Reflection and Refraction of Anti-Plane Shear Waves from a Moving Phase Boundary

The reflection and refraction of anti-plane shear waves from an interface separating half-spaces with different moduli is well understood in the linear theory of elasticity. Namely, an oblique incident wave gives rise to a reflected wave that departs at the same angle and to a refracted wave that, after transmission through the interface, departs at a possibly different angle. Here we study similar issues for a material that admits mobile elastic phase boundaries in anti-plane shear. We consider an energy minimal equilibrium state in anti-plane shear involving a planar phase boundary that is perturbed due to an incident wave of small magnitude. The phase boundary is allowed to move under this perturbation. As in the linear theory, the perturbation gives rise to a reflected and a refracted wave. The orientation of these waves is independent of the phase boundary motion and determined as in the linear theory. However, the phase boundary motion affects the amplitudes of the departing waves. Perturbation analysis gives these amplitudes for general small phase boundary motion, and also permits the specification of the phase boundary motion on the basis of additional criteria such as a kinetic relation. A standard kinetic relation is studied to quantify the subsequent energy partitioning and dissipation on the basis of the properties of the incident wave.     

Kinetic relations and the propagation of phase boundaries in solids

This paper treats the dynamics of phase transformations in elastic bars. The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition. A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed. For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities.     

Non-isothermal kinetics of a moving phase boundary

In this paper we derive an explicit formula for a kinetic relation governing the motion of a phase boundary in a bilinear thermoelastic material capable of undergoing solid-solid phase transitions. To obtain the relation, we study traveling wave solutions of a regularized problem that includes viscosity, heat conduction and convective heat exchange with an ambient medium. Both inertia and latent heat of transformation are taken into account. We investigate the effect of material parameters on the kinetic relation and show that in a certain range of parameters the driving force becomes a non-monotone function of the interface velocity. The model also predicts a nonzero resistance to phase boundary motion, part of which is caused by the thermal trapping. Received: November 15, 2001 / Published online September 4, 2002 RID="*" ID="*" e-mail: annav@math.pitt.edu Communicated by Lev Truskinovsky, Minneapolis     

Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.     

基于广义应变梯度理论的纳米梁挠曲电效应研究

挠曲电效应是应变梯度与电极化的耦合,它存在于所有的电介质材料中。在纳米电介质结构的挠曲电效应研究中,应变梯度弹性对挠曲电响应的影响一直以来被低估甚至被忽略了。根据广义应变梯度理论,应变梯度弹性中独立的尺度参数只有三个,而文献中所采用的一个或两个尺度参数的应变梯度理论只是它的简化形式。基于该理论,论文建立了考虑广义应变梯度弹性的三维电介质结构的理论模型,并以一维纳米梁为例研究了其弯曲问题的挠曲电响应及其能量俘获特性。结果表明,纳米梁的挠曲电响应存在尺寸效应,并且弹性应变梯度会影响结构挠曲电的尺寸效应,特别是当结构的特征尺寸低于尺度参数时。论文的工作为更进一步理解纳米尺度下的挠曲电机理和能量俘获特性提供理论基础和设计依据。     

Crack tip field andJ-integral with strain gradient effect

The mode I plane strain crack tip field with strain gradient effects is presented in this paper based on a simplified strain gradient theory within the framework proposed by Acharya and Bassani. The theory retains the essential structure of the incremental version of the conventionalJ ₂ deformation theory. No higher-order stress is introduced and no extra boundary value conditions beyond the conventional ones are required. The strain gradient effects are considered in the constitutive relation only through the instantaneous tangent modulus. The strain gradient measures are included into the tangent modulus as internal parameters. Therefore the boundary value problem is the same as that in the conventional theory. Two typical crack problems are studied: (a) the crack tip field under the small scale yielding condition induced by a linear elastic mode-IK-field and (b) the complete field for a compact tension specimen. The calculated results clearly show that the stress level near the crack tip with strain gradient effects is considerable higher than that in the classical theory. The singularity of the strain field near the crack tip is nearly equal to the square-root singularity and the singularity of the stress field is slightly greater than it. Consequently, theJ-integral is no longer path independent and increases monotonically as the radius of the calculated circular contour decreases. The project supported by the National Natural Science Foundation of China (19704100 and 10202023) and the Natural Science Foundation of Chinese Academy of Sciences (KJ951-1-20)     

