Showing papers on "Continuum mechanics published in 2020"

A fractal variational theory for one-dimensional compressible flow in a microgravity space

Abstract: The semi-inverse method is adopted to establish a family of fractal variational principles of the one-dimensional compressible flow under the microgravity condition, and Cauchy–Lagrange integral is...

Buckling loads of nano-beams in stress-driven nonlocal elasticity

Abstract: Size-dependent buckling of compressed Bernoulli-Euler nano-beams is investigated by stress-driven nonlocal continuum mechanics. The nonlocal elastic strain is obtained by convoluting the stress fie...

High order ADER schemes for continuum mechanics

Abstract: In this paper we first review the development of high order ADER finite volume and ADER discontinuous Galerkin schemes on fixed and moving meshes, since their introduction in 1999 by Toro et al. We show the modern variant of ADER based on a space-time predictor-corrector formulation in the context of ADER discontinuous Galerkin schemes with a posteriori subcell finite volume limiter on fixed and moving grids, as well as on space-time adaptive Cartesian AMR meshes. We then present and discuss the unified symmetric hyperbolic and thermodynamically compatible (SHTC) formulation of continuum mechanics developed by Godunov, Peshkov and Romenski (GPR model), which allows to describe fluid and solid mechanics in one single and unified first order hyperbolic system. In order to deal with free surface and moving boundary problems, a simple diffuse interface approach is employed, which is compatible with Eulerian schemes on fixed grids as well as direct Arbitrary-Lagrangian-Eulerian methods on moving meshes. We show some examples of moving boundary problems in fluid and solid mechanics.

Thermoelastic damping in nonlocal nanobeams considering dual-phase-lagging effect:

Abstract: This paper aims to present an explicit relation for thermoelastic damping in nanobeams capturing the small-scale effects on both the continuum mechanics and heat conduction domains. To incorporate ...

Nonlocal strain gradient torsion of elastic beams: variational formulation and constitutive boundary conditions

Abstract: Nonlocal strain gradient continuum mechanics is a methodology widely employed in the literature to assess size effects in nano-structures. Notwithstanding this, improper higher-order boundary conditions (HOBC) are prescribed to close the corresponding elastostatic problems. In the present study, it is proven that HOBC have to be replaced with univocally determined boundary conditions of constitutive type, established by a consistent variational formulation. The treatment, developed in the framework of torsion of elastic beams, provides an effective approach to evaluate scale phenomena in smaller and smaller devices of engineering interest. Both elastostatic torsional responses and torsional-free vibrations of nano-beams are investigated by applying a simple analytical method. It is also underlined that the nonlocal strain gradient model, if equipped with the inappropriate HOBC, can lead to torsional structural responses which unacceptably do not exhibit nonlocality. The presented variational strategy is instead able to characterize significantly peculiar softening and stiffening behaviors of structures involved in modern nano-electro-mechanical systems.

Non-standard and constitutive boundary conditions in nonlocal strain gradient elasticity

Abstract: Zaera et al. (Int J Eng Sci 138:65–81, 2019) recently showed that the nonlocal strain gradient theory (NSGT) is not consistent when it is applied to finite solids, since all boundary conditions associated to the corresponding problems cannot be simultaneously satisfied. Given the large number of works using the NSGT being currently published in the field of generalized continuum mechanics, it is pertinent to evince the shortcomings of the application of this theory. Some authors solved the problem omitting the constitutive boundary conditions. In the current paper we show that, in this case, the equilibrium fields are not compatible with the constitutive equation of the material. Other authors solved it omitting the non-standard boundary conditions. Here we show that, in this case, the solution does not fulfil conservation of energy. In conclusion, the inconsistency of the NSGT is corroborated, and its application must be prevented in the analysis of the mechanical behaviour of nanostructures.

