Showing papers in "Continuum Mechanics and Thermodynamics in 2016"

A hyperbolic model for viscous Newtonian flows

Abstract: We discuss a pure hyperbolic alternative to the Navier–Stokes equations, which are of parabolic type. As a result of the substitution of the concept of the viscosity coefficient by a microphysics-based temporal characteristic, particle settled life (PSL) time, it becomes possible to formulate a model for viscous fluids in a form of first-order hyperbolic partial differential equations. Moreover, the concept of PSL time allows the use of the same model for flows of viscous fluids (Newtonian or non-Newtonian) as well as irreversible deformation of solids. In the theory presented, a continuum is interpreted as a system of material particles connected by bonds; the internal resistance to flow is interpreted as elastic stretching of the particle bonds; and a flow is a result of bond destructions and rearrangements of particles. Finally, we examine the model for simple shear flows, arbitrary incompressible and compressible flows of Newtonian fluids and demonstrate that Newton’s viscous law can be obtained in the framework of the developed hyperbolic theory as a steady-state limit. A basic relation between the viscosity coefficient, PSL time, and the shear sound velocity is also obtained.

Granular micromechanics based micromorphic model predicts frequency band gaps

Abstract: Granular materials are typically characterized by complex structure and composition. Continuum modeling, therefore, remains the mainstay for describing properties of these material systems. In this paper, we extend the granular micromechanics approach by considering enhanced kinematic analysis. In this analysis, a decomposition of the relative movements of interacting grain pairs into parts arising from macro-scale strain as well as micro-scale strain measures is introduced. The decomposition is then used to formulate grain-scale deformation energy functions and derive inter-granular constitutive laws. The macro-scale deformation energy density is defined as a summation of micro-scale deformation energy defined for each interacting grain pair. As a result, a micromorphic continuum model for elasticity of granular media is derived and applied to investigate the wave propagation behavior. Dispersion graphs for different cases and different ratios between the microscopic stiffness parameters have been presented. It is seen that the model has the capability to present band gaps over a large range of wave numbers.

A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model

Abstract: A one-dimensional displacement second-gradient damage continuum theory has been already presented within the framework of a variational approach. Damage is associated with strain concentration. Therefore, not only non-local effects of displacement second-gradient modelling should be considered in a comprehensive model, but also any plastic effects. The aim of this paper is therefore to extend such a model to plasticity. The action is intended to depend not only with respect to first and second gradient of displacement field and to a scalar damage field, but also to further two internal variables, i.e. the accumulated plastic tension and the accumulated plastic compression. A constitutive prescription on the stiffness is given in terms of the scalar damage parameter in a usual way, i.e. as in many other works, it is prescribed to decrease as far as the damage increases. On the other hand, the microstructural material length (i.e. the square of the constitutive function in front of the squared displacement second-gradient term in the action functional) is prescribed to increase as far as the damage increases, being this last assumption connected to the interpretation that a damage state induces a microstructure in the continuum and that such a microstructure is more important as far as the damage increases. Initial damage threshold and yield stresses are naturally introduced in the action in front of linear terms, respectively, of damage and plastic internal variables. The hardening matrix is also introduced in a natural way as the coefficient matrix in front of the quadratic terms of the two plastic internal variables. At a given value of damage and plastic parameters, the behaviour is referred to second-gradient linear elastic material. However, the damage and plastic evolutions make the model not only nonlinear, but also inelastic. The second principle of thermodynamics is considered by assuming that the scalar damage and plastic parameters do not decrease their values in the process of deformation, and this implies a dissipation for the elastic strain energy. A novel result of this investigation, where displacement second-gradient and plastic effects are combined, is that the distributed and concentrated external double forces do not make work on the displacement gradient but only to its elastic part and this means that the displacement gradient cannot be prescribed, at the border, independently of the plastic internal variables.

