Showing papers on "Continuum mechanics published in 2014"

Theory of Mixtures

Abstract: In developing the balance equations for a multicomponent flow, there are two approaches available. Some authors (Whitaker 1967, Slattery 1967) start from the equations of fluid mechanics, valid on the particle scale, and then integrate these equations in regions sufficiently large to contain a representative mass of all components. In case of multiphase particulate flow, those regions must be much greater than the size of the particles contained in the system. The spatially averaged properties then become field variables and the new balance equations constitute a set of local equations describing the flow of a multicomponent mixture. The second approach is the theory of mixtures which uses the concepts of continuum mechanics, considering all components as superimposed interacting continua (Bowen 1976, Truesdell 1984, Dobran 1985). The macroscopic balances are established as the fundamental equations and, from them, the local balances and jump conditions are deduced. The field variables in the continuum approach are equivalent to the averaged variables in the first approach, so that both methods give the same results (Drew 1983). In both cases the local variables cannot be experimentally measured and are not to be confused with the experimental variables of fluid mechanics. In the work that follows, we use the continuum approach of the theory of mixtures.

A Cahn–Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials

Abstract: We formulate a unified framework of balance laws and thermodynamically-consistent constitutive equations which couple Cahn–Hilliard-type species diffusion with large elastic deformations of a body. The traditional Cahn–Hilliard theory, which is based on the species concentration c and its spatial gradient ∇ c , leads to a partial differential equation for the concentration which involves fourth-order spatial derivatives in c; this necessitates use of basis functions in finite-element solution procedures that are piecewise smooth and globally C 1 - continuous . In order to use standard C 0 - continuous finite-elements to implement our phase-field model, we use a split-method to reduce the fourth-order equation into two second-order partial differential equations (pdes). These two pdes, when taken together with the pde representing the balance of forces, represent the three governing pdes for chemo-mechanically coupled problems. These are amenable to finite-element solution methods which employ standard C 0 - continuous finite-element basis functions. We have numerically implemented our theory by writing a user-element subroutine for the widely used finite-element program Abaqus/Standard. We use this numerically implemented theory to first study the diffusion-only problem of spinodal decomposition in the absence of any mechanical deformation. Next, we use our fully coupled theory and numerical-implementation to study the combined effects of diffusion and stress on the lithiation of a representative spheroidal-shaped particle of a phase-separating electrode material.

From fractal media to continuum mechanics

Abstract: This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems.

Thermoelasticity in the framework of the fractional continuum mechanics

Abstract: Fractional continuum mechanics is the generalization of classical mechanics utilizing fractional calculus. Contrary to classical theory, the obtained description is non-local, which is inherently the consequence of the fractional derivative definition based on the interval. So, all fields obtained in the framework of this new formulation, such as temperature, thermal stresses, total stresses, displacements, etc., at the specific point of interest, depend on the information from its surroundings. The dimensions of these surroundings and the ways of influencing the results are governed by the fractional differential operator applied. In this article, the application of the fractional continuum mechanics to thermoelasticity is presented. A classical solution is obtained as a special case.

Elastic Theory of Nanomaterials Based on Surface-Energy Density

Abstract: Recent investigations into surface-energy density of nanomaterials lead to a ripe chance to propose, within the framework of continuum mechanics, a new theory for nanomaterials based on surface-energy density. In contrast to the previous theories, the linearly elastic constitutive relationship that is usually adopted to describe the surface layer of nanomaterials is not invoked and the surface elastic constants are no longer needed in the new theory. Instead, a surface-induced traction to characterize the surface effect in nanomaterials is derived, which depends only on the Eulerian surface-energy density. By considering sample-size effects, residual surface strain, and external loading, an explicit expression for the Lagrangian surface-energy density is achieved and the relationship between the Eulerian surface-energy density and the Lagrangian surface-energy density yields a conclusion that only two material constantsmthe bulk surface-energy density and the surface-relaxation parametermare needed in the new elastic theory. The new theory is further used to characterize the elastic properties of several fcc metallic nanofilms under biaxial tension, and the theoretical results agree very well with existing numerical results. Due to the nonlinear surface effect, nanomaterials may exhibit a nonlinearly elastic property though the inside of nanomaterials or the corresponding bulk one is linearly elastic. Moreover, it is found that externally applied loading should be responsible for the softening of the elastic modulus of a nanofilm. In contrast to the surface elastic constants required by existing theories, the bulk surface-energy density and the surface-relaxation parameter are much easy to obtain, which makes the new theory more convenient for practical applications.