Transport phenomena in electrolyte solutions: Nonequilibrium thermodynamics and statistical mechanics

Abstract: Author(s): Fong, KD; Bergstrom, HK; McCloskey, BD; Mandadapu, KK | Abstract: The theory of transport phenomena in multicomponent electrolyte solutions is presented here through the integration of continuum mechanics, electromagnetism, and nonequilibrium thermodynamics. The governing equations of irreversible thermodynamics, including balance laws, Maxwell's equations, internal entropy production, and linear laws relating the thermodynamic forces and fluxes, are derived. Green–Kubo relations for the transport coefficients connecting electrochemical potential gradients and diffusive fluxes are obtained in terms of the flux–flux time correlations. The relationship between the derived transport coefficients and those of the Stefan–Maxwell and infinitely dilute frameworks are presented, and the connection between the transport matrix and experimentally measurable quantities is described. To exemplify the application of the derived Green–Kubo relations in molecular simulations, the matrix of transport coefficients for lithium and chloride ions in dimethyl sulfoxide is computed using classical molecular dynamics and compared with experimental measurements.

Wave dispersion characteristics of high-speed-rotating laminated nanocomposite cylindrical shells based on four continuum mechanics theories

Abstract: This paper investigates the wave propagation behavior of a high-speed rotating laminated nanocomposite cylindrical shell. The small-scale effects are analyzed based on nonlocal strain gradient theo...

Frequency- and angle-dependent scattering of a finite-sized meta-structure via the relaxed micromorphic model

Abstract: In this paper, we explore the use of micromorphic-type interface conditions for the modeling of a finite-sized metamaterial. We show how finite-domain boundary value problems can be approached in the framework of enriched continuum mechanics (relaxed micromorphic model) by imposing continuity of macroscopic displacement and of generalized tractions, as well as additional conditions on the micro-distortion tensor and on the double-traction. The case of a metamaterial slab of finite width is presented, its scattering properties are studied via a semi-analytical solution of the relaxed micromorphic model and compared to a direct finite-element simulation encoding all details of the selected microstructure. The reflection and transmission coefficients obtained via the two methods are presented as a function of the frequency and of the direction of propagation of the incident wave. We find excellent agreement for a large range of frequencies going from the long-wave limit to frequencies beyond the first band-gap and for angles of incidence ranging from normal to near-parallel incidence. The present paper sets the basis for a new viewpoint on finite-size metamaterial modeling enabling the exploration of meta-structures at large scales.

From quantum to continuum mechanics in the delamination of atomically-thin layers from substrates.

Abstract: Anomalous proximity effects have been observed in adhesive systems ranging from proteins, bacteria, and gecko feet suspended over semiconductor surfaces to interfaces between graphene and different substrate materials. In the latter case, long-range forces are evidenced by measurements of non-vanishing stress that extends up to micrometer separations between graphene and the substrate. State-of-the-art models to describe adhesive properties are unable to explain these experimental observations, instead underestimating the measured stress distance range by 2–3 orders of magnitude. Here, we develop an analytical and numerical variational approach that combines continuum mechanics and elasticity with quantum many-body treatment of van der Waals dispersion interactions. A full relaxation of the coupled adsorbate/substrate geometry leads us to conclude that wavelike atomic deformation is largely responsible for the observed long-range proximity effect. The correct description of this seemingly general phenomenon for thin deformable membranes requires a direct coupling between quantum and continuum mechanics. The unexpectedly long-ranged interface stress observed in recent delamination experiments is yet to be clarified. Here, the authors develop an analytical approach to show the wavelike atomic deformation as the origin for the observed ultra long-range stress in delamination of graphene from various substrates.

The computational framework for continuum-kinematics-inspired peridynamics

Abstract: Peridynamics (PD) is a non-local continuum formulation. The original version of PD was restricted to bond-based interactions. Bond-based PD is geometrically exact and its kinematics are similar to classical continuum mechanics (CCM). However, it cannot capture the Poisson effect correctly. This shortcoming was addressed via state-based PD, but the kinematics are not accurately preserved. Continuum-kinematics-inspired peridynamics (CPD) provides a geometrically exact framework whose underlying kinematics coincide with that of CCM and captures the Poisson effect correctly. In CPD, one distinguishes between one-, two- and three-neighbour interactions. One-neighbour interactions are equivalent to the bond-based interactions of the original PD formalism. However, two- and three-neighbour interactions are fundamentally different from state-based interactions as the basic elements of continuum kinematics are preserved precisely. The objective of this contribution is to elaborate on computational aspects of CPD and present detailed derivations that are essential for its implementation. Key features of the resulting computational CPD are elucidated via a series of numerical examples. These include three-dimensional problems at large deformations. The proposed strategy is robust and the quadratic rate of convergence associated with the Newton–Raphson scheme is observed.

