Showing papers on "Linear elasticity published in 2018"

Strain gradient and generalized continua obtained by homogenizing frame lattices

Abstract: We determine the effective behavior of periodic structures made of welded elastic bars. Taking into account the fact that flexural and torsional stiffnesses are much smaller than the extensional one we overpass classical homogenization formula and obtain totally different types of effective energies. We work in the framework of linear elasticity. We give different examples of two dimensional or three dimensional micro-structures which lead to generalized 1D, 2D or 3D continua like Timoshenko beam, Mindlin-Reissner plate, strain gradient, Cosserat, or micromorphic continua.

Grain Boundary Softening: A Potential Mechanism for Lithium Metal Penetration through Stiff Solid Electrolytes.

Abstract: Models based on linear elasticity suggest that a solid electrolyte with a high shear modulus will suppress “dendrite” formation in batteries that use metallic lithium as the negative electrode. Nev...

Two-dimensional strain gradient damage modeling: a variational approach

Abstract: In this paper, we formulate a linear elastic second gradient isotropic two-dimensional continuum model accounting for irreversible damage. The failure is defined as the condition in which the damage parameter reaches 1, at least in one point of the domain. The quasi-static approximation is done, i.e., the kinetic energy is assumed to be negligible. In order to deal with dissipation, a damage dissipation term is considered in the deformation energy functional. The key goal of this paper is to apply a non-standard variational procedure to exploit the damage irreversibility argument. As a result, we derive not only the equilibrium equations but, notably, also the Karush–Kuhn–Tucker conditions. Finally, numerical simulations for exemplary problems are discussed as some constitutive parameters are varying, with the inclusion of a mesh-independence evidence. Element-free Galerkin method and moving least square shape functions have been employed.

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 state-based peridynamic model for quantitative fracture analysis

Abstract: A new state-based peridynamic model is proposed to quantitatively analyze fracture behavior (crack initiation and propagation) of materials. In this model, the general relationship of the critical stretch and the critical energy release rate is for the first time obtained for the state-based peridynamic model of linear elastic brittle materials, and the released energy density is defined to quantitatively track the energy released during crack propagation. The three-dimensional (3D) and two-dimensional (2D) (for both plane stress and plane strain) cases are all considered. As illustrations, the compact tension and double cantilever beam tests are analyzed using the proposed model, which is capable of successfully capturing fracture behaviors (e.g., crack path and concentration of strain energy density) of the considered fracture tests. The characteristic parameters (i.e., critical load, critical energy release rate, etc.) are calculated and compared with available experimental and numerical data in the literature to demonstrate validity of the proposed model.

Predictions of the electro-mechanical response of conductive CNT-polymer composites

Abstract: We present finite element simulations to predict the conductivity, elastic response and strain-sensing capability of conductive composites comprising a polymeric matrix and carbon nanotubes. Realistic representative volume elements (RVE) of the microstructure are generated and both constituents are modelled as linear elastic solids, with resistivity independent of strain; the electrical contact between nanotubes is represented by a new element which accounts for quantum tunnelling effects and captures the sensitivity of conductivity to separation. Monte Carlo simulations are conducted and the sensitivity of the predictions to RVE size is explored. Predictions of modulus and conductivity are found in good agreement with published results. The strain-sensing capability of the material is explored for multiaxial strain states.

Dislocation dynamics and crystal plasticity in the phase-field crystal model

Abstract: A phase-field model of a crystalline material is introduced to develop the necessary theoretical framework to study plastic flow due to dislocation motion. We first obtain the elastic stress from the phase-field crystal free energy under weak distortion and show that it obeys the stress-strain relation of linear elasticity. We focus next on dislocations in a two-dimensional hexagonal lattice. They are composite topological defects in the weakly nonlinear amplitude equation expansion of the phase field, with topological charges given by the standard Burgers vector. This allows us to introduce a formal relation between the dislocation velocity and the evolution of the slowly varying amplitudes of the phase field. Standard dissipative dynamics of the phase-field crystal model is shown to determine the velocity of the dislocations. When the amplitude expansion is valid and under additional simplifications, we find that the dislocation velocity is determined by the Peach-Koehler force. As an application, we compute the defect velocity for a dislocation dipole in two setups, pure glide and pure climb, and compare it with the analytical predictions.

