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Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2014"


Journal ArticleDOI
TL;DR: Numerical analyses allowed for finding the most desirable situations in which a gradual resorption of the artificial graft occurs together with the simultaneous formation of new bone, finally leading to an almost complete substitution of the bio‐resorbable material with living tissue.
Abstract: The bio-mechanical phenomena occurring in bones grafted with the inclusion of artificial materials demand the formulation of mathematical models which are refined enough to describe their not trivial behavior. A 3D theoretical model, previously developed and used in 1D space, is employed to investigate and explain possible effects resulting from 2D interactions, which may not be present in 1D case so more realistic situations are approached and discussed. The enhanced model was used to numerically analyze the physiological balance between the processes of bone apposition and resorption and material resorption in a bone sample under plain stress state. The specimen was constituted by a portion of bone living tissue and one of bio-resorbable material and was acted by an in-plane loading condition. The signal intensity between sensor cells and actor cells was assumed to decrease exponentially with their distance; the effects of adopting two different laws, namely an absolute and a quadratic functions, were compared. Ranges of load magnitudes were identified within which physiological states are established. A parametric analysis was carried out to evaluate the sensitivity of the model to changes of some critical quantities within physiological ranges , namely resorption rate of bio-material, load level and homeostatic strain. In particular the spatial distribution of mass densities of bone tissue and of resorbable bio-material and their time evolution were considered in order to analyze the biological effects due to the parameter's changes. Synthetically, these biological effects can be associated to different ratios between bone and bio-material densities at the end of the process and to different delays in the bone growth and material resorption. These numerical analyses allowed for finding the most desirable situations in which a gradual resorption of the artificial graft occurs together with the simultaneous formation of new bone, finally leading to an almost complete substitution of the bio-resorbable material with living tissue.

95 citations


Journal ArticleDOI
TL;DR: In this article, the authors present an overview of fractal media by continuum mechanics using the method of dimensional regularization and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains).
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.

79 citations


Journal ArticleDOI
Antonio Rinaldi1, Luca Placidi
TL;DR: In this article, the authors explore the connection between the series of critical strains at which the microcracks form and the second gradient of the microscale displacement field and support the new view that the damage evolution is a three regimes process (I dilute damage, II homogeneous interaction, III localization).
Abstract: Lattice models are powerful tools to investigate damage processes in quasi-brittle material by a microscale perspective. Starting from prior work on a novel rational damage theory for a 2D heterogenous lattice, this paper explores the connection between the series of critical strains at which the microcracks form (i.e. lattice links fail) and the second gradient of the microscale displacement field. Taking a simple tensile test as a representative case study for this endeavour, the analysis of accurate numerical results provides evidence that the second gradient of the microscale displacement field (notably the quantity | ∇ (∂ ux/∂ x)| for the specific example elaborated here) conveys indeed crucial information about the microcracks formation process and can be conveniently used to introduce simplifications of the rational theory that are of relevance by practical purposes as full field strain measurements become routinely possible with digital imaging correlation techniques. Note worthy, the results support the new view that the damage evolution is a three regimes process (I dilute damage, II homogeneous interaction, III localization.) The featured connection with the second gradient of the microscale displacement field is applicable in regions II–III, where microcracks interactions grow stronger and the lattice transitions to the softening regime. The potential impact of these findings towards the formulation of new and physically based CDM models, which are consistent with the reference discrete microscale theory, cannot be overlooked and is pointed out.

78 citations


Journal ArticleDOI
TL;DR: In this article, the authors prove the convergence of the approximate saturation and the approximate pressures and approximate pressure gradients thanks to monotony and compactness arguments under an assumption of non-degeneracy of the phase relative permeabilities.
Abstract: The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and approximate pressure gradients thanks to monotony and compactness arguments under an assumption of non-degeneracy of the phase relative permeabilities. Strong convergence results stem from the convergence of the norms of the gradients of pressures, which demand handling the nonlinear time term. Numerical results show the efficiency on these problems of a particular gradient scheme, called the Vertex Approximate Gradient scheme.