Nonlocal vibration and buckling analysis of single and multi-layered graphene sheets using finite strip method including van der Waals effects

Abstract: Detailed studies on the nanoscale vibration and buckling characteristics of rectangular single and multi-layered graphene sheets (SLGSs and MLGSs) are carried out using semi-analytical finite strip method (FSM), based on the classical plate theory (CPT). The displacement functions of the sheets are evaluated using continuous harmonic function series which satisfy the boundary conditions in one direction and a piecewise interpolation polynomial in the other direction. Nonlocal continuum mechanics is employed to derive the differential equation of the system. The weighted residual method is employed to obtain stiffness, stability and mass matrices of the graphene sheets. The effects of van der Waals (vdW) forces which are present as bonding forces between the layers are considered in the stiffness matrix of the system. The analysis of MLGSs is much more complex due to the influence of vdW forces. The mechanical properties of the graphene sheet are assumed in two ways as orthotropic or isotropic materials. A matrix eigenvalue problem is solved to find the natural frequency and critical stress of GSs subjected to different types of in-plane loadings including uniform and non-uniform uniaxial loadings. The accuracy of the proposed model is validated by comparing the results with those reported by the available references. Furthermore, a comprehensive parametric study is performed to investigate the effects of various parameters such as boundary conditions, nonlocal parameter, aspect ratio and the type of loading on the results.

Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM

Abstract: In this paper, a nonlocal continuum plate model is developed for the transverse vibration of double-piezoelectric-nanoplate systems (DPNPSs) with initial stress under an external electric voltage. The Pasternak foundation model is employed to take into account the effect of shearing between the two piezoelectric nanoplates in combination with normal behavior of coupling elastic medium. Size effects are taken into consideration using nonlocal continuum mechanics. Hamilton׳s principle is used to derive the differential equations of motion. The governing equations are solved for various boundary conditions by using the differential quadrature method (DQM). In addition, exact solutions are presented for the natural frequencies and critical electric voltages of DPNPS under biaxial prestressed conditions in in-phase and out-of-phase vibrational modes. It is shown that the natural frequencies of the DPNPS are quite sensitive to both nonlocal parameter and initial stress. The effects of in-plane preload and small scale are very important in the resonance mode of smart nanostructures using piezoelectric nanoplates.

The morphing method as a flexible tool for adaptive local/non-local simulation of static fracture

Abstract: We introduce a framework that adapts local and non-local continuum models to simulate static fracture problems. Non-local models based on the peridynamic theory are promising for the simulation of fracture, as they allow discontinuities in the displacement field. However, they remain computationally expensive. As an alternative, we develop an adaptive coupling technique based on the morphing method to restrict the non-local model adaptively during the evolution of the fracture. The rest of the structure is described by local continuum mechanics. We conduct all simulations in three dimensions, using the relevant discretization scheme in each domain, i.e., the discontinuous Galerkin finite element method in the peridynamic domain and the continuous finite element method in the local continuum mechanics domain.

Time-dependent elastic response to a local shear transformation in amorphous solids

Abstract: The elastic response of a two-dimensional amorphous solid to induced local shear transformations, which mimic the elementary plastic events occurring in deformed glasses, is investigated via molecular-dynamics simulations. We show that for different spatial realizations of the transformation, despite relative fluctuations of order one, the long-time equilibrium response averages out to the prediction of the Eshelby inclusion problem for a continuum elastic medium. We characterize the effects of the underlying dynamics on the propagation of the elastic signal. A crossover from a propagative transmission in the case of weakly damped dynamics to a diffusive transmission for strong damping is evidenced. In the latter case, the full time-dependent elastic response is in agreement with the theoretical prediction, obtained by solving the diffusion equation for the displacement field in an elastic medium.

Fractional continua for linear elasticity

Abstract: Fractional continua is a generalisation of the classical continuum body. This new concept shows the application of fractional calculus in continuum mechanics. The advantage is that the obtained description is non-local. This natural non-locality is inherently a consequence of fractional derivative definition which is based on the interval, thus variates from the classical approach where the definition is given in a point. In the paper, the application of fractional continua to one-dimensional problem of linear elasticity under small deformation assumption is presented.

Understanding Topographic Dependence of Friction with Micro- and Nano-Grooved Surfaces

Abstract: The work reported in this paper aims at understanding sliding friction anisotropy at the nano-, micro-, and macroscales with respect to surface asperity orientation and exploring the mechanisms behind this phenomenon. Experiments were conducted by probing surfaces with grooves parallel or perpendicular to the direction of relative motion. Continuum mechanics analyses with the FEM and a semi-analytical static friction model and the atomic molecular dynamics simulation were performed for the mechanism exploration. Friction anisotropy was understood from the differences in contact area, surface stiffness, stiction length, and energy barrier from the continuum mechanics prospective and from that in the stick–slip phenomena at the atomic level.

