Showing papers on "Continuum mechanics published in 2018"

Stress-driven nonlocal integral model for Timoshenko elastic nano-beams

Abstract: Size-dependent structural behavior of inflected Timoshenko elastic nano-beams is investigated by nonlocal continuum mechanics . An innovative stress-driven integral model of elasticity is conceived by swapping source and output fields of Eringen's strain-driven theory. Unlike Eringen's model, the stress-driven nonlocal integral formulation leads to well-posed structural problems. Solution uniqueness and continuous dependence on data are thus ensured. Selected case-studies of technical interest are examined and exact nonlocal solutions of Timoshenko nano-beams are provided, detecting thus also new benchmarks for numerical analyses. The contributed results are compared with those obtained by the gradient theory of elasticity and by the differential constitutive model consequent (not equivalent) to Eringen's strain-driven theory equipped with Helmholtz's kernel.

Wave propagation characteristics of a cylindrical laminated composite nanoshell in thermal environment based on the nonlocal strain gradient theory

Abstract: In this article, the wave propagation behavior of a size-dependent laminated composite cylindrical nanoshell in a thermal environment is presented. The small-scale effects are analyzed based on nonlocal strain gradient theory (NSGT). The governing equations of the cylindrical laminated composite nanoshell in a thermal environment were obtained using Hamilton’s principle and solved by the analytical method. The novelty of this study is considering the effects of the composite layers and NSGT in addition to considering the thermal environment of the cylindrical composite nanoshell. Finally, the investigation was performed on the influence of temperature difference, wave number, angular velocity and the different types of laminated composite on the phase velocity using the mentioned continuum mechanics theory. The results show that wave number, ply angle, shear correction factor and thermal environment play an important role on the phase velocity of the laminated composite nanostructure. Another significant result is that, in a specific temperature difference, there is an inverse relation between the number of layers in a laminate and the dynamic behavior of the nanostructure. The outcome of the present work can be used in a structural health monitoring and ultrasonic inspection techniques.

Incompressible fluid problems on embedded surfaces: Modeling and variational formulations

Abstract: Governing equations of motion for a viscous incompressible material surface are derived from the balance laws of continuum mechanics. The surface is treated as a time-dependent smooth orientable manifold of codimension one in an ambient Euclidian space. We use elementary tangential calculus to derive the governing equations in terms of exterior differential operators in Cartesian coordinates. The resulting equations can be seen as the Navier-Stokes equations posed on an evolving manifold. We consider a splitting of the surface Navier-Stokes system into coupled equations for the tangential and normal motions of the material surface. We then restrict ourselves to the case of a geometrically stationary manifold of codimension one embedded in $\Bbb{R}^n$. For this case, we present new well-posedness results for the simplified surface fluid model consisting of the surface Stokes equations. Finally, we propose and analyze several alternative variational formulations for this surface Stokes problem, including constrained and penalized formulations, which are convenient for Galerkin discretization methods.

Finite Element Method-Based Kinematics and Closed-Loop Control of Soft, Continuum Manipulators.

Abstract: This article presents a modeling methodology and experimental validation for soft manipulators to obtain forward kinematic model (FKM) and inverse kinematic model (IKM) under quasi-static conditions (in the literature, these manipulators are usually classified as continuum robots. However, their main characteristic of interest in this article is that they create motion by deformation, as opposed to the classical use of articulations). It offers a way to obtain the kinematic characteristics of this type of soft robots that is suitable for offline path planning and position control. The modeling methodology presented relies on continuum mechanics, which does not provide analytic solutions in the general case. Our approach proposes a real-time numerical integration strategy based on finite element method with a numerical optimization based on Lagrange multipliers to obtain FKM and IKM. To reduce the dimension of the problem, at each step, a projection of the model to the constraint space (gathering actuators, sensors, and end-effector) is performed to obtain the smallest number possible of mathematical equations to be solved. This methodology is applied to obtain the kinematics of two different manipulators with complex structural geometry. An experimental comparison is also performed in one of the robots, between two other geometric approaches and the approach that is showcased in this article. A closed-loop controller based on a state estimator is proposed. The controller is experimentally validated and its robustness is evaluated using Lypunov stability method.

Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations

Abstract: Continuum mechanics with dislocations, with the Cattaneo-type heat conduction, with mass transfer, and with electromagnetic fields is put into the Hamiltonian form and into the form of the Godunov-type system of the first-order, symmetric hyperbolic partial differential equations (SHTC equations). The compatibility with thermodynamics of the time reversible part of the governing equations is mathematically expressed in the former formulation as degeneracy of the Hamiltonian structure and in the latter formulation as the existence of a companion conservation law. In both formulations the time irreversible part represents gradient dynamics. The Godunov-type formulation brings the mathematical rigor (the local well posedness of the Cauchy initial value problem) and the possibility to discretize while keeping the physical content of the governing equations (the Godunov finite volume discretization).

Continuum Mechanics: Elasticity, Plasticity, Viscoelasticity

Abstract: FUNDAMENTALS OF CONTINUUM MECHANICS Material Models Classical Space-Time Material Bodies Strain Rate of Strain Curvilinear Coordinate Systems Conservation of Mass Balance of Momentum Balance of Energy Constitutive Equations Thermodynamic Dissipation Objectivity: Invariance for Rigid Motions Coleman-Mizel Model Fluid Mechanics Problems for Chapter 1 Bibliography NONLINEAR ELASTICITY Thermoelasticity Material Symmetries Isotropic Materials Incompressible Materials Conjugate Measures of Stress and Strain Some Symmetry Groups Rate Formulations for Elastic Materials Energy Principles Geometry of Small Deformations Linear Elasticity Special Constitutive Models for Isotropic Materials Mechanical Restrictions on the Constitutive Relations Problems for Chapter 2 Bibliography LINEAR ELASTICITY Basic Equations Plane Strain Plane Stress Properties of Solutions Potential Energy Special Matrix Notation The Finite Element Method of Solution General Equations for an Assembly of Elements Finite Element Analysis for Large Deformations Problems for Chapter 3 Bibliography PLASTICITY Classical Theory of Plasticity Work Principle von Mises-Type Yield Criterion Hill Yield Criterion for Orthotropic Materials Isotropic Hardening Kinematic Hardening Combined Hardening laws General Equations of Plasticity Strain Formulation of Plasticity Finite Element Analysis Large Deformations Thermodynamics of Elastic-Plastic Materials Problems for Chapter 4 Bibliography VISCOELASTICITY Linear Viscoelasticity Effect of Temperature Nonlinear Viscoelasticity Thermodynamics of Materials with Fading Memory Problems for Chapter 5 Bibliography FRACTURE AND FATIGUE Fracture Criterion Plane Crack through a Sheet Fracture Modes Calculation of the Stress Intensity Factor Crack Growth Problems for Chapter 6 Bibliography MATHEMATICAL TOOLS FOR CONTINUUM MECHANICS Sets of Real Numbers Matrices Vector Analysis Tensors Isotropic Functions Abstract Derivatives Some Basic Mathematical Definitions and Theorems Problems for Chapter 7 Bibliography INDEX

A nonlocal continuum model for the biaxial buckling analysis of composite nanoplates with shape memory alloy nanowires

Abstract: In this study, the buckling behavior of a three-layered composite nanoplate reinforced with shape memory alloy (SMA) nanowires is examined. Whereas the upper and lower layers are reinforced with typical nanowires, SMA nanoscale wires are used to strengthen the middle layer of the system. The composite nanoplate is assumed to be under the action of biaxial compressive loading. A scale-dependent mathematical model is presented with the consideration of size effects within the context of the Eringen's nonlocal continuum mechanics. Using the one-dimensional Brinson's theory and the Kirchhoff theory of plates, the governing partial differential equations of SMA nanowire-reinforced hybrid nanoplates are derived. Both lateral and longitudinal deflections are taken into consideration in the theoretical formulation and method of solution. In order to reduce the governing differential equations to their corresponding algebraic equations, a discretization approach based on the differential quadrature method is employed. The critical buckling loads of the hybrid nanosystem with various boundary conditions are obtained with the use of a standard eigenvalue solver. It is found that the stability response of SMA composite nanoplates is strongly sensitive to the small scale effect.

