Showing papers on "Continuum mechanics published in 2019"

On modeling of wave propagation in a thermally affected GNP-reinforced imperfect nanocomposite shell

Abstract: Due to rapid development of process manufacturing, composite materials with porosity have attracted commercial attention in promoting engineering applications. For this regard, in this research wave propagation-thermal characteristics of a size-dependent graphene nanoplatelet-reinforced composite (GNPRC) porous cylindrical nanoshell in thermal environment are investigated. The effects of small scale are analyzed based on nonlocal strain gradient theory (NSGT). The governing equations of the laminated composite cylindrical nanoshell in thermal environment have been evolved using Hamilton’s principle and solved with the assistance of the analytical method. For the first time, wave propagation-thermal behavior of a GNPRC porous cylindrical nanoshell in thermal environment based on NSGT is examined. The results show that by increasing the thickness, the effect of porosity on the phase velocity decreases. Another important result is that by increasing the value of the radius, the difference between the minimum and maximum values of the phase velocity increases. Finally, influence of temperature change, wave number, angular velocity and different types of porosity distribution on phase velocity are investigated using the mentioned continuum mechanics theory. As a useful suggestion, for designing of a GPLRC nanostructure should be attention to the GNP weight function and radius, simultaneously.

Thermodynamically consistent data-driven computational mechanics

Abstract: In the paradigm of data-intensive science, automated, unsupervised discovering of governing equations for a given physical phenomenon has attracted a lot of attention in several branches of applied sciences. In this work, we propose a method able to avoid the identification of the constitutive equations of complex systems and rather work in a purely numerical manner by employing experimental data. In sharp contrast to most existing techniques, this method does not rely on the assumption on any particular form for the model (other than some fundamental restrictions placed by classical physics such as the second law of thermodynamics, for instance) nor forces the algorithm to find among a predefined set of operators those whose predictions fit best to the available data. Instead, the method is able to identify both the Hamiltonian (conservative) and dissipative parts of the dynamics while satisfying fundamental laws such as energy conservation or positive production of entropy, for instance. The proposed method is tested against some examples of discrete as well as continuum mechanics, whose accurate results demonstrate the validity of the proposed approach.

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 minimizes 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 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 in their full three-dimensional generality, accounting for finite curvatures and finite shape changes.

Diffraction-based misorientation mapping: A continuum mechanics description

Abstract: As the typical intragranular misorientation parameters, Kernel Averaged Misorientation (KAM) and Grain Reference Orientation Deviation (GROD) are widely used in diffraction-based misorientation mapping, while their evolution laws under various conditions and physical meanings in continuum mechanics description are rarely investigated systematically. Therefore, we designed several comparative experiments considering the influences of grain boundaries (single-crystalline vs. poly-crystalline), sample geometries (smooth vs. notched) and plastic strain modes (tension vs. buckling) on KAM & GROD evolution, and captured the intragranular misorientation, dislocation density, and material distortion synchronously based on coupled EBSD-ECCI-DIC mapping in this research. Meanwhile, we also discussed the physical meanings of KAM & GROD based on continuum mechanics description and provided the theoretical explanations to phenomena observed in the above experiments. KAM results from three in-surface invariants ( ρ GND I , ρ GND II & ρ GND III ) of GND density tensor ρGND induced by plastic strain distribution incompatible with the activated slip systems in unloaded elastic-plastic condition, which cause the same lattice curvature effects as three elastic strain modes with non-zero curl (buckling, in-surface bending & torsion) in purely elastic condition. GROD reflects neither local plastic strain nor local material rotation alone, but its “V-type” distribution near the neutral surface reflects the buckling curvature of single-crystal. Besides, KAM ¯ & GROD ¯ averaged over multiple grains can be used to estimate the nominal plastic strain applied in poly-crystal.

A finite strain electro-chemo-mechanical theory for ion transport with application to binary solid electrolytes

Abstract: An electro-chemo-mechanical formulation of ion transport in solid electrolytes, in particular for binary systems, is presented. Starting with conservation laws and the second law of thermodynamic, we state a consistent Helmholtz-energy-based framework taking electrostatics, component transport and nonlinear elastic mechanical interaction into account. With the help of finite strain continuum mechanics, we include the effect of geometry changes on ion transport. Changes of local concentration cause swelling and shrinkage and hence stress assisted diffusion. Further coupling originates via an osmotic pressure. Since binary systems are of special interest in battery applications, we formulate both, a fully resolved and an electroneutral model for ion transport. The latter turns out to be an extended version of Newman’s concentrated solution theory taking mechanical effects into account. We demonstrate the importance of these mechanical effects by means of double layers adjacent to blocking electrodes and concentration profiles during galvanostatic charging. Further, we investigate the effect of external deformation as, e.g. found in dendrite growth.