A semi-infinite hydraulic fracture driven by a shear-thinning fluid

Abstract: We use the Carreau rheological model which properly accounts for the shear-thinning behaviour between the low and high shear rate Newtonian limits to investigate the problem of a semi-infinite hydraulic fracture propagating at a constant velocity in an impermeable linearly elastic material. We show that the solution depends on four dimensionless parameters: a dimensionless toughness (function of the fracture velocity, confining stress, material and fluid parameters), a dimensionless transition shear stress (related to both fluid and material behaviour), the fluid shear-thinning index and the ratio between the high and low shear rate viscosities. We solve the complete problem numerically combining a Gauss–Chebyshev method for the discretization of the elasticity equation, the quasi-static fracture propagation condition and a finite difference scheme for the width-averaged lubrication flow. The solution exhibits a complex structure with up to four distinct asymptotic regions as one moves away from the fracture tip: a region governed by the classical linear elastic fracture mechanics behaviour near the tip, a high shear rate viscosity asymptotic and power-law asymptotic region in the intermediate field and a low shear rate viscosity asymptotic far away from the fracture tip. The occurrence and order of magnitude of the extent of these different viscous asymptotic regions are estimated analytically. Our results also quantify how shear thinning drastically reduces the size of the fluid lag compared to a Newtonian fluid. We also investigate simpler rheological models (power law, Ellis) and establish the small domain where they can properly reproduce the response obtained with the complete rheology.

Data-Driven Problems in Elasticity

Abstract: We consider a new class of problems in elasticity, referred to as Data-Driven problems, defined on the space of strain-stress field pairs, or phase space. The problem consists of minimizing the distance between a given material data set and the subspace of compatible strain fields and stress fields in equilibrium. We find that the classical solutions are recovered in the case of linear elasticity. We identify conditions for convergence of Data-Driven solutions corresponding to sequences of approximating material data sets. Specialization to constant material data set sequences in turn establishes an appropriate notion of relaxation. We find that relaxation within this Data-Driven framework is fundamentally different from the classical relaxation of energy functions. For instance, we show that in the Data-Driven framework the relaxation of a bistable material leads to material data sets that are not graphs.

Validation of Numerical Simulations of Thoracic Aorta Hemodynamics: Comparison with In Vivo Measurements and Stochastic Sensitivity Analysis

Abstract: Computational fluid dynamics (CFD) and 4D-flow magnetic resonance imaging (MRI) are synergically used for the simulation and the analysis of the flow in a patient-specific geometry of a healthy thoracic aorta. CFD simulations are carried out through the open-source code SimVascular. The MRI data are used, first, to provide patient-specific boundary conditions. In particular, the experimentally acquired flow rate waveform is imposed at the inlet, while at the outlets the RCR parameters of the Windkessel model are tuned in order to match the experimentally measured fractions of flow rate exiting each domain outlet during an entire cardiac cycle. Then, the MRI data are used to validate the results of the hemodynamic simulations. As expected, with a rigid-wall model the computed flow rate waveforms at the outlets do not show the time lag respect to the inlet waveform conversely found in MRI data. We therefore evaluate the effect of wall compliance by using a linear elastic model with homogeneous and isotropic properties and changing the value of the Young’s modulus. A stochastic analysis based on the polynomial chaos approach is adopted, which allows continuous response surfaces to be obtained in the parameter space starting from a few deterministic simulations. The flow rate waveform can be accurately reproduced by the compliant simulations in the ascending aorta; on the other hand, in the aortic arch and in the descending aorta, the experimental time delay can be matched with low values of the Young’s modulus, close to the average value estimated from experiments. However, by decreasing the Young’s modulus the underestimation of the peak flow rate becomes more significant. As for the velocity maps, we found a generally good qualitative agreement of simulations with MRI data. The main difference is that the simulations overestimate the extent of reverse flow regions or predict reverse flow when it is absent in the experimental data. Finally, a significant sensitivity to wall compliance of instantaneous shear stresses during large part of the cardiac cycle period is observed; the variability of the time-averaged wall shear stresses remains however very low. In summary, a successful integration of hemodynamic simulations and of MRI data for a patient-specific simulation has been shown. The wall compliance seems to have a significant impact on the numerical predictions; a larger wall elasticity generally improves the agreement with experimental data.

Do uniform tangential interfacial stresses enhance adhesion

Abstract: We present theoretical arguments, based on linear elasticity and thermodynamics, to show that interfacial tangential stresses in sliding adhesive soft contacts may lead to a significant increase of the effective energy of adhesion. A sizable expansion of the contact area is predicted in conditions corresponding to such scenario. These results are easily explained and are valid under the assumptions that: (i) sliding at the interface does not lead to any loss of adhesive interaction and (ii) spatial fluctuations of frictional stresses can be considered negligible. Our results are seemingly supported by existing experiments, and show that frictional stresses may lead to an increase of the effective energy of adhesion depending on which conditions are established at the interface of contacting bodies in the presence of adhesive forces.

