Higher-order Continuum Theory Research Articles

Determination of flexoelectric coefficients of higher-order continuum theories from CsCl lattice model

Flexoelectricity is an electromechanical coupling phenomenon between inhomogeneous deformation and electric polarization. The classical theory, which relies on the strain, is insufficient for representing flexoelectricity. Instead, a higher-order continuum theory that employs a measure capable of expressing local inhomogeneous deformation is necessary. In the present work, we concatenate higher-order continuum theories with a CsCl lattice structure consisting of charged particles and determine the resulting constitutive coefficients for higher-order continuum theories. First, we derived continuum internal energy density functions for dielectrics from the conservation laws. We consider two different higher-order continuum theories: coupled stress and strain gradient theories. Second, we present a procedure for determining the coefficients of higher-order continuum theories by using a discrete lattice model under the assumption that the internal energies of the lattice and continuum models are the same for identical deformations. To validate the coefficients, three sample problems are examined by presenting their analytical solutions. The results of sample problems demonstrate that the two higher-order theories can provide quantitative information about the size and flexoelectric effects. In particular, the strain gradient theory predicts the deformation of the lattice structure more accurately.

Strong shock as a stringent test for Onsager-Burnett equations.

A new set of thermodynamically consistent Onsager-Burnett equations [Singh, Jadhav, and Agrawal, Phys. Rev. E 96, 013106 (2017)2470-004510.1103/PhysRevE.96.013106] has recently been derived. In this work, we subject these equations to a severe test case of strong shock (Mach number = 134) for a dilute gas system composed of hard-sphere molecules. The numerical results of OBurnett equations for conserved and nonconserved variables are compared against the molecular dynamics and direct simulation Monte Carlo results available in the literature. With no tweaking of the equations in any way, we establish several fundamental aspects of OBurnett equations which other higher-order continuum theories like Burnett and Grad equations lack. In particular, evidence is put forward for smooth shock structures, the existence of heteroclinic trajectory, and positive entropy generation inside the shock at all Mach numbers. With respect to shock profiles of hydrodynamic variables, it is observed that OBurnett equations significantly improve upon the results of Navier-Stokes equations. Further comparison with regularized 13 (R13) equations at lower Mach numbers shows that OBurnett equations capture the more rarefied upstream part better than the R13 equations. These evidences suggest that the OBurnett equations are accurate and form a reliable set of higher-order transport equations. Further, it should now be possible to describe the complex structure of the shock wave correctly, even at a very large Mach number.

Evaluation of Grad's Second Problem Using Different Higher Order Continuum Theories

Abstract In our earlier work (Jadhav, and Agrawal, 2020, “Grad's second problem and its solution within the framework of Burnett hydrodynamics,” ASME J. Heat Transfer, 142(10), p. 102105), we proposed Grad's second problem (examination of steady-state solution for a gas at rest upon application of a one-dimensional heat flux) as a potential benchmark problem for testing the accuracy of different higher order continuum theories and solved the problem within the framework of Burnett hydrodynamics. In this work, we solve this problem within the moment framework and also examine two variants, Bhatnagar–Gross–Krook (BGK)–Burnett and regularized 13 moment equations, for this problem. It is observed that only the conventional form of Burnett equations which are derived retaining the full nonlinear collision integral are able to capture nonuniform pressure profile observed in case of hard-sphere molecules. On the other hand, BGK–Burnett equations derived using BGK-kinetic model predict uniform pressure profile in both the cases. It seems that the variants based on BGK-kinetic model do not distinguish between hard-sphere and Maxwell molecules at least for the problem considered. With respect to moment equations, Grad 13 and regularized 13 moment equations predict consistent results for Maxwell molecules. However, for hard-sphere molecules, since the exact closed form of moment equations is not known, it is difficult to comment upon the results of moment equations for hard-sphere molecules. The present results for this relatively simple problem provide valuable insights into the nature of the equations and important remarks are made in this context.