Continuum mechanical modeling of liquid crystal elastomers as dissipative ordered solids

Abstract: Liquid crystal elastomers (LCEs) are special cross-linked polymer materials combining the large elastic deformability of elastomers with the orientational orders of liquid crystals. Here we develop a general framework of LCEs in the language of continuum mechanics to consider the interaction of polymer backbone and Liquid Crystal (LC) microstructure, which is suitable for large deformation and large director rotation. Based on the dissipation principle, the balance of momentum and the evolution equations of orientational order are obtained. In addition to the deformation and its time derivative, the basic kinematic ingredients of this theory are identical to those arising in the director theories and the order tensor theories for nematic fluids. The Cauchy stress consists of not only the bulk stress contribution of the backbone but also the Ericksen–Leslie stress of LC and the obtained rotational momentum balance implies the asymmetric Cauchy stress due to the inhomogeneous director rotation. Based on the principles of objectivity and symmetry, we present the general form of free energy densities and Rayleigh dissipation function and give some possible invariants of constitutive functions. Further, we propose a simple model to study the rate dependence of stretch induced reorientation processes for thin LCE films. Semi-analytical method is utilized to obtain the solutions of constrained uniaxial stretches and stretches with shear for homogenous deformations. The results indicate that the stress-deformation response and the director rotation are rate dependent and can be non-monotonic depending on the initial orientations. Finite element simulations are carried out to study the process of uniaxial stretches with fixed grips. Two different types of stress induced director reorientation processes are observed, one via stripe domains and the other via uniform rotations. We find that the appearance of the stripe domains has a strong dependence on aspect ratios and initial director orientation, which show good agreement with experiment results.

Geometrically incompatible confinement of solids

Abstract: The complex morphologies exhibited by spatially confined thin objects have long challenged human efforts to understand and manipulate them, from the representation of patterns in draped fabric in Renaissance art to current-day efforts to engineer flexible sensors that conform to the human body We introduce a theoretical principle, broadly generalizing Euler’s elastica —a core concept of continuum mechanics that invokes the energetic preference of bending over straining a thin solid object and that has been widely applied to classical and modern studies of beams and rods We define a class of geometrically incompatible confinement problems, whereby the topography imposed on a thin solid body is incompatible with its intrinsic (“target”) metric and, as a consequence of Gauss’ Theorema Egregium , induces strain By focusing on a prototypical example of a sheet attached to a spherical substrate, numerical simulations and analytical study demonstrate that the mechanics is governed by a principle, which we call the “Gauss–Euler elastica ” This emergent rule states that—despite the unavoidable strain in such an incompatible confinement—the ratio between the energies stored in straining and bending the solid may be arbitrarily small The Gauss–Euler elastica underlies a theoretical framework that greatly simplifies the daunting task of solving the highly nonlinear equations that describe thin solids at mechanical equilibrium This development thus opens possibilities for attacking a broad class of phenomena governed by the coupling of geometry and mechanics

Implementation of peridynamic beam and plate formulations in finite element framework

Abstract: Peridynamic (PD) theory is a new continuum mechanics formulation introduced to overcome the limitations of classical continuum mechanics such as predicting crack initiation and propagation, and capturing nonlocal effects. PD theory is based on integro-differential equations and these equations are generally difficult to be solved by using analytical techniques. Therefore, numerical approximations, especially with meshless method, have been widely used. Numerical solution of three-dimensional models is usually computationally expensive and structural idealization can be utilized to reduce the computational time significantly. In this study, two of such structural idealization types are considered, namely Timoshenko beam and Mindlin plate, and their peridynamic formulations are briefly explained. Moreover, the implementation of these formulations in finite element framework is presented. To demonstrate the capability of the present approach, several case studies are considered including beam and plate bending due to transverse loading, buckling analysis and propagation of an initial crack in a plate under bending loading.

