Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations

Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations, including crack propagation and strain localization is developed. By this approach, a peridynamic model capable of effectively treating strong and weak discontinuities is superimposed in the critical regions over an underlying finite element mesh placed over the entire problem domain. A rigorous variational framework of coupling local finite element and nonlocal peridynamics approximations that is free of blending parameters is developed. Several numerical examples involving mixed-model fracture, three-dimensional adaptive crack propagation and strain localization induced ductile failure demonstrate the rational and efficiency of the proposed superposition-based coupling approach.

     

Bond-based peridynamic modelling of singular and nonsingular crack-tip fields

A static meshfree implementation of the bond-based peridynamics formulation for linearly elastic solids is applied to the study of the transition from local to nonlocal behavior of the stress and displacement fields in the vicinity of a crack front and other sources of stress concentration. The long-range nature of the interactions between material points that is intrinsic to and can be modulated within peridynamics enables the smooth transition from the square-root singular stress fields predicted by the classical (local) linear theory of elasticity, to the nonsingular fields associated with nonlocal theories. The accuracy of the peridynamics scheme and the transition from local to nonlocal behavior, which are dictated by the lattice spacing and micromodulus function, are assessed by performing an analysis of the boundary layer that surrounds the front of a two dimensional crack subjected to mode-I loading and of a cracked plate subjected to far-field tension.     

近场动力学框架下的键基对应模型

传统键基近场动力学模型存在泊松比限制的问题,为了解决这一问题发展了态基近场动力学模型。其中非常规态的近场动力学模型通过定义非局部的变形梯度将近场力和传统应力关联起来,方便使用传统本构,但是态基近场动力学计算效率低于键基近场动力学。结合态基模型和键基模型的优势,提出键基对应模型,定义了基于键的变形梯度,参考连续介质力学中变形梯度的极分解过程,将键的变形分为转动部分和伸长部分。从而进一步定义了应变,通过物理方程求应力,进而计算键传递的近场力。键基对应模型解决了键基近场动力学的泊松比限制问题,也不需要进行近场动力学微观材料常数的计算。数值算例和理论推导证明了键变形梯度定义以及近场力计算方式的正确性。     

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 nonlocal strain gradient plasticity theory for finite deformations

Strain gradient plasticity for finite deformations is addressed within the framework of nonlocal continuum thermodynamics, featured by the concepts of (nonlocality) energy residual and globally simple material. The plastic strain gradient is assumed to be physically meaningful in the domain of particle isoclinic configurations (with the director vector triad constant both in space and time), whereas the objective notion of corotational gradient makes it possible to compute the plastic strain gradient in any domain of particle intermediate configurations. A phenomenological elastic–plastic constitutive model is presented, with mixed kinematic/isotropic hardening laws in the form of PDEs and related higher order boundary conditions (including those associated with the moving elastic/plastic boundary). Two fourth-order projection tensor operators, functions of the elastic and plastic strain states, are shown to relate the skew-symmetric parts of the Mandel stress and back stress to the related symmetric parts. Consistent with the thermodynamic restrictions therein derived, the flow laws for rate-independent associative plasticity are formulated in a six-dimensional tensor space in terms of symmetric parts of Mandel stresses and related work-conjugate generalized plastic strain rates. A simple shear problem application is presented for illustrative purposes.     

A nonhomogeneous nonlocal elasticity model

Nonlocal elasticity with nonhomogeneous elastic moduli and internal length is addressed within a thermodynamic framework suitable to cope with continuum nonlocality. The Clausius–Duhem inequality, enriched by the energy residual, is used to derive the state equations and all other thermodynamic restrictions upon the constitutive equations. A phenomenological nonhomogeneous nonlocal (strain difference-dependent) elasticity model is proposed, in which the stress is the sum of two contributions, local and nonlocal, respectively governed by the standard elastic moduli tensor and the (symmetric positive-definite) nonlocal stiffness tensor. The inhomogeneities of the elastic moduli and of the internal length are conjectured to be each the cause of additional attenuation effects upon the long distance particle interactions. The increased attenuation effects are accounted for by means of the standard attenuation function, but with the standard spatial distance replaced by a suitably larger equivalent distance, and with the spatially variable internal length replaced by the largest value within the domain. Formulae for the computation of the equivalent distance are heuristically suggested and illustrated with numerical examples. The solution uniqueness of the continuum boundary-value problem is proven and the related total potential energy principle given and employed for possible nonlocal-FEM discretizations. A bar in tension is considered for a few numerical applications, showing perfect numerical stability, provided the free energy potential is positive definite.     

