Shape sensitivity analysis for non-differentiable performance measures in multiscale crack propagation simulation using regression hybrid method

In this paper, we present a regression hybrid method that calculates shape sensitivity coe?cients for multiscale crack propagation problems with performance measures that are non-differentiable in numerical implementation. These measures are crack propagation speed (or crack speed) defined at atomistic level obtained by solving coupled atomistic/continuum structures using the bridging scale method (BSM). The major contributions of this paper are: first, by analyzing the characteristics of the performance measures of crack speed in design space, this paper verifies for the first time that these measures are theoretically continuous and differentiable with respect to design variables, and as a result, the sensitivity coe?cients exist in theory; second, to overcome the non-differentiability of the performance measures in numerical computation due to the finite size of integration time step, this paper proposes a regression hybrid method that calculates the shape sensitivity coe?cients of crack speed through polynomial regression analysis based on the sensitivity of atomic responses, which is calculated through analytical shape design sensitivity analysis (DSA). And finally, the proposed method supports for 3D crack propagation problems with periodic boundary condition in one direction. A nano-beam example is used to demonstrate numerically the feasibility, accuracy, and e?ciency of the proposed method.     

Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics

A computational method (CADD) is presented whereby a continuum region containing dislocation defects is coupled to a fully atomistic region. The model is related to previous hybrid models in which continuum finite elements are coupled to a fully atomistic region, with two key advantages: the ability to accomodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. The resulting CADD model allows for the study of 2d problems involving large numbers of defects where the system size is too big for fully atomistic simulation, and improves upon existing discrete dislocation techniques by preserving accurate atomistic details of dislocation nucleation and other atomic scale phenomena. Applications to nanoindentation, atomic scale void growth under tensile stress, and fracture are used to validate and demonstrate the capabilities of the model.     

Multiscale modeling of the dynamics of solids at finite temperature

We develop a general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we develop an additional strategy for defect tracking. This method naturally decouples the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented.     

Estimation of anisotropic elastic properties of nanocomposites using atomistic-continuum interphase model

We have revised classical micromechanics by accounting for the effect of interface to predict the effective anisotropic elastic properties of heterogeneous materials containing nano-inhomogeneities. In contrast to sharp interface between the matrix and inhomogeneity, we introduce the concept of interphase to account for the interfacial-stress effect at the nano-scale. The interphase’s constitutive properties are derived from atomistic simulations and then incorporated in a micromechanics-based interphase model to compute effective properties of nanocomposites. This scale transition approach bridges the gap between discrete atomic level interactions and continuum mechanics. An advantage of this approach is that it combines atomistic with continuum models that consider inhomogeneity and interphase morphology. It thereby enables us to account simultaneously for both the shape and the anisotropy of a nano-inhomogeneity and interphase at the continuum level when we compute material’s overall properties. In so doing, it frees us from making any assumptions about the interface characteristics between matrix and the nano-inhomogeneity.     

Coarse-graining atomistic dynamics of brittle fracture by finite element method

In this work we present the finite element (FE) implementation of an atomistic formulation of balance equations and its application to coarse-grained (CG) simulation of dynamic fracture. First, we simulate a notched specimen that contains about 1.8 million atoms by the CG-FE method, and we compare the CG-FE results with that by all-atom molecular dynamics (MD) simulations. We find that CG-FE simulations with about 5% degrees of freedom of the MD simulation can capture the essential dynamic features, not in exact correspondence, but qualitatively and quantitatively similar to that obtained by MD simulations. We then proceed to simulate a series of micron-sized specimens by the CF-FE method. We find that it is the interaction of the forward propagating crack with the stress waves being reflected back by the boundaries of the specimen that triggers the dynamic instability and hence the branching of cracks in micron-sized specimens. The potential application of the method and future work for improvements are discussed.     

Design optimization for manufacturability of axisymmetric continuum structures using metamorphic development

In this study, optimal shapes/profiles of axisymmetric continuum structures optimized for performance and manufacturability are sought, using a topology/shape optimization method called metamorphic development (MD). The optimization seeks to find an optimum cross-sectional profile and minimum weight design, subjected to von-Mises stress constraints under coupled thermal and pressure loadings. Both quadrilateral and triangular finite elements (FE) were used to ‘metamorphically’ develop the structure. Two different design optimization approaches were taken, by defining a set of finite ‘restricted’ and ‘unrestricted’ design domains. Manufacturability of the optimized structures was considered. Prior to manufacturing, a post-optimization process was performed. A comparison was then performed for both sets of optimized solutions to demonstrate whether the ‘restricted’ design domain solution gave greater or lesser performance characteristics compared to the ‘unrestricted’ design domain solution which could only be manufactured by additive manufacturing technologies (AMT). The results of the optimization demonstrated the success of the MD method in generating practical design solutions which meet both performance requirements and manufacturing considerations.     

