On Dual Configurational Forces

The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman–Noll–Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) and the dual configuration force (dual energy–momentum tensor) ,

A New Interpretation of Configurational Forces

In a recent paper (Gupta and Markenscoff in C. R., Méc. 336:126?C131, 2008) we interpreted configurational forces as necessary and sufficient dissipative mechanisms such that the corresponding Euler-Lagrange equations are satisfied. We now extend this argument for a dynamic elastic medium, and show that the energy flux obtained from the dynamic J integral ensures that the equations of motion hold throughout the body.     

Configurational Forces and the Dynamics of Planar Cracks in Three-Dimensional Bodies

This paper develops a three-dimensional framework for the evolution of planar cracks, concentrating on the derivation of balances and constitutive equations that describe the motion of the crack tip. The theory is based on the notion of configurational forces in conjunction with a mechanical version of the second law. This revised version was published online in August 2006 with corrections to the Cover Date.     

结构内力概率密度演化的向量马氏过程方法

本文提出了位移加载机制下求解无启发持性结构结点力的概率密度演化规律的方法。该方法将力与非线性构形状态演化联合起来。组成力-状态混合向量马氏过程。采用分离式分析方法进行状态转移概率速度分析。然后建立力-状态联合概率密度演化方程。求解这一方程可分别得到非线性构形状态演化和结点力随机演化的概率结构。意味深长的是，非线性构形状态的演化方程可以直接由力-状态联合演化方程推导出来。而力的概率演化方程则不能实现力与状态之间的解耦。     

On Cavitation, Configurational Forces and Implications for Fracture in a Nonlinearly Elastic Material

Experiments on polymers indicate that large tensile stress can induce cavitation, that is, the appearance of voids that were not previously evident in the material. This phenomenon can be viewed as either the growth of pre-existing infinitesimal holes in the material or, alternatively, as the spontaneous creation of new holes in an initially perfect body. In this paper our approach is to adopt both views concurrently within the framework of the variational theory of nonlinear elasticity. We model an elastomer on a macroscale as a void-free material and, on a microscale, as a material containing certain defects that are the only points at which hole formation can occur. Mathematically, this is accomplished by the use of deformations whose point singularities are constrained. One consequence of this viewpoint is that cavitation may then take place at a point that is not energetically optimal. We show that this disparity will generate configurational forces, a type of force identified previously in dislocations in crystals, in phase transitions in solids, in solidification, and in fracture mechanics. As an application of this approach we study the energetically optimal point for a solitary hole to form in a homogeneous and isotropic elastic ball subject to radial boundary displacements. We show, in particular, that the center of the ball is the unique optimal point. Finally, we speculate that the configurational force generated by cavitation at a non-optimal material point may be sufficient to result in the onset of fracture. The analysis utilizes the energy-momentum tensor, the asymptotics of an equilibrium solution with an isolated singularity, and the linear theory of elasticity at the stressed configuration that the body occupies immediately prior to cavitation. This revised version was published online in June 2006 with corrections to the Cover Date.     

Displacement Convexity and Minimal Fronts at Phase Boundaries

We show that certain free energy functionals that are not convex with respect to the usual convex structure on their domain of definition are strictly convex in the sense of displacement convexity under a natural change of variables.We use this to show that, in certain cases, the only critical points of these functionals are minimizers. This approach based on displacement convexity permits us to treat multicomponent systems as well as single component systems. The developments produce new examples of displacement convex functionals and, in the multi-component setting, jointly displacement convex functionals.     

A Sharp Interface Model for the Propagation of Martensitic Phase Boundaries

A model for the quasistatic evolution of martensitic phase boundaries is presented. The model is essentially the gradient flow of an energy that can contain elastic energy due to the underlying change in crystal structure in the course of the phase transformation and surface energy penalizing the area of the phase boundary. This leads to a free boundary problem with a nonlocal velocity that arises from a coupling to the elasticity equation. We show existence of solutions under a technical convergence condition using an implicit time-discretization.     

Stability of Subsonic Planar Phase Boundaries in a van der Waals Fluid

We are concerned with the structural stability of dynamic phase changes occurring across sharp interfaces in a multidimensional van der Waals fluid. Such phase transitions can be viewed as propagating discontinuities. However, they are usually subsonic, and thus undercompressive. The lacking information lies in an additional jump condition, which may be derived from the viscosity-capillarity criterion. This condition is rather simple in the case of reversible phase transitions, since it reduces to a generalized equal area rule. In a previous work, I proved that reversible planar phase boundaries are weakly linearly stable, in the sense introduced by Majda for shock fronts. This means that they satisfy a generalized Lopatinsky condition but not a uniform one. The aim of this paper is to point out the influence of viscosity on the stability analysis, in order to deal with the more realistic case of dissipative phase transitions. The main difficulty lies in the additional jump condition, which is no longer explicit and depends on the (unknown) internal structure of the interface. We overcome it by using bifurcation arguments on the nondimensional parameter measuring the competition between viscosity and capillarity. We show by perturbation that the positivity of this parameter stabilizes the phase transitions. As a conclusion, we find that dissipative planar phase boundaries are uniformly linearly stable, in the sense of the uniform Lopatinsky condition. Accepted December 14, 1998     