Isogeometric analysis: a powerful numerical tool for the elastic analysis of historical masonry arches

Abstract: We illustrate a numerical tool for analyzing plane arches such as those frequently used in historical masonry heritage. It is based on a refined elastic mechanical model derived from the isogeometric approach. In particular, geometry and displacements are modeled by means of non-uniform rational B-splines. After a brief introduction, outlining the basic assumptions of this approach and the corresponding modeling choices, several numerical applications to arches, which are typical of masonry structures, show the performance of this novel technique. These are discussed in detail to emphasize the advantage and potential developments of isogeometric analysis in the field of structural analysis of historical masonry buildings with complex geometries.

Existence and stability results for thermoelastic dipolar bodies with double porosity

Abstract: This paper is concerned with the theory of thermoelastic dipolar bodies which have a double porosity structure. In contrast with previous papers dedicated to classical elastic bodies, in our context the double porosity structure of the body is influenced by the displacement field, which is consistent with real models. In this setting, we show instability of solution as the initial energy is negative while under an appropriated (and realistic) condition, we prove existence and uniqueness of solution using semi-group theory.

Extension, inflation and torsion of a residually stressed circular cylindrical tube

Abstract: In this paper, we provide a new example of the solution of a finite deformation boundary-value problem for a residually stressed elastic body. Specifically, we analyse the problem of the combined extension, inflation and torsion of a circular cylindrical tube subject to radial and circumferential residual stresses and governed by a residual-stress dependent nonlinear elastic constitutive law. The problem is first of all formulated for a general elastic strain-energy function, and compact expressions in the form of integrals are obtained for the pressure, axial load and torsional moment required to maintain the given deformation. For two specific simple prototype strain-energy functions that include residual stress, the integrals are evaluated to give explicit closed-form expressions for the pressure, axial load and torsional moment. The dependence of these quantities on a measure of the radial strain is illustrated graphically for different values of the parameters (in dimensionless form) involved, in particular the tube thickness, the amount of torsion and the strength of the residual stress. The results for the two strain-energy functions are compared and also compared with results when there is no residual stress.

Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity

Abstract: Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin–Murdoch model to the Steigmann–Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and methods of functional analysis. The Steigmann–Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin–Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin–Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann–Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected.

A non-classical Kirchhoff plate model incorporating microstructure, surface energy and foundation effects

Abstract: A new non-classical Kirchhoff plate model is developed using a modified couple stress theory, a surface elasticity theory and a two-parameter elastic foundation model. A variational formulation based on Hamilton’s principle is employed, which leads to the simultaneous determination of the equations of motion and the complete boundary conditions and provides a unified treatment of the microstructure, surface energy and foundation effects. The new plate model contains a material length scale parameter to account for the microstructure effect, three surface elastic constants to describe the surface energy effect, and two foundation moduli to represent the foundation effect. The current non-classical plate model reduces to its classical elasticity-based counterpart when the microstructure, surface energy and foundation effects are all suppressed. In addition, the newly developed plate model includes the models considering the microstructure dependence or the surface energy effect or the foundation influence alone as special cases and recovers the Bernoulli–Euler beam model incorporating the microstructure, surface energy and foundation effects. To illustrate the new model, the static bending and free vibration problems of a simply supported rectangular plate are analytically solved by directly applying the general formulas derived. For the static bending problem, the numerical results reveal that the deflection of the simply supported plate with or without the elastic foundation predicted by the current model is smaller than that predicted by the classical model. Also, it is observed that the difference in the deflection predicted by the new and classical plate models is very large when the plate thickness is sufficiently small, but it is diminishing with the increase of the plate thickness. For the free vibration problem, it is found that the natural frequency predicted by the new plate model with or without the elastic foundation is higher than that predicted by the classical plate model, and the difference is significant for very thin plates. These predicted trends of the size effect at the micron scale agree with those observed experimentally. In addition, it is shown both analytically and numerically that the presence of the elastic foundation reduces the plate deflection and increases the plate natural frequency, as expected.

