Showing papers on "Continuum mechanics published in 2011"

Thermodynamically Consistent, Frame Indifferent Diffuse Interface Models for Incompressible Two-Phase Flows with Different Densities

Abstract: A new diffuse interface model for a two-phase flow of two incompressible fluids with different densities is introduced using methods from rational continuum mechanics. The model fulfills local and global dissipation inequalities and is frame indifferent. Moreover, it is generalized to situations with a soluble species. Using the method of matched asymptotic expansions we derive various sharp interface models in the limit when the interfacial thickness tends to zero. Depending on the scaling of the mobility in the diffusion equation we either derive classical sharp interface models or models where bulk or surface diffusion is possible in the limit. In the latter case a new term resulting from surface diffusion appears in the momentum balance at the interface. Finally, we show that all sharp interface models fulfill natural energy inequalities.

Bending ultrathin graphene at the margins of continuum mechanics.

Abstract: Deviations from continuum mechanics are always expected in nanoscale structures. We investigate the validity of the plate idealization of ultrathin graphene by gaining insight into the response of chemical bonds to bending deformations. In the monolayer, a bond orbital model reveals the breakdown of the plate phenomenology. In the multilayer, objective molecular dynamics simulations identify the validity margin and the role of discreteness in the plate idealization. Our result has implications for a broad class of phenomena where the monolayer easily curves, and for the design of mass and force detection devices.

Multiscale modeling and characterization of granular matter: From grain kinematics to continuum mechanics

Abstract: Granular sands are characterized and modeled here by explicitly exploiting the discrete-continuum duality of granular matter. Grain-scale kinematics, obtained by shearing a sample under triaxial compression, are coupled with a recently proposed multiscale computational framework to model the behavior of the material without resorting to phenomenological evolution (hardening) laws. By doing this, complex material behavior is captured by extracting the evolution of key properties directly from the grain-scale mechanics and injecting it into a continuum description (e.g., elastoplasticity). The effectiveness of the method is showcased by two examples: one linking discrete element computations with finite elements and another example linking a triaxial compression experiment using computed tomography and digital image correlation with finite element computation. In both cases, dilatancy and friction are used as the fundamental plastic variables and are obtained directly from the grain kinematics. In the case of the result linked to the experiment, the onset and evolution of a persistent shear band is modeled, showing—for the first time—three-dimensional multiscale results in the post-bifurcation regime with real materials and good quantitative agreement with experiments.

A Brief Review of Elasticity and Viscoelasticity for Solids

Abstract: There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain pro- posed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a com- putational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.

Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics

Abstract: This paper presents an investigation on the buckling characteristics of nanoscale rectangular plates under bi-axial compression considering non-uniformity in the thickness. Based on the nonlocal continuum mechanics, governing differential equations are derived. Numerical solutions for the buckling loads are obtained using the Galerkin method. The present study shows that the buckling behaviors of single-layered graphene sheets (SLGSs) are strongly sensitive to the nonlocal and non-uniform parameters. The influence of percentage change of thickness on the stability of SLGSs is more significant in the strip-type nonoplates (nanoribbons) than in the square-type nanoplates.

Frame-based elastic models

Abstract: We present a new type of deformable model which combines the realism of physically-based continuum mechanics models and the usability of frame-based skinning methods. The degrees of freedom are coordinate frames. In contrast with traditional skinning, frame positions are not scripted but move in reaction to internal body forces. The displacement field is smoothly interpolated using dual quaternion blending. The deformation gradient and its derivatives are computed at each sample point of a deformed object and used in the equations of Lagrangian mechanics to achieve physical realism. This allows easy and very intuitive definition of the degrees of freedom of the deformable object. The meshless discretization allows on-the-fly insertion of frames to create local deformations where needed. We formulate the dynamics of these models in detail and describe some precomputations that can be used for speed. We show that our method is effective for behaviors ranging from simple unimodal deformations to complex realistic deformations comparable with Finite Element simulations. To encourage its use, the software will be freely available in the simulation platform SOFA.