A conventional theory of mechanism-based strain gradient plasticity

There exist two frameworks of strain gradient plasticity theories to model size effects observed at the micron and sub-micron scales in experiments. The first framework involves the higher-order stress and therefore requires extra boundary conditions, such as the theory of mechanism-based strain gradient (MSG) plasticity [J Mech Phys Solids 47 (1999) 1239; J Mech Phys Solids 48 (2000) 99; J Mater Res 15 (2000) 1786] established from the Taylor dislocation model. The other framework does not involve the higher-order stress, and the strain gradient effect come into play via the incremental plastic moduli. A conventional theory of mechanism-based strain gradient plasticity is established in this paper. It is also based on the Taylor dislocation model, but it does not involve the higher-order stress and therefore falls into the second strain gradient plasticity framework that preserves the structure of conventional plasticity theories. The plastic strain gradient appears only in the constitutive model, and the equilibrium equations and boundary conditions are the same as the conventional continuum theories. It is shown that the difference between this theory and the higher-order MSG plasticity theory based on the same dislocation model is only significant within a thin boundary layer of the solid.     

温度梯度对热致型形状记忆聚合物板力学性能的影响

形状记忆聚合物作为一种新型的智能材料,由于质量轻、成本低以及变形回复率大等优势,已经在航空、医学等领域得到广泛应用。当前对于热致型形状记忆聚合物力学行为的研究,大都集中在整体变温的情况下,随着应用环境的越来越广泛,温度梯度对材料力学性能的影响效果越发重要。本文在均匀应力的假设下,给出材料在不同初始和传热条件下的温度梯度分布情况,结合传热学和热致型形状记忆聚合物相变理论本构模型,分别讨论了不同温度梯度对存储应变、弹性模量等力学性能的影响,通过理论分析和实验数据对比验证了模型正确性。本文研究可为不同导热情况下,对热致型形状记忆聚合物力学行为监测提供思路,也为形状记忆聚合物的进一步工程应用提供理论依据。     

Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux , where e and are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach that was introduced and evaluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to the Rankine Hugoniot relation. Received October 6, 1998     

On kinetics of hysteresis

An initiation criterion and a kinetic relation describing hysteresis in shape memory alloys consistent with non-equilibrium thermodynamics are proposed. The initiation of phase transition begins always at zero driving force, defined as the negative derivative of free energy with respect to the volume fraction of high strain phase. The kinetic relation is obtained from the dissipation potential which is proportional to the magnitude of the volume fraction rate times the newly formed volume fraction. The proposed theory turns out to be rate-independent, but history dependent, and can describe all features of hysteresis loops observed in experiments.      

Nonlinear vibration analysis of micro-plates based on strain gradient elasticity theory

In this study, non-linear free vibration of micro-plates based on strain gradient elasticity theory is investigated. A general form of Mindlin’s first-strain gradient elasticity theory is employed to obtain a general Kirchhoff micro-plate formulation. The von Karman strain tensor is used to capture the geometric non-linearity. The governing equations of motion and boundary conditions are obtained in a variational framework. The Homotopy analysis method is employed to obtain an accurate analytical expression for the non-linear natural frequency of vibration. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on some special forms of strain gradient elasticity theory. Accordingly, three different micro-plate formulations are introduced, which are based on three special strain gradient elasticity theories. It is found that both geometric non-linearity and size effect increase the natural frequency of vibration. In a micro-plate having a thickness comparable with the material length scale parameter, the strain gradient effect on increasing the non-linear natural frequency is higher than that of the geometric non-linearity. By increasing the plate thickness, the strain gradient effect decreases or even diminishes. In this case, geometric non-linearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for micro-plates with some specific thickness to length scale parameter ratios, both geometric non-linearity and size effect have significant role on increasing the frequency of non-linear vibration.     

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.     

Temperature analysis of one-dimensional NiTi shape memory alloys under different loading rates and boundary conditions

Superelastic polycrystalline NiTi shape memory alloys under tensile loading accompany the strain localization and propagation phenomena. Experiments showed that the number of moving phase fronts and the mechanical behavior are very sensitive to the loading rate due to the release/absorption of latent heat and the material’s inherent temperature sensitivity of the transformation stress. In this paper, the moving heat source method based on the heat diffusion equation is used to study the temperature evolution of one-dimensional superelastic NiTi specimen under different loading rates and boundary conditions with moving heat sources or a uniform heat source. Comparisons of temperature variations with different boundary conditions show that the heat exchange at the boundaries plays a major role in the nonuniform temperature profile that directly relates to the localized deformation. Analytical relation between the front temperature of a single phase front, the inherent Clausius–Clapeyron relation (sensitivity of the material’s transformation stress with temperature), heat transfer boundary conditions and the loading rate is established to analyze the nucleation of new phase fronts. Finally, the rate-dependent stress hysteresis is also simply discussed by using the results of temperature analyses.     