An asymmetric theory of nonlocal elasticity—Part 1. Quasicontinuum theory

In this article, an asymmetric theory of nonlocal elasticity is developed on the basis of three dimensional atomic lattice model, the Galileo invariance for constitutive equations and by use of Fourier transformation of generalized function and energy method. It is shown that nonlocal characteristic functions (or constitutive parameters of internal elastic energy) can be explicitly expressed in terms of interacting forces connecting atoms, and the general model of nonlocal theory with rotation effects is asymmetric. Both symmetric stress and anti-symmetric stress is a nonlocal function of strain and local rotation for anisotropic materials. For isotropic materials, symmetric stress is only a nonlocal function of strain, while antisymmetric stress is only a nonlocal function of local rotation.     

A consistent nonlocal scheme based on filters for the homogenization of heterogeneous linear materials with non-separated scales

In this work, the question of homogenizing linear elastic, heterogeneous materials with periodic microstructures in the case of non-separated scales is addressed. A framework if proposed, where the notion of mesoscopic strain and stress fields are defined by appropriate integral operators which act as low-pass filters on the fine scale fluctuations. The present theory extends the classical linear homogenization by substituting averaging operators by integral operators, and localization tensors by nonlocal operators involving appropriate Green functions. As a result, the obtained constitutive relationship at the mesoscale appears to be nonlocal. Compared to nonlocal elastic models introduced from a phenomenological point of view, the nonlocal behavior has been fully derived from the study of the microstructure. A discrete version of the theory is presented, where the mesoscopic strain field is approximated as a linear combination of basis functions. It allows computing the mesoscopic nonlocal operator by means of a finite number of transformation tensors, which can be computed numerically on the unit cell.     

A link between the residual-based gradient plasticity theory and the analogous theories based on the virtual work principle

A link is shown to exist between the so-called residual-based strain gradient plasticity theory and the analogous theories based on the (extended) virtual work principle (VWP). To this aim, the former theory is reformulated and cast in a residual-free form, whereby the insulation condition and the (nonlocal) Clausius–Duhem inequality, on which the theory is grounded, are substituted with equivalent residual-free ingredients, namely the energy balance condition and the residual-free form of the Clausius–Duhem inequality. The equivalence of the residual-free formulation to the original one is shown, also in their ability to cope with energetic size effects and interfacial energy ones. It emerges that the residual-free form of the Clausius–Duhem inequality coicides with the way the second thermodynamics principle is enforced within the VWP-based theories, and also that the energy balance condition amounts to the extended VWP enforced in the whole body. This makes the residual-free formulation possess strong similarities with the more general VWP-based theories, such as it constitutes an assessment of the existing link, which can be synthetized by the statement: the insulation condition is equivalent to the extended VWP deprived by the content of the standard principle.     

Nonlocal and gradient rate plasticity

This paper deals with a formulation of nonlocal and gradient plasticity with internal variables. The constitutive model complies with local internal variables which govern kinematic hardening and isotropic softening and with a nonlocal corrective internal variable defined either as the sum between a new internal variable and its spatial weighted average or as the gradient of a measure of plastic strain. The rate constitutive problem is cast in the framework provided by the convex analysis and the potential theory for monotone multivalued operators which provide the suitable tools to perform a theoretical analysis of such nonlocal and gradient problems. The validity of the maximum dissipation theorem is assessed and constitutive variational formulations of the rate model are provided. The structural rate problem for an assigned load rate is then formulated. The related variational formulation in the complete set of state variable is contributed and the methodology to derive variational formulations, with different combinations of the state variables, is explicitly provided. In particular the generalization to the present nonlocal and gradient model of the principles of Prager–Hodge, Greenberg and Capurso–Maier is presented. Finally nonlocal variational formulations provided in the literature are derived as special cases of the proposed model.     

