Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2011"

Generalized continua and non-homogeneous boundary conditions in homogenisation methods

Abstract: Extensions of classical homogenization methods are presented that are used to replace a composite material by an effective generalized continuum model. Homogeneous equivalent second gradient and micromorphic models are considered, establishing links between the macroscopic generalized stress and strain measures and the fields of displacement, strain and stress inside a volume element of composite material. Recently proposed non-homogeneous boundary conditions to be applied to the unit cell, are critically reviewed. In particular, it is shown that such polynomial expansions of the local displacement field must be complemented by a generally non-periodic fluctuation field. A computational strategy is introduced to unambiguously determine this fluctuation. The approach is well-suited for elastic as well as elastoplastic composites.

Thermal buckling of clamped thin rectangular FGM plates resting on Pasternak elastic foundation (Three approximate analytical solutions)

Abstract: Buckling of rectangular plates made of functionally graded material (FGM) under various types of thermal loading is considered. It is assumed that the plate is in contact with an elastic foundation during deformation. The derivation of equations is based on the classical plate theory. It is assumed that the mechanical and thermal non-homogeneous properties of FGM plate vary smoothly by distribution of power law across the plate thickness. Using the non-linear strain-displacement rela- tions, the equilibrium and stability equations of plates made of FGMs are derived. The boundary conditions for the plate are assumed to be clamped for all edges. The elastic foundation is modelled by two parameters Pasternak model, which is obtained by adding a shear layer to the Winkler model. Three distinct analytical solutions are presented to study the thermal buckling problem of thin FGM plates. These methods may be useful to study the eigenvalue problems for other loading types of FGM plates. Closed-form solutions are presented and effects of various parameters, such as geometric characteristics and Pasternak elastic coefficients, are presented comprehensively.

On the spectrum and stiffness of an elastic body with surface stresses

Abstract: A mathematical investigation of the eigenvalue problems for elastic bodies including surface stresses is presented. Weak setup of the problems is based on the Rayleigh variational principle. Certain spectral properties are established for the problems under consideration. In particular, bounds for the eigenfrequencies of an elastic body with surface stresses are presented. These bounds demonstrate increases in both the rigidity of the body and of the eigenfrequencies over those of the body with surface stresses neglected.

A formulation of the linear discrete Coulomb friction problem via convex optimization

Abstract: This paper presents a new formulation of the dynamical Coulomb friction problem in finite dimension with discretized time. The novelty of our approach is to capture and treat directly the friction model as a parametric quadratic optimization problem with second-order cone constraints coupled with a fixed point equation. This intrinsic formulation allows a simple existence proof under reasonable assumptions, as well as a variety of solution algorithms. We study mechanical interpretations of these assumptions, showing in particular that they are actually necessary and sufficient for a basic example similar to the so-called ''paradox of Painleve''. Finally, we present some implementations and experiments to illustrate the practical aspect of our work.

Existence theorems in the linear theory of micropolar shells

Abstract: Theorems regarding existence and uniqueness of weak solutions to mixed boundary value problems in the linear theory of micropolar shells in statics and dynamics are proved. Convergence of FEM for the static mixed problems is established. Eigenvalue problems for micropolar shells are studied and properties of the spectrum and eigenmodes are formulated.

Optimal control of cracks in elastic bodies with thin rigid inclusions

Abstract: In this paper we consider the control of cracks in elastic bodies with rigid inclusion. We first describe the problem statement, provide an equivalent formulation as a variational inequality and prove existence and uniqueness of solutions. Furthermore, we consider this problem as a limiting problem when the elasticity parameters of the inclusion tend to infinity. Then we formulate the optimal control problems and derive an explicit formula for the crack sensitivity and for the energy release rate. We show existence of optimal solutions.

Homogenization of a 1D nonlinear dynamical problem for periodic composites

Abstract: In this paper we study wave propagation through a composite material built up of a periodically repeated one-dimensional structure of coated inclusions and matrix material by the application of the asymptotic homogenization method. We take into account geometrical nonlinearity, which is described by the Cauchy-Green strain tensor and physical nonlinearity by the Murnaghan elastic potential. We take into account structural nonlinearity by considering the bonding between two materials to be imperfect. As a result we obtain homogenized equations for the low-frequency range.

A phenomenological model of finite strain viscoplasticity with distortional hardening

Abstract: In this paper we suggest a thermodynamically consistent approach to the simulation of a rate dependent material response at finite strains. The nonlinear mechanical phenomena which are covered by the proposed material model include distortional, kinematic, and isotropic hardening. Firstly, we present a new two-dimensional rheological model of distortional hardening, which predicts the yield curve in the stress space to be a limacon of Pascal. Such effects like the distortion of the yield surface in the stress space and its orientation depending on the loading path are captured by the rheological model in a vivid way. Next, the rheological model serves as a guideline for the construction of the constitutive equations. In particular, the kinematic assumptions, the ansatz for the free energy, and the form of the yield function are motivated by the rheological model. Further, two types of flow rules are considered in this study: a normality rule and a radial rule, both thermodynamically consistent. Moreover, we formulate explicitly the constraints on the material parameters, which guarantee the convexity of the yield surface. Furthermore, implicit time-stepping methods are considered which exactly preserve the incompressibility of the inelastic flow. Finally, the basic features of the predicted material response are illustrated by a series of numerical simulations. In particular, the simulation results are compared to the real experimental data.