77 citations


Journal ArticleDOI
TL;DR: In this article, the buckling problem of a column which is modeled by some finite rigid segments and elastic rotational springs is analyzed by introducing a Lagrange multiplier. And the bucking load of this column is exactly obtained by a recursive formula involving Chebyschev polynomials.
Abstract: This paper deals with the buckling of a column which is modeled by some finite rigid segments and elastic rotational springs and relating its solution to continuum nonlocal elasticity. This problem, which can be referred to Hencky's chain, can serve as a basic model to rigorously investigate the effect of the microstructure on the buckling behaviour of a simple equivalent continuum structural model. The buckling problem of the pinned-pinned discretized column is analytically investigated by introducing a Lagrange multiplier. Such a buckling problem is mathematically treated as an iterative eigenvalue problem. It is shown that the buckling load of this finite degree-of-freedom system is exactly obtained by a recursive formula involving Chebyschev polynomials. Euler's buckling load is asymptotically obtained at larger scales. However, at smaller scales, the buckling model highlights some scale effect that can be only captured by nonlocal elasticity for the equivalent continuum. We show that Eringen's nonlocal continuum is well suited to capture this scale effect. The small scale coefficient of the equivalent nonlocal continuum is then identified from the specific microstructure features, namely the length of each cell. It is shown that the small length scale coefficient valid for this buckling problem is very close to the one already identified from a comparison with the Born-Karman model of lattice dynamics using dispersive wave properties.

67 citations


Journal ArticleDOI
TL;DR: In this paper, the steady boundary layer flow of Casson fluid over a porous stretching/shrinking sheet is studied and the governing equations are transformed using similarity transformations and then solved analytically.
Abstract: In this investigation, the steady boundary layer flow of Casson fluid over a porous stretching/shrinking sheet is studied. The governing equations are transformed using similarity transformations and then solved analytically. In both stretching and shrinking sheet cases, the closed form exact solutions are obtained. The solution is always unique for stretching sheet case. On the other hand, in shrinking sheet case, the solution may exist or may not and if exists it may be unique or may be of dual nature; these all depend on the value of Casson parameter and wall mass transfer parameter. Also, the analysis reveals that for steady flow of Casson fluid stronger mass suction is needed.

66 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered an equilibrium problem for an elastic transversely isotropic Timoshenko's plate with a curvilinear crack and derived the formula of the derivative of the plate energy functional with respect to the perturbation parameter.
Abstract: An equilibrium problem for an elastic transversely isotropic Timoshenko’s plate witha curvilinear crack is considered. On the crack faces, the nonpenetration conditions,which have the form of inequalities (conditions of the Signorini type), are given. Byusing a suciently smooth perturbation determined in the middle plate plane, thevariation of plate geometry is speci ed. The formula of the derivative of the plateenergy functional with respect to the perturbation parameter is deduced. MSC 2010: 74G65, 74B05, 74R10, 35Q74, 35J50KEY WORDS: Timoshenko’ plate, nonpenetration condition, crack, varia-tional inequality, energy functional 1 Introduction The work presents the model describing the equilibrium of an elastic homoge-neous transversally isotropic Timoshenko’s plate containing a through crack.On the curve corresponding to the crack, we impose a condition in the formof inequality describing the nonpenetration of the crack faces. For the familyof variational equilibrium problems of plates depending on parameter ", de-pendence of energy functionals and solutions ˘

53 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that this restriction may no longer apply to large deformations due to a number of defects and that these defects can be eliminated by introducing a specific objective time derivative.
Abstract: At the beginning of the last century two different types of constitutive relations to describe the complex behavior of elasto-plastic material were presented. These were the deformation theory originally developed by Hencky and the Prandtl-Reuss theory. Whereas the former provides a direct solid-like relation of stress as function of strain, the latter has been based on an additive composition of elastic and plastic parts of the increments of strains. These in turn were taken as a solid- and fluid-like combination of the de Saint-Venant/Levy theory with an incremental form of Hooke's law. Even nowadays this Prandtl-Reuss theory is still accepted – within the restriction of small elastic deformations, i.e. it is generally stated in most textbooks on plasticity that this theory due to a number of defects can not be applied to large deformations. In the present article it is shown that this restrictive statement may be no longer true. Introducing a specific objective time derivative it could be shown that these defects disappear.