Modeling of chemo-hydromechanical behavior of unsaturated porous media: a nonlocal approach based on integral equations

Abstract: Unsaturated clay is a heterogeneous porous medium consisting of three phases, namely solid soil skeleton, pore water, and pore air. It has been well recognized that the variation of the chemical property of pore fluid in clay can affect the hydromechanical behavior of this material remarkably. In this study, we formulate a non-local chemo-hydromechanical model for unsaturated clay via the constitutive correspondence principle in the state-based peridynamics—a reformulation of classical continuum mechanics using integral equations instead of partial differential equations. We numerically implement this non-local constitutive model through the implicit return mapping algorithm at the material particle level and then integrate the material subroutine into a computational peridynamics code. We conduct a series of numerical simulations of unsaturated clay samples under different chemical loading rates. The numerical results demonstrate that the proposed non-local model can capture the dramatic impact of organic chemicals on the mechanical behavior of unsaturated clay. The numerical results also show that the proposed non-local numerical model can simulate localized deformation in chemically active unsaturated clay because of the intrinsic length scale embedded in the integral equations.

A general framework for the modeling of porous ferrogels at finite strains

Abstract: Porous ferrogels are a new class of magneto-active composite materials that deform and alter their material characteristics under the influence of magnetic fields. In the future such materials could find a wide range of application in biomedicine and microfluidics. In this work we present a theoretical and computational framework for the macroscopic, continuum-based modeling of porous ferrogels at finite strains. Departing from the balance laws of continuum mechanics, we derive a thermodynamically consistent model based on Biot’s consolidation theory. Regarding constitutive modeling we limit our attention to isotropic materials. Furthermore we discuss details of the numerical implementation of the coupled three field problem within a nonlinear finite element algorithm. The modeling capabilities and algorithmic performance are demonstrated by means of two representative initial boundary value problems.

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 literature to assess size effects in nanostructures. 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 behaviours of structures involved in modern Nano-Electro-Mechanical-Systems (NEMS).

Soft metamaterials with dynamic viscoelastic functionality tuned by pre-deformation.

Abstract: The small amplitude dynamic response of materials can be tuned by employing inhomogeneous materials capable of large deformation. However, soft materials generally exhibit viscoelastic behaviour, i.e. loss and frequency-dependent effective properties. This is the case for inhomogeneous materials even in the homogenization limit when propagating wavelengths are much longer than phase lengthscales, since soft materials can possess long relaxation times. These media, possessing rich frequency-dependent behaviour over a wide range of low frequencies, can be termed metamaterials in modern terminology. The sub-class that are periodic are frequently termed soft phononic crystals although their strong dynamic behaviour usually depends on wavelengths being of the same order as the microstructure. In this paper we describe how the effective loss and storage moduli associated with longitudinal waves in thin inhomogeneous rods are tuned by pre-stress. Phases are assumed to be quasi-linearly viscoelastic, thus exhibiting time-deformation separability in their constitutive response. We illustrate however that the effective incremental response of the inhomogeneous medium does not exhibit time-deformation separability. For a range of nonlinear materials it is shown that there is strong coupling between the frequency of the small amplitude longitudinal wave and initial large deformation. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

Continuum mechanics with torsion

Abstract: This paper is an attempt to introduce methods and concepts of the Riemann–Cartan geometry largely used in such physical theories as general relativity, gauge theories, solid dynamics to fluid dynamics in general and to studying and modeling turbulence in particular. Thus, in order to account for the rotational degrees of freedom of the irregular dynamics of small-scale vortexes, we further generalize our unified first-order hyperbolic formulation of continuum fluid and solid mechanics which treats the flowing medium as a Riemann–Cartan manifold with zero curvature but non-vanishing torsion. We associate the rotational degrees of freedom of the main field of our theory, the distortion field, to the dynamics of microscopic (unresolved) vortexes. The distortion field characterizes the deformation and rotation of the material elements and can be viewed as anholonomic basis triad with non-vanishing torsion. The torsion tensor is then used to characterize distortion’s spin and is treated as an independent field with its own time evolution equation. This new governing equation has essentially the structure of the nonlinear electrodynamics in a moving medium and can be viewed as a Yang–Mills-type gauge theory. The system is closed by providing an example of the total energy potential. The extended system describes not only irreversible dynamics (which raises the entropy) due to the viscosity or plasticity effect, but it also has dispersive features which are due to the reversible energy exchange (which conserves the entropy) between micro- and macroscales. Both the irreversible and dispersive processes are represented by relaxation-type algebraic source terms so that the overall system remains first-order hyperbolic. The turbulent state is then treated as an excitation of the equilibrium (laminar) state due to the nonlinear interplay between dissipation and dispersion.