Micromorphic continua: non-redundant formulations

Abstract: The kinematics of generalized continua is investigated and key points concerning the definition of overall tangent strain measure are put into evidence. It is shown that classical measures adopted in the literature for micromorphic continua do not obey a constraint qualification requirement, to be fulfilled for well-posedness in optimization theory, and are therefore termed redundant. Redundancy of continua with latent microstructure and of constrained Cosserat continua is also assessed. A simplest, non-redundant, kinematic model of micromorphic continua, is proposed by dropping the microcurvature field. The equilibrium conditions and the related variational linear elastostatic problem are formulated and briefly discussed. The simplest model involves a reduced number of state variables and of elastic constitutive coefficients, when compared with other models of micromorphic continua, being still capable of enriching the Cauchy continuum model in a significant way.

Modeling rammed earth wall using discrete element method

Abstract: Rammed earth is attracting renewed interest throughout the world thanks to its “green” characteristics in the context of sustainable development. Several research studies have thus recently been carried out to investigate this material. Some of them attempted to simulate the rammed earth’s mechanical behavior by using analytical or numerical models. Most of these studies assumed that there was a perfect cohesion at the interface between earthen layers. This hypothesis proved to be acceptable for the case of vertical loading, but it could be questionable for horizontal loading. To address this problem, discrete element modeling seems to be relevant to simulate a rammed earth wall. To our knowledge, no research has been conducted thus far using discrete element modeling to study a rammed earth wall. This paper presents an assessment of the discrete element modeling’s robustness for rammed earth walls. Firstly, a brief description of the discrete element modeling is presented. Then the parameters necessary for discrete element modeling of the material law of the earthen layers and their interfaces law following the Mohr–Coulomb model with a tension cut-off and post-peak softening were given. The relevance of the model and the material parameters were assessed by comparing them with experimental results from the literature. The results showed that, in the case of vertical loading, interfaces did not have an important effect. In the case of diagonal loading, model with interfaces produced better results. Interface characteristics can vary from 85 to 100% of the corresponding earthen layer’s characteristics.

The influence of different loads on the remodeling process of a bone and bioresorbable material mixture with voids

Abstract: A model of a mixture of bone tissue and bioresorbable material with voids was used to numerically analyze the physiological balance between the processes of bone growth and resorption and artificial material resorption in a plate-like sample. The adopted model was derived from a theory for the behavior of porous solids in which the matrix material is linearly elastic and the interstices are void of material. The specimen—constituted by a region of bone living tissue and one of bioresorbable material—was acted by different in-plane loading conditions, namely pure bending and shear. Ranges of load magnitudes were identified within which physiological states become possible. Furthermore, the consequences of applying different loading conditions are examined at the end of the remodeling process. In particular, maximum value of bone and material mass densities, and extensions of the zones where bone is reconstructed were identified and compared in the two different load conditions. From the practical view point, during surgery planning and later rehabilitation, some choice of the following parameters is given: porosity of the graft, material characteristics of the graft, and adjustment of initial mixture tissue/bioresorbable material and later, during healing and remodeling, optimal loading conditions.

Wave reflection at a free interface in an anisotropic pyroelectric medium with nonclassical thermoelasticity

Abstract: In this paper, the well-established two-dimensional mathematical model for linear pyroelectric materials is employed to investigate the reflection of waves at the boundary between a vacuum and an elastic, transversely isotropic, pyroelectric material. A comparative study between the solutions of (a) classical thermoelasticity, (b) Cattaneo–Lord–Shulman theory and (c) Green–Lindsay theory equations, characterised by none, one and two relaxation times, respectively, is presented. Suitable boundary conditions are considered in order to determine the reflection coefficients when incident elasto–electro–thermal waves impinge the free interface. It is established that, in the quasi-electrostatic approximation, three different classes of waves: (1) two principally elastic waves, namely a quasi-longitudinal Primary (qP) wave and a quasi-transverse Secondary (qS) wave; and (2) a mainly thermal (qT) wave. The observed electrical effects are, on the other hand, a direct consequence of mechanical and thermal phenomena due to pyroelectric coupling. The computed reflection coefficients of plane qP waves are found to depend upon the angle of incidence, the elastic, electric and thermal parameters of the medium, as well as the thermal relaxation times. The special cases of normal and grazing incidence are also derived and discussed. Finally, the reflection coefficients are computed for cadmium selenide observing the influence of (1) the anisotropy of the material, (2) the electrical potential and (3) temperature variations and (4) the thermal relaxation times on the reflection coefficients.