A Continuum Mechanical Approach to Geodesics in Shape Space

Abstract: In this paper concepts from continuum mechanics are used to define geodesic paths in the space of shapes, where shapes are implicitly described as boundary contours of objects. The proposed shape metric is derived from a continuum mechanical notion of viscous dissipation. A geodesic path is defined as the family of shapes such that the total amount of viscous dissipation caused by an optimal material transport along the path is minimized. The approach can easily be generalized to shapes given as segment contours of multi-labeled images and to geodesic paths between partially occluded objects. The proposed computational framework for finding such a minimizer is based on the time discretization of a geodesic path as a sequence of pairwise matching problems, which is strictly invariant with respect to rigid body motions and ensures a 1---1 correspondence along the induced flow in shape space. When decreasing the time step size, the proposed model leads to the minimization of the actual geodesic length, where the Hessian of the pairwise matching energy reflects the chosen Riemannian metric on the underlying shape space. If the constraint of pairwise shape correspondence is replaced by the volume of the shape mismatch as a penalty functional, one obtains for decreasing time step size an optical flow term controlling the transport of the shape by the underlying motion field. The method is implemented via a level set representation of shapes, and a finite element approximation is employed as spatial discretization both for the pairwise matching deformations and for the level set representations. The numerical relaxation of the energy is performed via an efficient multi-scale procedure in space and time. Various examples for 2D and 3D shapes underline the effectiveness and robustness of the proposed approach.

A new locking-free formulation for planar, shear deformable, linear and quadratic beam finite elements based on the absolute nodal coordinate formulation

Abstract: Many widely used beam finite element formulations are based either on Reiss- ner's classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the ben- efits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or in- elastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper. In many applications, a nonlinear large deformation beam element with bending, ax- ial and shear deformation properties is needed. In the present paper, linear and quadratic ANCF shear deformable beam finite elements are presented. A new locking-free continuum mechanics based formulation is compared to the classical Simo and Vu-Quoc formulation based on Reissner's virtual work of internal forces. Additionally, the introduced linear and quadratic ANCF elements are compared to a fully parameterized ANCF element from the literature. The performance of the respective elements is evaluated through analysis of con- ventional static and dynamic example problems. The investigation shows that the obtained linear and quadratic ANCF elements are advantageous compared to the original fully para- meterized ANCF element.

Inf–sup conditions for twofold saddle point problems

Abstract: Necessary and sufficient conditions for existence and uniqueness of solutions are developed for twofold saddle point problems which arise in mixed formulations of problems in continuum mechanics. This work extends the classical saddle point theory to accommodate nonlinear constitutive relations and the twofold saddle structure. Application to problems in incompressible fluid mechanics employing symmetric tensor finite elements for the stress approximation is presented.

Validation of a non-orthogonal constitutive model for woven composite fabrics via hemispherical stamping simulation

Abstract: A non-orthogonal constitutive model was previously developed to characterize the anisotropic material behavior of woven composite fabrics under large shear deformation. This paper presents a validation of the constitutive model via hemispherical stamping simulation of a square woven composite fabric by a fully continuum mechanics-based approach with finite element (FE) method. The constitutive model is imposed on conventional shell elements to equivalently characterize the global mechanical behavior of woven composite fabric during forming. A balanced plain woven composite is taken as an example. The stamping results from the non-orthogonal model and the corresponding orthogonal constitutive model are compared with experimental data. It is shown that the results predicted by the non-orthogonal model are in a good agreement with the experimental results, while those from the orthogonal model have large discrepancies. The numerical simulation demonstrates the necessity and efficiency of the non-orthogonal constitutive model in capturing the anisotropic material behavior that woven composite fabrics render in forming.

Sparse meshless models of complex deformable solids

Abstract: A new method to simulate deformable objects with heterogeneous material properties and complex geometries is presented. Given a volumetric map of the material properties and an arbitrary number of control nodes, a distribution of the nodes is computed automatically, as well as the associated shape functions. Reference frames attached to the nodes are used to apply skeleton subspace deformation across the volume of the objects. A continuum mechanics formulation is derived from the displacements and the material properties. We introduce novel material-aware shape functions in place of the traditional radial basis functions used in meshless frameworks. In contrast with previous approaches, these allow coarse deformation functions to efficiently resolve non-uniform stiffnesses. Complex models can thus be simulated at high frame rates using a small number of control nodes.