46 citations


Journal ArticleDOI
TL;DR: In this paper, the fractional order theory of thermoelasticity was applied to a 1D thermal shock problem for a half-space, and the predictions of the theory were discussed and compared with those for the coupled and generalized theories of thermodynamic properties.
Abstract: In this work, we apply the fractional order theory of thermoelasticity to a 1D thermal shock problem for a half-space. Laplace transform techniques are used. The predictions of the theory are discussed and compared with those for the coupled and generalized theories of thermoelasticity. Numerical results are computed and represented graphically for the temperature, displacement and stress distributions.

44 citations


Journal ArticleDOI
TL;DR: In this paper, the problem of determining the number and type of symmetry classes of an odd-order tensor space has been studied for higher-order elasticity tensors, including photoelectricity, piezoelectricity, and strain-gradient elasticity.
Abstract: Anisotropy, symmetry classes, higher order elasticity.We give a complete general answer to the problem, recurrent in continuum mechanics, of determining the number andtype of symmetry classes of an odd-order tensor space. This kind of investigation was initiated for the space of elasticitytensors. Since then, this problem has been solved for other kinds of physics such as photoelectricity, piezoelectricity,flexoelectricity, and strain-gradient elasticity. In all the aforementioned papers, the results are obtained after some lengthycomputations. In a former contribution we provide general theorems that solve the problem for even-order tensor spaces.In this paper we extend these results to the situation of odd-order tensor spaces. As an illustration of this method, and forthe first time, the symmetry classes of all odd-order tensors of Mindlin second strain-gradient elasticity are provided.

37 citations


Journal ArticleDOI
TL;DR: In this paper, a second-order plate theory for homogeneous monoclinic materials is presented, which is a system of two coupled PDEs of differentiation order six in two variables.
Abstract: By Fourier-series expansion in thickness direction of the plate with respect to a basis of scaled Legendre polynomials, several equivalent (and therefore exact) two-dimensional formulations of the three-dimensional boundary-value problem of linear elasticity in weak formulation for a plate with constant thickness are derived. These formulations are sets of countably many PDEs, which are power series in the squared plate parameter. For the special case of a homogeneous monoclinic material, we obtain an approximative plate theory in finitely many PDEs and unknown variables by the truncation approach of the uniform-approximation technique. The PDE system is reduced to a scalar PDE expressed in the mid-plane displacement. The resulting second-order theory, considered as a first-order theory, is equivalent to the classical Kirchhoff theory for the special case of an isotropic material and equivalent to Huber's classical theory for an actual monoclinic material. However it remains shear-rigid as a second-order theory. Therefore, it is modified by an a-priori assumption to a theory for monoclinic materials, that presumes the former equivalences, considered as a first-order theory, but is in addition equivalent to Kienzler's theory as a second-order theory for the special case of isotropy, which implies further equivalences to established shear-deformable theories, especially the Reissner-Mindlin theory and Zhilin's plate theory. The presented new second-order plate theory for monoclinic materials is finally a system of two coupled PDEs of differentiation order six in two variables.

