Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2013"

Propagation of linear compression waves through plane interfacial layers and mass adsorption in second gradient fluids

Abstract: Institut fu¨r Mechanik, Otto-von-Guericke-Universita¨tMagdeburg, 39106 Magdeburg, Germany, andSouth Scientific Center of RASci & South Federal University, Rostov on Don, RussiaReceived XXXX, revised XXXX, accepted XXXXPublished online XXXXKey words Linear waves, interfacial layer, gradient elasticity, second gradient, capillary fluid, Cahn-Hilliard fluid, massadsorptionThis paper addresses the problem of reflection and transmiss ion of compression waves at the phase transition layer betweenthe vapour and liquid phases of the same fluid. Within the fram ework of second gradient fluid modeling, we use a non-convex free energy in order to describe the phase transition phenomenon. A stationary solution for the fluid density isfo undfor an infinite domain, and an analytical expression for the p hase transition is presented. Then the propagation of linearwaves superposed to this stationary solution is discussed, with particular attention to the behaviour in correspondence ofthe interfacial layer. The reflection and transmission of wa ves is studied and analized with the aid of numerical simulations,and an interesting phenomenon of mass adsorption at the interface is observed and discussed.

Unsteady magnetohydrodynamic free convection flow past an accelerated vertical plate with constant heat flux and heat generation or absorption

Abstract: The unsteady free convection flow of an electrically conducting fluid past an accelerated infinite vertical plate with constant heat flux is investigated under the influence of uniform transverse magnetic field fixed relative to the fluid or to the plate in the presence of heat generation or absorption. Two important cases, (i) exponentially accelerated plate (EAP) and (ii) uniformly accelerated plate (UAP), have been considered. The governing partial differential equations have been solved analytically using the Laplace transform technique and closed form solutions are obtained for the velocity and temperature fields without any restriction. The effects of system parameters such as the Prandtl number, heat generation or absorption, Grashof number, and magnetic field parameter on the flow fields are analyzed through graphs and tables. Further, the solution of the problem involves inverse Laplace transforms of some new exponential forms and these formulas are provided.

Drug release from collagen matrices including an evolving microstructure

Abstract: Biodegradable collagen matrices have become a promising alternative to traditional drug delivery systems. The relevant mechanisms in controlled drug release are the diffusion of water into the collagen matrix, the swelling of the matrix coming along with drug release, and enzymatic degradation of the matrix with additional simultaneous drug release. These phenomena have been extensively studied in the past experimentally, via numerical simulations as well as analytically. However, a rigorous derivation of the macroscopic model description, which includes the evolving microstructure due to the degradation process, is still lacking. Since matrix degradation leads to the release of physically entrapped active agent, a good understanding of these phenomena is very important. We present such a derivation using formal twoscale asymptotic expansion in a level set framework and complete our results with numerical simulations in comparison with experimental data. c

Quasistatic adhesive contact of visco-elastic bodies and its numerical treatment for very small viscosity

Abstract: An adhesive unilateral contact of elastic bodies with a small viscosity in the linear Kelvin-Voigt rheology at small strains is scrutinized. The flow-rule for debonding the adhesive is considered rate-independent and unidirectional, and inertia is neglected. The asymptotics for the viscosity approaching zero towards purely elastic material involves a certain defect-like measure recording in some sense natural additional energy dissipated in the bulk due to (vanishing) viscosity, which is demonstrated on particular 2-dimensional computational simulations based on a semi-implicit time discretisation and a spacial discretisation implemented by boundary-element method.

