Studying thin film damping in a micro-beam resonator based on non-classical theories

In this paper, a mathematical model is presented for studying thin film damping of the surrounding fluid in an in-plane oscillating micro-beam resonator. The proposed model for this study is made up of a clamped-clamped micro-beam bound between two fixed layers. The micro-gap between the micro-beam and fixed layers is filled with air. As classical theories are not properly capable of pre-dicting the size dependence behaviors of the micro-beam, and also behavior of micro-scale fluid media, hence in the presented model, equation of motion governing longitudinal displacement of the micro-beam has been extracted based on non-local elasticity theory. Furthermore, the fluid field has been modeled based on micro-polar theory. These coupled equations have been simplified using Newton-Laplace and continuity equations. After transforming to non-dimensional form and linearizing, the equations have been discretized and solved simultaneously using a Galerkin-based reduced order model. Considering slip boundary conditions and applying a complex frequency approach, the equivalent damping ratio and quality factor of the micro-beam resonator have been obtained. The obtained values for the quality factor have been compared to those based on classical theories. We have shown that applying non-classical theories underestimate the values of the quality factor obtained based on classical theo-ries. The effects of geometrical parameters of the micro-beam and micro-scale fluid field on the quality factor of the res-onator have also been investigated.     

Non-local continuum mechanics and fractional calculus

The present work introduces fractional calculus into the continuum mechanics area describing non-local constitutive relations. Considering a one-dimensional body and assuming total stored energy depending not only upon the local strain but also upon a fractional derivative of the stain, an elastic model with non-local stress–strain behavior is introduced. Fractional calculus provides a natural framework for describing non-local constitutive relations and requires no assumptions for the interval of non-local influence. Furthermore, the proposed method works in finite intervals contrary to the existing theories requiring infinite domains.     

A novel Volume-Compensated Particle method for 2D elasticity and plasticity analysis

A novel Volume-Compensated Particle model (VCPM) is proposed for the modeling of deformation and fracture in solids. In this proposed method, two potentials are introduced to model the interactions between material particles, i.e., a local pair-wise potential and a non-local multi-body potential. The local pair-wise potential is utilized to account for the constitutive relationship within the connecting bonds between particles while the non-local multi-body potential is employed for considering the volumetric effects under general mechanical loadings. The potential coefficients are determined by matching the potential energy stored in a discrete unit cell to the strain energy at the classical continuum level. A volume conservation scheme is proposed to model the plastic deformation. The validity of the proposed model is tested against the classical elasticity and elasto-plasticity benchmarks before its application to fracture problems. Several conclusions are drawn based on the proposed study.     

An atomistic-continuum Cosserat rod model of carbon nanotubes

The focus of the present work is an atomistic-continuum model of single-walled carbon nanotubes (CNTs) based on an elastic rod theory which can exhibit geometric as well as material nonlinearity [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405-420]. In particular, the single-walled carbon nanotube (SWNT) is modeled as a one-dimensional elastic continuum with some finite thickness bounded by the lateral surface. Exploitation of certain symmetries in the underlying atomic structure leads to suitable representations of the continuum elastic strain energy density in terms of strain measures that capture extension, twist, bending, and shear deformations [Healey, T.J., 2002. Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7, 405-420]. Bridging between the atomic scale and the effective continuum is carried out by parameterization of the continuum elastic energy and determination of the parameters using unit cell atomistic simulations over a range of deformation magnitudes and types. Specifically, the proposed model takes into account (a) bending, (b) twist, (c) shear, (d) extension, (e) coupled extension and twist, and (f) coupled bending and shear deformations. The extracted parameters reveal benefits of accounting for important anisotropic and large-strain effects as improvements over employing traditional, linearly elastic, isotropic, small-strain, continuum models to simulate deformations of atomic systems such as SWNTs. It is envisioned that the proposed approach and the extracted model parameters can serve as a useful input to simulations of SWNT deformations using existing nonlinearly elastic continuum codes based, for example, on the finite element method (FEM).     

