Force flux and the peridynamic stress tensor

The peridynamic model is a framework for continuum mechanics based on the idea that pairs of particles exert forces on each other across a finite distance. The equation of motion in the peridynamic model is an integro-differential equation. In this paper, a notion of a peridynamic stress tensor derived from nonlocal interactions is defined. At any point in the body, this stress tensor is obtained from the forces within peridynamic bonds that geometrically go through the point. The peridynamic equation of motion can be expressed in terms of this stress tensor, and the result is formally identical to the Cauchy equation of motion in the classical model, even though the classical model is a local theory. We also establish that this stress tensor field is unique in a certain function space compatible with finite element approximations.     

A new modified couple stress theory for anisotropic elasticity and microscale laminated Kirchhoff plate model

A new modified couple stress theory for anisotropic elasticity is proposed. This theory contains three material length scale parameters. Differing from the modified couple stress theory, the couple stress constitutive relationships are introduced for anisotropic elasticity, in which the curvature (rotation gradient) tensor is asymmetric and the couple stress moment tensor is symmetric. However, under isotropic case, this theory can be identical to modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The differences and relations of standard, modified and new modified couple stress theories are given herein. A detailed variational formulation is provided for this theory by using the principle of minimum total potential energy. Based on the new modified couple stress theory, composite laminated Kirchhoff plate models are developed in which new anisotropic constitutive relationships are defined. The First model contains two material length scale parameters, one related to fiber and the other related to matrix. The curvature tensor in this model is asymmetric; however, the couple stress moment tensor is symmetric. Under isotropic case, this theory can be identical to the modified couple stress theory proposed by Yang et al. (Int J Solids Struct 39:2731–2743, 2002). The present model can be viewed as a simplified couple stress theory in engineering mechanics. Moreover, a more simplified model of couple stress theory including only one material length scale parameter for modeling the cross-ply laminated Kirchhoff plate is suggested. Numerical results show that the proposed laminated Kirchhoff plate model can capture the scale effects of microstructures.     

A linearized theory of elastic dielectrics which couples electrically to flexural vibrations

Assuming that the free energy depends on the deformation gradient and the spatial electric field, we derive the expressions for the Cauchy stress tensor and the spatial electric displacement from an observer invariant quadratic form of the free energy via the strict definitions of these quantities. Specific forms of the Piola-Kirchhoff stress tensor and the material electric displacement are then deduced and linearized in a particular sense. As an application of the resulting theory, we formulate the problem of an electrically driven disc within the context of the classical bending theory of thin plates. The material of the disc is assumed to have at most the symmetry of a hexagonal system of classC _6v.The resulting coupled differential equations for the axial mechanical displacement of the middle surface and the material electric potential indicate that the problem is not empty. This result is of particular interest in view of the fact that it is generally held that the classical theory of piezoelectricity does not permit such couplings to occur.     

Peridynamic States and Constitutive Modeling

A generalization of the original peridynamic framework for solid mechanics is proposed. This generalization permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. This extends the types of material response that can be reproduced by peridynamic theory to include an explicit dependence on such collectively determined quantities as volume change or shear angle. To accomplish this generalization, a mathematical object called a deformation state is defined, a function that maps any bond onto its image under the deformation. A similar object called a force state is defined, which contains the forces within bonds of all lengths and orientation. The relation between the deformation state and force state is the constitutive model for the material. In addition to providing a more general capability for reproducing material response, the new framework provides a means to incorporate a constitutive model from the conventional theory of solid mechanics directly into a peridynamic model. It also allows the condition of plastic incompressibility to be enforced in a peridynamic material model for permanent deformation analogous to conventional plasticity theory.      

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.     

