Higher-order Gradient Continuum Research Articles

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

AbstractA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

A Multiscale Model to Study the Mechanical Properties of the Graphene, Boron Nitride and Silicon Carbide Hexagonal Nanosheets

Background: It is very important to precisely comprehend nanosheet’s mechanical properties for their future application, and the continuum-based methods play a vital role in this research domain. But, most of continuum models doesn’t provide a systematical theory, and just display certain property of nanostructures. The Cauchy-Born rule provides an alternative multiscale method, the resulted model is not only less accurate, and but also doesn’t describe the bending effect. Methods: A nanosheet is viewed as a higher-order gradient continuum planar sheet, and the strain energy density is thus a function of both the first- and second-order deformation gradient. The higher- order Cauchy-Born rule is used to approximate the bond vectors in the representative cell, the multiscale model is established by minimizing the cell energy, and the structural and mechanical properties are thus obtained. Results: The obtained bond lengths are respectively 0.14507 nm, 0.14489 nm, 0.1816 nm for the graphene, boron nitride and silicon carbide hexagonal nanosheets. The elastic constants, including Young’s modulus, shear modulus, Poisson’s ratio and bending rigidity, are calculated by analyzing the physical meaning of the first- and second-order strain gradients. The developed model can also be used to study the nonlinear behavior of nanosheets under some simple loading situations, such as the uniform tension, torsion and bending. The stress-strain relationship of nanosheets is presented for the uniform tension/compression, and the three types of nannosheets exhibit better compressive resistance far greater than tensile resistance. Conclusion: A reasonable multiscale model is established for the nanosheets by using the higherorder Cauchy-Born rule that provides a good interlinking between the microscale and continuum descriptions. It is proved that all three types of nannosheets shows the isotropic mechanical property. The current model can be used to establish a global nonlinear numerical modeling method in which the bending rigidity is the basic elastic constants same as the elastic modulus and Poisson’s ratio.

Free vibration analysis of single-walled boron nitride nanotubes based on a computational mechanics framework

Free vibration behaviors of single-walled boron nitride nanotubes are investigated using a computational mechanics approach. Tersoff-Brenner potential is used to reflect atomic interaction between boron and nitrogen atoms. The higher-order Cauchy-Born rule is employed to establish the constitutive relationship for single-walled boron nitride nanotubes on the basis of higher-order gradient continuum theory. It bridges the gaps between the nanoscale lattice structures with a continuum body. A mesh-free modeling framework is constructed, using the moving Kriging interpolation which automatically satisfies the higher-order continuity, to implement numerical simulation in order to match the higher-order constitutive model. In comparison with conventional atomistic simulation methods, the established atomistic-continuum multi-scale approach possesses advantages in tackling atomic structures with high-accuracy and high-efficiency. Free vibration characteristics of single-walled boron nitride nanotubes with different boundary conditions, tube chiralities, lengths and radii are examined in case studies. In this research, it is pointed out that a critical radius exists for the evaluation of fundamental vibration frequencies of boron nitride nanotubes; opposite trends can be observed prior to and beyond the critical radius. Simulation results are presented and discussed.

A multiscale computational framework for the analysis of graphene involving geometrical and material nonlinearities

An atomistic–continuum approach, in which the constitutive model is derived from the lattice structure of graphene, is developed to simulate the mechanical behaviors of graphene. The chirality of graphene can be reflected by introducing a representative cell and calculated results reveal that the chirality of graphene has little effect on structural parameters and elastic properties. Since the constitutive model has incorporated the information in connection with atomistic structure, the material nonlinearity can be exactly reflected by iteratively updating the constitutive relationship in the present approach. Moreover, geometrical nonlinearity has also been considered under the higher-order gradient continuum theory. The bending deflections of rectangular and circular graphene, with both geometrical and material nonlinearities, having simply supported and clamped constraints are investigated. Based on the constitutive model, the definition of graphene thickness in building the stiffness matrix can be avoided by using the current atomistic–continuum approach. Computational results reveal that the atomistic–continuum approach can accurately capture geometrical and material nonlinearities of graphene and provide a good prediction of the full atomistic simulation even with a small number of nodes.