Identification of multiple open and fatigue cracks in beam-like structures using wavelets on deflection signals

Abstract: A novel method for damage detection of multi-cracked beam-like structures by analyzing the static deflection is presented. The damage incurred produces a change in the stiffness of the beam. This causes a localized singularity which can be identified by a wavelet analysis of the displacement response. The existence and location of the cracks can be revealed by positions of the peaks in the continuous wavelet transform (CWT). To achieve this, the static profile of beams is analyzed with Gauss2 wavelet to identify the cracks. Beams under some ideal boundary and prescribed load conditions are considered. The deflected shape of the beam with open and fatigue cracks has been simulated under static loading using lumped crack models adopted from fracture mechanics and involving various degrees of complexity. The deflection of cracked beam in closed form for several cases of loads, crack sizes, and crack locations is calculated, and an explicit expression for the damage index (DI), based on CWT, is developed; it is demonstrated that the proposed damage index does not depend on mechanical properties of a homogeneous beam, and that the DI of one crack does not depend on the size and location of other cracks in a multiple cracked beam. Hence, the obtained expression for the DI can be used to find the size of each crack independently. Numerical results show that the method can detect cracks of small depth and is also applicable under the presence of measurement noise.

An energy-based method to determine material constants in nonlinear rheology with applications

Abstract: Many polymer-type materials show a rate-dependent and nonlinear rheological behavior. Such a response may be modeled by using a series of spring-dashpot systems. However, in order to cover different time scales the number of systems may become unreasonably large. A more appropriate treatment based on continuum mechanics will be presented herein. This approach uses representation theorems for deriving material equations and allows for a systematic increase in modeling complexity. Moreover, we propose an approach based on energy to determine thematerial parameters.This method results in a simple linear regression problemeven for highly nonlinearmaterial equations. Therefore, the inverse problem leads to a unique solution. The significance of the proposed method is that the stored and dissipated energies necessary for the procedure are measurable quantities. We apply the proposed method to a “semi-solid” material and measure its material parameters by using a simple-shear rheometer.

Closure conditions for non-equilibrium multi-component models

Abstract: A class of non-equilibrium models for compressible multi-component fluids in multi-dimensions is investigated taking into account viscosity and heat conduction. These models are subject to the choice of interfacial pressures and interfacial velocity as well as relaxation terms for velocity, pressure, temperature and chemical potentials. Sufficient conditions are derived for these quantities that ensure meaningful physical properties such as a non-negative entropy production, thermodynamical stability, Galilean invariance and mathematical properties such as hyperbolicity, subcharacteristic property and existence of an entropy–entropy flux pair. For the relaxation of chemical potentials, a two-component and a three-component models for vapor–water and gas–water–vapor, respectively, are considered.

A strain-consistent elastic plate model with surface elasticity

Abstract: A strain-consistent elastic plate model is formulated in which both initial surface tension and the induced residual stress are treated as finite values, and the exactly same strain expressions are consistently employed for both the surface and the bulk plate. Different than most of previous related models which follow the original Gurtin–Murdoch model and include some non-strain displacement gradient terms (which cannot be expressed in terms of the surface infinitesimal strains or the von Karman-type strains) in the surface stress–strain relations, the present model does not include any non-strain displacement gradient terms in the surface stress–strain relations. For a free elastic plate with in-plane movable edges, the present model predicts that initial surface tension exactly cancels out the induced residual compressive stress. On the other hand, if all edges are in-plane immovable, residual stress cannot develop in the plate and the initial surface tension causes a tensile net membrane force. In addition, the present model predicts that initial surface tension reduces the effective bending rigidity of the plate, while this reduction does not depend on Poisson ratio. In particular, self-buckling of a free elastic plate under tensile surface tension cannot occur unless the effective bending rigidity of plate vanishes or becomes negative.