Non-local elastic effects in electroactive polymers

In this work we study the onset of inhomogeneous deformations in thin electroactive polymers (EAPs) under voltage control. In order to account for the regularizing effects due to both the constitutive nature of the film and to its mechanical interaction with the compliant electrodes, we introduce a non-local energy term depending on the second gradient of deformation. We prove that very small non-local effects are sufficient to find realistic inhomogeneous deformations at the onset of the bifurcation, which are characterized by periodic thickness undulations with finite wavelength. Finally we prove that strong regularizing effects can suppress the onset of inhomogeneous deformations.     

基于近场动力学的薄板弯曲大变形及断裂分析

薄板结构仅在较小的荷载下就能产生较大的位移、旋转,甚至引发结构产生裂纹并扩展,进而发生结构整体断裂,因此,建立薄板结构在大变形过程中的裂纹扩展及断裂仿真模型,具有重要的工程实际意义.文章建立了用于薄板结构几何大变形和断裂分析的近场动力学(PD)和连续介质力学(CCM)耦合模型.首先基于冯·卡门假设,采用更新的拉格朗日法得到薄板在几何大变形增量步下的虚应变能密度增量公式,并利用虚功原理和均质化假设求出几何大变形微梁键的本构模型参数;接着分别建立几何大变形薄板PD模型与CCM模型的虚应变能密度增量,并建立了薄板几何大变形PD-CCM耦合模型;最后模拟了薄板结构在横向变形作用下的渐进断裂过程,得到与实验结果高度一致的仿真结果,验证了所提出的几何非线性PD-CCM耦合模型的精度.结果表明:本文所提出的薄板PD-CCM耦合模型具有简单高效,无需考虑材料参数限制和边界效应的特点,可以很好地用于预测薄板结构在几何大变形过程中的局部损伤和结构断裂,有利于薄板结构的断裂安全评价和理论发展.     

A large strain plasticity model for anisotropic materials — composite material application

In this work a generalized anisotropic model in large strains based on the classical isotropic plasticity theory is presented. The anisotropic theory is based on the concept of mapped tensors from the anisotropic real space to the isotropic fictitious one. In classical orthotropy theories it is necessary to use a special constitutive law for each material. The proposed theory is a generalization of classical theories and allows the use of models and algorithms developed for isotropic materials. It is based on establishing a one-to-one relationship between the behavior of an anisotropic real material and that of an isotropic fictitious one. Therefore, the problem is solved in the isotropic fictious space and the results are transported to the real field. This theory is applied to simulate the behavior of each material in the composite. The whole behavior of the composite is modeled by incorporating the anisotropic model within a model based on a modified mixing theory.     

Elliptic inhomogeneities and inclusions in anti-plane couple stress elasticity with application to nano-composites

It is well-known that classical continuum theory has certain deficiencies in predicting material’s behavior at the micro- and nanoscales, where the size effect is not negligible. Higher order continuum theories introduce new material constants into the formulation, making the interpretation of the size effect possible. One famous version of these theories is the couple stress theory, invoked to study the anti-plane problems of the elliptic inhomogeneities and inclusions in the present work. The formulation in elliptic coordinates leads to an exact series solution involving Mathieu functions. Subsequently, the elastic fields of a single inhomogeneity in conjunction with the Mori–Tanaka theory is employed to estimate the overall anti-plane shear moduli of composites with uni-directional elliptic cylindrical fibers. The dependence of the anti-plane elastic moduli on several important physical parameters such as size, aspect ratio and rigidity of the fiber, the characteristic length of the constituents, and the orientation of the reinforcements is analyzed. Based on the available data in the literature, certain nano-composite models have been proposed and their overall behavior estimated using the present theory.     

A continuum model with microstructures for wave propagation in ultra-thin films

Ultrasonic waves are powerful and popular methods for measuring mechanical properties of solids even at nanoscales. The extraction of material constants from the measured wave data requires the use of a model that can accurately describe the wave motion in the solid. The objective of this paper is to develop a continuum theory with microstructures that can capture the effect of the microstructure or nanostructure in ultra-thin films when waves of short wavelengths are used. This continuum theory is developed from assumed displacement fields for microstructures. Local kinematic variables are introduced to express these local displacements and are subjected to internal continuity conditions. The accuracy of the present theory is verified by comparing the results with those of the lattice model for the thin film. Specifically, dispersion curves for surface wave propagation and wave propagation in a thin film supported by an elastic homogeneous substrate are studied. The inadequacy of the conventional continuum theory is discussed.     