A Natural Generalization of Hypoelasticity and Eulerian Rate Type Formulation of Hyperelasticity

According to the classical hypoelasticity theory, the hypoelasticity tensor, i.e. the fourth order Eulerian constitutive tensor, characterizing the linear relationship between the stretching and an objective stress rate, is dependent on the current stress and must be isotropic. Although the classical hypoelasticity in this sense includes as a particular case the isotropic elasticity, it fails to incorporate any given type of anisotropic elasticity. This implies that one can formulate the isotropic elasticity as an integrable-exactly classical hypoelastic relation, whereas one can in no way do the same for any given type of anisotropic elasticity. A generalization of classical theory is available, which assumes that the material time derivative of the rotated stress is dependent on the rotated Cauchy stress, the rotated stretching and a Lagrangean spin, linear and of the first degree in the latter two. As compared with the original idea of classical hypoelasticity, perhaps the just-mentioned generalization might be somewhat drastic. In this article, we show that, merely replacing the isotropy property of the aforementioned stress-dependent hypoelasticity tensor with the invariance property of the latter under an R-rotating material symmetry group R⋆ G ₀, one may establish a natural generalization of classical theory, which includes all of elasticity. Here R is the rotation tensor in the polar decomposition of the deformation gradient and G ₀ any given initial material symmetry group. In particular, the classical case is recovered whenever the material symmetry is assumed to be isotropic. With the new generalization it is demonstrated that any two non-integrable hypoelastic relations based on any two objective stress rates predict quite different path-dependent responses in nature and hence can in no sense be equivalent. Thus, the non-integrable hypoelastic relations based on any given objective stress rate constitute an independent constitutive class in its own right which is disjoint with and hence distinguishes itself from any class based on another objective stress rate. Only for elasticity, equivalent hypoelastic formulations based on different stress rates may be established. Moreover, universal integrability conditions are derived for all kinds of objective corotational stress rates and for all types of material symmetry. Explicit, simple, integrable-exactly hypoelastic relations based on the newly discovered logarithmic stress rate are presented to characterize hyperelasticity with any given type of material symmetry. It is shown that, to achieve the latter goal, the logarithmic stress rate is the only choice among all infinitely many objective corotational stress rates. This revised version was published online in August 2006 with corrections to the Cover Date.     

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。     

Linear Constitutive Relations in Isotropic Finite Elasticity

For simple shearing and simple extension deformations of a homogeneous and isotropic elastic body, it is shown that a linear relation between the second Piola-Kirchhoff stress tensor and the Green-St. Venant strain tensor does not predict a physically reasonable response of the body. This constitutive relation implies that the slope of the curve between an appropriate component of the first Piola-Kirchhoff stress tensor and a deformation measure is an increasing functions of the deformation measure. This revised version was published online in August 2006 with corrections to the Cover Date.     

Microstructure-dependent couple stress theories of functionally graded beams

A microstructure-dependent nonlinear Euler-Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler-Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.     

Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void

The effects of void size and hardening in a hexagonal close-packed single crystal containing a cylindrical void loaded by a far-field equibiaxial tensile stress under plane strain conditions are studied. The crystal has three in-plane slip systems oriented at the angle 60° with respect to one another. Finite element simulations are performed using a strain gradient crystal plasticity formulation with an intrinsic length scale parameter in a non-local strain gradient constitutive framework. For a vanishing length scale parameter the non-local formulation reduces to a local crystal plasticity formulation. The stress and deformation fields obtained with a local non-hardening constitutive formulation are compared to those obtained from a local hardening formulation and to those from a non-local formulation. Compared to the case of the non-hardening local constitutive formulation, it is shown that a local theory with hardening has only minor effects on the deformation field around the void, whereas a significant difference is obtained with the non-local constitutive relation. Finally, it is shown that the applied stress state required to activate plastic deformation at the void is up to three times higher for smaller void sizes than for larger void sizes in the non-local material.     

Acoustomechanical constitutive theory for soft materials

Acoustic wave propagation from surrounding medium into a soft material can generate acoustic radiation stress due to acoustic momentum transfer inside the medium and material, as well as at the interface between the two. To analyze acoustic-induced deformation of soft materials, we establish an acoustomechanical constitutive theory by com-bining the acoustic radiation stress theory and the nonlinear elasticity theory for soft materials. The acoustic radiation stress tensor is formulated by time averaging the momen-tum equation of particle motion, which is then introduced into the nonlinear elasticity constitutive relation to construct the acoustomechanical constitutive theory for soft materials. Considering a specified case of soft material sheet subjected to two counter-propagating acoustic waves, we demonstrate the nonlinear large deformation of the soft material and ana-lyze the interaction between acoustic waves and material deformation under the conditions of total reflection, acoustic transparency, and acoustic mismatch.     