A continuum mechanics constitutive framework for transverse isotropic soft tissues

Abstract: In this work, a continuum constitutive framework for the mechanical modelling of soft tissues that incorporates strain rate and temperature dependencies as well as the transverse isotropy arising from fibres embedded into a soft matrix is developed. The constitutive formulation is based on a Helmholtz free energy function decoupled into the contribution of a viscous-hyperelastic matrix and the contribution of fibres introducing dispersion dependent transverse isotropy. The proposed framework considers finite deformation kinematics, is thermodynamically consistent and allows for the particularisation of the energy potentials and flow equations of each constitutive branch. In this regard, the approach developed herein provides the basis on which specific constitutive models can be potentially formulated for a wide variety of soft tissues. To illustrate this versatility, the constitutive framework is particularised here for animal and human white matter and skin, for which constitutive models are provided. In both cases, different energy functions are considered: Neo-Hookean, Gent and Ogden. Finally, the ability of the approach at capturing the experimental behaviour of the two soft tissues is confirmed.

From quantum to continuum mechanics: studying the fracture toughness of transition metal nitrides and oxynitrides

Abstract: We show here, based on VAlN, TiAlN and the related oxynitrides, that the (brittle) fracture and elastic properties may be consistently modelled from quantum- to continuum mechanics using micromechanical testing to link both scales. The measured elastic moduli match closely with those predicted by density functional theory calculations. Good agreement was also observed between the micro-cantilever bending experiments and cohesive-zone-finite element modelling. These scale-bridging data serve as a baseline for future improvements of the fracture toughness of these coating systems based on microstructure and coating architecture optimization.

Mechanics and statistics of the worm-like chain

Abstract: The worm-like chain model is a simple continuum model for the statistical mechanics of a flexible polymer subject to an external force. We offer a tutorial introduction to it using three approaches. First, we use a mesoscopic view, treating a long polymer (in two dimensions) as though it were made of many groups of correlated links or “clinks,” allowing us to calculate its average extension as a function of the external force via scaling arguments. We then provide a standard statistical mechanics approach, obtaining the average extension by two different means: the equipartition theorem and the partition function. Finally, we work in a probabilistic framework, taking advantage of the Gaussian properties of the chain in the large-force limit to improve upon the previous calculations of the average extension.

Dynamic analysis of nanoscale beams including surface stress effects

Abstract: In this article, an analytic non-classical model for the free vibrations of nanobeams accounting for surface stress effects is developed The classical continuum mechanics fails to capture the surface energy effects and hence is not directly applicable at nanoscale A general beam model based on Gurtin-Murdoch continuum surface elasticity theory is developed for the analysis of thin and thick beams Thus, surface energy has a significant effect on the response of nanoscale structures, and is associated with their size-dependent behavior To check the validity of the present analytic solution, the numerical results are compared with those obtained in the scientific literature The influences of beam thickness, surface density, surface residual stress and surface elastic constants on the natural frequencies of nanobeams are also investigated It is indicated that the effect of surface stress on the vibrational response of a nanobeam is dependent on its aspect ratio and thickness

Continuum modeling of twinning, amorphization, and fracture: theory and numerical simulations

Abstract: A continuum mechanical theory is used to model physical mechanisms of twinning, solid-solid phase transformations, and failure by cavitation and shear fracture. Such a sequence of mechanisms has been observed in atomic simulations and/or experiments on the ceramic boron carbide. In the present modeling approach, geometric quantities such as the metric tensor and connection coefficients can depend on one or more director vectors, also called internal state vectors. After development of the general nonlinear theory, a first problem class considers simple shear deformation of a single crystal of this material. For homogeneous fields or stress-free states, algebraic systems or ordinary differential equations are obtained that can be solved by numerical iteration. Results are in general agreement with atomic simulation, without introduction of fitted parameters. The second class of problems addresses the more complex mechanics of heterogeneous deformation and stress states involved in deformation and failure of polycrystals. Finite element calculations, in which individual grains in a three-dimensional polycrystal are fully resolved, invoke a partially linearized version of the theory. Results provide new insight into effects of crystal morphology, activity or inactivity of different inelasticity mechanisms, and imposed deformation histories on strength and failure of the aggregate under compression and shear. The importance of incorporation of inelastic shear deformation in realistic models of amorphization of boron carbide is noted, as is a greater reduction in overall strength of polycrystals containing one or a few dominant flaws rather than many diffusely distributed microcracks.