The computational framework for continuum-kinematics-inspired peridynamics

Abstract: Peridynamics (PD) is a non-local continuum formulation. The original version of PD was restricted to bond-based interactions. Bond-based PD is geometrically exact and its kinematics are similar to classical continuum mechanics (CCM). However, it cannot capture the Poisson effect correctly. This shortcoming was addressed via state-based PD, but the kinematics are not accurately preserved. Continuum-kinematics-inspired peridynamics (CPD) provides a geometrically exact framework whose underlying kinematics coincide with that of CCM and captures the Poisson effect correctly. In CPD, one distinguishes between one-, two- and three-neighbour interactions. One-neighbour interactions are equivalent to the bond-based interactions of the original PD formalism. However, two- and three-neighbour interactions are fundamentally different from state-based interactions as the basic elements of continuum kinematics are preserved precisely. The objective of this contribution is to elaborate on computational aspects of CPD and present detailed derivations that are essential for its implementation. Key features of the resulting computational CPD are elucidated via a series of numerical examples. These include three-dimensional problems at large deformations. The proposed strategy is robust and the quadratic rate of convergence associated with the Newton--Raphson scheme is observed.

A Coupled Mass Transport and Deformation Theory of Multi-constituent Tumor Growth.

Abstract: We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and incompressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.

Encyclopedia of continuum mechanics

Abstract: Plasticity Continuum Damage Mechanics Biographies of Scientists Continuum Mechanics Basics Fracture Mechanics Shells Variational Principles Generalized Continua Numerical Continuum Mechanics Plane Problems Plates Nanomechanics Impact Mechanics Biomechanics Tensor Calculus Tensor Calculus Lightweight Structures Lightweight Structures Lightweight Structures Continuum Thermodynamics Composites Continuum Waves Creep Mechanics Optimization Fractional Calculus in Continuum Mechanics Fractional Calculus in Continuum Mechanics Ray Expansion in Continua Mechanical Behavior and Performance of Electronic and Photonic Materials, Packages and Systems Rods and Beams Adaptive Structures Random and Fractal Media

Nonlinear coupled electro-mechanical behavior of a novel anisotropic fiber-reinforced dielectric elastomer

Abstract: Dielectric elastomers are smart materials which produce mechanical actuation when subjected to an electrical field. In this study, we created an efficient framework to model the electromechanical coupling behavior of fiber-reinforced dielectric elastomers using large deformation continuum mechanics, electroelasticity and finite element method. We derived a consistent tangent modulus based on the Cauchy stress definition by incorporating dielectric effect into the tangent modulus. We used the Newton–Raphson method to solve the nonlinear finite element problem at hand with Abaqus and a new user defined material. In the next step, we calibrated the model with experimental tensile tests data on Cotton-reinforced Silicone rubber and check its validity by simulating the exact uniaxial tensile tests and comparing the results to experiments. We further checked the ability of the model to take the dielectric effect into account by comparing simulation results to available experimental data. We showed that the present model can trace the behavior of fiber-reinforced dielectric elastomers with proper accuracy. Also, we presented an example of a fiber-reinforced bending actuator and analyzed the effect of fiber orientation on its actuation and blocking force. We used the Maxwell–Garnett effective medium model to consider the effect of fiber inclusions on the dielectric constant of the material.

A Review of Computational Models for the Flow of Milled Biomass Part II: Continuum-Mechanics Models

Abstract: The design of efficient material-handling systems for milled lignocellulosic biomass is challenging due to their complex particle morphologies and frictional interactions. Computational modeling, i...