Journal ArticleDOI
TL;DR: In this article, the authors study flow problems in unsaturated porous media and show that fingering effects can be observed in various models and discuss the importance of the static hysteresis term.
Abstract: We study flow problems in unsaturated porous media. Our main interest is the gravity driven penetration of a dry material, a situation in which fingering effects can be observed experimentally and numerically. The flow is described by either a Richards or a two-phase model. The important modelling aspect regards the capillary pressure relation which can include static hysteresis and dynamic corrections. We report on analytical existence and instability results for the corresponding models and present numerical calculations. We show that fingering effects can be observed in various models and discuss the importance of the static hysteresis term. (C) 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Journal ArticleDOI
TL;DR: In this paper, the spectral properties of the eigenvalue problems with account for surface effects are determined, and a variational minimal principle is constructed which is similar to the well-known variational principle for problems f or pure elastic and piezoelectric media.
Abstract: We consider the harmonic and eigenvalue problems for piezoelectric nanodimensional bodies with account for surface stresses and surface electric charges. For harmonic problem new mathematical model is suggested, which generalizes the model of the piezoelectric medium with damping properties, boundary conditions of contact type and surface effects. The classical and generalized (weak) statements for harmonic and eigenvalue problems are formulated in the extended and reduced forms. The spectral properties of the eigenvalue problems with account for surface effects are determined. A variational minimal principle is constructed which is similar to the well-known variational principle for problems f or pure elastic and piezoelectric media. The discreteness of t he spectrum and completeness of the eigenvectors are proved. As a consequence of variational principle, the properties of the natural frequencies increase or decrease are establis hed for changing the mechanical, electric and “surface” boundary conditions and the moduli of piezoelectric body. The finit e element approaches are described for determination of the natural frequencies, the resonance and antiresonance frequencies and harmonic behavior of nanosized piezoelectric bodies with account for surface effects. Copyright line will be provided by the publisher

Journal ArticleDOI
TL;DR: In this paper, the authors consider large deformations of curved thin shells in the framework of a classical Kirchhoff-love theory for material surfaces, where the geometry of the element is approximated via the position vector and its derivatives with respect to the material coordinates at the four nodes, and C1 continuity of the surface over the interfaces between the elements is guaranteed.
Abstract: We consider large deformations of curved thin shells in the framework of a classical Kirchhoff-Love theory for material surfaces. The geometry of the element is approximated via the position vector and its derivatives with respect to the material coordinates at the four nodes, and C1 continuity of the surface over the interfaces between the elements is guaranteed. Theoretical background provides certainty concerning the boundary conditions, the range of applicability of the model, extensions to multi-field problems, etc. Robust convergence and accuracy of the resulting simple numerical scheme is demonstrated by the analysis of benchmark problems in comparison with other solutions.

Journal ArticleDOI
TL;DR: In this paper, the authors used the Green-Naghdi theory of thermomechanics of continua to derive a nonlinear theory of thermoelasticity with diffusion of types II and III, which permits propagation of both thermal and diffusion waves at finite speeds.
Abstract: In this paper, we use the Green-Naghdi theory of thermomechanics of continua to derive a nonlinear theory of thermoelasticity with diffusion of types II and III. This theory permits propagation of both thermal and diffusion waves at finite speeds. The equations of the linear theory are also obtained. With the help of the semigroup theory of linear operators we establish that the linear anisotropic problem is well posed and we study the asymptotic behavior of the solutions. Finally, we investigate the impossibility of the localization in time of solutions. c

Journal ArticleDOI
TL;DR: In this article, it was shown that the Legendre-Hadamard condition and elliptic-ity condition for the quadratic form |DuP +(DuP) T | 2 is in general false, even if P 2 SO(3).
Abstract: We show that the following generalized version of Korn's second inequality with non- constant measurable matrix valued coefficientsP : � R 3 ! R 3×3 ||DuP + (DuP) T ||q + ||u||qc||Du||q for u 2 W 1,q 0 (;R 3 ), 1 < q < 1 is in general false, even if P 2 SO(3), while the Legendre-Hadamard condition and elliptic- ity on C n for the quadratic form |DuP +(DuP) T | 2 is satisfied. Thus Gu arding's inequality may be violated for formally positive quadratic forms.