Higher-order nonlocal theory of Updated Lagrangian Particle Hydrodynamics (ULPH) and simulations of multiphase flows

In this work, we present a higher-order nonlocal continuum theory of a recently developed updated Lagrangian particle hydrodynamics (ULPH) (see: Tu and Li, 2017 and Yan et al., 2019) and its applications to multiphase flows. The original nonlocal differential operators used in ULPH have only the first-order accuracy, and they may not be able to simulate some complicated and delicate three-dimensional flow problems. In order to improve the computational accuracy and stability, we adopt the nonlocal synchronized differential operator from the reproducing kernel particle method (RKPM), which is regarded as the higher-order nonlocal differential operator, and we apply them to build higher-order ULPH formulations. The main advantage of the higher-order nonlocal differential operator over the original one is its high accuracy in unstructured particle distribution, and it is fully controlled by the approximation polynomial basis.Numerical verifications have been carried out to validate the accuracy of the proposed approach by using various polynomial bases. Several challenging and delicate three-dimensional multiphase flow benchmark problems are solved to demonstrate the capability of the proposed method. The numerical results of the higher-order ULPH show good agreement with theoretical and numerical solutions in the literature, demonstrating the promising potential of higher-order nonlocal ULPH formulation in modeling and simulating multiphase flows with substantial topological variations of interfaces and high density ratios.

Existence of second spectrums of Timoshenko beam and Mindlin–Herrmann rod theories on the basis of atomistic studies

This paper considers a single-layer 2D triangular lattice system as an ideal higher-order rod and/or beam. The objective is to compare the solutions of these continuum models with the standard atomistic solutions. The accuracy of these higher-order rod and beam theories, in modeling several nano-systems, depends on the frequency of vibration. The existence of a second spectrum associated with these higher-order continuum theories and its contribution to the continuum solutions are studied on the basis of molecular dynamics (MD) results. It is shown that the second spectrum of a Timoshenko beam is less accurate with respect to the MD results but cannot be disregarded totally. Also, the existence of a second propagating mode associated with a Mindlin–Herrmann rod is substantiated by MD simulation results. The accuracy of these higher order rod and beam theories deteriorates beyond their cut-on frequencies. However, the use of higher-order rod and beam theories in several high-frequency dynamic and multiscale analyses is still justifiable.

Two novel Kirchhoff plate finite elements for the modified strain gradient theory

AbstractMicroplates are vastly used in MEMS and NEMS devices. Due to size effect, classical theories of continuum mechanics can not be used to model mechanical behaviour of them. Instead, higher order continuum theories should be implemented. Modified Strain Gradient Theory (MSGT) is considered to be one of the most accurate methods for this, considering both theoretical compatibility and numerical accuracy. In this study, two sets of Kirchhoff and twist‐Kirchhoff bending rectangular plate finite elements for MSGT are developed as novel aspects. This is done by first obtaining the weak forms by using variational methods and then deriving the set of algebraic equations for the finite element formulation starting with shape functions. Although non‐conforming, the finite elements are in alignment with the existing models with the given length scale parameters. The plate finite elements are then implemented to existing models. The length scale parameters for MSGT are also identified for gold using the existing experimental results in literature. This study proposes the use of higher order theories for gold microplates for thicknesses less than 40 µm.

Hemitropic properties of thin non-centrosymmetric 3D woven textile composites

Due to their microstructural architecture, 3D woven textile composites, (3DWTC), exhibit complex macroscopic behavior, which cannot be effectively captured using classical continuum theories. One reason for this can be attributed to the ratio of the characteristic microscopic and macroscopic length scales, (d and L, respectively), which violates the separation of length scales requirement d/L<<1 in micromechanics. In these instances, higher order continuum theories, such as micropolar theory, are considered to effectively describe the macroscopic behavior of a material. In this study, the 3D macroscopic constitutive relation of 3D woven textile composite (3DWTC) is reported. The three different 3DWTC considered are referred to as thin asymmetric, thick asymmetric, and thick symmetric. The lack of centro-symmetry in the asymmetric microstructures produces an interesting anisotropic response, such as stress-curvature (or couple stress-strain) coupling effect. This is analogous to asymmetric plate structures and the coupling between the resultant forces and curvatures through the B-matrix. At the macroscopic (continuum) scale, the material is modeled as a micropolar continuum, where internal couple-stresses and curvature strains are introduced. To capture the stress-curvature coupling effect, a generalized Aero-Kuvshinski type constitutive model is considered. With this, the anisotropic micropolar properties and the additional hemitropic material constants of the 3DWTC are determined numerically and their results qualitatively summarized.