Multiscale coupling of molecular dynamics and peridynamics

We propose a multiscale computational model to couple molecular dynamics and peridynamics. The multiscale coupling model is based on a previously developed multiscale micromorphic molecular dynamics (MMMD) theory, which has three dynamics equations at three different scales, namely, microscale, mesoscale, and macroscale. In the proposed multiscale coupling approach, we divide the simulation domain into atomistic region and macroscale region. Molecular dynamics is used to simulate atom motions in atomistic region, and peridynamics is used to simulate macroscale material point motions in macroscale region, and both methods are nonlocal particle methods. A transition zone is introduced as a messenger to pass the information between the two regions or scales. We employ the “supercell” developed in the MMMD theory as the transition element, which is named as the adaptive multiscale element due to its ability of passing information from different scales, because the adaptive multiscale element can realize both top-down and bottom-up communications. We introduce the Cauchy–Born rule based stress evaluation into state-based peridynamics formulation to formulate atomistic-enriched constitutive relations. To mitigate the issue of wave reflection on the interface, a filter is constructed by switching on and off the MMMD dynamic equations at different scales. Benchmark tests of one-dimensional (1-D) and two-dimensional (2-D) wave propagations from atomistic region to macro region are presented. The mechanical wave can transit through the interface smoothly without spurious wave deflections, and the filtering process is proven to be efficient.     

近场动力学在断裂力学领域的研究进展

近场动力学采用非局部积分计算节点内力, 利用统一数学框架描述空间连续与非连续, 避免了非连续区局部空间导数引起的应力奇异, 数值上具有无网格属性, 可自然模拟材料结构的断裂问题. 本文概述了近场动力学的弹性本构力模型, 系统介绍了近场动力学临界伸长率、临界能量密度以及材料强度相关的键失效准则. 详细介绍了近场动力学在断裂力学领域的研究进展, 包括断裂参数能量释放率与应力强度因子的求解、J积分、混合型裂纹、弹塑性断裂、黏聚力模型、动态断裂、材料界面断裂以及疲劳裂纹扩展等. 最后讨论了断裂问题近场动力学研究的发展方向.      

Finite viscoplasticity of amorphous glassy polymers in the logarithmic strain space

The paper outlines a new constitutive model and experimental results of rate-dependent finite elastic–plastic behavior of amorphous glassy polymers. In contrast to existing kinematical approaches to finite viscoplasticity of glassy polymers, the formulation proposed is constructed in the logarithmic strain space and related to a six-dimensional plastic metric. Therefore, it a priori avoids difficulties concerning with the uniqueness of a plastic rotation. The constitutive framework consists of three major steps: (i) A geometric pre-processing defines a total and a plastic logarithmic strain measures determined from the current and plastic metrics, respectively. (ii) The constitutive model describes the stresses and the consistent moduli work-conjugate to the logarithmic strain measures in an analogous structure to the geometrically linear theory. (iii) A geometric post-processing maps the stresses and the algorithmic tangent moduli computed in the logarithmic strain space to their nominal, material or spatial counterparts in the finite deformation space. The analogy between the formulation of finite plasticity in the logarithmic strain space and the geometrically linear theory of plasticity makes this framework very attractive, in particular regarding the algorithmic implementation. The flow rule for viscoplastic strains in the logarithmic strain space is adopted from the celebrated double-kink theory. The post-yield kinematic hardening is modeled by different network models. Here, we compare the response of the eight chain model with the newly proposed non-affine micro-sphere model. Apart from the constitutive model, experimental results obtained from both the homogeneous compression and inhomogeneous tension tests on polycarbonate are presented. Besides the load–displacement data acquired from inhomogeneous experiments, quantitative three-dimensional optical measurements of the surface strain fields are carried out. With regard to these experimental data, the excellent predictive quality of the theory proposed is demonstrated by means of representative numerical simulations.     

A one-dimensional theory of strain-gradient plasticity: Formulation, analysis, numerical results

This study develops a one-dimensional theory of strain-gradient plasticity based on: (i) a system of microstresses consistent with a microforce balance; (ii) a mechanical version of the second law that includes, via microstresses, work performed during viscoplastic flow; (iii) a constitutive theory that allows

•

the free-energy to depend on the gradient of the plastic strain, and

•

the microstresses to depend on the gradient of the plastic strain-rate.

The constitutive equations, whose rate-dependence is of power-law form, are endowed with energetic and dissipative gradient length-scales L and l, respectively, and allow for a gradient-dependent generalization of standard internal-variable hardening. The microforce balance when augmented by the constitutive relations for the microstresses results in a nonlocal flow rule in the form of a partial differential equation for the plastic strain. Typical macroscopic boundary conditions are supplemented by nonstandard microscopic boundary conditions associated with flow, and properties of the resulting boundary-value problem are studied both analytically and numerically. The resulting solutions are shown to exhibit three distinct physical phenomena:

(i)

standard (isotropic) internal-variable hardening;

(ii)

energetic hardening, with concomitant back stress, associated with plastic-strain gradients and resulting in boundary layer effects;

(iii)

dissipative strengthening associated with plastic strain-rate gradients and resulting in a size-dependent increase in yield strength.