Journal ArticleDOI
TL;DR: In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the Stochastic space.
Abstract: The continuous sedimentation process in a clarifier-thickener can be described by a scalar nonlinear conservation law for the local solids volume fraction. The flux density function is discontinuous with respect to spatial position due to feed and discharge mechanisms. Typically, the feed flow cannot be given deterministically and efficient numerical simulation requires a concept for quantifying uncertainty. In this paper uncertainty quantification is expressed by a new hybrid stochastic Galerkin (HSG) method that extends the classical polynomial chaos approximation by multiresolution discretization in the stochastic space. The new approach leads to a deterministic hyperbolic system for a finite number of stochastic moments which is however partially decoupled and thus allows efficient parallelisation. The complexity of the problem is further reduced by stochastic adaptivity. For the approximate solution of the resulting high-dimensional system a finite volume scheme is introduced. Numerical experiments cover one- and two-dimensional situations.

Journal ArticleDOI
TL;DR: In this paper, a system of partial differential equations describing the steady flow of a compressible heat conducting Newtonian fluid in a three-dimensional channel with inflow and outflow part is considered.
Abstract: We consider a system of partial differential equations describing the steady flow of a compressible heat conducting Newtonian fluid in a three-dimensional channel with inflow and outflow part. We show the existence of a strong solution provided the data are close to a constant, but nontrivial flow with sufficiently large dissipation in the energy equation.

Journal ArticleDOI
Abstract: A direct time integration method is presented for the solution of the equations of motion describing the dynamic response of structural linear and nonlinear multi-degree-of-freedom systems. It applies also to large systems of second order differential equations with fully populated, non symmetric coefficient matrices as well as to equations with variable coefficients. The proposed method is based on the concept of the analog equation, which converts the coupled N equations into a set of single term uncoupled second order ordinary quasi-static differential equations under appropriate fictitious loads, unknown in the first instance. The fictitious loads are established from the integral representation of the solution of the substitute single term equations. The method is simple to implement. It is self starting, unconditionally stable and accurate and conserves energy. It performs well when large deformations and long time durations are considered and it can be used as a practical method for integration of the equations of motion in cases where widely used time integration procedures, e.g. Newmark's, become unstable. Several examples are presented, which demonstrate the efficiency of the method. The method can be straightforward extended to evolution equations of order higher than two.

Journal ArticleDOI
TL;DR: In this article, a rate-independent finite elastoplastic equations are proposed in unified forms applicable to all loading-unloading cases, which are not subjected to extrinsic restrictive conditions, including the yield condition as well as the loadingunloading conditions.
Abstract: New rate-independent finite elastoplastic equations are proposed in unified forms applicable to all loading-unloading cases. A departure from the classical elastoplastic equations is that these new equations are not subjected to and hence free from the usual extrinsic restrictive conditions, including the yield condition as well as the loading-unloading conditions. Such free equations are of Eulerian rate type and assume the same smooth form for all possible stresses and for all strain rates. It is demonstrated that the essential representative features of finite elastoplastic deformations, namely, the yield behavior and the loading-unloading behavior in the traditional sense, may be derived from and hence naturally incorporated as intrinsic physical characteristics into the free elastoplastic equations proposed in a more realistic sense and, in particular, the classical notions characterizing these features are found to exhibit novel, perhaps more profound physical meanings in the new equations. Furthermore, the strong discontinuity in tangent moduli at transition from elastic to plastic state, involved in the traditional formulation, is shown to be replaced by a smooth transition. Implications are discussed in respects of constitutive implications and numerical treatment.

Journal ArticleDOI
TL;DR: In this paper, a general constitutive elastoplastic model for associated plasticity is investigated, which can include basic plastic criteria with a combination of kinematic hardening and non-linear isotropic hardening.
Abstract: In the paper, a general constitutive elastoplastic model for associated plasticity is investigated. The model is based on the thermodynamical framework with internal variables and can include basic plastic criteria with a combination of kinematic hardening and non-linear isotropic hardening. The corresponding initial value constitutive elastoplastic problem is discretized by the implicit Euler method. The discretized one-time-step constitutive problem defines the elastoplastic operator, which is formulated by a simple generalization of a projection onto a convex set. Properties of the so-called generalized projection are used for deriving basic properties of the elastoplastic operator like potentiality, monotonicity, Lipschitz continuity and local semismoothness. Further, hardening variables are eliminated from the projective definition of the elastoplastic operators, which yields relations among the models with hardening variables and the perfect plasticity model. Also a simplification of the operator for plastic criteria in eigenvalue forms is introduced. The simplifications are useful for a numerical implementation and can be used for deriving other properties like strong semismoothness of the elastoplastic operators for the classical plastic criteria or strong monotonicity of the stress-strain operator for some models with hardening. The derived properties can be important for convergence analyses of Newton-like methods and other mathematical and numerical analyses.