Characterization and behaviors of single walled carbon nanotube by equivalent-continuum mechanics approach

Abstract: This paper focuses on two main objectives. The first one is to exploit an energy equivalent model and finite element method to evaluate the equivalent Young's modulus of single walled carbon nanotubes (SWCNTs) at any orientation angle by using tensile test. The calculated Young's modulus is validated with published experimental results. The second target is to exploit the finite element simulation to investigate mechanical buckling and natural frequencies of SWCNTs. Energy equivalent model is presented to describe the atomic bonding interactions and their chemical energy with mechanical structural energies. A Program of Nanotube modeler is used to generate a geometry of SWCNTs structure by defining its chirality angle, overall length of nanotube and bond length between two adjacent nodes. SWCNTs are simulated as a frame like structure; the bonds between each two neighboring atoms are treated as isotropic beam members with a uniform circular cross section. Carbon bonds is simulated as a beam and the atoms as nodes. A finite element model using 3D beam elements is built under the environment of ANSYS MAPDL environment to simulate a tensile test and characterize equivalent Young

Static chiral Willis continuum mechanics for three-dimensional chiral mechanical metamaterials

Abstract: Recent static experiments on twist effects in chiral three-dimensional mechanical metamaterials have been discussed in the context of micropolar Eringen continuum mechanics, which is a generalization of linear Cauchy elasticity. For cubic symmetry, Eringen elasticity comprises nine additional parameters with respect to linear Cauchy elasticity, of which three directly influence chiral effects. Here, we discuss the behavior of the static case of an alternative generalization of linear Cauchy elasticity, the Willis equations. We show that in the homogeneous static cubic case, only one additional parameter with respect to linear Cauchy elasticity results, which directly influences chiral effects. We show that the static Willis equations qualitatively describe the experimentally observed chiral twist effects, too. We connect the behavior to a characteristic length scale.

Thermodynamics-based stability criteria for constitutive equations of isotropic hyperelastic solids

Abstract: A generalized thermodynamic stability criterion for isotropic finite elastic solids is derived using the fundamental balance laws and field equations of continuum mechanics, which is then used to formulate constitutive inequalities for the polynomial form of hyperelastic constitutive equations. Individual thermodynamic constitutive inequalities (called T-C inequalities) are derived for the neo-Hookean, Mooney Rivlin, and three-parameter generalized Rivlin models under three pure homogeneous deformation modes, namely, uniaxial compression, uniaxial tension and shear (simple and pure), and are compared against two commonly used adscititious inequalities, the Baker-Ericksen (B-E) and E-inequalities. The range of stable model constants as defined by the T-C inequalities is represented by a region in an N-dimensional coordinate space (N is the total number of model constants), which is defined as the Region of Stability (ROS). It is shown that the ROS is a function of material deformation and evolves with the limiting strain, shrinking from an initially large region representing the necessary condition of thermodynamic stability to a converged region under infinite limiting strain that is equivalent to the ROS defined by the E-inequalities. By investigating the evolution of the ROS under different deformation modes, the implication of T-C inequalities on the selection of experimental routines and filtering of erroneous test data and model constants is discussed. It is also demonstrated that while the E-inequalities are over-restrictive for hyperelastic materials with small to moderate limiting strains, an observation supported by recent experimental evidence, the B-E inequalities are inaccurate under moderate to large limiting strain conditions. The applicability of the proposed mathematical framework to other hyperelastic strain energy density forms, such as exponential/logarithmic functions, is demonstrated by investigating the thermodynamic stability of the Fung-Demiray model. It is shown that the commonly assumed restriction that the Fung-Demiray model constants must be positive can be relaxed so that some typical material behaviors under small to moderate limiting strains can also be modeled.

