Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.      

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.     

Linearized Theory of Peridynamic States

A state-based peridynamic material model describes internal forces acting on a point in terms of the collective deformation of all the material within a neighborhood of the point. In this paper, the response of a state-based peridynamic material is investigated for a small deformation superposed on a large deformation. The appropriate notion of a small deformation restricts the relative displacement between points, but it does not involve the deformation gradient (which would be undefined on a crack). The material properties that govern the linearized material response are expressed in terms of a new quantity called the modulus state. This determines the force in each bond resulting from an incremental deformation of itself or of other bonds. Conditions are derived for a linearized material model to be elastic, objective, and to satisfy balance of angular momentum. If the material is elastic, then the modulus state is obtainable from the second Fréchet derivative of the strain energy density function. The equation of equilibrium with a linearized material model is a linear Fredholm integral equation of the second kind. An analogue of Poincaré’s theorem is proved that applies to the infinite dimensional space of all peridynamic vector states, providing a condition similar to irrotationality in vector calculus.     

Deformation of a Peridynamic Bar

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venant's principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Green's function for general loading problems.     

A new numerical algorithm for viscoelastic fluid flows : The grid-by-grid inversion method

We present a new algorithm for solving viscoelastic flows with a general constitutive equation. In our approach the hyperbolic constitutive equation is split such that the term for the convective transport of stress tensor is treated as a source. This allows the stress tensor at each grid point to be expressed mainly in terms of the velocity gradient tensor at the same point. Then, the set of six stress tensor components is found after inverting a six by six matrix at each grid point. Thus we call this algorithm the grid-by-grid inversion method. The convective transport of stress tensor in the constitutive equation, which has been treated as a source, is updated iteratively. The present algorithm can be combined with finite volume method, finite element method or the spectral methods. To corroborate the accuracy and robustness of the present algorithm we consider viscoelastic flow past a cylinder placed at the center between two plates, which has served as a benchmark problem. Also considered is the investigation of the pattern and strength of the secondary flows in the viscoelastic flows through a rectangular pipe. It is found that the present method yields accurate results even for large relaxation times.     

修正的偶应力线弹性理论及广义线弹性体的有限元方法

以含偶应力的弹性理论为基础,考虑小变形情况下变形体的平动变形和旋转变形,提出关于偶应力与曲率张量的线性本构关系,建立一般弹性体的线性模型。为满足有限单元C1连续性要求,考虑转角为独立变量,利用罚方法引入约束条件,构造一般弹性体的约束变分形式。应用8节点48个自由度的实体等参元,建立一般弹性体力学响应分析的有限元方程。对悬臂梁的静力和动力分析表明,一般弹性体模型较之经典弹性力学更适合结构分析;较之Timoshenko梁模型,一般弹性体模型能够计及结构尺度对结构动力特性和动力响应造成的显著影响。     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

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.      

The effect of long-range forces on the dynamics of a bar

The one-dimensional dynamic response of an infinite bar composed of a linear “microelastic material” is examined. The principal physical characteristic of this constitutive model is that it accounts for the effects of long-range forces. The general theory that describes our setting, including the accompanying equation of motion, was developed independently by Kunin (Elastic Media with Microstructure I, 1982), Rogula (Nonlocal Theory of Material Media, 1982) and Silling (J. Mech. Phys. Solids 48 (2000) 175), and is called the peridynamic theory. The general initial-value problem is solved and the motion is found to be dispersive as a consequence of the long-range forces. The result converges, in the limit of short-range forces, to the classical result for a linearly elastic medium. Explicit solutions in elementary form are given in a broad class of special cases. The most striking observations arise in the Riemann-like problem corresponding to a constant initial displacement field and a piecewise constant initial velocity field. Even though, initially, the displacement field is continuous, it involves a jump discontinuity for all later times, the Lagrangian location of which remains stationary. For some materials the magnitude of the discontinuity-jump oscillates about an average value, while for others it grows monotonically, presumably fracturing the material when it exceeds some critical level.     

Derivation of a dual-mixed <Emphasis Type="Italic">hp</Emphasis>-finite element model for axisymmetrically loaded cylindrical shells

The two-field dual-mixed Fraeijs de Veubeke variational formulation of three-dimensional elasticity serves as the starting point of the derivation of a dimensionally reduced shell model presented in this paper. The fundamental variables of this complementary energy-based variational principle are the not a priori symmetric stress tensor and the skew-symmetric rotation tensor. The tensor of first-order stress functions is applied to satisfy translational equilibrium, while the rotation tensor plays the role of a Lagrange multiplier to ensure rotational equilibrium. The volumetric locking-free shell model uses unmodified three-dimensional constitutive equations, and no classical kinematical hypotheses are employed during the derivation. The numerical performance of the related low-order h-, and higher-order p-version finite elements developed for axisymmetrically loaded cylindrical shells is investigated by two representative model problems. It is numerically proven that no negative effect can be experienced when the thickness is small and tends to zero.     