Journal ArticleDOI
TL;DR: In this article, an approach for constructing semi-analytical solutions in contact problems of the theory of elasticity for inhomogeneous layers is developed, which is efficient for the layer of arbitrary thickness which is either continuously inhomogenous (functionally graded) or piecewise homogeneous (i.e. presented as a set of homogeneous layers with different elastic properties).
Abstract: An approach for constructing semi-analytical solutions in contact problems of the theory of elasticity for inhomogeneous layers is developed. The approach is efficient for the layer of arbitrary thickness which is either continuously inhomogeneous (functionally graded) or piecewise homogeneous (i.e. presented as a set of homogeneous layers with different elastic properties). The foundation is also assumed to be elastic, but much stiffer than the layer. The loads considered address the case of axisymmetric contact problems under torsion and indentation of a rigid circular punch with the flat base. The technique based on integral transforms is used to reduce the problems to the integral equations. Special approximations for the kernel transforms are used to obtain analytical solutions of the integral equations. The main results include computations of the profiles of contact stresses under the punch and the dependences of displacements with depth for different types of variation of elastic properties in the layer. The results are also compared with those obtained by other methods. c

Journal ArticleDOI
TL;DR: In this paper, the exact analytical solution of stress intensity factors (SIFs) for an infinite plane containing three collinear cracks under biaxial compression has been obtained.
Abstract: The fracture behaviour of multi-cracked materials has become a key issue in fracture mechanics and has received large attention recently. In this paper, an infinite plane containing three collinear cracks under biaxial compression has been investigated. Considering crack surface friction and using complex stress function theory, the exact analytical solution of stress intensity factors (SIFs) for an infinite plane containing three collinear cracks is obtained. The corresponding finite element code of Abaqus is employed to validate the theoretical results, and its results agree very well with the theoretical results. The effects of confining stresses, crack distances and the crack surface frictions on SIFs are analyzed through the theoretical results and the Abaqus code. A photoelastic experiment was conducted to validate the theoretical result about the effect of confining stresses.

Journal ArticleDOI
TL;DR: In this article, a geometrically nonlinear, continuum thermomechanical framework for pulsed laser heating in crystalline matter is introduced, which is characterized by a non-Fourier like heat propagation and defect diffusion.
Abstract: This contribution introduces a geometrically nonlinear, continuum thermomechanical framework for pulsed laser heating in crystalline matter: a physical process which is characterized by a non-Fourier like heat propagation and defect diffusion. The key objective of this work is to derive the highly nonlinear and strongly coupled system of governing equations describing the multi-physical behavior from fundamental balance principles. A general form for the Helmholtz energy is proposed and the resulting constitutive laws are derived from logical, thermodynamically consistent argumentation. The approach adopted to derive the governing equations is not entirely specific to laser induced heating, rather it encompasses a wide range of applications wherein heat conduction, species diffusion, and finite elastic effects are coupled. The present theory is thus applicable to the generality of models for thermal and mechanical waves: an area of increasing research interest. A numerical example is presented for the fully coupled, nonlinear and transient theory.

Journal ArticleDOI
TL;DR: In this paper, the authors studied grain boundaries in the Swift-Hohenberg equation and showed that such stationary interfaces exist near onset of instability for arbitrary angles between the roll solutions, and developed a singular perturbation approach to treat resonances.
Abstract: We study grain boundaries in the Swift-Hohenberg equation. Grain boundaries arise as stationary interfaces between roll solutions of different orientations. Our analysis shows that such stationary interfaces exist near onset of instability for arbitrary angles between the roll solutions. This extends prior work in [6] where the analysis was restricted to large angles, that is, weak bending near the grain boundary. The main new difficulty stems from possible interactions of the primary modes with other resonant modes. We generalize the normal form analysis in [6] and develop a singular perturbation approach to treat resonances.

