Sharif Digital Repository

The Cauchy-Born hypothesis (CB) provides a hierarchical approach in the molecular theory of crystal elasticity to relate the continuum and atomic deformations. This kinematic theory has been extensively used as the constitutive law of continuum regions in multi-scale models. In these models, the fine scale is proposed to describe the real behavior of crystalline structure wherever the continuum description fails. The main objective of this article is to investigate the stability and size-dependency of CB hypothesis in three-dimensional applications by direct comparison of information between atomistic and continuous description of a medium. The Sutton-Chen many-body potential is used for the... 

We have used a hierarchical multiscale modeling scheme for the analysis of cortical bone considering it as a nanocomposite. This scheme consists of definition of two boundary value problems, one for macroscale, and another for microscale. The coupling between these scales is done by using the homogenization technique. At every material point in which the constitutive model is needed, a microscale boundary value problem is defined using a macroscopic kinematical quantity and solved. Using the described scheme, we have studied elastic properties of cortical bone considering its nanoscale microstructural constituents with various mineral volume fractions. Since the microstructure of bone... 

We have used a hierarchical multiscale modeling scheme for the analysis of carbon nanotube reinforced nanocomposites. This scheme consists of definition of two boundary value problems, one for macroscale (the scale in which the material exists homogeneously and we are interested in modeling the material behavior on that scale), and another for microscale (the scale in which the material becomes heterogeneous and microstructural constituents emerge). The coupling between these scales is done by using homogenization techniques. Using the presented scheme, we have studied carbon nanotube (CNT) reinforced composites behavior and the effects of an interphase layer between CNT and matrix material.... 

This article presents a multi-scale progressive micro-mechanical fatigue model. The model employs fundamental equation of the kinetic theory of fracture to calculate damage parameters of both fiber and matrix during cyclic loading. In order to adapt the equation, required material coefficients of the constituents can be achieved from fatigue test results of longitudinal and transverse unidirectional composites, only. Sharing stress capacities of fiber and matrix are determined using a modified progressive micro-mechanical bridging model in the presence of damage. The damage parameters in the constituents are calculated employing two different equivalent scalars. However, during sinusoidal... 

In nano-structures, the influence of surface effects on the properties of material is highly important because the ratio of surface to volume at the nano-scale level is much higher than that of the macro-scale level. In this paper, a novel temperature-dependent multi-scale model is presented based on the modified boundary Cauchy-Born (MBCB) technique to model the surface, edge, and corner effects in nano-scale materials. The Lagrangian finite element formulation is incorporated into the heat transfer analysis to develop the thermo-mechanical finite element model. The temperature-related Cauchy-Born hypothesis is implemented by using the Helmholtz free energy to evaluate the temperature... 

In this paper, a new multi-scale technique is developed for concurrent coupling of atomistic-continuum domains in modeling nano-mechanical behavior of atomic structures. A Lagrange multiplier method is employed over an overlapping domain to coupling the continuum nodal velocities with atomic lattice velocities. The Hamiltonian method is applied to combine the continuum and molecular Hamiltonians with the same weight in the overlapping domain. The mass and stiffness matrices of the continuum domain are obtained using a linear bridging map of the atomic lattice displacement laid underneath the continuum grid to the element displacements. Numerical examples are performed by presenting the... 

The band gap offset is an effect of coordination numbers (CNs) of atom reduction at the edge of transversal cross-section of Silicon nanowires (SiNWs). In this paper, a hierarchical multi-scale technique is developed to model the edge effect on the band gap shift of SiNWs since the geometric effect is dominant in the energy gap due to the appearance of strain in the self-equilibrium state. The multi-scale model is performed based on the molecular dynamics approach and finite element method for the micro- (atomistic) and macro-scale levels, respectively. The Cauchy-Born (CB) hypothesis is used to relate the atomic positions to the continuum field through the deformation gradient. Finally, the... 

In this paper, a new multi-scale approach is presented based on the modified boundary Cauchy-Born (MBCB) technique to model the surface effects of nano-structures. The salient point of the MBCB model is the definition of radial quadrature used in the surface elements which is an indicator of material behavior. The characteristics of quadrature are derived by interpolating data from atoms laid in a circular support around the quadrature, in a least-square scene. The total-Lagrangian formulation is derived for the equivalent continua by employing the Cauchy-Born hypothesis for calculating the strain energy density function of the continua. The numerical results of the proposed method are... 

In this paper, a novel temperature-dependent multi-scale method is developed to investigate the role of temperature on surface effects in the analysis of nano-scale materials. In order to evaluate the temperature effect in the micro-scale (atomic) level, the temperature related Cauchy-Born hypothesis is implemented by employing the Helmholtz free energy, as the energy density of equivalent continua relating to the inter-atomic potential. The multi-scale technique is applied in atomistic level (nano-scale) to exhibit the temperature related characteristics. The first Piola-Kirchhoff stress and tangential stiffness tensor are computed, as the first and second derivatives of the free energy... 

In this paper, a multi-scale molecular dynamics-finite element coupling is presented to study the mechanical behavior of heterogeneous nano-crystalline structures. The stiffness and mass matrices of the continuum sub-domain are generated by applying a linear transformation on the matrices obtained via the atomic structure underlying the FE mesh. A Lagrange multiplier method is employed to the transition zone imposing velocity resemblance of the coupling regions. The constraint equations of motion are solved by the multi-time-step decomposition thus giving the opportunity to ascribe different time steps to each individual zone. The molecular dynamics is performed by employing the... 

