound of the rank-one-convex hull. The semi-relaxed energy can be employed to investigate elasto-plastic models with a single as well as multiple active slip systems. Based on the minimization of a Lagrange functional (consisting of the sum of energy rate and dissipation potential), we outline an incremental strategy to model the time-continuous evolution of the laminate microstructure, then present a numerical scheme by means of which the microstructure development can be computed, and show numerical results for particular examples in single- and double-slip plasticity. We discuss the influence of hardening and of slip system orientations in the present model. In contrast to many approaches before, we do not minimize a condensed energy functional. Instead, we incrementally solve the evolution equations at each time step and account for the actual microstructural changes during each time step. Results indicate a reduction in energy when compared to those theories based on a condensed energy functional.     

The B-bar method and the limitation principles

The generalized elastic material provides a reference model to cast in a unitary framework many structural models which are based on nonlinear monotone multivalued relations such as viscoelasticity, plasticity and unilateral models. The modified forms of the Hu-Washizu and Hellinger-Reissner principles and the displacement-based variational formulation are recovered for the generalized elastic material starting from a functional in the complete set of state variables. The related limitation principles are derived and their specialization to elasticity and elastoplasticity with mixed hardening are provided. It is shown that the interpolating fields for the pressure and the volumetric strain usually adopted in the B-bar method lead to a limitation principle. Accordingly the same elastic and elastoplastic solutions can be obtained by means of an approximate mixed displacement⧸pressure variational principle. A second application is concerned with the conditions ensuring the coincidence of the solutions between an approximate two-field mixed formulation and the displacement-based method. Numerical examples are provided to show the coincidence of the solutions obtained from different mixed finite element formulations, in elasticity or elastoplasticity, under the validity of the limitation principles.     

Variational formulations and a consistent finite-element procedure for a class of nonlocal elastic continua

The structural boundary-value problem in the context of nonlocal (integral) elasticity and quasi-static loads is considered in a geometrically linear range. The nonlocal elastic behaviour is described by the so-called Eringen model in which the nonlocality lies in the constitutive relation. The diffusion processes of the nonlocality are governed by an integral relation containing a recently proposed symmetric spatial weight function expressed in terms of an attenuation function. A firm variational basis to the nonlocal model is given by providing the complete set of variational formulations, composed by ten functionals with different combinations of the state variables. In particular the nonlocal counterpart of the classical principles of the total potential energy, complementary energy and mixed Hu–Washizu principle and Hellinger–Reissner functional are recovered. Some examples concerning a piecewise bar in tension are provided by using the Fredholm integral equation and the proposed nonlocal FEM.     

耗散系统的虚功变分不等式及其应用

对于保守系统,能量变分原理为推导力学系统控制方程提供了简洁的途径.对于耗散系统,控制方程的建立往往需要引入经验的或理性的假定,增大了建模的难度.针对耗散系统,引入系统局部稳定的概念,并在此基础上,提出一类虚功变分不等式.这一不等式事实上揭示了耗散系统的一类虚功不等原理.该原理的物理含义为:使系统状态稳定的必要条件是,在该状态附近所有可能的虚拟路径上系统释放的势能不大于系统耗散的能量.研究表明:仅需结合虚功不等原理和能量守恒原理,即可导出准静态系统力学状态量的全部控制方程.作为应用,文章重新讨论了塑性力学,结合虚功不等原理与能量守恒原理,导出经典塑性力学的全部控制方程,并证明了经典的最大塑性耗散原理可以作为虚功不等原理的推论导出;同时,以Mohr-Coulomb强度准则为例,讨论了虚功不等原理在强度理论中的应用,说明基于应力的强度准则可以是基于能量的稳定性准则的推论.上述例子说明了虚功不等原理的广泛适用性和在建立耗散系统控制方程中的有效性.     

Mechanics and thermodynamics of fluid-saturated highly elastic materials

We consider a mixture that consists of a highly elastic material and a liquid dissolved in this material. Using the laws of classical thermodynamics, we state a variational principle describing the mixture equilibrium under static loading conditions. From this principle, we derive equilibrium equations and a system of constitutive relations characterizing the mixture elastic and thermodynamic properties. We state problems describing the stress-strain state of a swollen material and a statically loaded material in thermodynamic equilibrium with the liquid. We consider the case of incompressible mixture. The general theory is illustrated by examples concerned with the deformation behavior of inhomogeneously swollen cross-linked polymers and with their thermodynamics of strains and swelling in solvent media.     

相空间中非保守系统的Herglotz广义变分原理及其Noether定理

与经典变分原理相比,基于由微分方程定义的作用量的Herglotz广义变分原理给出了非保守动力学系统的一个变分描述,它不仅能够描述所有采用经典变分原理能够描述的动力学过程,而且能够应用于经典变分原理不能适用的非保守或耗散系统.将Herglotz广义变分原理拓展到相空间,研究相空间中非保守力学系统的Herglotz广义变分原理与Noether定理及其逆定理.首先,提出相空间中Herglotz广义变分原理,给出相空间中非保守系统的变分描述,导出相应的Hamilton正则方程;其次,基于非等时变分与等时变分之间的关系,导出相空间中Hamilton-Herglotz作用量变分的两个基本公式;再次,给出Noether对称变换的定义和判据,提出并证明相空间中非保守系统基于Herglotz变分问题的Noether定理及其逆定理,揭示了相空间中力学系统的Noether对称性与守恒量之间的内在联系.在经典条件下,Herglotz广义变分原理退化为经典变分原理,与之相应的相空间中的Noether定理退化为经典Hamilton系统的Noether定理.文末以著名的Emden方程和平方阻尼振子为例说明上述方法和结果的有效性.     