The role of several heat transfer mechanisms on the enhancement of thermal conductivity in nanofluids

Abstract: A modelling of the thermal conductivity of nanofluids based on extended irreversible thermodynamics is proposed with emphasis on the role of several coupled heat transfer mechanisms: liquid interfacial layering between nanoparticles and base fluid, particles agglomeration and Brownian motion. The relative importance of each specific mechanism on the enhancement of the effective thermal conductivity is examined. It is shown that the size of the nanoparticles and the liquid boundary layer around the particles play a determining role. For nanoparticles close to molecular range, the Brownian effect is important. At nanoparticles of the order of 1–100 nm, both agglomeration and liquid layering are influent. Agglomeration becomes the most important mechanism at nanoparticle sizes of the order of 100 nm and higher. The theoretical considerations are illustrated by three case studies: suspensions of alumina rigid spherical nanoparticles in water, ethylene glycol and a 50/50w% water/ethylene glycol mixture, respectively, good agreement with experimental data is observed.

Micropolar continuum in spatial description

Abstract: Within the spatial description, it is customary to refer thermodynamic state quantities to an elementary volume fixed in space containing an ensemble of particles. During its evolution, the elementary volume is occupied by different particles, each having its own mass, tensor of inertia, angular and linear velocities. The aim of the present paper is to answer the question of how to determine the inertial and kinematic characteristics of the elementary volume. In order to model structural transformations due to the consolidation or defragmentation of particles or anisotropic changes, one should consider the fact that the tensor of inertia of the elementary volume may change. This means that an additional constitutive equation must be formulated. The paper suggests kinetic equations for the tensor of inertia of the elementary volume. It also discusses the specificity of the inelastic polar continuum description within the framework of the spatial description.

The exponentiated Hencky-logarithmic strain energy: part III—coupling with idealized multiplicative isotropic finite strain plasticity

Abstract: We investigate an immediate application in finite strain multiplicative plasticity of the family of isotropic volumetric–isochoric decoupled strain energies $$F \mapsto W_{\rm eH}(F):= \widehat{W}_{\rm eH}(U) := \left\{ \begin{array}{lll} \frac{\mu}{k}\,e^{k\, \| {\rm dev}_n \log {U}\|^2}+\frac{\kappa}{2\, {\widehat{k}}}\,e^{\widehat{k}\,[ {\rm tr}(\log U)]^2}&\quad \text{if}& \det\, F > 0,\\ + \infty & \quad \text{if} & \det F \leq 0,\end{array} \right.$$ based on the Hencky-logarithmic (true, natural) strain tensor $${\log U}$$ . Here, $${\mu > 0}$$ is the infinitesimal shear modulus, $${\kappa=\frac{2 \mu+3\lambda}{3} > 0}$$ is the infinitesimal bulk modulus with λ the first Lame constant, $${k,\widehat{k}}$$ are additional dimensionless material parameters, $${F= abla \varphi}$$ is the gradient of deformation, $${U=\sqrt{F^T F}}$$ is the right stretch tensor, and dev n $${\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot{\mathbb{1}}}$$ is the deviatoric part of the strain tensor $${\log U}$$ . Based on the multiplicative decomposition $${F=F_e\, F_p}$$ , we couple these energies with some isotropic elasto-plastic flow rules $${F_p\,\frac{\rm d}{{\rm d t}}[F_p^{-1}]\in-\partial \chi({\rm dev}_3 \Sigma_{e})}$$ defined in the plastic distortion F p , where $${\partial \chi}$$ is the subdifferential of the indicator function $${\chi}$$ of the convex elastic domain $${\mathcal{E}_{\rm e}({\Sigma_{e}},\frac{1}{3}{\boldsymbol{\sigma}}_{\mathbf{y}}^2)}$$ in the mixed-variant $${\Sigma_{e}}$$ -stress space, $${\Sigma_{e}=F_e^T D_{F_e}W_{\rm iso}(F_e)}$$ , and $${W_{\rm iso}(F_e)}$$ represents the isochoric part of the energy. While $${W_{\rm eH}}$$ may loose ellipticity, we show that loss of ellipticity is effectively prevented by the coupling with plasticity, since the ellipticity domain of $${W_{\rm eH}}$$ on the one hand and the elastic domain in $${\Sigma_{e}}$$ -stress space on the other hand are closely related. Thus, the new formulation remains elliptic in elastic unloading at any given plastic predeformation. In addition, in this domain, the true stress–true strain relation remains monotone, as observed in experiments.