Cosserat Continuum Mechanics: With Applications to Granular Media

Abstract: This textbook explores the theory of Cosserat continuum mechanics, and covers fundamental tools, general laws and major models, as well as applications to the mechanics of granular media. While classical continuum mechanics is based on the axiom that the stress tensor is symmetric, theories such as that expressed in the seminal work of the brothers Eugene and Francois Cosserat are characterized by a non-symmetric stress tensor. The use of von Mises motor mechanics is introduced, for the compact mathematical description of the mechanics and statics of Cosserat continua, as the Cosserat continuum is a manifold of oriented “rigid particles” with 3 dofs of displacement and 3 dofs of rotation, rather than a manifold of points with 3 dofs of displacement. Here, the analysis is restricted to infinitesimal particle displacements and rotations. This book is intended as a valuable supplement to standard Continuum Mechanics courses, and graduate students as well as researchers in mechanics and applied mathematics will benefit from its self-contained text, which is enriched by numerous examples and exercises.

A unified numerical framework for rigid and compliant granular materials

Abstract: A numerical framework for the simulation of granular materials composed of mixed rigid and compliant grains is presented in this paper. This approach is based on a multibody meshfree technique, coupled in a very natural way with classic concepts from the discrete element method. The equations of motion (for the rigid grains) and of continuum mechanics (for the compliant ones) are solved using an adaptive explicit scheme, in fully dynamic conditions. The parallelization strategy is described and tested on an illustrative simulation involving both kinds of grains.

Formulation and validation of a reduced order model of 2D materials exhibiting a two-phase microstructure as applied to graphene oxide

Abstract: Novel 2D materials, e.g., graphene oxide (GO), are attractive building blocks in the design of advanced materials due to their reactive chemistry, which can enhance interfacial interactions while providing good in-plane mechanical properties. Recent studies have hypothesized that the randomly distributed two-phase microstructure of GO, which arises due to its oxidized chemistry, leads to differences in nano- vs meso‑scale mechanical responses. However, this effect has not been carefully studied using molecular dynamics due to computational limitations. Herein, a continuum mechanics model, formulated based on density functional based tight binding (DFTB) constitutive results for GO nano-flakes, is establish for capturing the effect of oxidation patterns on the material mechanical properties. GO is idealized as a continuum heterogeneous two-phase material, where the mechanical response of each phase, graphitic and oxidized, is informed from DFTB simulations. A finite element implementation of the model is validated via MD simulations and then used to investigate the existence of GO representative volume elements (RVE). We find that for the studied GO, an RVE behavior arises for monolayer sizes in excess to 40 nm. Moreover, we reveal that the response of monolayers with two main different functional chemistries, epoxide-rich and hydroxyl‑rich, present distinct differences in mechanical behavior. In addition, we explored the role of defect density in GO, and validate the applicability of the model to larger length scales by predicting membrane deflection behavior, in close agreement with previous experimental and theoretical observations. As such the work presents a reduced order modeling framework applicable in the study of mechanical properties and deformation mechanisms in 2D multiphase materials.

Application of an enriched FEM technique in thermo-mechanical contact problems

Abstract: In this paper, an enriched FEM technique is employed for thermo-mechanical contact problem based on the extended finite element method. A fully coupled thermo-mechanical contact formulation is presented in the framework of X-FEM technique that takes into account the deformable continuum mechanics and the transient heat transfer analysis. The Coulomb frictional law is applied for the mechanical contact problem and a pressure dependent thermal contact model is employed through an explicit formulation in the weak form of X-FEM method. The equilibrium equations are discretized by the Newmark time splitting method and the final set of non-linear equations are solved based on the Newton–Raphson method using a staggered algorithm. Finally, in order to illustrate the capability of the proposed computational model several numerical examples are solved and the results are compared with those reported in literature.