Greedy‐based approximation of frequency‐weighted Gramian matrices for model reduction in multibody dynamics

Abstract: The method of elastic multibody systems is frequently used to describe the dynamical behavior of the mechanical subsystems in multi-physics simulations. One important issue for the simulation of elastic multibody systems is the error-controlled reduction of the flexible body's degrees of freedom. By the use of second order frequency-weighted Gramian matrix based reduction techniques the distribution of the loads is taken into account a-priori and very accurate models can be obtained within a predefined frequency range and even a-priori error bounds are available. However, the calculation of the frequency-weighted Gramian matrices requires high computational effort. Hence, appropriate approximation schemes have to be used to find the dominant eigenspace of these matrices. In the current contribution, the matrix integral needed for calculating the Gramian matrices is approximated by quadratures using integral kernel snapshots. The number and location of these snapshots have a strong influence on the reduction results. Sophisticated snapshot selection methods based on Greedy algorithms from the reduced basis methods are used to construct the optimal location of snapshot frequencies. The method can be viewed as an automatic determination of optimal frequency weighting and as an adaptive learning of quadrature rules. One ingredient of Greedy algorithms is the need of error measures. To gain computational advantage two different error estimators are derived and used in the Greedy algorithm instead of the absolute or relative error.

Infiltration processes in cohesionless soils

Abstract: In this paper, a thermodynamically consistent four-phase continuum model in the framework of the mixture theory is presented describing infiltration processes of suspensions in cohesionless granular material. The paper focuses on the distinct form of the constitutive relation for the volume production term of the fluidized particles and its consequences on the infiltration process. To this end a constitutive equation describing infiltration phenomena is proposed which includes only one material parameter. Therefore we study numerically a boundary value problem, which is characterized by a homogeneous field of the hydraulic gradient in the reference configuration at the time t0 = 0. Infiltration is affecting the distribution of the hydraulic properties and illustrates the consequences of the proposed constitutive equation for specific parameter choices. Furthermore it is shown how the material parameter can be estimated without explicit numerical calculations.

On the ability of nanoindentation to measure anisotropic elastic constants of pyrolytic carbon

Abstract: We used cube corner, Berkovich, cono-spherical, and Vickers indenters to measure the indentation modulus of highly oriented bulk pyrolytic carbon both normal to and parallel to the plane of elastic isotropy. We compared the measurements with elastic constants previously obtained using strain gage methods and ultrasound phase spectroscopy. While no method currently exists to extract the anisotropic elastic constants from the indentation modulus, the method of Delafargue and Ulm (DU) [17] was used to predict the indentation modulus from the known elastic constants. The indentation modulus normal to the plane of isotropy was %sim; 20% higher than the DU predictions and was independent of indenter type. The indentation modulus parallel to the plane of isotropy was 2–3 times lower than DU predictions, was depth dependent, and was lowest for the cube corner indenter. We attribute the low indentation modulus to nanobuckling of the graphite-like planes and the indenter type dependence to the impact of differing degree of transverse stress on the tendency toward nanobuckling.

Model reduction of an elastic crankshaft for elastic multibody simulations

Abstract: System analysis and optimization of combustion engines and engine components are increasingly supported by digital simulations. In the simulation process of combustion engines multi physics simulations are used. As an example, in the simulation of a crank drive the mechanical subsystem is coupled to a hydrodynamic subsystem. As far as the modeling of the mechanical subsystems is concerned, elastic multibody systems are frequently used. During the simulation many equations must be solved simultaneously, the hydrodynamic equations as well as the equations of motion of each body in the elastic multibody system. Since the discretization of the elastic bodies, e.g with the help of the finite element method, introduces a large number of elastic degrees of freedom, an efficient simulation of the system becomes difficult. The linear model reduction of the elastic degrees of freedom is a key step for using flexible bodies in multibody systems and turning simulations more efficient from a computational point of view. In recent years, a variety of new reduction methods alongside the traditional techniques were developed in applied mathematics. Some of these methods are reviewed and compared for reducing the equations of motion of an elastic body used in multibody systems. The special focus of this work is on balanced truncation model order reduction, which is a singular value based reduction technique using the Gramian matrices of the system. We investigate a version of this method that is adapted to the structure of a special class of second order dynamical systems which is important for the particular application discussed here. The simulation of a crank drive with a flexible crankshaft is taken as technically relevant example. The results are compared to other methods like Krylov approaches or modal reduction.