Devising superconvergent HDG methods with symmetric approximate stresses for linear elasticity by M-decompositions

Abstract: We propose a new tool, which we call $M$-decompositions, for devising superconvergent hybridizable discontinuous Galerkin (HDG) methods and hybridized-mixed methods for linear elasticity with strongly symmetric approximate stresses on unstructured polygonal/polyhedral meshes. We show that for an HDG method, when its local approximation space admits an $M$-decomposition, optimal convergence of the approximate stress and superconvergence of an element-by-element postprocessing of the displacement field are obtained. The resulting methods are locking-free. Moreover, we explicitly construct approximation spaces that admit $M$-decompositions on general polygonal elements. We display numerical results on triangular meshes validating our theoretical findings.

Visualising elastic anisotropy: theoretical background and computational implementation

Abstract: In this article, we present the technical realisation for visualisations of characteristic parameters of the fourth-order elasticity tensor, which is classified by three-dimensional symmetry groups. Hereby, expressions for spatial representations of Young’s modulus and bulk modulus as well as plane representations of shear modulus and Poisson’s ratio are derived and transferred into a comprehensible form to computer algebra systems. Additionally, we present approaches for spatial representations of both latter parameters. These three- and two-dimensional representations are implemented into the software MATrix LABoratory. Exemplary representations of characteristic materials complete the present treatise.

A superconvergent hybridisable discontinuous Galerkin method for linear elasticity

Abstract: This is the peer reviewed version of the following article: Sevilla, R, Giacomini, M, Karkoulias, A, Huerta, A A superconvergent hybridisable discontinuous Galerkin method for linear elasticity "International journal for numerical methods in engineering", 12 Octubre 2018, vol 116, num 2, p 91-116, which has been published in final form at https://doiorg/101002/nme5916 This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving

A Comparison of Curing Process-Induced Residual Stresses and Cure Shrinkage in Micro-Scale Composite Structures with Different Constitutive Laws

Abstract: In this paper, three kinds of constitutive laws, elastic, “cure hardening instantaneously linear elastic (CHILE)” and viscoelastic law, are used to predict curing process-induced residual stress for the thermoset polymer composites. A multi-physics coupling finite element analysis (FEA) model implementing the proposed three approaches is established in COMSOL Multiphysics-Version 4.3b. The evolution of thermo-physical properties with temperature and degree of cure (DOC), which improved the accuracy of numerical simulations, and cure shrinkage are taken into account for the three models. Subsequently, these three proposed constitutive models are implemented respectively in a 3D micro-scale composite laminate structure. Compared the differences between these three numerical results, it indicates that big error in residual stress and cure shrinkage generates by elastic model, but the results calculated by the modified CHILE model are in excellent agreement with those estimated by the viscoelastic model.

Toward the constitutive modeling of epoxy matrix: Temperature-accelerated quasi-static molecular simulations consistent with the experimental test

Abstract: We propose an efficient simulation-based methodology to characterize the quasi-static (experimental low strain rate) yield stress of an amorphous thermoset polymer, which has generally been considered a limitation of molecular dynamics (MD) simulations owing to the extremely short time steps involved. In an effort to overcome this limitation, the temperature-accelerated method – in which temperature is treated as being equivalent to time in deformation kinetics – is employed to explore the experimental strain rate conditions. The mechanical tensile behavior of a highly crosslinked polymer is then investigated with MD simulations by considering different strain rates and temperatures below the glass transition temperature. The derived yield stress represents the time- and temperature-dependent characteristics, showing that the yield stress decreases with increasing temperature and decreasing strain rate. Changeable vertical and horizontal shift factors are introduced for the first time to reflect nonlinear characteristics of the yield stress across a broad range of strain rates and to quantify the correlation between increasing temperatures and decreasing strain rates. With the proposed method, the Eyring plot, which describes the rate effect on yield from quasi-static to high-rate conditions, is predicted from MD simulations, and agrees well with macroscopic experimental results. From the constructed Eyring plot, the experimentally validated quasi-static stress-strain response is also estimated by using linear elastic model and Ludwick's hardening model. The proposed method provides new avenues for the design of glassy polymers using only fully atomistic MD simulations, thus overcoming the existing temporal scale limitations.

Finite element dynamic analysis of beams on nonlinear elastic foundations under a moving oscillator

Abstract: This paper presents a study on the dynamic response of beams on elastic foundations, subjected to a uniformly moving oscillator. Using a finite element model programmed within a MATLAB environment the response of the system is studied for three different types of mechanical behaviour of the foundation: (a) linear elastic (classical Winkler model), (b) nonlinear elastic (in which the foundation reaction displays a cubic dependence on the beam displacement) and (c) bilinear elastic (with different compressive and tensile stiffnesses). The effects of the oscillator's natural frequency and velocity and of the foundation's stiffness and damping are investigated. In particular, critical velocities of the oscillator and ranges of velocities for which the system is dynamically unstable are numerically determined for the first time in the above mentioned nonlinear cases.