Modelling fluid deformable surfaces with an emphasis on biological interfaces

Abstract: Fluid deformable surfaces are ubiquitous in cell and tissue biology, including lipid bilayers, the actomyosin cortex, or epithelial cell sheets. These interfaces exhibit a complex interplay between elasticity, low Reynolds number interfacial hydrodynamics, chemistry, and geometry, and govern important biological processes such as cellular traffic, division, migration, or tissue morphogenesis. To address the modelling challenges posed by this class of problems, in which interfacial phenomena tightly interact with the shape and dynamics of the surface, we develop a general continuum mechanics and computational framework for fluid deformable surfaces. The dual solid-fluid nature of fluid deformable surfaces challenges classical Lagrangian or Eulerian descriptions of deforming bodies. Here, we extend the notion of Arbitrarily Lagrangian-Eulerian (ALE) formulations, well-established for bulk media, to deforming surfaces. To systematically develop models for fluid deformable surfaces, which consistently treat all couplings between fields and geometry, we follow a nonlinear Onsager formalism according to which the dynamics minimize a Rayleighian functional where dissipation, power input and energy release rate compete. Finally, we propose new computational methods, which build on Onsager's formalism and our ALE formulation, to deal with the resulting stiff system of higher-order of partial differential equations. We apply our theoretical and computational methodology to classical models for lipid bilayers and the cell cortex. The methods developed here allow us to formulate/simulate these models for the first time in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

Instability analysis of an electro-magneto-elastic actuator: A continuum mechanics approach

Abstract: The study of advanced artificial electro-magneto-elastic (EME) materials recently connects the material science with the electrodynamics. In particular, EME materials established a new research direction, which provides the fruitful ideas for the advanced engineering and medical field applications. In the present paper, we introduce a continuum mechanics-based method to analyze an electro-magneto-mechanical instability (EMMI) phenomenon of a smart actuator made of an EME material. The proposed method is based on the nonlinear theory of electro-magneto-elasticity followed by the second law of thermodynamics. We develop an analytical EMMI model for a smart actuator through a new amended energy function. This amended energy function accounts the electro-magnetostriction phenomenon for a class of an incompressible isotropic EME material. Additionally, the amended energy function successfully resolves the physical interpretation issue of the Maxwell stress tensor in large deformation. The formulated continuum mechanics-based EMMI model is also compared and validated with an energy-based EMMI model existing in the literature.

XFEM, strong discontinuities and second-order work in shear band modeling of saturated porous media

Abstract: We investigate shear band initiation and propagation in fully saturated porous media by means of a combination of strong discontinuities (discontinuities in the displacement field) and XFEM. As a constitutive behavior of the solid phase, a Drucker–Prager model is used within a framework of non-associated plasticity to account for dilation of the sample. Strong discontinuities circumvent the difficulties which appear when trying to model shear band formation in the context of classical nonlinear continuum mechanics and when trying to resolve them with classical numerical methods like the finite element method. XFEM, on the other hand, is well suited to deal with problems where a discontinuity propagates, without the need of remeshing. The numerical results are confirmed by the application of Hill’s second-order work criterion which allows to evaluate the material point instability not only locally but also for the whole domain.

Variational formulation of generalized interfaces for finite deformation elasticity

Abstract: The objective of this contribution is to formulate generalized interfaces in a variationally consistent manner within a finite deformation continuum mechanics setting. The general interface model i...

The Need to Use Generalized Continuum Mechanics to Model 3D Textile Composite Forming

Abstract: 3D textile composite reinforcements can generally be modelled as continuum media. It is shown that the classical continuum mechanics of Cauchy is insufficient to depict the mechanical behavior of textile materials. A Cauchy macroscopic model is not capable of exhibiting very low transverse shear stiffness, given the possibility of sliding between the fibers and simultaneously taking into account the individual stiffness of each fibre. A first solution is presented which consists in adding a bending stiffness to the tridimensional finite elements. Another solution is to supplement the potential of the hyperelastic model by second gradient terms. Another approach consists in implementing a shell approach specific to the fibrous medium. The developed Ahmad elements are based on the quasi-inextensibility of the fibers and the bending stiffness of each fiber.