Objective constitutive relations in elasticity and viscoelasticity

Summary The compatibility between the objectivity principle and affine constitutive equations for the elastic Cauchy and Piola-Kirchhoff stress tensors with non-zero residual stress is examined. It is found that the Cauchy stress is allowed to be only a constant tensor, proportional to the identity tensor, while the Piola-Kirchhoff stress may be a linear function on the deformation gradient thus generalizing previous results by Fosdick and Serrin. The same conclusions are arrived at also by starting from viscoelasticity. Finally, in the case of Maxwell-like materials, the solutions to the objective evolution equations are shown to be objective functionals.

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Sommario Si esamina la compatibilità tra il principio di obiettività ed equazioni costitutive affini per i tensori di stress elastici di Cauchy a Piola-Kirchhoff con stress residuo non nullo. Generalizzando risultati di Fosdick e Serrin si prova che il tensore di Cauchy può essere soltanto un tensore costante, proporzionale al tensore identità, mentre il tensore di Piola-Kirchhoff può essere una funzione lineare del gradiente di deformazione. Alle stesse conclusioni si perviene anche partendo dal funzionale della viscoelasticità. Infine si mostra che, per materiali tipo Maxwell, le soluzioni di equazioni di evoluzione obiettive sono funzionali obiettivi.

     

Multiscale Dynamics of Heterogeneous Media in?the?Peridynamic Formulation

A methodology is presented for investigating the dynamics of heterogeneous media using the nonlocal continuum model given by the peridynamic formulation. The approach presented here provides the ability to model the macroscopic dynamics while at the same time resolving the dynamics at the length scales of the microstructure. Central to the methodology is a novel two-scale evolution equation. The rescaled solution of this equation is shown to provide a strong approximation to the actual deformation inside the peridynamic material. The two scale evolution can be split into a microscopic component tracking the dynamics at the length scale of the heterogeneities and a macroscopic component tracking the volume averaged (homogenized) dynamics. The interplay between the microscopic and macroscopic dynamics is given by a coupled system of evolution equations. The equations show that the forces generated by the homogenized deformation inside the medium are related to the homogenized deformation through a history dependent constitutive relation.     

Discontinuous bifurcation analysis of thermodynamically consistent gradient poroplastic materials

In this work, analytical and numerical solutions of the condition for discontinuous bifurcation of thermodynamically consistent gradient-based poroplastic materials are obtained and evaluated. The main aim is the analysis of the potentials for localized failure modes in the form of discontinuous bifurcation in partially saturated gradient-based poroplastic materials as well as the dependence of these potentials on the current hydraulic and stress conditions. Also the main differences with the localization conditions of the related local theory for poroplastic materials are evaluated to perfectly understand the regularization capabilities of the non-local gradient-based one. Firstly, the condition for discontinuous bifurcation is formulated from wave propagation analyses in poroplastic media. The material formulation employed in this work for the spectral properties evaluation of the discontinuous bifurcation condition is the thermodynamically consistent, gradient-based modified Cam Clay model for partially saturated porous media previously proposed by the authors. The main and novel feature of this constitutive theory is the inclusion of a gradient internal length of the porous phase which, together with the characteristic length of the solid skeleton, comprehensively defined the non-local characteristics of the represented porous material. After presenting the fundamental equations of the thermodynamically consistent gradient based poroplastic constitutive model, the analytical expressions of the critical hardening/softening modulus for discontinuous bifurcation under both drained and undrained conditions are obtained. As a particular case, the related local constitutive model is also evaluated from the discontinuous bifurcation condition stand point. Then, the localization analysis of the thermodynamically consistent non-local and local poroplastic Cam Clay theories is performed. The results demonstrate, on the one hand and related to the local poroplastic material, the decisive role of the pore pressure and of the volumetric non-associativity degree on the location of the transition point between ductile and brittle failure regimes in the stress space. On the other hand, the results demonstrate as well the regularization capabilities of the non-local gradient-based poroplastic theory, with exception of a particular stress condition which is also evaluated in this work. Finally, it is also shown that, due to dependence of the characteristic lengths for the pore and skeleton phases on the hydraulic and stress conditions, the non-local theory is able to reproduce the strong reduction of failure diffusion that takes place under both, low confinement and low pore pressure of partially saturated porous materials, without loosing, however, the ellipticity of the related differential equations.     

