Finite element methods for the non‐linear mechanics of crystalline sheets and nanotubes

The formulation and finite element implementation of a finite deformation continuum theory for the mechanics of crystalline sheets is described. This theory generalizes standard crystal elasticity to curved monolayer lattices by means of the exponential Cauchy–Born rule. The constitutive model for a two‐dimensional continuum deforming in three dimensions (a surface) is written explicitly in terms of the underlying atomistic model. The resulting hyper‐elastic potential depends on the stretch and the curvature of the surface, as well as on internal elastic variables describing the rearrangements of the crystal within the unit cell. Coarse grained calculations of carbon nanotubes (CNTs) are performed by discretizing this continuum mechanics theory by finite elements. A smooth discrete representation of the surface is required, and subdivision finite elements, proposed for thin‐shell analysis, are used. A detailed set of numerical experiments, in which the continuum/finite element solutions are compared to the corresponding full atomistic calculations of CNTs, involving very large deformations and geometric instabilities, demonstrates the accuracy of the proposed approach. Simulations for large multi‐million systems illustrate the computational savings which can be achieved. Copyright © 2003 John Wiley & Sons, Ltd.     

Energy and momentum conserving elastodynamics of a non‐linear brick‐type mixed finite shell element

The paper presents aspects of the finite element formulation of momentum and energy conserving algorithms for the non‐linear dynamic analysis of shell‐like structures. The key contribution is a detailed analysis of the implementation of a Simó–Tarnow‐type conservation scheme in a recently developed new mixed finite shell element. This continuum‐based shell element provides a well‐defined interface to strain‐driven constitutive stress updates algorithms. It is based on the classic brick‐type trilinear displacement element and is equipped with specific gradient‐type enhanced strain modes and shell‐typical assumed strain modifications. The excellent performance of the proposed dynamic shell formulation with respect to conservation properties and numerical stability behaviour is demonstrated by means of three representative numerical examples of elastodynamics which exhibit complex free motions of flexible structures undergoing large strains and large rigid‐body motions. Copyright © 2001 John Wiley & Sons, Ltd.     

Buckling analysis of carbon nanotubes by a mixed atomistic and continuum model

A mixed atomistic and continuum model is applied to carbon nanotubes, in order to study their buckling behavior. Herein, the term “atomistic” refers to the underlying constitutive model that is formulated on the basis of interatomic potentials, whereas “continuum” means the application of the Cauchy–Born rule, which links the bond vectors before and after deformation via the deformation gradient of the continuum. Because the bond vectors are not infinitesimal and the continuum is modeled as surface, the Cauchy–Born rule has to be appropriately adapted to crystalline sheets. This is done via an exponential mapping in a new and surprisingly simple form such that in the analysis the current configuration has never to be left. The numerical buckling analysis of carbon nanotubes using the mixed atomistic and continuum model is carried out by means of the finite element method. For this purpose, the linearization of the equilibrium equations is provided.     

Multi‐scale modeling of surface effect via the boundary Cauchy–Born method

In this paper, a novel multi‐scale approach is developed for modeling of the surface effect in crystalline nano‐structures. The technique is based on the Cauchy–Born hypothesis in which the strain energy density of the equivalent continua is calculated by means of inter‐atomic potentials. The notion of introducing the surface effect in the finite element method is based on the intrinsic function of quadratures, called as an indicator of material behavior. The information of quadratures is derived by interpolating the data from probable representative atoms in their proximity. The technique is implemented by the definition of reference boundary CB elements, which enable to capture not only the surface but also the edge and corner effects. As the surface effect is important in small‐scale simulation, the relative number of boundary CB elements increases which leads to predomination of boundary effects in the model. In order to implement the equivalent continua in boundary value problems, the updated‐Lagrangian formulation of nonlinear finite element is derived. The numerical simulation of the proposed model together with the direct comparison with fully atomistic model indicates that the technique provides promising results for facile modeling of boundary effects and investigating its effect on the mechanical response of metallic nano‐scale devices. Copyright © 2010 John Wiley & Sons, Ltd.     

Multi‐scale modeling of surface effects in nano‐materials with temperature‐related Cauchy‐Born hypothesis via the modified boundary Cauchy‐Born model

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 effect in the atomistic level. The thermo‐mechanical multi‐scale model is applied to determine the temperature related characteristics at the nano‐scale level. The first and second derivatives of free energy density are computed using the first Piola‐Kirchhoff stress and tangential stiffness tensor at the macro‐scale level. The concept of MBCB is introduced to capture the surface, edge, and corner effects. The salient point of MBCB model is the definition of radial quadrature used at the surface, edge, and corner elements as an indicator of material behavior. The characteristics of quadrature are derived by interpolating the data from the atomic level laid in a circular support around the quadrature in a least‐square approach. Finally, numerical examples are modeled using the proposed computational algorithm, and the results are compared with the fully atomistic model to illustrate the performance of MBCB multi‐scale model in the thermo‐mechanical analysis of metallic nano‐scale devices. Copyright © 2013 John Wiley & Sons, Ltd.     