Journal ArticleDOI
TL;DR: In this paper, a formulation within the theory of porus media for continuum multicomponent modeling of bacterial driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase is presented.
Abstract: This study focuses on a formulation within the theory of porus media for continuum multicomponent modeling of bacterial driven methane oxidation in a porous landfill cover layer which consists of a porous solid matrix (soil and bacteria) saturated by a liquid (water) and gas phase. The solid, liquid, and gas phases are considered as immiscible constituents occupying spatially their individual volume fraction. However, the gas phase is composed of three components, namely methane (CH4), oxygen (O2), and carbon dioxide (CO2). A thermodynamically consistent constitutive framework is derived by evaluating the entropy inequality on the basis of Coleman and Noll [8], which results in constitutive relations for the constituent stress and pressure states, interaction forces, and mass exchanges. For the final set of process variables of the derived finite element calculation concept we consider the displacement of the solid matrix, the partial hydrostatic gas pressure and osmotic concentration pressures. For simplicity, we assume a constant water pressure and isothermal conditions. The theoretical formulations are implemented in the finite element code FEAP by Taylor [29]. A new set of experimental batch tests has been created that considers the model parameter dependencies on the process variables; these tests are used to evaluate the nonlinear model parameter set. After presenting the framework developed for the finite element calculation concept, including the representation of the governing weak formulations, we examine representative numerical examples.

Journal ArticleDOI
TL;DR: In this paper, a macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing and thawing processes, and the model is capable of simulating the temperature development and energetic effects during phase change.
Abstract: In civil engineering, the frost durability of partly liquid saturated porous media under freezing and thawing conditions is a point of great discussion. Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this paper, a macroscopic model based on the Theory of Porous Media (TPM) is presented which describes energetic effects of freezing and thawing processes. For simplification a ternary model consisting of the phases solid, ice and liquid is used. Attention is paid to the description of the temperature development, the determination of energy, enthalpy and mass supply as well as volume deformations due to ice formation during a freezing and thawing cycle. For the detection of energetic effects regarding the characterization and control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange between ice and liquid is presented. Comparing experimental data with numerical examples shows that the simplified model is indeed capable of simulating the temperature development and energetic effects during phase change.

Journal ArticleDOI
TL;DR: In this paper, an anisotropic, unbounded elastic body containing three collinear, equal cracks subjected to asymmetrically tangential stresses, case corresponding to Mode II of classical Fracture, was determined using the formalism of Riemann-Hilbert problem and representation of elastic stresses and displacements fields due to Lekhnitskii.
Abstract: One consider an anisotropic, unbounded elastic body containing three collinear, equal cracks subjected to asymmetrically tangential stresses, case corresponding to Mode II of classical Fracture. We determine the elastic state produced in the body using the formalism of Riemann-Hilbert problem and the representation of elastic stresses and displacements fields due to Lekhnitskii. Using the asymptotical analysis, we obtain the asymptotic values of the stress and the displacement fields in a vicinity of the cracks tips.

Journal ArticleDOI
TL;DR: In this article, an effective variational formalism of construction of the thick anisotropic linear shell theory of Vekua-type of an arbitrary order is presented, which allows one for computer-aided derivation of all equations using computer algebra software supporting main tensor operations.
Abstract: This paper presents an effective variational formalism of construction of the thick anisotropic linear shell theory of Vekua-type of an arbitrary order. The dynamic equations are formulated as the Lagrange equations of the second kind, independent from expansion functions. The asymmetric stress tensor is used to obtain the compact form of the dynamic equations similar to the classical shell theories. The proposed approach allows one for computer-aided derivation of all equations using computer algebra software supporting main tensor operations. Some solutions of test problems are presented.