     

A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations

This study develops a gradient theory of single-crystal plasticity that accounts for geometrically necessary dislocations. The theory is based on classical crystalline kinematics; classical macroforces; microforces for each slip system consistent with a microforce balance; a mechanical version of the second law that includes, via the microforces, work performed during slip; a rate-independent constitutive theory that includes dependences on a tensorial measure of geometrically necessary dislocations. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems. The field equations consist of the yield conditions coupled to the standard macroscopic force balance; these are supplemented by classical macroscopic boundary conditions in conjunction with nonstandard boundary conditions associated with slip. As an aid to solution, a weak (virtual power) formulation of the nonlocal yield conditions is derived. To make contact with classical dislocation theory, the microstresses are shown to represent counterparts of the Peach-Koehler force on a single dislocation.     

Thermodynamically consistent nonlocal theory of ductile damage

In this paper a thermodynamically consistent, weakly nonlocal theory of ductile damage is presented. The theory is based on the classical dynamical balance laws of forces and couples in the physical space and dynamical balance laws of material forces on evolving defects and on the first and second law of thermodynamics formulated for physical and material space. Assuming general constitutive equations their frame-invariant and thermodynamically admissible form is determined. It is shown that physical and material forces and stresses consist of two parts, a nondissipative part derivable from a free energy potential, and a dissipative part, which can be obtained from a dissipation pseudo-potential, if such a pseudo-potential exists.The theory can be considered as a framework with gradient elastoplasticity, isotropic and anisotropic brittle and ductile gradient damage at finite strain as special cases.     

Stress gradient versus strain gradient constitutive models within elasticity

A stress gradient elasticity theory is developed which is based on the Eringen method to address nonlocal elasticity by means of differential equations. By suitable thermodynamics arguments (involving the free enthalpy instead of the free internal energy), the restrictions on the related constitutive equations are determined, which include the well-known Eringen stress gradient constitutive equations, as well as the associated (so far uncertain) boundary conditions. The proposed theory exhibits complementary characters with respect to the analogous strain gradient elasticity theory. The associated boundary-value problem is shown to admit a unique solution characterized by a Hellinger–Reissner type variational principle. The main differences between the Eringen stress gradient model and the concomitant Aifantis strain gradient model are pointed out. A rigorous formulation of the stress gradient Euler–Bernoulli beam is provided; the response of this beam model is discussed as for its sensitivity to the stress gradient effects and compared with the analogous strain gradient beam model.     

A unified residual-based thermodynamic framework for strain gradient theories of plasticity

A unified thermodynamic framework for gradient plasticity theories in small deformations is provided, which is able to accommodate (almost) all existing strain gradient plasticity theories. The concept of energy residual (the long range power density transferred to the generic particle from the surrounding material and locally spent to sustain some extra plastic power) plays a crucial role. An energy balance principle for the extra plastic power leads to a representation formula of the energy residual in terms of a long range stress, typically of the third order, a macroscopic counterpart of the micro-forces acting on the GNDs (Geometrically Necessary Dislocations). The insulation condition (implying that no long range energy interactions are allowed between the body and the exterior environment) is used to derive the higher order boundary conditions, as well as to ascertain a principle of the plastic power redistribution in which the energy residual plays the role of redistributor and guarantees that the actual plastic dissipation satisfies the second thermodynamics principle. The (nonlocal) Clausius-Duhem inequality, into which the long range stress enters aside the Cauchy stress, is used to derive the thermodynamic restrictions on the constitutive equations, which include the state equations and the dissipation inequality. Consistent with the latter inequality, the evolution laws are formulated for rate-independent models. These are shown to exhibit multiple size effects, namely (energetic) size effects on the hardening rate, as well as combined (dissipative) size effects on both the yield strength (intrinsic resistance to the onset of plastic strain) and the flow strength (resistance exhibited during plastic flow). A friction analogy is proposed as an aid for a better understanding of these two kinds of strengthening effects. The relevant boundary-value rate problem is addressed, for which a solution uniqueness theorem and a minimum variational principle are provided. Comparisons with other existing gradient theories are presented. The dissipation redistribution mechanism is illustrated by means of a simple shear model.