Continuum‐based design sensitivity analysis and optimization of nonlinear shell structures using meshfree method

A continuum‐based shape and configuration design sensitivity analysis (DSA) method for a finite deformation elastoplastic shell structure has been developed. Shell elastoplasticity is treated using the projection method that performs the return mapping on the subspace defined by the zero‐normal stress condition. An incrementally objective integration scheme is used in the context of finite deformation shell analysis, wherein the stress objectivity is preserved for finite rotation increments. The material derivative concept is used to develop a continuum‐based shape and configuration DSA method. Significant computational efficiency is obtained by solving the design sensitivity equation without iteration at each converged load step using the same consistent tangent stiffness matrix. Numerical implementation of the proposed shape and configuration DSA is carried out using the meshfree method. The accuracy and efficiency of the proposed method is illustrated using numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

Four‐node semi‐EAS element in six‐field nonlinear theory of shells

We propose a new four‐node C⁰ finite element for shell structures undergoing unlimited translations and rotations. The considerations concern the general six‐field theory of shells with asymmetric strain measures in geometrically nonlinear static problems. The shell kinematics is of the two‐dimensional Cosserat continuum type and is described by two independent fields: the vector field for translations and the proper orthogonal tensor field for rotations. All three rotational parameters are treated here as independent. Hence, as a consequence of the shell theory, the proposed element has naturally six engineering degrees of freedom at each node, with the so‐called drilling rotation. This property makes the element suitable for analysis of shell structures containing folds, branches or intersections. To avoid locking phenomena we use the enhanced assumed strain (EAS) concept. We derive and linearize the modified Hu–Washizu principle for six‐field theory of shells. What makes the present approach original is the combination of EAS method with asymmetric membrane strain measures. Based on literature, we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry. To gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element, we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches. Copyright © 2006 John Wiley & Sons, Ltd.     

Application of the higher‐order Cauchy–Born rule in mesh‐free continuum and multiscale simulation of carbon nanotubes

This paper investigates the application of a recently proposed higher‐order Cauchy–Born rule in the continuum simulation and multiscale analysis of carbon nanotubes (CNTs). A mesh‐free computational framework is developed to implement the numerical computation of the hyper‐elastic constitutive model that is derived from the higher‐order Cauchy–Born rule. The numerical computation reveals that the buckling pattern of a single‐walled carbon nanotube (SWCNT) can be accurately displayed by taking into consideration the second‐order deformation gradient, and fewer mesh‐free nodes can provide a good simulation of homogeneous deformation. The bridging domain method is employed to couple the developed mesh‐free method and the atomistic simulation. The coupling method is used to simulate the bending buckling of an SWCNT and the tensile failure of an SWCNT with a single‐atom vacancy defect, and good computational results are obtained. Copyright © 2008 John Wiley & Sons, Ltd.     

Consistent coupling of beam and shell models for thermo‐elastic analysis

In this paper, the finite element formulation of a transition element for consistent coupling between shell and beam finite element models of thin‐walled beam‐like structures in thermo‐elastic problems is presented. Thin‐walled beam‐like structures modelled only with beam elements cannot be used to study local stress concentrations or to provide local mechanical or thermal boundary conditions. For this purpose, the structure has to be modelled using shell elements. However, computations using shell elements are a lot more expensive as compared to beam elements. The finite element model can be more efficient when the shell elements are used only in regions where the local effects are to be studied or local boundary conditions have to be provided. The remaining part of the structure can be modelled with beam elements. To couple these two models (i.e. shell and beam models) at transitional cross‐sections, transition elements are derived here for thermo‐elastic problems. The formulation encloses large displacement and rotational behaviour, which is important in case of thin‐walled beam‐like structures. Copyright © 2004 John Wiley & Sons, Ltd.     