The Cauchy-Born (CB) hypothesis has been widely used in multi-scale modeling of crystalline nano-structures. The violation of CB hypothesis in stress space and the transition to plasticity, which is equivalent to the violation of CB hypothesis in strain space, are generally confused and it becomes crucial to differentiate between the two distinct phenomena; the violation of the former usually occurs at high values of stress and at regions where the surface effects are manifest while the violation of the latter occurs at low stresses when the material loses its strength to tolerate the applied loading. In this paper, a novel technique is developed to investigate the validity of CB hypothesis... 

The Cauchy–Born (CB) hypothesis has been widely used in multi-scale modeling of crystalline nano-structures. The violation of CB hypothesis in stress space and the transition to plasticity, which is equivalent to the violation of CB hypothesis in strain space, are generally confused and it becomes crucial to differentiate between the two distinct phenomena; the violation of the former usually occurs at high values of stress and at regions where the surface effects are manifest while the violation of the latter occurs at low stresses when the material loses its strength to tolerate the applied loading. In this paper, a novel technique is developed to investigate the validity of CB hypothesis... 

A large amount of research in computational mechanics has biased toward atomistic simulations. This trend, on one hand, is due to the increased demand to perform computations in nanoscale and, on the other hand, is due to the rather simple applications of pairwise potentials in modeling the interactions between atoms of a given crystal. The Cauchy–Born (CB) hypothesis has been used effectively to model the behavior of crystals under different loading conditions, in which the comparison with molecular dynamics simulations presents desirable coincidence between the results. A number of research works have been devoted to the validity of CB hypothesis and its application in post-elastic limit.... 

In this paper, a multiscale finite element framework is developed based on the first-order homogenization method for fully coupled saturated porous media using an extension of the Hill-Mandel theory in the presence of microdynamic effects. The multiscale method is employed for the consolidation problem of a 2-dimensional saturated soil medium generated from the periodic arrangement of circular particles embedded in a square matrix, which is compared with the direct numerical simulation method. The effects of various issues, including the boundary conditions, size effects, particle arrangements, and the integral domain constraints for the microscale boundary value problem, are numerically... 

The band gap offset is an effect of coordination numbers (CNs) of atom reduction at the edge of transversal cross-section of Silicon nanowires (SiNWs). In this paper, a hierarchical multi-scale technique is developed to model the edge effect on the band gap shift of SiNWs since the geometric effect is dominant in the energy gap due to the appearance of strain in the self-equilibrium state. The multi-scale model is performed based on the molecular dynamics approach and finite element method for the micro- (atomistic) and macro-scale levels, respectively. The Cauchy–Born (CB) hypothesis is used to relate the atomic positions to the continuum field through the deformation gradient. Finally, the... 

In this paper, a novel temperature-dependent multi-scale method is developed to investigate the role of temperature on surface effects in the analysis of nano-scale materials. In order to evaluate the temperature effect in the micro-scale (atomic) level, the temperature related Cauchy–Born hypothesis is implemented by employing the Helmholtz free energy, as the energy density of equivalent continua relating to the inter-atomic potential. The multi-scale technique is applied in atomistic level (nano-scale) to exhibit the temperature related characteristics. The first Piola–Kirchhoff stress and tangential stiffness tensor are computed, as the first and second derivatives of the free energy... 

In this article, a multiscale modeling procedure is implemented to study the effect of interphase on the Young's modulus of CNT/polymer composites. For this purpose, a three-phase RVE is introduced which consists of three components, i.e., a carbon nanotube, an interphase layer, and an outer polymer matrix. The nanotube is modeled at the atomistic scale using molecular structural mechanics. Moreover, three-dimensional elements are employed to model the interphase layer and polymer matrix. The nanotube and polymer matrix are assumed to be bonded by van der Waals interactions based on the Lennard-Jones potential at the interface. Using this Molecular Structural Mechanics/Finite Element... 

In this article, a Molecular Structural Mechanics/Finite Element (MSM/FE) multiscale modeling of carbon nanotube/polymer composites with viscoelastic (VE) polymer matrix is introduced. The nanotube is modeled at the atomistic scale using structural molecular mechanics. The matrix deformation is analyzed by nonlinear finite element method considering VE behavior. The nanotube and matrix are assumed to be bonded by van der Waals interactions based on the Lennard-Jones potential at the interface. Using the MSM/FE multiscale model, we investigate the effect of carbon nanotube (CNT) on the improvement of mechanical stability of the nanocomposite. Also, the buckling behavior of these... 

CNT-reinforced polymer composites have attracted attention due to their exceptional high strength. The high strength can be affected by the presence of defects in the nanotubes used as reinforcements in the practical nanocomposites. In this paper, a Molecular Structural Mechanics / Finite Element (MSM/FE) multiscale modeling is used to study the effect of carbon nanotube geometrical defects on the stability of SWCNT-polymer composites. Here, two types of representative volume elements (RVEs) for these nanocomposites are considered with perfect and defected CNT. These RVEs have the same dimensions. The nanotube is modeled at the atomistic scale using molecular structural mechanics whereas the... 

Exploring efficient chemotherapy would benefit from a deeper understanding of the tumor microenvironment (TME) and its role in tumor progression. As in vivo experimental methods are unable to isolate or control individual factors of the TME, and in vitro models often cannot include all the contributing factors, some questions are best addressed with mathematical models of systems biology. In this study, we establish a multi-scale mathematical model of the TME to simulate three-dimensional tumor growth and angiogenesis and then implement the model for an array of chemotherapy approaches to elucidate the effect of TME conditions and drug scheduling on controlling tumor progression. The...