On bending of strain gradient elastic micro-plates

Bending of strain gradient elastic thin plates is studied, adopting Kirchhoff’s theory of plates. Simple linear strain gradient elastic theory with surface energy is employed. The governing plate equation with its boundary conditions are derived through a variational method. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient plate theories; however, they strongly increase the stiffness of the thin plate.     

Interfacial energy effects within the framework of strain gradient plasticity

In the framework of strain gradient plasticity, a solid body with boundary surface playing the role of a dissipative boundary layer endowed with surface tension and surface energy, is addressed. Using the so-called residual-based gradient plasticity theory, the state equations and the higher order boundary conditions are derived quite naturally for both the bulk material and the boundary layer. A phenomenological constitutive model is envisioned, in which the bulk material and the boundary layer obey (rate independent associative) coupled plasticity evolution laws, with kinematic hardening laws of differential nature for the bulk material, but of nondifferential nature for the layer. A combined global maximum dissipation principle is shown to hold. The higher order boundary conditions are discussed in details and categorized in relation to some peculiar features of the boundary surface, and their basic role in the coupling of the bulk/layer plasticity evolution laws is pointed out. The case of an internal interface is also studied. An illustrative example relating to a shear model exhibiting energetic size effects is presented. The theory provides a unified view on gradient plasticity with interfacial energy effects.     

Analytical solutions for buckling of size-dependent Timoshenko beams

The inconsistences of the higher-order shear resultant expressed in terms of displacement(s) and the complete boundary value problems of structures modeled by the nonlocal strain gradient theory have not been well addressed. This paper develops a size-dependent Timoshenko beam model that considers both the nonlocal effect and strain gradient effect. The variationally consistent boundary conditions corresponding to the equations of motion of Timoshenko beams are reformulated with the aid of the weighted residual method. The complete boundary value problems of nonlocal strain gradient Timoshenko beams undergoing buckling are solved in closed forms. All the possible higher-order boundary conditions induced by the strain gradient are selectively suggested based on the fact that the buckling loads increase with the increasing aspect ratios of beams from the conventional mechanics point of view. Then, motivated by the expression for beams with simply-supported(SS) boundary conditions, some semiempirical formulae are obtained by curve fitting procedures.     

Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models

Mindlin, in his celebrated papers of Arch. Rat. Mech. Anal. 16, 51–78, 1964 and Int. J. Solids Struct. 1, 417–438, 1965, proposed two enhanced strain gradient elastic theories to describe linear elastic behavior of isotropic materials with micro-structural effects. Since then, many works dealing with strain gradient elastic theories, derived either from lattice models or homogenization approaches, have appeared in the literature. Although elegant, none of them reproduces entirely the equation of motion as well as the classical and non-classical boundary conditions appearing in Mindlin theory, in terms of the considered lattice or continuum unit cell. Furthermore, no lattice or continuum models that confirm the second gradient elastic theory of Mindlin have been reported in the literature. The present work demonstrates two simple one dimensional models that conclude to first and second strain gradient elastic theories being identical to the corresponding ones proposed by Mindlin. The first is based on the standard continualization of the equation of motion taken for a sequence of mass-spring lattices, while the second one exploits average processes valid in continuum mechanics. Furthermore, Mindlin developed his theory by adding new terms in the expressions of potential and kinetic energy and introducing intrinsic micro-structural parameter without however providing explicit expressions that correlate micro-structure with macro-structure. This is accomplished in the present work where in both models the derived internal length scale parameters are correlated to the size of the considered unit cell.     

On the boundary conditions of the geometrically nonlinear Kirchhoff–Love shell theory

The present paper addresses the problem of establishing the boundary conditions of a geometrically nonlinear thin shell model, especially the kinematic ones. Our model is consistently derived from general 3D continuum mechanics statements. Generalized cross-sectional strains and stresses are based on the deformation gradient and the first Piola–Kirchhoff stress tensor. Since only the bending deformation is included in this model, no special technique needs to be adopted in order to avoid shear-locking. The theory is derived in such a way that any material model can be considered as a constitutive relation, once the zero transverse normal stress assumption is properly taken into account.     

Stress gradient continuum theory

A stress gradient continuum theory is presented that fundamentally differs from the well-established strain gradient model. It is based on the assumption that the deviatoric part of the gradient of the Cauchy stress tensor can contribute to the free energy density of solid materials. It requires the introduction of so-called micro-displacement degrees of freedom in addition to the usual displacement components. An isotropic linear elasticity theory is worked out for two-dimensional stress gradient media. The analytical solution of a simple boundary value problem illustrates the essential differences between stress and strain gradient models.