The deformation work rate can be expressed by the time rate of pair functional potentials which describe the energy of materi- als in terms of atomic bonds and atom embedding interactions. According to Cauchy-Born rul...The deformation work rate can be expressed by the time rate of pair functional potentials which describe the energy of materi- als in terms of atomic bonds and atom embedding interactions. According to Cauchy-Born rule, the relations between the micro- scopic deformations of atomic bonds and electron gas and macroscopic deformation are established. Further, atomic bonds are grouped according to their directions, and atomic bonds in the same direction are simplified as a spring-bundle component. Atom embedding interactions in unit reference volume are simplified as a cubage component. Consequently, a material model com- posed of spring-bundle components and a cubage component is established. Since the essence of damage is the decrease and loss of atomic bonding forces, the damage effect can be reflected by the response functions of these two kinds of components. For- mulating the mechanical responses of two kinds of components, the corresponding elasto-damage constitutive equations are de- rived. Considering that slip is the main plastic deformation mechanism of polycrystalline metals, the slip systems of crystal are extended to polycrystalline, and the slip components are proposed to describe the plastic deformation. Based on the decomposition of deformation gradient and combining the plastic response with the elasto-damage one, the elasto-plastic damage constitutive equations are derived. As a result, a material model iormulated with spring-bundle components, a cubage component and slip components is established. Different from phenomenological constitutive theories, the mechanical property of materials depends on the property of components rather than that directly obtained on the representative volume element. The effect of finite deformation is taken into account in this model. Parameter calibration procedure and the basic characteristics of this model are discussed.展开更多
In this paper,a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant.The numerical scheme consists of a 3D immersed interface method(IIM)for solving Stokes equations with jum...In this paper,a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant.The numerical scheme consists of a 3D immersed interface method(IIM)for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface.The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface.This assumption is convenient in conjunction with the level-set techniques.It allows standard Lagrangian interpolation for quantities at the projection points on the interface.The interface jump relations are re-derived accordingly.A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation.Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy.A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.展开更多
We study continuum and atomistic models for the elastodynamics of crystalline solids at zero temperature. We establish sharp criterion for the regime of validity of the nonlinear elastic wave equations derived from th...We study continuum and atomistic models for the elastodynamics of crystalline solids at zero temperature. We establish sharp criterion for the regime of validity of the nonlinear elastic wave equations derived from the well-known Cauchy-Born rule.展开更多
In the present research, a simple quasi-continuum model, the Cauchy-Born rule model, is used to investigate the size effects of elastic modulus for fcc metals. By considering a nanoplate model and calculating the stra...In the present research, a simple quasi-continuum model, the Cauchy-Born rule model, is used to investigate the size effects of elastic modulus for fcc metals. By considering a nanoplate model and calculating the strain energy for the nano-sized plate under tension and bending, the relationship between the elastic modulus and the plate thickness is found. Size effects of the elastic modulus are displayed by the relative differences of the elastic modulus between the nano-sized plate sample and the bulk sample. By comparing the present results with those of others, the effectiveness of the Cauchy-Born rule model in studying the size effects of material properties are shown.展开更多
We propose a multigrid method to solve the molecular mechanics model(molecular dynamics at zero temperature).The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state ...We propose a multigrid method to solve the molecular mechanics model(molecular dynamics at zero temperature).The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state as the initial guess of the molecular mechanics model.The efficiency of the algorithm is demonstrated by three examples with homogeneous deformation,namely,one dimensional chain under tensile deformation and aluminum under tension and shear deformations.The method exhibits linear-scaling computational complexity,and is insensitive to parameters arising from iterative solvers.In addition,we study two examples with inhomogeneous deformation:vacancy and nanoindentation of aluminum.The results are still satisfactory while the linear-scaling property is lost for the latter example.展开更多
We introduce a new multigrid method to study the lattice statics model arising from nanoindentation.A constrained Cauchy-Born elasticity model is used as the coarse-grid operator.This method accelerates the relaxation...We introduce a new multigrid method to study the lattice statics model arising from nanoindentation.A constrained Cauchy-Born elasticity model is used as the coarse-grid operator.This method accelerates the relaxation process and considerably reduces the computational cost.In particular,it saves quite a bit when dislocations nucleate and move,as demonstrated by the simulation results.展开更多
The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity mode...The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices.展开更多
基金National Natural Science Foundation of China (10572140,10721202)
文摘The deformation work rate can be expressed by the time rate of pair functional potentials which describe the energy of materi- als in terms of atomic bonds and atom embedding interactions. According to Cauchy-Born rule, the relations between the micro- scopic deformations of atomic bonds and electron gas and macroscopic deformation are established. Further, atomic bonds are grouped according to their directions, and atomic bonds in the same direction are simplified as a spring-bundle component. Atom embedding interactions in unit reference volume are simplified as a cubage component. Consequently, a material model com- posed of spring-bundle components and a cubage component is established. Since the essence of damage is the decrease and loss of atomic bonding forces, the damage effect can be reflected by the response functions of these two kinds of components. For- mulating the mechanical responses of two kinds of components, the corresponding elasto-damage constitutive equations are de- rived. Considering that slip is the main plastic deformation mechanism of polycrystalline metals, the slip systems of crystal are extended to polycrystalline, and the slip components are proposed to describe the plastic deformation. Based on the decomposition of deformation gradient and combining the plastic response with the elasto-damage one, the elasto-plastic damage constitutive equations are derived. As a result, a material model iormulated with spring-bundle components, a cubage component and slip components is established. Different from phenomenological constitutive theories, the mechanical property of materials depends on the property of components rather than that directly obtained on the representative volume element. The effect of finite deformation is taken into account in this model. Parameter calibration procedure and the basic characteristics of this model are discussed.