Finite element modeling of micromorphic continua in the context of three-dimensional elasticity

The micromorphic theory (MMT) is one of the most general higher-order continuum theories capable of describing the behavior of materials when the microstructure of body is important, in which the micro-deformation degrees of freedom (DOFs) of material particles are taken into account. In this article, a new size-dependent finite element approach is developed for the mechanical analysis of micromorphic continua based on the three-dimensional (3D) elasticity. To this end, the linear MMT is first formulated within the framework of 3D elasticity. Relations are matricized in order to use in the finite element method. Then, a 3D size-dependent element is developed with taking the effects of micro-deformation and micro-rotation DOFs of material particles into account. Micromorphic rectangular plates subject to different sets of boundary conditions are considered as the problem under study whose free vibration is investigated. The effects of thickness-to-length scale parameter and side length-to-thickness ratios on the resonant frequencies of plates are studied in the given results. Also, the results of MMT are compared to those of micropolar theory and classical elasticity.

Second Strain Gradient Finite Element Analysis of Vibratory Nanostructures Based on the Three-Dimensional Elasticity Theory

By introducing the $$C^{2}$$ continuous hexahedral element, the free vibration finite element analysis of the nanobeam and nanoplate structures is reported based on the second strain gradient (SSG) theory and three-dimensional elasticity model. The SSG elasticity presents the powerful higher-order continuum theory which can be efficiently used to capture the size-effect on the nano-objects. The finite element discretization procedure is performed within Hamilton’s principle where the quadratic matrix version of the strain and kinetic energies are derived on the basis of the three-dimensional SSG elasticity model. In the proposed $$C^{2}$$ continuous hexahedral element, the values of the displacement field and the associated higher-order derivatives are considered as the nodal values to satisfy the continuity conditions. As the case studies, the free vibration of the nanobeams and rectangular nanoplates is investigated. Different results are outlined to show the efficiency and convergence of the present model. The influences of the involved parameters on the natural frequencies of nanobeams and plates are also investigated. It is realized that with the increase in the thickness-to-lattice parameter ratio, the difference of the results related to the SSG theory and classical theory decreases.

Size-dependent two-node axisymmetric shell element for buckling analysis with couple stress theory

In this paper, two-node size-dependent axisymmetric shell element formulation is developed by using thin conical shell model in the place of the beam model, which is used in previous research and using the modified couple stress theory in the place of the classical continuum theory. Since classical continuum theory is unable to correctly compute stiffness and account for size effects in micro/nanostructures, higher order continuum theories such as modified couple stress theory have become quite popular. The mass stiffness matrix and geometric stiffness matrix for axisymmetric shell element are developed in this paper, and by means of size-dependent finite element formulation it is extended to more precisely account for nanotube buckling. The results have indicated using the two-node axisymmetric shell element, where the rigidity of the nano-shell is greater than that in the classical, and the critical axial buckling loads obtained from couple stress theory are greater than that of classical, which is due to the presence of one size parameter in couple stress theory. The findings also indicate that the developed size-dependent axisymmetric shell element is able to analyze the buckling of cylindrical and conical shells and also circular plate, which is reliable for simulating micro/nanostructures and can be used for the analysis of size effect and has desirable convergence characteristic. Besides, in addition to reducing the number of elements required, using axisymmetric shell element also increases convergence speed and accuracy.