Small-scale Timoshenko beam element

Abstract Higher-order gradient continuum mechanics theories are of critical importance as they can afford to describe the size-dependent mechanical behavior of micro-scale structures. In this article, based on the most general form of strain gradient elasticity theory, a new extended Timoshenko beam element capable of accommodating size effects is introduced. To this end, the higher-order tensors of energy pairs in the energy functional are vectorized and represented in the quadratic form first. This gives way to realize the stiffness and mass matrices of the newly proposed element. Compared to the standard Timoshenko beam element, the new element requires two additional nodal degrees of freedom (d.o.f.) consisting of derivatives of lateral translation and rotation, which means a total of 4 d.o.f. per node. Therefore, the Hermite functions are employed to construct the shape functions of this new element. The proposed element is indicated to exhibit stiffer character, making it desirable when dealing with the problems at the microscale. Also, the standard Timoshenko beam element is recovered when the small scale factor tends to zero. Using this new element, the free vibration and bending of Timoshenko microbeams are investigated. The results are compared with those available in the literature and excellent agreement is achieved.

Analytical applications and effective properties of a second gradient isotropic elastic material model

Recently, the works by Toupin, Mindlin, Sokolowski and Germain have been developed following two research streams. In the first one, higher-order gradient continuum models were developed based on the Cauchy tetrahedron argument (see, e.g., dell’Isola and Seppecher in Comptes Rendus de l Academie de Sciences 17 Serie IIb: Mecanique, Physique, Chimie, Astronomie 321:303–308, 1995, Meccanica 32:33–52 1997, Zeitschrift fr Angewandte Mathematik und Physik 63(6):1119–1141, 2012). In the second one, the structure of higher-order gradient models is developed with a view to the applications. In particular in the model of linear isotropic solids proposed by Dell’Isola, Sciarra and Vidoli (DSV), the main constitutive equation is obtained for the case of second gradient models. This model introduces in addition to the two well-known Lame’s elastic constants five constitutive constants. The practical applications of this model remain in its infancy since the issue of determining the new moduli it introduces is not yet completely addressed. Also, analytical solutions of simple boundary value problems that can be helpful to grasp some of the physical foundations of this model are missing. This paper aims to address these two issues by providing the analytical solutions for two model problems, a spherical shell subjected to axisymmetric loading conditions and the circular bending of a beam in plane strain, both the beam and the shell obeying the DSV second gradient isotropic elastic model. The solution of the circular bending of a beam has served to grasp some of the physical soundness of the model. A framework based on homogenization under inhomogeneous boundary conditions is also suggested to determine the unknown constitutive constants, which are provided in the particular case of elastic porous heterogeneous materials.

A computational framework for transverse compression of microtubules based on a higher-order Cauchy–Born rule

This work presents a computational framework for the transverse compression of microtubules using the Cauchy–Born rule. Atomistic-continuum simulation and mesh-free method are employed in the computation for the theoretical scheme of the higher-order gradient continuum. Elastic properties and mechanical responses of microtubules under transverse compression are intensively studied. Without tracing each atom in large protein structures, a homogenization technique is proposed to evaluate interatomic energy stored among macromolecules. The concept of fictitious bonds is proposed for microtubules to bridge the gap between the atomistic simulation and the continuum approach, which is of great significance for application of the continuum approach to macromolecular structures. To reflect the inhomogeneous deformations of a cylindrical structure, the higher-order Cauchy–Born rule is employed to calculate the fictitious bond vectors emanating from a given evaluation point during the deformation process. By selecting a representative unit, a higher-order gradient continuum constitutive relationship is established to take atomistic interactions into consideration. Elastic modulus and transverse mechanics, including critical hydrostatic pressure and transverse compression-induced structure transitions, are numerically simulated. Example problems are carefully selected and the obtained results are discussed in detail.