The finite element method for the mechanically based model of non‐local continuum

In this paper the finite element method (FEM) for the mechanically based non‐local elastic continuum model is proposed. In such a model each volume element of the domain is considered mutually interacting with the others, beside classical interactions involved by the Cauchy stress field, by means of central body forces that are monotonically decreasing with their inter‐distance and proportional to the product of the interacting volume elements. The constitutive relations of the long‐range interactions involve the product of the relative displacement of the centroids of volume elements by a proper, distance‐decaying function, which accounts for the decrement of the long‐range interactions as long as distance increases. In this study, the elastic problem involving long‐range central interactions for isotropic elastic continuum will be solved with the aid of the FEM. The accuracy of the solution obtained with the proposed FEM code is compared with other solutions obtained with Galerkins' approximation as well as with finite difference method. Moreover, a parametric study regarding the effect of the material length scale in the mechanically based model and in the Kr”oner–Eringen non‐local elasticity has been investigated for a plane elasticity problem. Copyright © 2011 John Wiley & Sons, Ltd.     

A temperature‐related homogenization technique and its implementation in the meshfree particle method for nanoscale simulations

A new homogenization technique, the temperature‐related Cauchy–Born (TCB) rule, is proposed in this paper with the consideration of the free energy instead of the potential energy. Therefore, temperature effects at the nanoscale can be investigated using continuum approximation with the implementation of the TCB rule. The TCB rule is verified via stress analyses of several crystalline solids. Temperature‐related material instability is also studied. In addition, a new hierarchical multiscale method is developed through implementing the TCB rule into meshfree particle methods. Quasicontinuum meshfree particle simulations are conducted to investigate bending of nanobeams, crack propagation in nanoplates and a three‐dimensional nanoindentation problem. Copyright © 2006 John Wiley & Sons, Ltd.     

Stress resultant geometrically non-linear shell theory with drilling rotations. Part III: Linearized kinematics

A consistent formulation of the geometrically linear shell theory with drilling rotations is obtained by the consistent linearization of the geometrically non-linear shell theory considered in Parts I and II of this work. It was also shown that the same formulation can be recovered by linearizing the governing variational principle for the three-dimensional geometrically non-linear continuum with independent rotation field. In the finite element implementation of the presented shell theory, relying on the modified method of incompatible modes, we were able to construct a four-node shell element which delivers a very high-level performance. In order to simplify finite element implementation, a shallow reference configuration is assumed over each shell finite element. This approach does not impair the element performance for the present four-node element. The results obtained herein match those obtained with the state-of-the-art implementations based on the classical shell theory, over the complete set of standard benchmark problems.     

An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials

We present a new approach based on coupling the extended finite element method (XFEM) and level sets to study surface and interface effects on the mechanical behavior of nanostructures. The coupled XFEM‐level set approach enables a continuum solution to nanomechanical boundary value problems in which discontinuities in both strain and displacement due to surfaces and interfaces are easily handled, while simultaneously accounting for critical nanoscale surface effects, including surface energy, stress, elasticity and interface decohesion. We validate the proposed approach by studying the surface‐stress‐driven relaxation of homogeneous and bi‐layer nanoplates as well as the contribution from the surface elasticity to the effective stiffness of nanobeams. For each case, we compare the numerical results with new analytical solutions that we have derived for these simple problems; for the problem involving the surface‐stress‐driven relaxation of a homogeneous nanoplate, we further validate the proposed approach by comparing the results with those obtained from both fully atomistic simulations and previous multiscale calculations based upon the surface Cauchy–Born model. These numerical results show that the proposed method can be used to gain critical insights into how surface effects impact the mechanical behavior and properties of homogeneous and composite nanobeams under generalized mechanical deformation. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability and size-dependency of temperature-related Cauchy–Born hypothesis

In continuum mechanics, the constitutive models are usually based on the Cauchy–Born (CB) hypothesis which seeks the intrinsic characteristics of the material via the atomistic information and it is valid in small deformation. The main purpose of this paper is to investigate the temperature effect on the stability and size-dependency of Cauchy–Born hypothesis. Three-dimensional temperature-related Cauchy–Born formulations are developed for crystalline structure and the stability and size-dependency of temperature-related Cauchy–Born hypothesis are investigated by means of direct comparison between atomistic and continuous mediums. In order to control the temperature effect, the Nose–Hoover thermostat is employed. Since the Helmholtz free energy is temperature dependent; the first Piola–Kirchhoff stresses are explicitly computed as the first derivative of the Helmholtz free energy density to the deformation gradient. It is numerically shown that the validity surfaces become smaller at higher temperature, which is significant in larger specimen. It is also presented that the material stability decreases with increasing the ambient temperature.     