A comparative study of modified strain gradient theory and modified couple stress theory for gold microbeams

Microbeams are common structures encountered in micro- and nano-electromechanical systems. Their mechanical response cannot be modelled by local theories of continuum mechanics due to size effect, which becomes more pronounced as the structural length scale approaches the microstructural length scale. The size effect can be circumvented by higher-order continuum theories. In this study, Euler–Bernoulli microbeams are analysed with modified strain gradient theory (MSGT) and modified couple stress theory (MCST). The weak forms for the numerical implementation are obtained by using variational methods. Then, the set of algebraic equations for the finite element method are derived. As a novel aspect, the performance of MSGT and MCST is compared and the length scale parameters of these theories are identified for gold microbeams from the existing experimental results in the literature. With the help of the identified parameters, the cut-off point for the applicability of the classical beam theories for gold microbeams is assessed. The study suggests use of higher-order theories for the state-of-the-art gold microbeam structures having thickness $$t<30\,\upmu {\hbox {m}}$$ .

Toupin–Mindlin first strain gradient theory revisited for cubic crystals of hexoctahedral class: Analytical expression of the material parameters in terms of the atomic force constants and evaluation via ab initio DFT

Capture of the discrete nature of crystalline solids for the purpose of the determination of their mechanical behavior with high precision is of interest. To achieve this objective, two fundamental contributing factors are on top of the list: (1) formulation in the mathematical framework of an appropriate higher order continuum theory rather than using classical treatment, and (2) incorporation of the true anisotropy of the media. The present work revisits Toupin–Mindlin first strain gradient theory for media with general anisotropy, and then specialize it to cubic crystals of hexoctahedral class. This formulation in addition to 3 classical material constants encountered in classical theory of elasticity, gives rise to 11 additional material parameters peculiar to first strain gradient theory. To date, there is no experimental method in the literature for the measurement of these parameters. A methodology incorporating lattice dynamics is proposed, by which all the material parameters including the classic ones are analytically expressed in terms of the atomic force constants. Subsequently, the analytical expressions for the nonzero components of the 4th and 6th order elastic moduli tensors as well as 6 characteristic lengths are derived. Finally, with the aid of ab initio calculations all the material properties in Toupin–Mindlin first strain gradient theory are numerically obtained with high precision. In this work the transformation matrices of cubic crystals of diploidal class which also falls under centrosymmetric point groups are discussed as well.

Analysis of Gold Microbeams with Higher Order Continuum Theories

AbstractMicrobeams are building blocks for many microstructures as well as microelectromechanical systems (MEMS) and cannot accurately be modelled by classical continuum theories due to size effects based on their micro‐scale. These size effects can be taken into account by the so‐called higher order continuum theories. Modified Strain Gradient Theory (MSGT) and Modified Couple Stress Theory (MCST) are two commonly used theories, which extend the classical local continuum theories of grade one with the introduction of additional length scale parameters. In this contribution, the variational problem governing the elasticity of higher order beam formulation and the finite element implementation based upon, are briefly introduced. To this end, well known Euler‐Bernoulli beam formulation assumptions are used. The size effect for gold‐micro beams is demonstrated and the length scale parameters of gold microbeams for MSGT and MCST are identified form the existing experimental data from literature for the first time. As a novel aspect, significant size effect is demonstrated for the length‐scales associated with the state of the art gold microbeam structures developed for MEMS applications, which reveals the necessity of the use of higher order theories at these length scales. Advantages and drawbacks of these theories are also identified. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

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.

Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix

In the present paper, buckling analysis of silicon carbide nanowires has been investigated including size effect. The size effect has been taken into consideration by using different size-dependent continuum theories. These theories are modified couple stress theory, modified strain gradient theory, nonlocal elasticity theory, surface elasticity theory, and nonlocal surface elasticity theory. Analyses have been made for a continuum model which is embedded in double-parameter elastic foundation. The foundation has been modeled by using both Winkler- and Pasternak-type elastic foundation models. Simply supported boundary conditions have been used. Buckling equations have been obtained by using energy principle and solved via Navier’s solution procedure. Results are given and compared in figures and tables.