A poroplastic model of structural reorganisation in porous media of biomechanical interest

Abstract: We present a poroplastic model of structural reorganisation in a binary mixture comprising a solid and a fluid phase. The solid phase is the macroscopic representation of a deformable porous medium, which exemplifies the matrix of a biological system (consisting e.g. of cells, extracellular matrix, collagen fibres). The fluid occupies the interstices of the porous medium and is allowed to move throughout it. The system reorganises its internal structure in response to mechanical stimuli. Such structural reorganisation, referred to as remodelling, is described in terms of “plastic” distortions, whose evolution is assumed to obey a phenomenological flow rule driven by stress. We study the influence of remodelling on the mechanical and hydraulic behaviour of the system, showing how the plastic distortions modulate the flow pattern of the fluid, and the distributions of pressure and stress inside it. To accomplish this task, we solve a highly nonlinear set of model equations by elaborating a previously developed numerical procedure, which is implemented in a non-commercial finite element solver.

Quadratic and rate-independent limits for a large-deviations functional

Abstract: We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth–death process on a lattice, with rates derived from Kramers’ law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits, we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via ‘L log L’ gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.

Neglected transport equations: extended Rankine–Hugoniot conditions and J -integrals for fracture

Abstract: Transport equations in integral form are well established for analysis in continuum fluid dynamics but less so for solid mechanics. Four classical continuum mechanics transport equations exist, which describe the transport of mass, momentum, energy and entropy and thus describe the behaviour of density, velocity, temperature and disorder, respectively. However, one transport equation absent from the list is particularly pertinent to solid mechanics and that is a transport equation for movement, from which displacement is described. This paper introduces the fifth transport equation along with a transport equation for mechanical energy and explores some of the corollaries resulting from the existence of these equations. The general applicability of transport equations to discontinuous physics is discussed with particular focus on fracture mechanics. It is well established that bulk properties can be determined from transport equations by application of a control volume methodology. A control volume can be selected to be moving, stationary, mass tracking, part of, or enclosing the whole system domain. The flexibility of transport equations arises from their ability to tolerate discontinuities. It is insightful thus to explore the benefits derived from the displacement and mechanical energy transport equations, which are shown to be beneficial for capturing the physics of fracture arising from a displacement discontinuity. Extended forms of the Rankine–Hugoniot conditions for fracture are established along with extended forms of J -integrals.

Material degradation due to moisture and temperature. Part 1: mathematical model, analysis, and analytical solutions

Abstract: The mechanical response, serviceability, and load-bearing capacity of materials and structural components can be adversely affected due to external stimuli, which include exposure to a corrosive chemical species, high temperatures, temperature fluctuations (i.e., freezing–thawing), cyclic mechanical loading, just to name a few. It is, therefore, of paramount importance in several branches of engineering—ranging from aerospace engineering, civil engineering to biomedical engineering—to have a fundamental understanding of degradation of materials, as the materials in these applications are often subjected to adverse environments. As a result of recent advancements in material science, new materials such as fiber-reinforced polymers and multi-functional materials that exhibit high ductility have been developed and widely used, for example, as infrastructural materials or in medical devices (e.g., stents). The traditional small-strain approaches of modeling these materials will not be adequate. In this paper, we study degradation of materials due to an exposure to chemical species and temperature under large strain and large deformations. In the first part of our research work, we present a consistent mathematical model with firm thermodynamic underpinning. We then obtain semi-analytical solutions of several canonical problems to illustrate the nature of the quasi-static and unsteady behaviors of degrading hyperelastic solids.