考虑尺度影响的平面正交各向异性功能梯度微梁的屈曲分析

基于新修正偶应力理论，建立了能描述尺度效应的各向异性功能梯度微梁的屈曲分析模型。基于最小势能原理推导了控制方程及边界条件，并以简支梁为例分析了屈曲载荷及尺度效应受材料尺度参数和几何尺寸的影响。算例结果表明，在材料几何尺寸较小时，本文模型预测到的屈曲载荷明显大于传统理论的结果，有效地反映了尺度效应。几何尺寸较大时，尺度效应消失，本文模型将自动退化为传统宏观模型。模型反映出不同方向上的尺度参数对各向异性材料影响的效果不同。     

Analysis of the Volume-Constrained Peridynamic Navier Equation of Linear Elasticity

Well-posedness results for the state-based peridynamic nonlocal continuum model of solid mechanics are established with the help of a nonlocal vector calculus. The peridynamic strain energy density for an elastic constitutively linear anisotropic heterogeneous solid is expressed in terms of the field operators of that calculus, after which a variational principle for the equilibrium state is defined. The peridynamic Navier equilibrium equation is then derived as the first-order necessary conditions and are shown to reduce, for the case of homogeneous materials, to the classical Navier equation as the extent of nonlocal interactions vanishes. Then, for certain peridynamic constitutive relations, the peridynamic energy space is shown to be equivalent to the space of square-integrable functions; this result leads to well-posedness results for volume-constrained problems of both the Dirichlet and Neumann types. Using standard results, well-posedness is also established for the time-dependent peridynamic equation of motion.     

A thermodynamical gradient theory for deformation and strain localization of porous media

In this work, a thermodynamically consistent gradient formulation for partially saturated cohesive-frictional porous media is proposed. The constitutive model includes a classical or local hardening law and a softening formulation with state parameters of non-local character based on gradient theory. Internal characteristic length in softening regime accounts for the strong shear band width sensitivity of partially saturated porous media regarding both governing stress state and hydraulic conditions. In this way the variation of the transition point (TP) of brittle-ductile failure mode can be realistically described depending on current confinement condition and saturation level. After describing the thermodynamically consistent gradient theory the paper focuses on its extension to the case of partially saturated porous media and, moreover, on the formulation of the gradient-based characteristic length in terms of stress and hydraulic conditions. Then the localization indicator for discontinuous bifurcation is formulated for both drained and undrained conditions.     

Investigation of the size effects in Timoshenko beams based on the couple stress theory

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.     

新修正偶应力理论Reddy型层合板稳定分析

基于新修正偶应力理论建立了一个Reddy型复合材料层合板稳定性模型。该理论中曲率张量不对称,而偶应力矩张量对称。Reddy型层合板模型能够满足横向剪切应力为0的自由表面条件,而且横向剪切为二次函数,避免了常剪力一阶理论需要引入的剪力修正系数。为了便于工程应用,通过虚功原理推导了只含纤维材料尺度参数正交铺设的Reddy型层合板偶应力模型的稳定性方程,并以微尺度正交铺设四边简支层合方板为例,分析了不同铺设角和轴向载荷作用时临界载荷的细观尺度效应,并且与一阶剪切变形和Kirchhoff板理论结果对比。结果表明,本文建立的新修正偶应力Reddy型层合板模型更适合分析较厚的复合材料层合板稳定性的尺度效应。     

VIBRATION ANALYSIS OF MICROSCALE PLATES BASED ON MODIFIED COUPLE STRESS THEORY

A non-classical Kirchhoff plate model is developed for the dynamic analysis of microscale plates based on the modified couple stress theory in which an internal material length scale parameter is included. Unlike the classical Kirchhoff plate model, the newly developed model can capture the size effect of microscale plates. Two boundary value problems of rectangular micro- plates are solved and the size effect on the lowest two natural frequencies is investigated. It is shown that the natural frequencies of the microscale plates predicted by the current model are size-dependent when the plate thickness is comparable to the material length scale parameter.