The non-uniqueness of the atomistic stress tensor and its relationship to the generalized Beltrami representation

The non-uniqueness of the atomistic stress tensor is a well-known issue when defining continuum fields for atomistic systems. In this paper, we study the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). We demonstrate the validity of these analogies using a numerical test. A program for performing the decomposition of the atomistic stress tensor called MDStressLab is available online at http://mdstresslab.org.     

颗粒材料多尺度分析的连接尺度方法

本文提出了耦合细尺度上基于离散颗粒集合体模型的离散单元法（DEM）和粗尺度上基于Cosserat连续体模型的有限元法（FEM）的连接尺度方法（BSM）以研究颗粒材料的力学行为。采用Cosserat连续体模型和FEM模拟的粗尺度域覆盖全域,而采用离散颗粒集合体模型的DEM模拟的细尺度域仅限于需特别关注材料微结构演变和非连续变形行为的局部区域。对这两个区域间的界面提出了适当的界面条件及其实施方案。通过采用适当的连接尺度投影算子,空间离散的粗、细尺度耦合系统多尺度运动方程具有解耦和允许分别求解、因而也允许分别采用不同时间步长对粗、细尺度计算的特点,可极大地提高BSM的计算效率。文中二维地基数值算例结果说明了所陈述方法的可应用性,以及相对基于Cosserat连续体模型的FEM和基于离散颗粒集合体模型的DEM的优越性。      

A Bridging Scale Method for Multi-scale Analysis of Granular Materials

本文提出了耦合细尺度上基于离散颗粒集合体模型的离散单元法（DEM）和粗尺度上基于Cosserat连续体模型的有限元法（FEM）的连接尺度方法（BSM）以研究颗粒材料的力学行为。采用Cosserat连续体模型和FEM模拟的粗尺度域覆盖全域,而采用离散颗粒集合体模型的DEM模拟的细尺度域仅限于需特别关注材料微结构演变和非连续变形行为的局部区域。对这两个区域间的界面提出了适当的界面条件及其实施方案。通过采用适当的连接尺度投影算子,空间离散的粗、细尺度耦合系统多尺度运动方程具有解耦和允许分别求解、因而也允许分别采用不同时间步长对粗、细尺度计算的特点,可极大地提高BSM的计算效率。文中二维地基数值算例结果说明了所陈述方法的可应用性,以及相对基于Cosserat连续体模型的FEM和基于离散颗粒集合体模型的DEM的优越性。     

Multiscale modeling of crack initiation and propagation at the nanoscale

Fracture occurs on multiple interacting length scales; atoms separate on the atomic scale while plasticity develops on the microscale. A dynamic multiscale approach (CADD: coupled atomistics and discrete dislocations) is employed to investigate an edge-cracked specimen of single-crystal nickel, Ni, (brittle failure) and aluminum, Al, (ductile failure) subjected to mode-I loading. The dynamic model couples continuum finite elements to a fully atomistic region, with key advantages such as the ability to accommodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. An ad hoc computational technique is also applied to dissipate localized waves formed during crack advance in the atomistic zone, whereby an embedded damping zone at the atomistic/continuum interface effectively eliminates the spurious reflection of high-frequency phonons, while allowing low-frequency phonons to pass into the continuum region.The simulations accurately capture the essential physics of the crack propagation in a Ni specimen at different temperatures, including the formation of nano-voids and the sudden acceleration of the crack tip to a velocity close to the material Rayleigh wave speed. The nanoscale brittle fracture happens through the crack growth in the form of nano-void nucleation, growth and coalescence ahead of the crack tip, and as such resembles fracture at the microscale. When the crack tip behaves in a ductile manner, the crack does not advance rapidly after the pre-opening process but is blunted by dislocation generation from its tip. The effect of temperature on crack speed is found to be perceptible in both ductile and brittle specimens.     