Characterization and statistical modeling of irregular porosity in carbon/carbon composites based on X‐ray microtomography data

Abstract: moments of inertia are constructed. Based on this information, a statistically significant range of pore geometry parameters is determined for evaluation of their contribution to the effective elastic properties of the material. Using the design of experiment approach, a subset of 53 pores is selected for the finite element simulations. The results are analyzed to construct a 3-factor stochastic model of a pore contribution to the overall elastic response based on its geometric parameters. It is determined that the non-dimensionalized surface to volume ratio factor plays an important role for pores in this type of material. A 4-factor model incorporating this ratio is proposed. The model is validated by direct finite element simulations for a set of 150 randomly selected pores not included in the initial subset. The accuracy of the proposed approach is compared with the traditionally used approximation of pores by equivalent ellipsoids.

DGFEM for the analysis of airfoil vibrations induced by compressible flow

Abstract: The subject of the paper is the numerical simulation of the interaction of two-dimensional compressible viscous flow and a vibrating airfoil. A solid airfoil with two degrees of freedom performs rotation around an elastic axis and oscillations in the vertical direction. The numerical simulation consists of the solution of the Navier-Stokes system by the discontinuous Galerkin method coupled with a system of nonlinear ordinary differential equations describing the airfoil motion. The time-dependent computational domain and a moving grid are taken into account by the arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations. The developed method is robust with respect to the magnitude of the Mach number and Reynolds number. Its applicability is demonstrated by numerical experiments.

A two scale model for liquid phase epitaxy with elasticity: An iterative procedure

Abstract: Epitaxy is a technically relevant process since it gives the possibility to generate microstructures of different morphologies. These microstructures can be influenced by elastic effects in the epitaxial layer. We consider a two scale model including elasticity, introduced in [8]. The coupling of the microscopic and the macroscopic equations is described by an iterative procedure. We concentrate on the microscopic equations and study their solvability in appropriate function spaces. As the main results we prove the existence and uniqueness of solutions of the three single parts of the microscopic problem. The composition of the corresponding solution operators maps a suitable function space into itself. These results are a first step in the proof of existence of solutions via suitable fixed point arguments of the fully coupled two scale model.

A contact problem for electro-elastic materials

Abstract: We analyze the frictionless unilateral contact between an electro-elastic body and a rigid electrically conductive foundation. On the potential contact zone, we use the Signorini condition with non-zero gap and an electric contact condition with a conductivity depending on the Cauchy vector. We provide a weak variationally consistent formulation and show existence, uniqueness and stability of the solution. Our analysis is based on fixed point techniques for weakly sequentially continuous maps. We conclude by a numerical example that illustrates the applicability of the model.

On Cosserat–Bingham fluids

Abstract: We formulate constitutive laws for micro-polar visco-plastic fluids which support yield stresses. The material is shown to be a generalization of Bingham and Herschel–Bulkley fluids in which local micro-rotations are taken into account. In contrast to the classical visco-plastic fluids, micro-polar visco-plastic fluids support two types of plug zones. To illustrate the derived equations, we consider a pressure driven flow between two parallel planes.

Is the one-equation coupling of finite and boundary element methods always stable?

Abstract: In this paper we present a sufficient and necessary condition to ensure the ellipticity of the bilinear form which is related to the one-equation coupling of finite and boundary element methods to solve a scalar free space transmission problem for a second order uniform elliptic partial differential equation in the case of general Lipschitz interfaces. This condition relates the minimal eigenvalue of the coefficient matrix in the bounded interior domain to the contraction constant of the shifted double layer integral operator which reflects the shape of the interface. This paper extends and improves earlier results [12] on sufficient conditions, but now includes also necessary conditions. Numerical examples confirm the theoretical results on the sharpeness of the presented estimates.