Thermal effects on the stability of circular graphene sheets via nonlocal continuum mechanics

Abstract: Recently, graphene sheets have shown significant potential for environmental engineering applications such as wastewater treatment. Different non-classical theories have been used for modeling of such nano-sized systems to take account of the effect of small length scale. Among all size-dependent theories, the nonlocal elasticity theory has been commonly used to examine the stability of nano-sized structures. Some research works have been reported about the mechanical behavior of rectangular nanoplates with the consideration of thermal effects. However, in comparison with the rectangular graphene sheets, research works about the nanoplates of circular shape are very limited, especially for the buckling properties with thermal effects. Hence, in this paper, an axisymmetric buckling analysis of circular single-layered graphene sheets (SLGS) is presented by decoupling the nonlocal equations of Eringen theory. Constitutive relations are modified to describe the nonlocal effects. The governing equations are derived using equilibrium equations of the circular plate in polar coordinates. Numerical solutions for buckling loads are computed using Galerkin method. It is shown that nonlocal effects play an important role in the buckling of circular nanoplates. The effects of the small scale on the buckling loads considering various parameters such as the radius of the plate, radius-to-thickness ratio, temperature change and mode numbers are investigated.

Differential Geometry And Kinematics Of Continua

Abstract: Introduction Geometric Fundamentals Kinematics of Integrable Deformation Geometry of Anholonomic Deformation Kinematics of Anholonomic Deformation List of Symbols References Index

Crackling noise in plasticity

Abstract: Plastic deformation is a paradigmatic problem of multiscale materials modelling with relevant processes ranging from the atomistic scale up to macroscopic scales where deformation is treated by continuum mechanics. Recent experiments, investigating deformation fluctuations under conditions where plastic deformation was expected to occur in a smooth and stable manner, demonstrate that deformation is spatially heterogeneous and temporally intermittent, not only on atomic scales, where spatial heterogeneity is expected, but also on mesoscopic scales where plastic fluctuations involve collective events of widely different amplitudes. Evidence for crackling noise in plastic deformation comes from acoustic emission measurements and from deformation of micron-scale samples both in crystalline and amorphous materials. Here we provide a detailed account of our current understanding of crackling noise in crystal and amorphous plasticity stemming from experiments, computational models and scaling theories. We focus our attention on the scaling properties of plastic strain bursts and their interpretation in terms of non-equilibrium critical phenomena.

Theoretical modeling of the Casimir force-induced instability in freestanding nanowires with circular cross-section

Abstract: The Casimir force can induce instability and adhesion in freestanding nanostructures. Previous research efforts in this area have exclusively focused on modeling the instability in structures with planar or rectangular cross-section, while, to the best knowledge of the authors, no attention has been paid to investigate this phenomenon for nanowires with circular cross-section. In this study, effects of the Casimir force on the instability and adhesion of freestanding Cylinder–Plate and Cylinder–Cylinder geometries are investigated, which are commonly encountered in real nanodevices. To compute the Casimir force, two approaches, i.e. the proximity force approximation (PFA) for small separations and Dirichlet asymptotic approximation (scattering theory) for large separations, are considered. A continuum mechanics theory is employed, in conjunction with the Euler-beam model, to obtain constitutive equations of the systems. The governing nonlinear constitutive equations of the nanostructures are solved using two different approaches, i.e. the analytical modified Adomian decomposition (MAD) and the numerical finite difference method (FDM). The detachment length and minimum gap, both of which prevent the Casimir force-induced adhesion, are computed for both configurations.

From coherent to incoherent mismatched interfaces: A generalized continuum formulation of surface stresses

Abstract: The equilibrium of coherent and incoherent mismatched interfaces is reformulated in the context of continuum mechanics based on the Gibbs dividing surface concept. Two surface stresses are introduced: a coherent surface stress and an incoherent surface stress, as well as a transverse excess strain. The coherent surface stress and the transverse excess strain represent the thermodynamic driving forces of stretching the interface while the incoherent surface stress represents the driving force of stretching one crystal while holding the other fixed and thereby altering the structure of the interface. These three quantities fully characterize the elastic behavior of coherent and incoherent interfaces as a function of the in-plane strain, the transverse stress and the mismatch strain. The isotropic case is developed in detail and particular attention is paid to the case of interfacial thermo-elasticity. This exercise provides an insight on the physical significance of the interfacial elastic constants introduced in the formulation and illustrates the obvious coupling between the interface structure and its associated thermodynamics quantities. Finally, an example based on atomistic simulations of Cu/Cu2O interfaces is given to demonstrate the relevance of the generalized interfacial formulation and to emphasize the dependence of the interfacial thermodynamic quantities on the incoherency strain with an actual material system.