A non-ordinary state-based peridynamic method to model solid material deformation and fracture

In this paper, we develop a new non-ordinary state-based peridynamic method to solve transient dynamic solid mechanics problems. This new peridynamic method has advantages over the previously developed bond-based and ordinary state-based peridynamic methods in that its bonds are not restricted to central forces, nor is it restricted to a Poisson’s ratio of 1/4 as with the bond-based method. First, we obtain non-local nodal deformation gradients that are used to define nodal strain tensors. The deformation gradient tensors are used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. The polar decomposition of the deformation gradient tensors are then used to obtain the nodal rotation tensors which are used to rotate the rate of deformation tensors and previous Cauchy stress tensors into an unrotated configuration. These are then used with conventional Cauchy stress constitutive models in the unrotated state where the unrotated Cauchy stress rate is objective. We then obtain the unrotated Cauchy nodal stress tensors and rotate them back into the deformed configuration where they are used to define the forces in the nodal connecting bonds. As a first example we quasi-statically stretch a bar, hold it, and then rotate it ninety degrees to illustrate the methods finite rotation capabilities. Next, we verify our new method by comparing small strain results from a bar fixed at one end and subjected to an initial velocity gradient with results obtained from the corresponding one-dimensional small strain analytical solution. As a last example, we show the fracture capabilities of the method using both a notched and un-notched bar.     

Membrane Curvatures and Stress-strain Full Fields of Axisymmetric Bulge Tests from 3D-DIC Measurements. Theory and Validation on Virtual and Experimental results

The bulge test is mostly used to analyze equibiaxial tensile stress state at the pole of inflated isotropic membranes. Three-dimensional digital image correlation (3D-DIC) technique allows the determination of three-dimensional surface displacements and strain fields. In this paper, a method is proposed to determine also the membrane stress tensor fields for in-plane isotropic materials, independently of any constitutive equation. Stress-strain state is then known at any surface point which enriches greatly experimental data deduced from the axisymmetric bulge tests. Our method consists, first in calculating from the 3D-DIC experimental data the membrane curvature tensor at each surface point of the bulge specimen. Then, curvature tensor fields are used to investigate axisymmetry of the test. Finally in the axisymmetric case, membrane stress tensor fields are determined from meridional and circumferential curvatures combined with the measurement of the inflating pressure. Our method is first validated for virtual 3D-DIC data, obtained by numerical simulation of a bulge test using a hyperelastic material model. Afterward, the method is applied to an experimental bulge test performed using as material a silicone elastomer. The stress-strain fields which are obtained using the proposed method are compared with results of the finite element simulation of this overall bulge test using a neo-Hookean model fitted on uniaxial and equibiaxial tensile tests.     

Finite Bending of a Rectangular Block of an Elastic Hencky Material

Hencky's elasticity model is an isotropic, finite hyperelastic equation obtained by simply replacing the Cauchy stress tensor and the infinitesimal strain tensor in the classical Hooke's law for isotropic infinitesimal elasticity with the Kirchhoff stress tensor and Hencky's logarithmic strain tensor. A study by Anand in 1979 and 1986 indicates that it is a realistic finite elasticity model that is in good accord with experimental data for a variety of engineering materials for moderate deformations. Most recently, by virtue of well-founded physical grounds and rigorous mathematical procedures it has been demonstrated by these authors that this model may be essential to achieving self-consistent Eulerian rate type theories of finite inelasticity, e.g., the J ₂-flow theory for metal plasticity, etc. Its predictions have been studied for some typical deformation modes, including extension, simple shear and torsion, etc. Here we are concerned with finite bending of a rectangular block. We show that a closed-form solution may be obtained. We present explicit expressions for the bending angle and the bending moment in terms of the maximum or minimum circumferential stretch in a general case of compressible deformations for any assigned stretch normal to the bending plane. In particular, simplified results are derived for the plane strain case and for the case of incompressibility. This revised version was published online in June 2006 with corrections to the Cover Date.     

Nonlocal Constrained Value Problems for a Linear Peridynamic Navier Equation

In this paper, we carry out further mathematical studies of nonlocal constrained value problems for a peridynamic Navier equation derived from linear state-based peridynamic models. Given the nonlocal interactions effected in the model, constraints on the solution over a volume of nonzero measure are natural conditions to impose. We generalize previous well-posedness results that were formulated for very special kernels of nonlocal interactions. We also give a more rigorous treatment to the convergence of solutions to nonlocal peridynamic models to the solution of the conventional Navier equation of linear elasticity as the horizon parameter goes to zero. The results are valid for arbitrary Poisson ratio, which is a characteristic of the state-based peridynamic model.     

Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation

Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived.Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption.Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential.     

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.      

正装结构非线性应力应变累加规律研究

基于有限位移理论的正装结构非线性分析理论对解决超大跨度、柔性结构的非线性计算有重大意义.文中探讨了正装结构非线性的分析特点,研究了其应变场与应力场的Kirchhoff应力张量与Lagrange应变张量的适用性,提出了正装结构非线性分析中应力场与应交场的累加规律,导出了拖动坐标法的虚功增量方程,以此对杆系结构非线性分析常用的CR法和UL列式进行了精度比较分析.文中研究成果可为正装结构的非线性分析提供理论指导.