Multi-scale modelling of elastic/viscoelastic compounds

Abstract: The present contribution is concerned with the requirements of an efficient multi-scale modelling approach for elastic/viscoelastic compounds such as bituminous asphalt concrete. Typically, this heterogeneous composite material consist of a mineral filler (e.g. crushed rock), a bituminous binding agent, pores and further additives. The contrast in stiffness between the different is extremely high and accounts for several orders of magnitude. Prediction of effective mechanical properties of such complex materials on the macroscopic level requires a detailed knowledge of the micro-scale behaviour of the particular constituents. In this study, we will focus on modelling aspects due to upscaling routines based on volume averaging. Particularly, we will show that the choice of micro-level boundary conditions not only influences the effective stiffness of the viscoelastic substitute material (upper/lower limit), but also the viscous contribution to the macro-model (shift of maximal attenuation in frequency space). In order to study these fundamental homogenization properties, we introduce a simplified compound consisting of homogeneous viscoelastic binder phase and spherical filler particles with a volume fraction low compared to realistic asphalt concrete. Depending on the chosen boundary condition, stress-relaxation and creep tests are considered. After transformation of the effective stress-strain-relations from time- to frequency space, the viscoelastic properties of the compound will be discussed in frequency domain.

High‐order finite elements compared to low‐order mixed element formulations

Abstract: High-order finite elements are commonly compared with linear element formulations showing that, in terms of the relation between computational time and achievable accuracy, linear element formulations are inferior to high-order elements. On the other hand, the finite element community follows the h-version approach due to its simplicity in implementation. This article compares high-order finite elements based on hierarchic shape functions with low-order mixed element formulations using finite strain hyperelasticity. These comparisons are conducted from the point of view of both accuracy and efficiency as well as highly deformed structures. It also investigates improvements to minimize the overall computational effort such as parallelizing the element assemblage procedure, choosing a starting vector estimator for Newton's method, and investigating the Newton-Chord method. The advantages and disadvantages of both finite element approaches are also discussed.

Numerical algorithm based on an implicit fully discrete local discontinuous Galerkin method for the time-fractional KdV-Burgers-Kuramoto equation

Abstract: In this paper, an implicit fully discrete local discontinuous Galerkin (LDG) finite element method is considered for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation. The scheme is based on a finite difference method in time and local discontinuous Galerkin methods in space. We prove that our scheme is unconditional stable and L2 error estimate for the linear case with the convergence rate O(hk+1 + (Delta t)2+ (Delta t)alpha/2hk+1/2). Numerical examples are presented to show the efficiency and accuracy of our scheme.

Index reduction for operator differential-algebraic equations in elastodynamics

Abstract: In space semi-discretized equations of elastodynamics with weakly enforced Dirichlet boundary conditions lead to differential algebraic equations (DAE) of index 3. We rewrite the continuous model as operator DAE and present an index reduction technique on operator level. This means that a semi-discretization leads directly to an index-1 system. We present existence results for the operator DAE with nonlinear damping term and show that the reformulated operator DAE is equivalent to the original equations of elastodynamics. Furthermore, we show that index reduction and semi-discretization in space commute if the discretization schemes are chosen in an appropriate way.

A numerical homogenisation method for sandwich plates based on a plate theory with thickness change

Abstract: Nowadays composite plates are widely used, especially in the transport industry. Their importance derives from their relatively low weight and good mechanical properties, but their behaviour is complex and cannot be obtained from a simple mixture rule properly. In the scope of this work, the considered composite plate is composed of a complex assembly of metal layers and carbon fibers reinforced polymers (CFRP) layers. Whereas the CFRP can be considered as anisotropic elastic, the metal layers have an elasto-plastic material behaviour. In this case, the possibility to use the classical plate theory is excluded, because the classical plate theory can only regard small deformations and linear material behaviour. For this reason, the modelling of the mechanical behaviour of the composite plate is based on a numerical homogenisation. The underlying principle is a numerical multi-scale consideration of the composite plate: on the one hand, the macroscale is considered as a Finite Element computation of a plate which is following a plate theory including a thickness change, i. e. a seven degrees of freedom approach. On the other hand, the mesoscale is obtained with a three-dimensional modelling which explicitly takes into account the stacking order and the material behaviour of the different layers. From each integration point of the macroscale, the deformations are projected on the boundaries of a Representative Volume Element (RVE) of the mesoscale. In this scale, a boundary value problem is solved, and the macroscopic forces, moments and shear forces are obtained as resultants from the mesoscopic stresses.