材料微结构演化的相场模拟

材料的各种宏观性能与其内部的微观结构密切相关,如何通过控制材料微结构的分布来提高其性能指标一直是工程界与学术界广泛关注的课题.由于场变量在界面的突变,传统的基于局部理论的突变界面模型在描述材料微结构演化方面存在一定的困难.基于非局部理论的相场法采用扩散界面的概念来描述界面,避开了理论上描述突变界面的困难,在模拟材料内部任意的组织形态和复杂的微结构演化方面具有独特的优点.论文首先介绍相场法的热力学理论基础,包括自由移动边界问题、扩散界面模型、非局部能量泛函、相场动力学方程及其常用求解方法.然后重点介绍铁电、铁磁和多铁性材料微结构演化的相场模拟,同时简要介绍相场法在软物质和锂离子电池材料微结构演化模拟中的应用,最后给出总结和展望.     

Normal Modes and Nonlinear Stability Behaviour of Dynamic Phase Boundaries in Elastic Materials

This paper considers an ideal nonthermal elastic medium described by a stored-energy function W. It studies time-dependent configurations with subsonically moving phase boundaries across which, in addition to the jump relations (of Rankine–Hugoniot type) expressing conservation, some kinetic rule g acts as a two-sided boundary condition. The paper establishes a concise version of a normal-modes determinant that characterizes the local-in-time linear and nonlinear (in)stability of such patterns. Specific attention is given to the case where W has two local minimizers U ^A ,U ^B which can coexist via a static planar phase boundary. Being dynamic perturbations of such interesting configurations, this paper shows that the stability behaviour of corresponding almost-static phase boundaries is uniformly controlled by an explicit expression that can be determined from derivatives of W and g at U ^A and U ^B .     

Viscosity of Strain Gradient Effects on the Kinetics of Propagating Phase Boundaries in Solids

The theory of thermoelastic materials undergoing solid-solid phase transformations requires constitutive information that governs the evolution of a phase boundary. This is known as a kinetic relation which relates a driving traction to the speed of propagation of a phase boundary. The kinetic relation is prescribed in the theory from the onset. Here, though, a special kinetic relation is derived from an augmented theory that includes viscous, strain gradient and heat conduction effects. Based on a special class of solutions, namely travelling waves, the kinetic relation is inherited from the augmented theory as the viscosity, strain gradient and heat conductivity are removed by a suitable limit process.     

Optimal Force Transmission by Flexure-Clamped Boundaries

ABSTRACT

Grillages of maximum strength and maximum stiffness and fiber-reinforced plates of maximum strength are considered. Using a technique developed earlier, optimal solutions are given for a large number of boundary shapes. In all problems discussed, the flexural systems are clamped along all boundaries and the loading is given by an arbitrary nonnegative function.     

Existence and Stability of the Inner Boundaries of the Domain of Multiple Hysteresis of Static Aerodynamic Forces and Moments

The results of experimental investigations of the multiple static hysteresis of the aerodynamic characteristics of a rectangular high-aspect-ratio wing are presented. Schematic wing-flow structure patterns, time dependences of the coefficients c _y(t), m _z(t), and m _x(t) and their frequency spectra obtained for a fixed model are given for different boundaries of the hysteresis domain. The time dependences of aerodynamic forces and moments are analyzed at angles of incidence at which sharp changes are observed. It is shown that the static hysteresis can be described by a mathematical model used in catastrophe theory.     

Role of Second Phase Powders on Microstructural Evolution During Sintering

The influence of second phase fine powders on solid state sintering was investigated in situ by synchrotron radiation X-ray computed tomography (SR-CT). The evolutions of microstructure during sintering of pure SiO₂ powders and SiO₂ powders mixed with Si₃N₄ powders were observed from reconstructed SR-CT images. Typical sintering parameters, which determine mechanical properties, were extracted from the experimental data, and the effects of the second phase powders were quantitatively analyzed. The experimental results showed that with second phase powders, 1) the specific surface area decreases more efficiently; 2) pore shape evolution is accelerated; 3) sintering necks grow faster and 4) densification process is notably accelerated. It was indicated that the Si₃N₄ fine powders enhanced the sintering of the original SiO₂ powders rather than hindering the process. A qualitative explanation based on the experimental results and existing powder sintering theory is provided.     