Evaporation-driven crumpling and assembling of two-dimensional (2D) materials: A rotational spring – mechanical slider model

Abstract: Crumpling of suspended two-dimensional (2D) materials by droplet evaporation creates a new form of aggregation-resistant ultrafine particles with more scalable properties such as high specific surface areas. However, the underpinned fundamental mechanics theory that addresses large deformation, severe instability and self-assembly of 2D sheets under dynamic solid-liquid interactions during liquid evaporation is lacking. In the present study, we propose a theoretical mechanics framework to quantitatively describe the simultaneous process of crumpling and self-assembling of 2D materials and their competition during droplet evaporation. In this theory, a rotational spring is developed to describe the out-of-plane deformation of crumpling sheets, and a mechanical slider is implemented to describe the interactive binding energy in self-folding of a single sheet or overlapping of neighboring sheets. The spring-slider mechanics model is calibrated with the energy-based continuum mechanics analysis by crumpling a single sheet, and is further extended to a network model to characterize the crumpling and assembling of multiple sheets in the droplet. An equivalent pressure model is developed to unify the resultant forces associated with liquid evaporation including capillary force, vapor pressure, gas pressure, vapor recoil pressure and capillary flow-induced force. A coarse-grained model of 2D materials is developed and its dynamic interaction with liquid molecules during evaporation is mimicked by proposing a controllable virtual van der Waal force field. Molecular dynamics simulation results show remarkable agreement with theoretical predictions, from crumpling and assembling energies of graphene during liquid evaporation to overall size and accessible area of the crumpled particles after the complete evaporation of liquid. Besides, both theoretical predictions and simulation results agree well with independent experiments. The effect of concentration, size, shape, number and size distribution of 2D material graphene sheets in liquid droplets on crumpling and self-assembling energy and shape, size and surface morphology of crumpled particles is also discussed. The mechanics theories and coarse-grained modeling established here are expected to offer immediate and quantitative application guidance to control the crumpling and self-assembling process of 2D materials by liquid solution evaporation processing to fine tune particle size and morphology. More importantly, the fundamental understanding of large deformation, instability, and self-assembly of 2D materials in such dynamic liquid environments could be extended to aerosol-like processing of a broad scope of other low-dimensional nanomaterials such as lipid membranes, nanowires, nanotubes, nanofibers and nanoparticles, for their emerging applications including ultrafine particle manufacturing and various printing processes.

Computational Continuum Mechanics of Nanoscopic Structures: Nonlocal Elasticity Approaches

Abstract: From the content: Fundamental tenets of nanomechanics -- Fundamental notions from classical continuum mechanics -- Essential concepts from nonlocal elasticity theory -- Nonlocal modelling of nanoscopic structures -- Elastic properties of carbon-based nanoscopic structures.

Quasi-Static and High Strain Rate Simple Shear Characterization of Soft Polymers

Abstract: The simple shear response of soft polymers under large deformation (>50%) and strain rates spanning 10−3 – 103 s−1 is characterized by developing quasi-static and split-Hopkinson pressure bar based single-pulse dynamic simple shear experiments rooted in continuum mechanics fundamentals. Cross-linked polydimethylsiloxane (PDMS) is chosen as a model material. By examining the evolution of stress, strain and strain rate, the latter two parameters measured using two-dimensional digital image correlation (DIC), it is demonstrated that dynamic simple shear deformation consists of four distinct stages: momentum diffusion, inertia effect, steady-state material response, and strain rate decay. By isolating the unsteady and steady-state deformation stages, inertia-free material response is captured under a uniform strain rate. It is shown that the shear response of PDMS is nearly linear with a weakly rate-sensitive shear modulus in the investigated strain rate range. Further, by analyzing the DIC strain-field and comparing the kinematic experimental results with those predicted by classical continuum mechanics, it is demonstrated that the proposed experiments not only achieve a nearly theoretical simple shear state that is uniform across the specimen, but also allow for post-test validation of individual experiments based on these criteria.

Mechanical modeling of deepwater flexible structures with large deformation based on absolute nodal coordinate formulation

Abstract: In this paper, a mechanical analysis model is proposed on basis of the absolute nodal coordinate formulation (ANCF) and the theories of continuum mechanics and finite element method to accurately analyze the statics and dynamics of deepwater flexible structures with large deformation. In this model, the traditional angle coordinate is replaced with slope coordinate under the frame of overall coordinate system. The mapping relation of the parameters under current and reference configurations is established, and the method of describing the nonlinear geometric relationship of the element with the current configuration parameters is discussed. Then, based on the energy variation principle, the generalized elastic force and stiffness matrix of the element are derived, and the mass matrix and external load matrix of the element are combined to perform the element assembling using the finite element method, and the static and dynamic equilibrium equations are then formed. The calculation programs are compiled by FORTRAN language, whose reliability and accuracy are checked by the cases of beam model with theoretical solutions. Finally, a kind of steel lazy wave catenary riser is taken as an example, and its static and dynamic characteristics are analyzed systematically, which further verifies the effectiveness and practicability of the mechanical model.