The three-hinged arch as an example of piezomechanic passive controlled structure

Abstract: Although piezoelectric transducers are employed in a variety of fields, their application for vibration control of civil or industrial structures has not yet been fully developed, at the best of authors’ knowledge. Thanks to a new generation of ever more performing piezoceramic materials and to the recent development of scientific proposals based on a very simple technology, this paper presents a step forward to engineering applications for the control of structural systems. A three-hinged arch controlled by piezoelectric stack actuators and passive RL electrical circuits is chosen as a simple structural model that may represent the starting point for a generalization to the most common typologies of civil and industrial engineering structures. Based on the concept of electromechanical analogy, the evolution equations are obtained through a consistent Lagrangian approach. A multimodal vibration suppression is guaranteed by the spectral analogy between the mechanical and electrical components. Preliminary applications related to free oscillations, with one or more actuators on each member, seem to lead to excellent performance in terms of multimodal damping and dissipated energy.

Double porosity in fluid-saturated elastic media: deriving effective parameters by hierarchical homogenization of static problem

Abstract: We propose a model of complex poroelastic media with periodic or locally periodic structures observed at microscopic and mesoscopic scales Using a two-level homogenization procedure, we derive a model coherent with the Biot continuum, describing effective properties of such a hierarchically structured poroelastic medium The effective material coefficients can be computed using characteristic responses of the micro- and mesostructures which are solutions of local problems imposed in representative volume elements describing the poroelastic medium at the two levels of heterogeneity In the paper, we discus various combinations of the interface between the micro- and mesoscopic porosities, influence of the fluid compressibility, or solid incompressibility Gradient of porosity is accounted for when dealing with locally periodic structures Derived formulae for computing the poroelastic material coefficients characterize not only the steady-state responses with static fluid, but are relevant also for quasistatic problems The model is applicable in geology, or in tissue biomechanics, in particular for modeling canalicular-lacunar porosity of bone which can be characterized at several levels

Dissipative heating of elastomers: a new modelling approach based on finite and coupled thermomechanics

Abstract: Especially in the automotive industries, elastomers take an important role. They are used in different types of bearings, where they inhibit vibration propagation and thereby significantly enhance driving performance and comfort. That is why several models have already been developed to simulate the material’s mechanical response to stresses and strains. In many cases, these models are developed under isothermal conditions. Others include the temperature-dependent mechanical behaviour to represent lower stiffness’s for higher temperatures. In this contribution it is shown by some exemplary experiments that viscoelastic material heats up under dynamic deformations. Hence, the material’s properties change due to the influence of the temperature without changing the surrounding conditions. With some of these experiments, it is shown that a fully coupled material model is necessary to predict the behaviour of bearings under dynamic loads. The focus of this contribution lies on the modelling of the thermoviscoelastic behaviour of elastomers. In a first step, a twofold multiplicative split of the deformation gradient is performed to be able to describe both mechanical and thermal deformations. This concept introduces different configurations. The stress tensors existing on these configurations are used to formulate the stress power in the first law of thermodynamics which allows to simulate the material’s self-heating. To formulate the temperature dependency of the mechanical behaviour, the non-equilibrium part of the Helmholtz free energy function is formulated as a function of the temperature and the deformation history. With the introduced model, some FE calculations are carried out to show the model’s capability to represent the thermoviscoelastic behaviour including the coupling in both directions.