Gradient Nanomechanics: Applications to Deformation, Fracture, and Diffusion in Nanopolycrystals

Abstract: The term “gradient nanomechanics” is used here to designate a generalized continuum mechanics framework accounting for “bulk-surface” interactions in the form of extra gradient terms that enter in the balance laws or the evolution equations of the relevant constitutive variables that govern behavior at the nanoscale. In the case of nanopolycrystals, the grain boundaries may be viewed either as sources/sinks of “effective” mass and internal force or as a separate phase, interacting with the bulk phase that it surrounds, and supporting its own fields, balance laws, and constitutive equations reflecting this interaction. In either view, a further common assumption introduced is that the constitutive interaction between bulk and “interface” phases enters in the form of higher order gradient terms, independently of the details of the underlying physical mechanisms that bring these terms about. The effectiveness of the approach is shown by considering certain benchmark problems for nanoelasticity, nanoplasticity, and nanodiffusion for which standard continuum mechanics theory fails to model the observed behavior. Its implications to interpreting size-dependent stress-strain curves for nanopolycrystals with varying grain size are also discussed.

Critical buckling temperature of single-walled carbon nanotubes embedded in a one-parameter elastic medium based on nonlocal continuum mechanics

Abstract: In this paper, the critical budding temperature of single-walled carbon nanotubes (SWCNTs), which are embedded in one-parameter elastic medium (Winkler foundation) is estimated under the umbrella of continuum mechanics theory. Nonlocal continuum theory is incorporated into Timoshenko beam model and the governing differential equations of motion are derived. An explicit expression for the non-dimensional critical buckling temperature is also derived in this work. The effect of the nonlocal small scale coefficient, the Winkler foundation parameter and the ratio of the length to the diameter on the critical buckling temperature is investigated in detail. It can be observed that the effects of nonlocal small scale parameter and the Winkler foundation parameter are significant and should be considered for thermal analysis of SWCNTs. The results presented in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the thermal buckling properties of embedded single-walled carbon nanotubes. (C) 2011 Elsevier B.V. All rights reserved.

A continuum theory of thermoelectric bodies and effective properties of thermoelectric composites

Abstract: We develop a continuum theory for thermoelectric bodies following the framework of continuum mechanics and conforming to general principles of thermodynamics. For steady states, the governing equations for local fields are intrinsically nonlinear. However, under conditions of small variations of electrochemical potential, temperature and their gradients, the governing equations may be reduced to a linear elliptic system, which can be conveniently solved to determine behaviors of thermoelectric bodies. The linear theory is further applied to predict effective properties of thermoelectric composites. In particular, explicit formula of effective properties are obtained for simple microstructures of laminates and periodic E-inclusions, which implies useful design principles for engineering thermoelectric composites.

Fluid Mechanics of Viscoelasticity: General Principles, Constitutive Modelling, Analytical and Numerical Techniques

Abstract: General Principles. 1. Kinematics of fluid flow. 2. Balance equations for smooth and non-smooth regions. Constitutive Modelling. 3. Formulation of constitutive equations - the simple fluid. 4. Constitutive equations derived from microstructures. Analytical and Numerical Techniques. 5. The shape and nature of general solutions. 6. Simple models and complex phenomena. 7. Computational viscoelastic fluid dynamics.

Damage and progressive failure of concrete structures using non-local peridynamic modeling

Abstract: Peridynamics (PD), a recently developed theory of solid mechanics, which employs a non-local model of force interaction and makes use of integral formulation rather than the spatial partial differential equations used in the classical continuum mechanics theory, has shown effectiveness and promise in solving discontinuous problems at both macro and micro scales In this paper, the peridynamics theory is used to analyze damage and progressive failure of concrete structures A non-local peridynamic model for a rectangular concrete plate is developed, and a central pairwise force function is introduced to describe the interior interactions between particles within some definite distance Damage initiation, evolution and crack propagation in the concrete model subject to in-plane uni-axial tension, in-plane uni-axial compression and out-of-plane impact load are investigated respectively The numerical results show that discontinuities appear and grow spontaneously as part of the solution to the peridynamic equations of motion, and no special failure criteria or re-meshing techniques are required, which proves the potential of peridynamic modeling as a promising technique for analyzing the progressive failure of concrete materials and structures

A non-local, ordinary-state-based viscoelasticity model for peridynamics.

Abstract: A non-local, ordinary-state-based, peridynamics viscoelasticity model is developed. In this model, viscous effects are added to deviatoric deformations and the bulk response remains elastic. The model uses internal state variables and is conceptually similar to linearized isotropic viscolelasticity in the local theory. The modulus state, which is used to form the Jacobian matrix in Newton-Raphson algorithms, is presented. The model is shown to satisfy the 2nd law of thermodynamics and is applicable to problems in solid continuum mechanics where fracture and rate effects are important; it inherits all the advantages for modeling fracture associated with peridynamics. By combining this work with the previously published ordinary-state-based plasticity model, the model may be amenable to viscoplasticity problems where plasticity and rate effects are simultaneously important. Also, the model may be extended to include viscous effects for spherical deformations as well. The later two extensions are not presented and may be the subject of further work.