时间尺度上Herglotz变分原理及其Noether定理

Herglotz变分原理的作用量是由微分方程定义的,不仅可以描述所有经典变分原理能够描述的动力学过程,还可以对经典变分原理不能适用的非保守系统或耗散系统进行变分描述．时间尺度上微积分理论提供了一种可同时研究离散系统和连续系统的有效方法．本文结合Herglotz变分原理和时间尺度微积分理论来研究时间尺度上的Herglotz变分原理及其Noether定理．首先,给出时间尺度上Lagrange系统的Herglotz变分原理．其次,根据Herglotz变分原理和Dubois-Reymond引理,推导出时间尺度上Lagrange系统的Herglotz变分问题的运动微分方程．再次,基于时间尺度上Hamilton-Herglotz作用量在群的无限小变换下的不变性,给出Noether对称性的定义并导出其Noether等式．最后,建立了时间尺度上Lagrange系统的Herglotz变分问题的Noether定理,给出了连续和离散两种情况下基于Herglotz变分问题的Noether守恒量．文末举例说明结果的应用．     

A variational solution of 2-D sound-structure interaction problems

Based on the fluid-structure coupling theory, a localized variational principle for analyzing the sound radiation from elastic structure submerged in water due to harmonic excitations is presented. It will be a powerful tool to formulate various numerical methods for steady response of structural-acoustic systems. By means of this variational principle a hybrid element method, in which an analytical solution valid in most of the surrounding water is incorporated with finite elements distributed in the structure and its neighboring water, is devised. Computational examples are then given to demonstrate its high accuracy and time saving. The project supported by the National Natural Science Foundation of China (10172038) and the Doctoral Program Foundation of Institution of Higher Education of China (20040487013). The English text was polished by Yunming Chen.     

ANALYSES ON NONLINEAR COUPLING OF MAGNETO-THERMO-ELASTICITY OF FERROMAGNETIC THIN SHELL I： GENERALIZED VARIATIONAL THEORETICAL MODELING

Based on the generalized variational principle of magneto-thermo-elasticity of the ferromagnetic elastic medium, a nonlinear coupling theoretical modeling for a ferromagnetic thin shell is developed. All governing equations and boundary conditions for the ferromagnetic shell are obtained from the variational manipulations on the magnetic scalar potential, temperature and the elastic displacement related to the total energy functional. The multi-field couplings and geometrical nonlinearity of the ferromagnetic thin shell are taken into account in the modeling. The general modeling can be further deduced to existing models of the magneto-elasticity and the thermo-elasticity of a ferromagnetic shell and magneto-thermo-elasticity of a ferromagnetic plate, which are coincident with the ones in literature.     

Extended variational principle in nonlinear theory of elasticity

From the extended energy functional, a unified variational principle and an extended variational principle are given and proved in this paper. It is about the non-conservative, abrupt and divided domains static (or kinetic) system and by using the undetermined energy functional, we can immediately deduce various variational principles.     

The combined energy variational principles in linear elasticity

In this paper, a new kind of mixed energy variational principles in linear elasticity—the combined energy variational principles is presented. First, we define the conjugate body of an elastic body, which is obtained by changing the boundary conditions of the elastic body. Next, we decompose the conjugate body into two component-states, construct functionals of potential energy and complementary energy, respectively, for the component-states and define the additional hybrid energy between the component-states. Thus the functionals of combined energy can be constructed. Three typical combined energy variational principles are demonstrated and the other forms of combined energy variational principles are given. The application of the proposed principles to the calculation of thin plates with complicated boundaries is shown.     

Local existence for Frémond’s model of damage in elastic materials

We consider a dissipative model recently proposed by M. Frémond to describe the evolution of damage in elastic materials. The corresponding PDEs system consists of an elliptic equation for the displacements with a degenerating elastic coefficient coupled with a variational dissipative inclusion governing the evolution of damage. We prove a local-in-time existence and uniqueness result for an associated initial and boundary value problem, namely considering the evolution in some subinterval where the damage is not complete. The existence result is obtained by a truncation technique combined with suitable a priori estimates. Finally, we give an analogous local-in-time existence and uniqueness result for the case in which we introduce viscosity into the relation for macroscopic displacements such that the macroscopic equilibrium equation is of parabolic type.Received: 31 July 2002, Accepted: 9 August 2003, Published online: 21 November 2003Correspondence to: E. Bonetti     

点蚀演化及腐蚀疲劳裂纹成核的能量原理

作为一种不可逆的热力学过程,腐蚀疲劳的点蚀演化受到体系应变能、表面能和电化学能的共同影响。基于能量学原理,对腐蚀疲劳点蚀演化过程中的能量问题进行探索性研究。引入双变量半椭球模型描述点蚀的演化过程,推导了点蚀形状参数在演化过程中的变化方程。基于应力强度因子准则建立点蚀向疲劳裂纹转化的临界条件,同时分析应力振幅对腐蚀疲劳裂纹成核寿命的影响。