Investigation of the size effects in Timoshenko beams based on the couple stress theory

In this paper, a size-dependent Timoshenko beam is developed on the basis of the couple stress theory. The couple stress theory is a non-classic continuum theory capable of capturing the small-scale size effects on the mechanical behavior of structures, while the classical continuum theory is unable to predict the mechanical behavior accurately when the characteristic size of structures is close to the material length scale parameter. The governing differential equations of motion are derived for the couple-stress Timoshenko beam using the principles of linear and angular momentum. Then, the general form of boundary conditions and generally valid closed-form analytical solutions are obtained for the axial deformation, bending deflection, and the rotation angle of cross sections in the static cases. As an example, the closed-form analytical results are obtained for the response of a cantilever beam subjected to a static loading with a concentrated force at its free end. The results indicate that modeling on the basis of the couple stress theory causes more stiffness than modeling by the classical beam theory. In addition, the results indicate that the differences between the results of the proposed model and those based on the classical Euler–Bernoulli and classical Timoshenko beam theories are significant when the beam thickness is comparable to its material length scale parameter.     

Direct calculations of material parameters for gradient plasticity

Length scale parameters introduced in gradient theories of plasticity are calculated in closed form with a continuum dislocation based theory. The similarity of the governing equations in both models for the evolution of plastic deformation of a constrained thin film makes it possible to identify parameters of the gradient plasticity theory with the dislocation based model. A one-to-one identification is not possible given that gradient plasticity does not account for individual dislocations. However, by comparing the mean plastic deformation across the film thickness we find that the length scale parameter, l, introduced in the gradient plasticity theory depends on the geometry as well as material constants.     

一类多孔固体的等效偶应力动力学梁模型

一维多孔固体结构可采用等效连续介质梁模型来研究其动力学行为. 当类梁结构的高度尺寸和多孔固体单胞结构尺寸相近时,等效模型的力学行为会产生尺寸效应现象. 等效经典模型由于不包含尺度参数而无法描述尺寸相关特点,而广义连续介质力学模型则可以准确地考虑尺寸效应的影响. 基于偶应力理论,对一类单胞含有圆形孔洞的周期性多孔固体类梁结构,给出了分析其横向自由振动的等效连续介质铁木辛柯梁模型. 通过对单胞分析,在应变能等价和几何平均的意义下,定义了等效偶应力介质的材料常数. 利用已有的材料常数,推导了等效铁木辛柯梁的动力学微分方程. 将实际多孔固体结构进行完全的动力学有限元离散计算,所获得的解作为精确解以检验等效梁模型所获得的频率和振型的精度. 振型的比较借助于模态置信准则矩阵方法. 大量算例表明,等效偶应力铁木辛柯梁模型在频率和振型两方面均具有较高的计算精度. 重点研究了单胞孔径的相对大小、类梁结构高度与单胞尺寸比以及类梁结构长高比对等效梁模型精度的影响. 在此基础上,偏保守地建议了多孔固体类梁结构自振分析方法.      

Surface effects in anti-plane deformations of a micropolar elastic solid: integral equation methods

The theory of linear micropolar elasticity is used in conjunction with a new representation of micropolar surface mechanics to develop a comprehensive model for the deformations of a linearly micropolar elastic solid subjected to anti-plane shear loading. The proposed model represents the surface effect as a thin micropolar film of separate elasticity, perfectly bonded to the bulk. This model captures not only the micro-mechanical behavior of the bulk which is known to be considerable in many real materials but also the contribution of the surface effect which has been experimentally well observed for bodies with significant size-dependency and large surface area to volume ratios. The contribution of the surface mechanics to the ensuing boundary-value problem gives rise to a highly nonstandard boundary condition not accommodated by classical studies in this area. Nevertheless, the corresponding interior and exterior mixed boundary-value problems are formulated and reduced to systems of singular integro-differential equations using a representation of solutions in the form of modified single-layer potentials. Analysis of these systems demonstrates that the classical Noether theorems reduce to Fredholms theorems leading to results on well-posedness of the corresponding mathematical model.     