基金supports by Hunan Provincial Education Department(10C1264),Xiangtan Univ.(10QDZ45),and Hunan NSFC(10JJ70)supported in part by NSFC key project 11031006supported in part by National Science Council of Taiwan under grant NSC98-2115-M-009-014-MY3 and NCTS.Z.Li was supported in part by the US ARO grant 550694-MA,the AFSOR grant FA9550-09-1-0520,the US NSF grant DMS-0911434,the NIH grant 096195-01,and CNSF11071123.
文摘In this paper,a numerical method is presented for simulating the 3D interfacial flows with insoluble surfactant.The numerical scheme consists of a 3D immersed interface method(IIM)for solving Stokes equations with jumps across the interface and a 3D level-set method for solving the surfactant convection-diffusion equation along a moving and deforming interface.The 3D IIM Poisson solver modifies the one in the literature by assuming that the jump conditions of the solution and the flux are implicitly given at the grid points in a small neighborhood of the interface.This assumption is convenient in conjunction with the level-set techniques.It allows standard Lagrangian interpolation for quantities at the projection points on the interface.The interface jump relations are re-derived accordingly.A novel rotational procedure is given to generate smooth local coordinate systems and make effective interpolation.Numerical examples demonstrate that the IIM Poisson solver and the Stokes solver achieve second-order accuracy.A 3D drop with insoluble surfactant under shear flow is investigated numerically by studying the influences of different physical parameters on the drop deformation.
基金an NSFgrant DMS 04-07866theproject"Research Team on Complex Systems"of the Chinese Academy of Sciences+1 种基金the National Basic Research Program(No.2005CB321704)the National Natural Science Foundation of China(No.10571172)
文摘We study continuum and atomistic models for the elastodynamics of crystalline solids at zero temperature. We establish sharp criterion for the regime of validity of the nonlinear elastic wave equations derived from the well-known Cauchy-Born rule.
基金Project supported by the National Natural Science Foundation of China(Nos.11021262,10932011 and 91216108)the National Basic Research Program of China(2012CB937500)
文摘In the present research, a simple quasi-continuum model, the Cauchy-Born rule model, is used to investigate the size effects of elastic modulus for fcc metals. By considering a nanoplate model and calculating the strain energy for the nano-sized plate under tension and bending, the relationship between the elastic modulus and the plate thickness is found. Size effects of the elastic modulus are displayed by the relative differences of the elastic modulus between the nano-sized plate sample and the bulk sample. By comparing the present results with those of others, the effectiveness of the Cauchy-Born rule model in studying the size effects of material properties are shown.
基金supported by National Natural Science Foundation of China under the grants 10871197 and 10932011by the funds for creative research group of China(grant No.11021101)the support of state center for mathematics and interdisciplinary sciences.
文摘We propose a multigrid method to solve the molecular mechanics model(molecular dynamics at zero temperature).The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state as the initial guess of the molecular mechanics model.The efficiency of the algorithm is demonstrated by three examples with homogeneous deformation,namely,one dimensional chain under tensile deformation and aluminum under tension and shear deformations.The method exhibits linear-scaling computational complexity,and is insensitive to parameters arising from iterative solvers.In addition,we study two examples with inhomogeneous deformation:vacancy and nanoindentation of aluminum.The results are still satisfactory while the linear-scaling property is lost for the latter example.
基金supported by National Science Foundation grant DMS-1217315supported by National Natural Science Foundation of China under the grant 10932011,91230203 and by the funds for creative research group of China(Grant No.11021101)+1 种基金by the support of CAS National Center for Mathematics and Interdisciplinary Sciencessupported by National Natural Science Foundation of China under the grants 11001210 and 11171305 and 91230203.
文摘We introduce a new multigrid method to study the lattice statics model arising from nanoindentation.A constrained Cauchy-Born elasticity model is used as the coarse-grid operator.This method accelerates the relaxation process and considerably reduces the computational cost.In particular,it saves quite a bit when dislocations nucleate and move,as demonstrated by the simulation results.
基金Project supported by the Natural Science Foundation(No. DMS 04-07866)the "Research Team on Complex Systems" of Chinese Academy of Sciences.
文摘The authors consider the simplest quantum mechanics model of solids, the tight binding model, and prove that in the continuum limit, the energy of tight binding model converges to that of the continuum elasticity model obtained using Cauchy-Born rule. The technique in this paper is based mainly on spectral perturbation theory for large matrices.