基于偶应力模型的连续体结构拓扑优化设计

经典连续介质理论不包含材料尺度参数,因而基于经典理论的结构拓扑优化无法显现尺度效应.本文在偶应力理论的框架下,构造了四节点四边形离散偶应力单元,将传统的SIMP方法推广至偶应力介质.结果表明,在以结构的最大刚度为目标的设计中,偶应力介质的最优结果取决于宏观结构尺寸与材料微结构尺寸(或者特征长度)的比值,最优结果具有明显的尺度效应,具体为,二者比值较大将产生与传统理论相似的结构,而二者比值相当则产生独特的偶应力主导的结构.     

An atomistically validated continuum model for strain relaxation and misfit dislocation formation

In this paper, molecular dynamics (MD) calculations have been used to examine the physics behind continuum models of misfit dislocation formation and to assess the limitations and consequences of approximations made within these models. Without compromising the physics of misfit dislocations below a surface, our MD calculations consider arrays of dislocation dipoles constituting a mirror imaged “surface”. This allows use of periodic boundary conditions to create a direct correspondence between atomistic and continuum representations of dislocations, which would be difficult to achieve with free surfaces. Additionally, by using long-time averages of system properties, we have essentially reduced the errors of atomistic simulations of large systems to “zero”. This enables us to deterministically compare atomistic and continuum calculations. Our work results in a robust approach that uses atomistic simulation to accurately calculate dislocation core radius and energy without the continuum boundary conditions typically assumed in the past, and the novel insight that continuum misfit dislocation models can be inaccurate when incorrect definitions of dislocation spacing and Burgers vector in lattice-mismatched systems are used. We show that when these insights are properly incorporated into the continuum model, the resulting energy density expression of the lattice-mismatched systems is essentially indistinguishable from the MD results.     

Nanoscale finite element models for vibrations of single-walled carbon nanotubes: atomistic versus continuum

By the atomistic and continuum finite element models, the free vibration behavior of single-walled carbon nanotubes （SWCNTs） is studied. In the atomistic finite element model, the bonds and atoms are modeled by the beam and point mass elements, respectively. The molecular mechanics is linked to structural mechanics to determine the elastic properties of the mentioned beam elements. In the continuum finite element approach, by neglecting the discrete nature of the atomic structure of the nanotubes, they are modeled with shell elements. By both models, the natural frequencies of SWCNTs are computed, and the effects of the geometrical parameters, the atomic structure, and the boundary conditions are investigated. The accuracy of the utilized methods is verified in comparison with molecular dynamic simulations. The molecular structural model leads to more reliable results, especially for lower aspect ratios. The present analysis provides valuable information about application of continuum models in the investigation of the mechanical behaviors of nanotubes.     

Discrete and non-local elastica

In this paper, the buckling and post-buckling behavior of an elastic lattice system referred to as the discrete elastica problem is investigated using an equivalent non-local continuum approach. The geometrically exact post-buckling analysis of the elastic chain, also called Hencky system, is first numerically solved using the shooting method. This discrete physical model is also mathematically equivalent to a finite difference formulation of the continuum elastica. Starting from the exact difference equations of the discrete problem, a continualization method is applied for approximating the difference operators by differential ones, in order to better characterize the discrete system by an enriched continuous one. It is shown that the new continuum associated with the discrete system exactly fits the discrete elastica post-buckling problem, where the non-locality is of Eringen׳s type (also called stress gradient non-local model). An asymptotic expansion is performed for both the discrete and the non-local continuum models, in order to approximate the post-buckling branches of the discrete system. Some numerical investigations show the efficiency of the non-local approach, especially for capturing the scale effects inherent to the cell size of the lattice model.     

Unified proof to oscillation property of discrete beam

The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discretized continuum model should keep the OP. In literatures, the OP of discrete beam models is discussed essentially by means of matrix factorization. The discussion is model-specific and boundary-condition- specific. Besides, matrix factorization is difficult in handling finite element （FE） models of beams. In this paper, according to a sufficient condition for the OP, a new approach to discuss the property is proposed. The local criteria on discrete displacements rather than global matrix factorizations are given to verify the OP. Based on the proposed approach, known results such as the OP for the 2-node FE beams via the Heilinger- Reissener principle （HR-FE beams） as well as the 5-point finite difference （FD） beams are verified. New results on the OP for the 2-node PE-FE beams and the FE Timoshenko beams with small slenderness are given. Through a simple manipulation, the qualitative property of discrete multibearing beams can also be discussed by the proposed approach.     