基于近场动力学理论的准脆性材料的本构力函数的构建

近场动力学理论(PD)是基于非局部思想的连续介质力学新理论,用于研究材料破坏问题。根据准脆性材料破坏的线性和非线性的力学行为,在初始微观弹脆性材料(PMB)的本构力函数中引入了键的损伤模型,将键的断裂过程分成了线性的弹性变形阶段和非线性的损伤变形阶段,以此构建了准脆性材料的本构力函数的基本形式。以典型的准脆性材料为例构建了其本构力函数,通过在压缩载荷下对含预制不同角度单裂纹缺陷的类岩材料的裂纹扩展进行PD数值模拟仿真,裂纹起裂位置和扩展方向与试样试验结果在一定程度上保持了一致,证明了该基于近场动力学理论的典型准脆性材料的本构力函数可用于该类材料的破坏分析。     

On the rheology of dilative granular media: Bridging solid- and fluid-like behavior

A new rate-dependent plasticity model for dilative granular media is presented, aiming to bridge the seemingly disparate solid- and fluid-like behavioral regimes. Up to date, solid-like behavior is typically tackled with rate-independent plasticity models emanating from Mohr–Coulomb and Critical State plasticity theory. On the other hand, the fluid-like behavior of granular media is typically treated using constitutive theories amenable to viscous flow, e.g., Bingham fluid. In our proposed model, the material strength is composed of a dilation part and a rate-dependent residual strength. The dilatancy strength plays a key role during solid-like behavior but vanishes in the fluid-like regime. The residual strength, which in a classical plasticity model is considered constant and rate-independent, is postulated to evolve with strain rate. The main appeal of the model is its simplicity and its ability to reconcile the classic plasticity and rheology camps. The applicability and capability of the model are demonstrated by numerical simulation of granular flow problems, as well as a classical shear banding problem, where the performance of the continuum model is compared to discrete particle simulations and physical experiment. These results shed much-needed light onto the mechanics and physics of granular media at various shear rates.     

Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green–Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant–Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.     

Efficient meshless SPH method for the numerical modeling of thick shell structures undergoing large deformations

The objective of this paper is to present an extension of the Lagrangian Smoothed Particle Hydrodynamics (SPH) method to solve three-dimensional shell-like structures undergoing large deformations. The present method is an enhancement of the classical stabilized SPH commonly used for 3D continua, by introducing a Reissner–Mindlin shell formulation, allowing the modeling of moderately thin structure using only one layer of particles in the shell mid-surface. The proposed Shell-based SPH method is efficient and very fast compared to the classical continuum SPH method. The Total Lagrangian Formulation valid for large deformations is adopted using a strong formulation of the differential equilibrium equations based on the principle of collocation. The resulting non-linear dynamic problem is solved incrementally using the explicit time integration scheme, suited to highly dynamic applications. To validate the reliability and accuracy of the proposed Shell-based SPH method in solving shell-like structure problems, several numerical applications including geometrically non-linear behavior are performed and the results are compared with analytical solutions when available and also with numerical reference solutions available in the literature or obtained using the Finite Element method by means of ABAQUS^© commercial software.     

Atomistic viewpoint of the applicability of microcontinuum theories

Microcontinuum field theories, including Micromorphic theory, Microstructure theory, Micropolar theory, Cosserat theory, nonlocal theory and couple stress theory, are the extensions of the classical field theories for the applications in microscopic space and time scales. They have been expected to overlap atomic model at microscale and encompass classical continuum mechanics at macroscale. This work provides an atomic viewpoint to examine the physical foundations of those well-established microcontinuum theories, and to justify their applicability through lattice dynamics and molecular dynamics.