A consistent variational formulation of Bishop nonlocal rods

Abstract: Thick rods are employed in Nanotechnology to build modern electro mechanical systems. Design and optimization of such structures can be carried out by nonlocal continuum mechanics which is computationally convenient when compared to atomistic strategies. Bishop's kinematics is able to describe small-scale thick rods if a proper mathematical model of nonlocal elasticity is formulated to capture size effects. In all papers on the matter, nonlocal contributions are evaluated by replacing Eringen's integral convolution with the consequent (but not equivalent) differential equation governed by Helmholtz's differential operator. As notorious in integral equation theory, this replacement is possible for convolutions, defined in unbounded domains, governed by averaging kernels which are Green's functions of differential operators. Indeed, Eringen himself, in order to study nonlocal problems defined in unbounded domains, such as screw dislocations and wave propagations, suggested to replace integro-differential equations with differential conditions. A different scenario appears in Bishop rod mechanics where nonlocal integral convolutions are defined in bounded structural domains, so that Eringen's nonlocal differential equation has to be supplemented with additional boundary conditions. The objective is achieved by formulating the governing nonlocal equations by a proper variational statement. The new methodology provides an amendment of previous contributions in literature and is illustrated by investigating the elastostatic behavior of simple structural schemes. Exact solutions of Bishop rods are evaluated in terms of nonlocal parameter and cross-section gyration radius. Both hardening and softening structural responses are predictable with a suitable tuning of the parameters.

A consistent variational formulation of Bishop nonlocal rods

Abstract: Thick rods are employed in nanotechnology to build modern electromechanical systems. Design and optimization of such structures can be carried out by nonlocal continuum mechanics which is computationally convenient when compared to atomistic strategies. Bishop’s kinematics is able to describe small-scale thick rods if a proper mathematical model of nonlocal elasticity is formulated to capture size effects. In all papers on the matter, nonlocal contributions are evaluated by replacing Eringen’s integral convolution with the consequent (but not equivalent) differential equation governed by Helmholtz’s differential operator. As notorious in integral equation theory, this replacement is possible for convolutions, defined in unbounded domains, governed by averaging kernels which are Green’s functions of differential operators. Indeed, Eringen himself, in order to study nonlocal problems defined in unbounded domains, such as screw dislocations and wave propagation, suggested to replace integro-differential equations with differential conditions. A different scenario appears in Bishop rod mechanics where nonlocal integral convolutions are defined in bounded structural domains, so that Eringen’s nonlocal differential equation has to be supplemented with additional boundary conditions. The objective is achieved by formulating the governing nonlocal equations by a proper variational statement. The new methodology provides an amendment of previous contributions in the literature and is illustrated by investigating the elastostatic behavior of simple structural schemes. Exact solutions of Bishop rods are evaluated in terms of nonlocal parameter and cross section gyration radius. Both hardening and softening structural responses are predictable with a suitable tuning of the parameters.

A continuum viscoelastic model of Timoshenko NSGT nanobeams

Abstract: Developed and solved in this article, for the first time, is a coupled continuum viscoelastic model for Timoshenko nonlocal strain gradient theory (NSGT)-based nanobeams; this is performed using a finite difference analysis (FDA). The viscosity of infinitesimal elements of the Timoshenko NSGT nanobeams is incorporated via the Kelvin–Voigt scheme (as a two-parameter rheological scheme) for both the transverse/longitudinal/rotational motions. The NSGT for the normal/shear stress fields is used for ultrasmall size influences on the continuum model. Rotary inertia is automatically present due to the Timoshenko-type rotation. The viscosity in the Timoshenko NSGT nanobeam is responsible for energy dissipation formulated via negative work. Hamilton’s balance scheme is utilised and resulted in the coupled continuum viscoelastic dynamic model for Timoshenko NSGT nanobeams, which is solved via a FDA for nonlinear mechanics. A nonlinear bending test is conducted via development of a finite element method model in the absence of the NSGT.