Optimum shape and topology design using the boundary element method

A procedure is developed for simultaneous shape and topology design optimization of linear elastic two-dimensional continuum structures. An intuitive approach is presented to treat such topological problems whereby material is eliminated from within the structure (by introducing holes at regions of low stress) through a sequence of shape optimization processes. A mathematical programming technique coupled with the boundary element (BE) method of response and sensitivity analyses enables the optimal positioning of these holes plus optimization of the overall structural shape. The analytical derivative BE formulation is explained together with the use of appropriate design velocity fields, and example problems are solved to demonstrate the optimization procedure.     

Design Sensitivity Analysis of Truss Structures with Elastoplastic Material*

ABSTRACT

A continuum-based design sensitivity analysis (DSA) method is presented for configuration (or layout) design of nonlinear structural systems with rate-independent elastoplastic material. Configuration design variables are characterized by shape and orientation changes of the structural component. A continuum-based shape DSA method that utilizes the material derivative of continuum mechanics is extended to account for effects of shape and orientation variations. The incremental analysis method, with updated Lagrangian formulation, is used to derive the design sensitivity for the nonlinear structural system.

To derive the design sensitivity, incremental energy and load forms are utilized. The first variations of energy and load forms and the static response with respect to configuration design variables are described using the material derivative. Direct differentiation is utilized to obtain the first variation of the performance measure explicitly in terms of variations of configuration design variables. With the consistent tangent stiffness matrix employed at the end of each load step to compute the design sensitivity, it is found that no iterations are necessary to compute design sensitivity. In addition, the linear design velocity is used to account for configuration design changes, with the velocity field being updated at each load step of the incremental analysis.     

一种从离散模拟到连续介质弹性模拟的过渡方法

提出了一种从离散分子动力学模拟(MD)到连续介质弹性有限元计算分析(FEA)的过渡方法, 简称MD-FEA方法. 首先通过MD计算获得晶体材料原子的移动位置, 然后根据晶体结构的周期性特征构造连续介质假设下的有限单元变形模型, 进一步结合材料的力学行为本构关系获得应变和应力场. 为了检验MD-FEA方法的有效性, 将该方法应用于详细分析Al-Ni软硬组合两相材料纳米柱体的拉伸变形问题和基底材料为Al球形压头材料为金刚石的纳米压痕问题. 采用MD-FEA方法获得了上述两种问题的应力?应变场, 并将计算结果分别与传统MD方法中通过变形梯度计算的原子应变以及原子的位力应力进行了比较, 详细讨论了用MD-FEA方法计算的应力?应变场与传统MD原子应变和位力应力的区别, 并对MD-FEA方法的有效性及其相较于传统MD方法所具有的优势进行了探讨. 结论显示, MD-FEA方法与传统MD方法在应力?应变变化平缓的区域得到的结果接近, 但在变化剧烈的区域以及材料的表/界面区域, MD-FEA方法能够得到更加精确的结果. 同时, MD-FEA方法避免了传统MD方法中, 需要人为选取截断半径以及加权函数所导致的误差. 另外, 当应变较大时, MD-FEA方法计算的小应变与传统MD方法计算的格林应变存在一定差异, 因此, MD-FEA方法更适合应变较小的情形.      

Evolution of nanoscale defects to planar cracks in a brittle solid

Fracture of a solid is a highly multiscale process that associates atomic scale bond breaking with macroscopic crack propagation, and the process can be dramatically influenced by the presence of defects in materials. In a nanomaterial, defect formation energy decreases with the reduction of material size, and therefore, the role of defects in crack formation and subsequent crack growth in such materials may not be understood from the classical laws of fracture mechanism. In this study, we investigated the crack formation process of a defective (with missing atoms) nanostructured material (NaCl) using a series of molecular dynamics (MD) simulations. It was demonstrated that simple defects in the form of several missing atoms in the material could develop into a planar crack. Subsequently, MD simulations on failures of nanosized NaCl with pre-defined planar atomistic cracks of two different lengths under prescribed tensile displacement loads were performed. These failure loads were then applied on the equivalent continuum models, separately, to evaluate the associated fracture toughness values using the finite element analysis. For small cracks, the fracture toughness thus obtained is cracksize dependent and the corresponding critical energy release rate is significantly smaller than Griffith’s theoretical value. Explanation for this discrepancy between LEFM and the atomistic model was attempted.