A mesh-free computational framework for predicting buckling behaviors of single-walled carbon nanocones under axial compression based on the moving Kriging interpolation

This paper employs a higher-order gradient continuum theory for studying mechanical properties of single-walled carbon nanocones (SWCNCs). The SWCNC is constructed by rolling up a fan-shaped graphite sheet and connecting their two sides to form a conical structure. A mesh-free computational framework based on the moving Kriging (MK) interpolation is developed to study the buckling behaviors of SWCNCs under axial compression. Mechanical behaviors of SWCNCs with five different apex angles, i.e. 19.2°, 38.9°, 60°, 84.6° and 112.9°, are studied. Critical strains are predicted with the SWCNC subjects to uniform axial compression on the two ends. Computational results demonstrate that the apex angle has an increasing effect on the critical strain but a decreasing effect on the elastic properties (such as axial Young’s modulus) of SWCNCs. The corresponding buckling patterns reveal that a larger apex angle developed more fins on the side surface of the CNC at critical strain. For some of the CNCs, it is found that the elastic property slightly increases as the cutting tip’s length increases. Besides, a sharp-decrease of the critical strain with an increase number of fins, which approach to ripples, indicates that the CNC becomes unstable.

Predicting mechanical properties of single-walled carbon nanocones using a higher-order gradient continuum computational framework

The higher-order gradient continuum theory is employed to study the structural parameters and elastic properties of single-walled carbon nanocones (SWCNCs), where the higher-order Cauchy–Born rule is used to link the deformation of the carbon atomic structure to that of the continuum level. Unlike single-walled carbon nanotubes (SWCNTs), mechanical properties of SWCNCs vary along its side edge’s direction, owing to the monotonically increasing radius. In the constitutive model, a representative cell in an initial graphite sheet is selected to study the mechanical property of this domain of SWCNCs. By minimizing the potential energy of the representative cell in the undeformed SWCNC, structural parameters and elastic properties of the domain are obtained. The varying chiralities seem to have a large impact on mechanical properties of SWCNCs. Five kinds of SWCNCs are chosen in our study to test the influence of the apex angle on mechanical properties. The computational results demonstrate that mechanical properties of SWCNCs trend to the constant of graphite sheet when the radius is extremely large but this trend becomes mild as the apex angle increases.

Analysis of single-walled carbon nanotubes using the moving Kriging interpolation

The higher-order Cauchy–Born rule is applied to predict the mechanical response of single-walled carbon nanotubes (SWCNTs). As second-order deformation gradients can describe the bending effect of C–C bond vectors involved in the theoretical scheme of the higher-order gradient continuum, the established constitutive model accords extremely well with physical behavior. From the constitutive relationship constructed, a novel computational method is proposed for numerical simulation of buckling behaviors of SWCNTs. In this study, a new mesh-free method developed from the moving Kriging (MK) interpolation is employed to implement numerical simulation of mechanical properties of SWCNTs under axis-symmetrical loadings. As the mesh-free shape function constructed using the MK interpolation has the delta function property, the shape functions satisfy the essential boundary conditions automatically. Therefore, the essential boundary conditions can be easily implemented. Several numerical examples of buckling behaviors of SWCNTs are presented to test the effectiveness and efficiency of this method. The numerical results are also compared with those obtained from the full atomistic simulation method, and are found to be in agreement. Moreover, this computational method can largely reduce the degrees of freedom of the system, and thus save a large amount of computational resources. As a result, this is a very attractive approach, which has great potential in the engineering field. This mesh-free method is further applied to the study of post-buckling of SWCNTs. The results are compared with those obtained from full atomistic simulation, and demonstrate that this method is truly effective and efficient.

Dynamic behaviors of long and curved microtubules based on an atomistic-continuum model

This paper analyses dynamic response and vibration characteristics of long microtubules. Mechanical aspects of material properties are described based on an atomistic-continuum model and the use of a higher-order Cauchy–Born rule that bridges the scale between microstructures and continuum description. Long microtubules are simulated with one-dimensional strips, which include microscale interaction among protein molecules and can be dealt with using continuum mechanical approaches under higher-order gradient continuum scheme. Based on Hamilton’s principle, the differential equation of motion is established and is incorporated in the higher-order gradient continuum mesh-free framework. The performances of structures of microtubules are determined by direct numerical integration in time domain and applying the Fourier transform technique in frequency domain. Time-histories and frequency spectrums are obtained to determine vibration characteristics. Curved shapes of microtubules are involved in the study. Different cases are considered and compared, and the results are presented and discussed.