Non‐linear exact geometry 12‐node solid‐shell element with three translational degrees of freedom per node

This paper presents the finite rotation exact geometry (EG) 12‐node solid‐shell element with 36 displacement degrees of freedom. The term ‘EG’ reflects the fact that coefficients of the first and second fundamental forms of the reference surface and Christoffel symbols are taken exactly at each element node. The finite element formulation developed is based on the 9‐parameter shell model by employing a new concept of sampling surfaces (S‐surfaces) inside the shell body. We introduce three S‐surfaces, namely, bottom, middle and top, and choose nine displacements of these surfaces as fundamental shell unknowns. Such choice allows one to represent the finite rotation higher order EG solid‐shell element formulation in a very compact form and to derive the strain–displacement relationships, which are objective, that is, invariant under arbitrarily large rigid‐body shell motions in convected curvilinear coordinates. The tangent stiffness matrix is evaluated by using 3D analytical integration and the explicit presentation of this matrix is given. The latter is unusual for the non‐linear EG shell element formulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Modeling arbitrary crack propagation in coupled shell/solid structures with X‐FEM

In this paper, the extended finite element method (X‐FEM) formulation for the modeling of arbitrary crack propagation in coupled shell/solid structures is developed based on the large deformation continuum‐based (CB) shell theory. The main features of the new method are as follows: (1) different kinematic equations are derived for different fibers in CB shell elements, including the fibers enriched by shifted jump function or crack tip functions and the fibers cut into two segments by the crack surface or connecting with solid elements. So the crack tip can locate inside the element, and the crack surface is not necessarily perpendicular to the middle surface. (2) The enhanced CB shell element is developed to realize the seamless transition of crack propagation between shell and solid structures. (3) A revised interaction integral is used to calculate the stress intensity factor (SIF) for shells, which avoids that the auxiliary fields for cracks in Mindlin–Reissner plates cannot satisfy exactly the equilibrium equations. Several numerical examples, including the calculation of SIF for the cracked plate under uniform bending and crack propagation between solid and shell structures are presented to demonstrate the performance of the developed method. Copyright © 2015 John Wiley & Sons, Ltd.     

An extended finite element method approach for structural‐acoustic problems involving immersed structures at arbitrary positions

Noise reduction for passengers' comfort in transport industry is now an important constraint to be taken into account during the design process. This process involves to study several configurations of the structures immersed in a given acoustic cavity in the context of an optimization, uncertainty, or reliability study for instance. The finite element method may be used to model this coupled fluid–structure problem but needs an interface conforming mesh for each studied configuration that may become time consuming. This work aims at avoiding this remeshing step by using noncompatible meshes between the fluid and the structures. The immersed structures are supposed to be thin shells and are localized in the fluid domain by a signed distance level‐set. To take into account the pressure discontinuity from one side of the structures to the other one, the fluid pressure approximation is enriched according to the structures positions by a Heaviside function using a partition of unity strategy (extended finite element method). The same fluid mesh of the empty cavity is then used during the whole parametric study. The method is implemented for a three‐dimensional fluid and tested on academic examples before being applied to an industrial‐like case. Copyright © 2012 John Wiley & Sons, Ltd.     

3D‐based hp‐adaptive first‐order shell finite element for modelling and analysis of complex structures—Part 1: The model and the approximation

The paper presents a 3D‐based adaptive first‐order shell finite element to be applied to hierarchical modelling and adaptive analysis of complex structures. The main feature of the element is that it is equipped with 3D degrees of freedom, while its mechanical model corresponds to classical first‐order shell theory. Other useful features of the element are its modelling and adaptive capabilities. The element is assigned to hierarchical modelling and hpq‐adaptive analysis of shell parts of complex structures consisting of solid, thick‐ and thin‐shell parts, as well as of transition zones, where h, p and q denote the mesh density parameter and the longitudinal and transverse orders of approximation, respectively. The proposed hp‐adaptive first‐order shell element can be joined with 3D‐based hpq‐adaptive hierarchical shell elements or 3D hpp‐adaptive solid elements by means of the family of 3D‐based hpq/hp‐ or hpp/hp‐adaptive transition elements. The main objective of the first part of our research, presented in this paper, is to provide non‐standard information on the original parts of the element algorithm. In order to do that, we present the definition of shape functions necessary for p‐adaptivity, as well as the procedure for imposing constraints corresponding to the lack of elongation of the straight lines perpendicular to the shell mid‐surface, which is the procedure necessary for q‐adaptivity. The 3D version of constrained approximation presented next is the basis for h‐adaptivity of the element. The second part of our research, devoted to methodology and results of the numerical research on application of the element to various plate and shell problems, are described in the second part of this paper. Copyright © 2006 John Wiley & Sons, Ltd.