Isotropic invariants of a completely symmetric third-order tensor

Abstract: In both theoretical and applied mechanics, the modeling of nonlinear constitutive relations of materials is a topic of prime importance. To properly formulate consistent constitutive laws some restrictions need to be impose on tensor functions. To that aim representations theorems for both isotropic and anisotropic functions have been extensively investigated since the middle of the XXth century. Nevertheless, in three-dimensional physical space, most of the results are restricted to sets of tensors up to second-order. The purpose of the present paper is thus to get one step further and to provide an integrity basis for isotropic polynomial functions of a completely symmetric third-order tensor. To explicitly construct this basis, the link that exists between the O(3)-action on harmonic tensors and the SL(2,C)-action on the space of binary forms is exploited. We believe that such an integrity basis may found interesting applications both in continuum mechanics and in other fields of theoretical physics.

An affine continuum mechanical model for cross-linked F-actin networks with compliant linker proteins.

Abstract: Cross-linked actin networks are important building blocks of the cytoskeleton. In order to gain deeper insight into the interpretation of experimental data on actin networks, adequate models are required. In this paper we introduce an affine constitutive network model for cross-linked F-actin networks based on nonlinear continuum mechanics, and specialize it in order to reproduce the experimental behavior of in vitro reconstituted model networks. The model is based on the elastic properties of single filaments embedded in an isotropic matrix such that the overall properties of the composite are described by a free-energy function. In particular, we are able to obtain the experimentally determined shear and normal stress responses of cross-linked actin networks typically observed in rheometer tests. In the present study an extensive analysis is performed by applying the proposed model network to a simple shear deformation. The single filament model is then extended by incorporating the compliance of cross-linker proteins and further extended by including viscoelasticity. All that is needed for the finite element implementation is the constitutive model for the filaments, the linkers and the matrix, and the associated elasticity tensor in either the Lagrangian or Eulerian formulation. The model facilitates parameter studies of experimental setups such as micropipette aspiration experiments and we present such studies to illustrate the efficacy of this modeling approach.

Molecular structural mechanics model for the mechanical properties of microtubules.

Abstract: The aim of this paper was to develop a structural mechanics (SM) model for the microtubules (MTs) in cells. The technique enables one to study the configuration effect on the mechanical properties of MTs and enjoys greatly improved computational efficiency as compared with molecular dynamics simulations. The SM model shows that the Young’s modulus has nearly a constant value around 0.83 GPa, whereas the shear modulus, two orders of magnitude lower, varies considerably with the protofilament number $$N$$ and helix-start number $$S$$ . The dependence of the bending stiffness and persistence length on the MT length and protofilament number $$N$$ is also examined and explained based on the continuum mechanics theories. Specifically, the SM model is found to be in good agreement with available simulation and experiment results, showing its robustness in studying the static deformation of MTs and the potential for characterizing the buckling and vibration of MTs as well as the mechanical behaviour of intermediate and actin filaments.

Geometric continuum mechanics

Abstract: Geometric Continuum Mechanics ( GCM) is a new formulation of Continuum Mechanics ( CM) based on the requirement of Geometric Naturality ( GN) According to GN, in introducing basic notions, governing principles and constitutive relations, the sole geometric entities of space-time to be involved are the metric field and the motion along the trajectory The additional requirement that the theory should be applicable to bodies of any dimensionality, leads to the formulation of the Geometric Paradigm ( GP) stating that push-pull transformations are the natural comparison tools for material fields This basic rule implies that rates of material tensors are Lie-derivatives and not derivatives by parallel transport The impact of the GP on the present state of affairs in CM is decisive in resolving questions still debated in literature and in clarifying theoretical and computational issues As a consequence, the notion of Material Frame Indifference ( MFI) is corrected to the new Constitutive Frame Invariance ( CFI) and reasons are adduced for the rejection of chain decompositions of finite elasto-plastic strains Geometrically consistent notions of Rate Elasticity ( RE) and Rate Elasto-Visco-Plasticity ( REVP) are formulated and consistent relevant computational methods are designed