Dynamic Forces Produced by Swinging Bells

The swinging of bells on belfries is a classical problem in structural dynamics and has been addressed in the Central European specialized bibliography. To carry out our study, the different modalities of swinging bells have been classified in “systems” according to their most relevant characteristics in three groups: Central European, English and Spanish systems. Each group presents some singular characteristics of frequency and oscillation, unbalance and/or turning rate, which give rise to different forces variable in time on their supporting structures. We have analyzed the three systems and, compared the maximum values of the horizontal and vertical forces that appear on the structure as well as its main harmonics. Besides the parameters analyzed, the complete dynamic study of the structure, and therefore the evaluation of its dynamic amplification factor (D.A.F.), requires the knowledge of its dynamic characteristics: main frequencies and damping ratio.     

Configurational mechanics of necking phenomena in engineering thermoplastics

Necking is a significant part of the yielding process in many thermoplastics. It starts as strain localization associated with microshear banding and/or cavitations and appears as a domain of oriented (drawn) material, i.e., a “neck”, separated from the domain of original (isotropic) material by a narrow transition zone, which appears as a distinct boundary of the neck region. On further increase of displacement, the neck propagates through the test specimen under constant draw stress. Strain localization such as crazing and shear bending is associated with necking on micro- and sub-microscales. As a result material toughness, i.e., resistance to cracking, as well as durability, i.e., service lifetime under various service conditions, are related to the material ability to necking and specific characteristics of necking process. Necking is manifested in significant changes in a characteristic length scale, e.g., the distance between equally spaced marks in the reference state may increases by factor of 2 in amorphous polymers and up to a factor of 10 in some semicrystalline thermoplastics. There is also a characteristic relaxation time change during the necking. Thus from continuum mechanics viewpoint, the changes of intrinsic material space-time metric are the most fundamental manifestation of necking. Therefore we model necking phenomena as space-time scales transformation and introduce a four-dimensional (4D) Riemannian metric tensor of a material space-time imbedded into 4D Newtonian (laboratory) space-time with a Euclidean metric. Kinetic equation of necking, i.e., evolution equation for material metric tensor is derived using extremal action principle. An example of traveling wave solution for neck propagation in a tensile bar is presented. Analysis of the solution and comparison with experimental observations are discussed.     

Regulation of Cell Adhesion Free Energy by External Sliding Forces

Cells experience a variety of forces and external velocities in vivo. While a number of continuum based models have addressed cell substrate interactions in these dynamic environments, a molecular picture of adhesion under external sliding forces and velocities is lacking. Using a molecular thermodynamic model, that incorporates entropic and steric repulsions, molecular conformations and constraints, penetrable substrates and explicit binding interactions, we study the effect of external sliding velocities on cell adhesion. We map the free energy landscapes under a broad range of external forces, binding energies and receptor surface coverages. Our calculations predict the regimes where free energy landscapes become resistant to external forces. Our model shows good agreement with experimental studies and lays out a scalable framework for analyzing and quantifying a broad spectrum of in vivo and in vitro adhesion studies.     

Wetting and Prewetting of Anti-Phase Boundaries

A degenerate Allen-Cahn/Cahn-Hilliard system which was developed to describe simultaneous ordering and phase separation, can also be viewed as a diffuse interface approximation for various problems in materials science in which surface diffusion and motion by mean curvature are coupled. In the original context a low temperature coarsening limit, yielding geometric motion in the limit, was designed to describe small particles of a disordered phase whose shape evolves by surface diffusion which are embedded along grain boundaries which partition the system into two ordered variants. While an early analysis focused on systems in the proximity of the complete wetting limit [9], a later analysis extended these results also to the partial wetting setting [11]. We outline here some features which determine whether a given degenerate Allen-Cahn/Cahn-Hilliard system corresponds to complete or partial wetting in the limit, giving some explicit examples of both possibilities.     

Boundaries of the initial melted area of a semiconductor film formed by floating-zone melting

Stationary hydrodynamic and temperature fields near the upper triple point of the floating-zone melting process are analyzed. Regularities determining the angular position and shape of the initial melted area as functions of thermal conditions on solid and liquid surfaces in the immediate vicinity of the triple point are established in the form of four analytical relations. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No, 3, pp. 139–148, May–June, 2000     

Effects of Body Forces on the Statistics of Flame Surface Density and Its Evolution in Statistically Planar Turbulent Premixed Flames

Flow, Turbulence and Combustion - Body forces such as buoyancy and externally imposed pressure gradients are expected to have a strong influence on turbulent premixed combustion due to the...