Analytical solutions to general anti-plane shear problems in finite elasticity

Abstract: This paper presents a pure complementary energy variational method for solving a general anti-plane shear problem in finite elasticity. Based on the canonical duality–triality theory developed by the author, the nonlinear/nonconvex partial differential equations for the large deformation problem are converted into an algebraic equation in dual space, which can, in principle, be solved to obtain a complete set of stress solutions. Therefore, a general analytical solution form of the deformation is obtained subjected to a compatibility condition. Applications are illustrated by examples with both convex and nonconvex stored strain energies governed by quadratic-exponential and power-law material models, respectively. Results show that the nonconvex variational problem could have multiple solutions at each material point, the complementary gap function and the triality theory can be used to identify both global and local extremal solutions, while the popular convexity conditions (including rank-one condition) provide mainly local minimal criteria and the Legendre–Hadamard condition (i.e., the so-called strong ellipticity condition) does not guarantee uniqueness of solutions. This paper demonstrates again that the pure complementary energy principle and the triality theory play important roles in finite deformation theory and nonconvex analysis.

Microstructure-based modelling of multiphase materials and complex structures

Abstract: Micromechanical approaches are frequently employed to monitor local and global field quantities and their evolution under varying mechanical and/or thermal loading scenarios. In this contribution, an overview on important methods is given that are currently used to gain insight into the deformational and failure behaviour of multiphase materials and complex structures. First, techniques to represent material microstructures are reviewed. It is common to either digitise images of real microstructures or generate virtual 2D or 3D microstructures using automated procedures (e.g. Voronoi tessellation) for grain generation and colouring algorithms for phase assignment. While the former method allows to capture exactly all features of the microstructure at hand with respect to its morphological and topological features, the latter method opens up the possibility for parametric studies with respect to the influence of individual microstructure features on the local and global stress and strain response. Several applications of these approaches are presented, comprising low and high strain behaviour of multiphase steels, failure and fracture behaviour of multiphase materials and the evolution of surface roughening of the aluminium top metallisation of semiconductor devices.

Thermodynamical properties of triangular quantum wires: entropy, specific heat, and internal energy

Abstract: In the present work, thermodynamical properties of a GaAs quantum wire with equilateral triangle cross section are studied. First, the energy levels of the system are obtained by solving the Schrodinger equation. Second, the Tsallis formalism is applied to obtain entropy, internal energy, and specific heat of the system. We have found that the specific heat and entropy have certain physically meaningful values, which mean thermodynamic properties cannot take any continuous value, unlike classical thermodynamics in which they are considered as continuous quantities. Maximum of entropy increases with increasing the wire size. The specific heat is zero at special temperatures. Specific heat decreases with increasing temperature. There are several peaks in specific heat, and these are dependent on quantum wire size.

A thermodynamic approach to model the caloric properties of semicrystalline polymers

Abstract: It is well known that the crystallisation and melting behaviour of semicrystalline polymers depends in a pronounced manner on the temperature history. If the polymer is in the liquid state above the melting point, and the temperature is reduced to a level below the glass transition, the final degree of crystallinity, the amount of the rigid amorphous phase and the configurational state of the mobile amorphous phase strongly depend on the cooling rate. If the temperature is increased afterwards, the extents of cold crystallisation and melting are functions of the heating rate. Since crystalline and amorphous phases exhibit different densities, the specific volume depends also on the temperature history. In this article, a thermodynamically based phenomenological approach is developed which allows for the constitutive representation of these phenomena in the time domain. The degree of crystallinity and the configuration of the amorphous phase are represented by two internal state variables whose evolution equations are formulated under consideration of the second law of thermodynamics. The model for the specific Gibbs free energy takes the chemical potentials of the different phases and the mixture entropy into account. For simplification, it is assumed that the amount of the rigid amorphous phase is proportional to the degree of crystallinity. An essential outcome of the model is an equation in closed form for the equilibrium degree of crystallinity in dependence on pressure and temperature. Numerical simulations demonstrate that the process dependences of crystallisation and melting under consideration of the glass transition are represented.