Moving dislocations in finite plasticity: a topological approach

Abstract: Plastic deformation of crystals is mostly mediated by the motion of dislocations. During the last two decades a lot of effort was directed towards including more knowledge about dislocations in continuum descriptions of plasticity. Promising approaches towards building continuum plasticity theories on averages of the behavior of many single dislocations have been formulated under the assumption of small deformations. In the current paper we derive the kinematics of single dislocations moving inside a dislocated crystal simultaneously deforming by the motion of other dislocations in the language of large deformation plasticity. The evolution equation of a single dislocation is connected to the formation of kinks and jogs due to cutting by other dislocations and is shown to parallel the evolution equation of the dislocation density tensor in finite deformation formulation. Implications for dislocation based modeling of plasticity are discussed.

Parameter re‐identification in nanoindentation problems of viscoelastic polymer layers: small deformation

Abstract: Recently, nanoindentation became a new but all the same a primary testing technique of thin layers. A wide application of nanoindentation in polymeric layers is obstructed by the analysis method, which is used to extract the rate-dependent properties. In the present paper, the inverse method based on the finite element simulation and numerical optimisation is used to characterise the viscoelastic properties of polymers from nanoindentation. First of all, the boundary value problems using nanoindentation of polymer layers considering real geometry, is simulated with the FE code ABAQUS®. A linear viscoelastic model for small strain, based on a general Maxwell rheological model is currently applied to describe the rate-dependent material behaviour. The rate-dependent properties of the polymer layer under nanoindentation is investigated with various loading histories: cyclic testing, single step relaxation, monotonic testing, and sinusoidal oscillatory testing. A parameter re-identification strategy offers a deep insight into the relationship between the accuracy of identification and the loading history associated with the rate-dependent material model. A method to choose a suitable loading history to identify the parameters more accurately is recommended.

An extended Discontinuous Galerkin and Spectral Difference Method with modal filtering

Abstract: We give a short overview of an extended Discontinuous Galerkin and Spectral Difference Method using PKD polynomials on triangular grids. A corresponding exponential filter is used to avoid oscillations near discontinuous solutions and to give some stabilization for nonsmooth testcases.

Dynamic response of a cracked magnetoelectroelastic layer sandwiched between two elastic layers

Abstract: The dynamic response of a cracked magnetoelectroelastic layer sandwiched between dissimilar elastic layers under anti-plane shear and in-plane electric and magnetic impacts is investigated by the integral transform method. Fourier transforms and Laplace transforms are applied to reduce the mixed boundary value problem of the impermeable crack to simultaneous dual integral equations, which are then expressed in terms of simultaneous Fredholm integral equations of the second kind. The stress field, electric field and magnetic field near the crack tip are obtained asymptotically, and the corresponding field intensity factors are further determined. Numerical results show that the stress intensity factors are influenced by the material properties, the electric and magnetic loadings, and the geometry. The crack initiation can be enhanced or retarded depending on the electric and magnetic loading, and the crack may propagate along its original crack line when the criterion of maximum hoop stress is applied.

On the stable discretization of strongly anisotropic phase field models with applications to crystal growth

Abstract: We introduce unconditionally stable finite element approximations for anisotropic Allen– Cahn and Cahn–Hilliard equations. These equations frequently feature in phase field models that appear in materials science. On introducing the novel fully practical finite element approximations we prove their stability and demonstrate their applicability with some numerical results. We dedicate this article to the memory of our colleague and friend Christof Eck (1968– 2011) in recognition of his fundamental contributions to phase field models.

Global weak solutions for coupled transport processes in concrete walls at high temperatures

Abstract: We consider an initial-boundary value problem for a fully nonlinear coupled parabolic system with nonlinear boundary conditions modeling hygro-thermal behavior of concrete at high temperatures. We prove a global existence of a weak solution to this system on any physically relevant time interval. The main result is proved by an approximation procedure. This consists in proving the existence of solutions to mollified problems using the Leray-Schauder theorem, for which a priori estimates are obtained. The limit then provides a weak solution for the original problem. A practical example illustrates a performance of the model for a problem of a concrete segment exposed to transient heating according to three different fire scenarios. Here, the focus is on the short-term pore pressure build up, which can lead to explosive spalling of concrete and catastrophic failures of concrete structures.