Free vibration analysis of microtubules based on an atomistic-continuum model

Analysis of free vibration of microtubules, based on an atomistic-continuum model, is presented. The theory bridges the polyatomic structure of microtubules with a macroscopic continuum approach under mesh-free computing scheme. A higher-order gradient continuum constitutive relationship is developed and incorporated into a higher-order Cauchy–Born rule. Instead of considering the atomic interaction between every atom pair in such polyatomic bio-composite textures, interatomic energy between subcomponents in microtubules is evaluated by a homogenization technique, represented by the fictitious bond. Material properties and stiffness matrix are determined, depending on deformation of fictitious bonds. Since the mesh-free approximation automatically satisfies the higher-order continuity and possesses intrinsic nonlocal property, the established quasicontinuum theory takes into consideration both atomic interaction and the size effect for modeling this microscale structure. Natural vibration frequencies of microtubules of different lengths and boundary restrictions are predicted and compared with existing solutions.

Investigation of temperature effect on the mechanical properties of single-walled carbon nanotubes

A temperature-related higher-order gradient continuum theory is proposed for predicting the mechanical properties of single-walled carbon nanotubes (SWCNTs) at various temperatures. It is found that the axial elastic moduli of zigzag (21, 0), armchair (12, 12) and chiral (15, 9) SWCNTs with similar radii approach 0.7 TPa when T = 0 K, but decline slightly on different slopes. These results indicate that the temperature effect influences the axial Young moduli of zigzag SWCNTs less than those of the other types. Moreover, the parameters λ 1 and λ 2 corresponding to the uniform longitudinal and circumferential stretches at different temperatures are also examined, and the results show that with an increasing temperature, all SWCNTs are stretched in the longitudinal direction, while in the circumferential direction, only the zigzag SWCNTs are stretched, whereas the others are compressed.

Predicting buckling behavior of microtubules based on an atomistic-continuum model

An atomistic-continuum model is proposed for microtubules. A higher-order gradient continuum constitutive relationship is established, and elasticity and global buckling of microtubules are studied intensively. As a typical macromolecular bio-system, atomic components and structures are much more complicated. Traditional atomistic simulation methods and classical continuum approaches have their own fundamental drawbacks in dealing with this large atomic system. Adopting a homogenization technique, this paper proposes a concept of fictitious bonds for microtubules to link the large atomic structure with continuum description. After selecting a representative unit, the fictitious bond energy equals to the energy stored in the continuum model. The higher-order Cauchy–Born rule is used to approximate the deformation of fictitious bonds under arbitrary loading conditions. A mesh-free numerical scheme is specifically developed for modeling computation. The elastic modulus and critical compressive force are predicted. Representative case studies are presented, and some results are obtained and discussed.

Mesh-Free Numerical Simulation for Carbon Nanotubes upon Bending

This paper presents a mesh-free numerical modeling approach for carbon nanotubes (CNTs) subjected to bending loads. The higher-order Cauchy-Born rule was employed to construct the higher-order continuum constitutive model. An initial equilibrium single-walled CNT (SWCNT) was viewed as been formed by rolling up a graphite sheet into a cylindrical shape. The deformation from an original SWCNT to the current configuration was approximated with the moving least-square (MLS) approximation, and the mesh-free computational framework was established in the theoretical scheme of higher-order gradient continuum. Mesh-free numerical simulations were carried out for SWCNTs, and the accuracy and convergence were discussed in comparison with the results of atomistic simulation. The buckling behavior was studied for various types of SWCNTs upon bending, and the buckling mechanism was investigated in virtue of the continuum variables, which showed that the maximum axial compressive strain played a vital role in the development of kinking.

Elastic properties and pressure-induced structural transitions of single-walled carbon nanotubes

The mechanical properties and pressure-induced structural transitions of single-walled carbon nanotubes (SWCNTs) are studied in this paper in the theoretical scheme of the higher-order gradient continuum. Both the first- and second-order deformation gradients are involved in the approximation of the deformations of microscale lattice vectors, and the continuum elastic potential is obtained by equating the deformation energy of a representative cell with that of an equivalent volume of the continuum, and thus, the established continuum constitutive model is more reasonable. By setting an appropriate equation as the deformation map from a graphite sheet to an initial undeformed SWCNT, the elastic constants of SWCNTs are obtained by minimizing the energy of a representative cell. The response of SWCNTs under axis-symmetrical loadings can also be obtained by changing the values of the set geometrical parameters. Based on the derived constitutive model, a computational method is developed to study the response of SWCNTs under hydrostatic pressure.