Terahertz wave propagation in uniform nanorods: A nonlocal continuum mechanics formulation including the effect of lateral inertia

Abstract: The dynamic testing of materials and components often involves predicting the propagation of stress waves in slender rods. The present work deals with the analysis of the wave propagation characteristics of nanorods. The nonlocal elasticity theory and also the lateral inertia are incorporated into classical/local rod model to capture unique features of the nanorods under the umbrella of continuum mechanics theory. The strong effect of the nonlocal scale has been obtained which leads to substantially different wave behaviors of nanorods from those of macroscopic rods. Nonlocal rod/bar model is developed for nanorods including the lateral inertia effects. The analysis shows that the wave characteristics are highly over estimated by the classical rod model, which ignores the effect of small-length scale. The wave propagation properties of the nanorod obtained from the present formulations are compared with the continuum rod model, nonlocal second and fourth order strain gradient models, Born-K a ´ rm a ´ n model and the nonlocal stress gradient model. It has also been shown that, the unstable second order strain gradient model can be replaced by considering the inertia gradient terms in the formulations. The effects of both the nonlocal scale and the diameter of the nanorod on spectrum curves are highlighted in the present manuscript. The results provided in this article are useful guidance for the study and design of the next generation of nanodevices that make use of the wave propagation properties of single-walled carbon nanotubes.

Continuum Damage Mechanics Models for the Analysis of Progressive Failure in Open-Hole Tension Laminates

Abstract: The performance of a state-of-the-art continuum damage mechanics model for interlaminar damage, coupled with a cohesive zone model for delamination is examined for failure prediction of quasi-isotropic open-hole tension laminates. Limitations of continuum representations of intra-ply damage and the effect of mesh orientation on the analysis predictions are discussed. It is shown that accurate prediction of matrix crack paths and stress redistribution after cracking requires a mesh aligned with the fiber orientation. Based on these results, an aligned mesh is proposed for analysis of the open-hole tension specimens consisting of different meshes within the individual plies, such that the element edges are aligned with the ply fiber direction. The modeling approach is assessed by comparison of analysis predictions to experimental data for specimen configurations in which failure is dominated by complex interactions between matrix cracks and delaminations. It is shown that the different failure mechanisms observed in the tests are well predicted. In addition, the modeling approach is demonstrated to predict proper trends in the effect of scaling on strength and failure mechanisms of quasi-isotropic open-hole tension laminates.

A nonlocal approach to modeling crack nucleation in AA 7075-T651.

Abstract: A critical stage in microstructurally small fatigue crack growth in AA 7075-T651 is the nucleation of cracks originating in constituent particles into the matrix material. Previous work has focused on a geometric approach to modeling microstructurally small fatigue crack growth in which damage metrics derived from an elastic-viscoplastic constitutive model are used to predict the nucleation event [1, 2]. While a geometric approach based on classical finite elements was successful in explicitly modeling the polycrystalline grain structure, singularities at the crack tip necessitated the use of a nonlocal sampling approach to remove mesh size dependence. This study is an initial investigation of the peridynamic formulation of continuum mechanics as an alternative approach to modeling microstructurally small fatigue crack growth. Peridynamics, a nonlocal extension of continuum mechanics, is based on an integral formulation that remains valid in the presence of material discontinuities. To capture accurately the material response at the grain scale, a crystal elastic-viscoplastic constitutive model is adapted for use in non-ordinary state-based peridynamics through the use of a regularized deformation gradient. The peridynamic approach is demonstrated on a baseline model consisting of a hard elastic inclusion in a single crystal. Coupling the elastic-viscoplastic material model with peridynamics successfully facilitates the modeling of plastic deformation and damage accumulation in the vicinity of the particle inclusion. Lattice orientation is shown to have a strong influence on material response.Copyright © 2011 by ASME

Microcanonical Entropy and Mesoscale Dislocation Mechanics and Plasticity

Abstract: A methodology is devised to utilize the statistical mechanical entropy of an isolated, constrained atomistic system to define constitutive response functions for the dissipative driving-force and energetic fields in continuum thermomechanics A thermodynamic model of dislocation mechanics is discussed as an example Primary outcomes are constitutive relations for the back-stress tensor and the Cauchy stress tensor in terms of the elastic distortion, mass density, polar dislocation density, and the scalar statistical density