Homogenization of thermoelasticity processes in composite materials with periodic structure of heterogeneities

Abstract: The majority of processes in composite materials involve a wide range of scales. Because of the scale disparity in multi scale problem, it's often impossible to resolve the effect of small scales directly. In this paper we perform multi scale modeling in order to analyze properties of composite materials with periodical structure under temperature and stresses influence. Each component has its own thermal and mechanical (elastic) properties. We replace differential equations with rapidly oscillating coefficients by homogenized equations having effective parameters, incorporating multi scale structure and properties of any component. The influence of any component's properties and of the layer thickness on the effective properties of whole sample has been studied. We perform comparison of results, obtained with and without account for thermostresses.

Dynamic contact problem for a von Kármán–Donnell shell

Abstract: The existence of solutions is proved for the unilateral dynamic contact of a von Karman-Donnell shell with a rigid obstacle. Both purely elastic material and a material with a singular memory are treated.

A perturbative model for predicting the high‐Reynolds‐number behaviour of the streamwise travelling waves technique in turbulent drag reduction

Abstract: The background of this work is the problem of reducing the aerodynamic turbulent friction drag, which is an important source of energy waste in innumerable technological fields (transportation being probably the most important). We develop a theoretical framework aimed at predicting the behaviour of existing drag reduction techniques when used at the large values of the Reynolds numbers Re which are typical of applications. We focus on one recently proposed and very promising technique, which consists in creating at the wall streamwise-travelling waves of spanwise velocity (M. Quadrio, P. Ricco, and C. Viotti, J. Fluid Mech. 627, 161–178, 2009). A perturbation analysis of the Navier-Stokes equations that govern the fluid motion is carried out, for the simplest wall-bounded flow geometry, i.e. the plane channel flow. The streamwise base flow is perturbed by the spanwise time-varying base flow induced by the travelling waves. An asymptotic expansion is then carried out with respect to the velocity amplitude of the travelling wave. The analysis, although based on several assumptions, leads to predictions of drag reduction that agree well with the measurements available in literature and mostly computed through Direct Numerical Simulations (DNS) of the full Navier–Stokes equations. New DNS data are produced on purpose in this work to validate our method further. The method is then applied to predict the drag-reducing performance of the streamwise-travelling waves at increasing Re, where comparison data are not available. The current belief, based on a Re-range of about one decade only above the transitional value, that drag reduction obtained at low Re is deemed to decrease as Re is increased is fully confirmed by our results. From a quantitative standpoint, however, our outlook based on several decades of increase in Re is much less pessimistic than other existing estimates, and motivates further, more accurate studies on the present subject.

On the temporal behaviour in the bending theory of porous thermoelastic plates

Abstract: In this paper we consider the bending theory of Mindlin type thermoelastic plates with voids. We study the temporal behaviour of the solution of the boundary-initial value problem of this theory. Assuming that the internal energy density is positive definite, relations describing the asymptotic behaviour of the Cesaro means of various parts of total energy are established. An extension of the results to a large class of thermoelastic materials with voids is given.

An infinite element for the solution of Galbrun equation

Abstract: In this article we present a method for the three-dimensional numerical simulation of the propagation of acoustic radiation inside and in the near field of long, slender, and hollow objects. While the fluid inside and close to the radiating body is meshed by Taylor-Hood tetrahedral finite elements, complex conjugated Astley-Leis infinite elements are added on the outer finite element boundary to present the effects in the far field. Flow is considered in the modal analysis, with the goal to determine the influence a moving fluid has on the eigenfrequencies of the model. Galbrun equation in a mixed, meaning pressure and displacement based, formulation is used for the numerical realization and its weak form is presented for the finite element domain as well as for the infinite element domain.