Dynamic pull-in stability of torsional nano/micromirrors with size-dependency, squeeze film damping and van der Waals effect

Recently, electrostatic torsional nanomirrors are applied widely in various industries. In previous lectures, coefficient of squeeze film air damping in the gap between main plate and substrate has been constant, while this coefficient is the function of viscosity and air pressure and has nonlinear equation. In this paper, the effect of nonzero initial conditions also, and the effect of nonlinear coefficient of squeeze film air damping on the stability of torsional nanomirrors using higher order continuum theories are investigated. First considering two-degree-of-freedom coupled bending/torsion system, the governing equations of system are derived, then using the fourth order Runge-Kutta method and nonzero initial conditions, the equations of motion are solved. Dynamic pull-in instability voltage is obtained and by plotting of response curves and phase portraits, the size effect is investigated. Study of phase portraits shows that size effect causes increasing in the range of stability of nanomirror and increasing of length scale parameter increases the mentioned surface area. As well as assuming constant squeeze film air damping causes the error in investigating of its behavior. Therefore in equation of motion, for getting more accurate response, this coefficient must be considered nonlinear.

Inverse scattering problem of reconstruction of an embedded micro-/nano-size scatterer within couple stress theory with micro inertia

As long as the size of the embedded scatterer in comparison to the internal length scale of its surrounding elastic matrix is large, then the linear sampling method (LSM) and singular sources method (SSM) can be used in conjunction with classical theory of elasticity to reconstruct the size of the scatterer with reasonable accuracy. On the other hand, for the micro-/nano-size scatterer this treatment ceases to hold due to the shortcomings of classical theory of elasticity. Moreover, in the realm of this theory, wave propagation through a homogeneous medium is nondispersive on the macro-scale even for high frequency waves. This outcome is incompatible with the practical observations. A remedy to circumvent these shortcomings is to resort to such a higher order continuum theory as couple stress theory with micro inertia. In particular, by accounting for the micro inertia term a new length scale referred to as ”dynamic characteristic length” enters the equation of motion. Association of the media with positive or negative dispersion is a clear manifestation of the new formulation. Therefore, with attention to the aforementioned, both the direct and inverse scattering of high frequency SH-waves by a micro-/nano-size inhomogeneity embedded in an unbounded elastic isotropic medium are addressed in the mathematical framework of couple stress elasticity with micro inertia. It will be shown that for positive (negative) dispersion materials the couple stress phase velocity is greater (less) than the corresponding classical phase velocity. For the purpose of inverse scattering reconstruction of the micro-/nano-size scatterer the LSM and SSM are developed in the context of the proposed theory for the first time. It will be shown that LSM and SSM reconstructions of the size of the scatterer are unique.

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.

Buckling analysis of cylindrical thin-shells using strain gradient elasticity theory

The stability problem of cylindrical shells is addressed using higher-order continuum theories in a generalized framework. The length-scale effect which becomes prominent at microscale can be included in the continuum theory using gradient-based nonlocal theories such as the strain gradient elasticity theories. In this work, expressions for critical buckling stress under uniaxial compression are derived using an energy approach. The results are compared with the classical continuum theory, which can be obtained by setting the length-scale parameters to zero. A special case is obtained by setting two length scale parameters to zero. Thus, it is shown that both the couple stress theory and classical continuum theory forms a special case of the strain gradient theory. The effect of various parameters such as the shell-radius, shell-length, and length-scale parameters on the buckling stress are investigated. The dimensions and constants corresponding to that of a carbon nanotube, where the length-scale effect becomes prominent, is considered for this investigation.

Elastostatic Fields of an Embedded Circular Rigid Nano/Micro-Fiber with Interfacial Damage in Anti-Plane Couple Stress Elasticity

It is well-known that classical continuum theory has certain deficiencies in predicting the nanoscopic behavior of materials in the vicinity of defects. Couple stress theory is one of the higher order continuum theories which can overcome such difficulties by introducing new characteristic length. An accurate analytical determination of the elastic fields of an embedded circular nano/micro-rigid fiber with damaged interface under remote anti-plane loading in the context of couple stress elasticity is of particular interest.