Experimental investigations and material modelling of curing processes under small deformations

Abstract: There are many applications of curing epoxy resins. The paste-like material can be applied to joint two structural parts of different materials. The curing reaction in the epoxy resin increases the stiffness of the material to fix the connected partners. Furthermore, the curing epoxy resin shrinks and the reaction generates exothermal energy. In order to characterise the exothermal reaction, experimental results of dynamic scanning calorimetry are presented. The change in the viscoelastic behaviour is measured with the rheometer AR-G2 of TA-Instruments. In addition, investigations of the shrinkage behaviour are presented. Based on the observed changes during the curing reaction a thermodynamic consistent material model for small deformations is introduced and the parameters of the material model are identified to reproduce the experimental data.

Disclinations in nonlinear elasticity

Abstract: The theory of line defects (dislocations and disclinations) in elastic bodies has been revisited. A consistent application of the formal limiting passage from isolated defects to the continuous distribution of these allows one to obtain a complete system of equations describing internal stresses in a body with distributed defects. Special emphasis is placed on disclinations in planar systems like graphene which very often demonstrate nonlinear elastic response. As a specific example an exact solution has been found for the problem of the eigenstresses in a disc of nonlinear elastic material due to a given density of continuously distributed disclinations.

Optimal control of static plasticity with linear kinematic hardening

Abstract: An optimal control problem for the static problem of infinitesimal elastoplasticity with linear kinematic hardening is considered. The variational inequality arising on the lower-level is regularized using a Yosida-type approach, and an optimal control problem for the so-called viscoplastic model is obtained. Existence of a global optimizer is proved for both the regularized and original problems, and strong convergence of the solutions is established.

Evaluation of the elastic properties of a functionally-graded coating from the indentation measurements

Abstract: The paper focuses on the development of a mathematical model connecting penetration profile of a rigid circular indenter with elastic properties of a functionally graded coating (FGC). We introduce the concept of stiffness function dependent on the radius of the contact zone for FGC coupled with a homogeneous half-space. This approach allows one to determine the properties variation across the coating. Several most typical numerical examples are discussed in detail to illustrate effect of the Young's modulus variation on the stiffness function. These examples also show that the proposed methodology allows one to identify and experimentally distinguish between different types of variation of FGC elastic properties.

The undulatory motion of a chain of particles in a resistive medium

Abstract: The motion of a finite chain of identical bodies along a straight line in a resistive medium is studied The major aim of this study is to investigate the fundamental properties of such systems, in particular, their ability to move from a state of rest and sustain the motion at constant average velocity in media with different resistance properties and the influence of the control strategy on the motion The motion is excited and controlled by changing the distances between the bodies of the chain For a given friction law, the necessary and sufficient conditions for the system to be able to move from rest are established

Investigation of the stress‐strain state of the laminated shallow shells by R‐functions method combined with spline‐approximation

Abstract: The bending behavior of the laminated shallow shells under static loading has been studied using the R-functions theory together with the spline-approximation Formulation is based on the first order shear deformation theory Due to usage of the R-functions theory the laminated shallow shells with complex shape and different types of the boundary conditions can be investigated Application of the spline-approximation allows getting reliable and validated results for non concave domains and domains with holes The proposed method is implemented in the appropriate software in framework of the mathematical package MAPLE The analysis of influence of certain factors (curvature, packing of layers, geometrical parameters, boundary conditions) on a stress-strain state is carried out for shallow shells with cut-outs The comparison of obtained results with those already known from literature and results obtained by using ANSYS are also presented

Analysis of a pre‐stressed bi‐material accelerated‐life‐test (ALT) specimen

Abstract: Application of mechanical pre-stressing could be an effective means for achieving a failure-mode-shift-free "destructive ALT effect" in electronic and photonic devices and micro-electro-mechanical systems (MEMS). A simple, physically meaningful and easy-to-use analytical ("mathematical") predictive model has been developed to assess the magnitude and the distribution of stresses in a bi-material assembly subjected to the combined action of thermally induced (considered by the ALT design) and external ("mechanical") pre-stressing. Such a compressive pre-stressing is applied to the assembly component that is expected to experience thermal compression. The model is an extension and a modification of the author's 1986 and 1989 "bi-metal thermostat" models suggested as a generalization of the 1925 Timoshenko's theory.

On a strain gradient elastic Timoshenko beam model

Abstract: Considering the influence of the microstructure, the Timoshenko beam model is revisited, invoking Mindlin's strain gradient strain energy density function. The equations of motion are derived and the bending equilibrium equations are discussed. The adopted strain energy density function includes new terms. Those terms introduce the strong effect of the beam cross-section area. The influence of those terms is more evident in thin beams where the cross-section area is far bigger than its moment of inertia. Applications have been worked out exhibiting the difference of the present theory not only from the classical Timoshenko beam, but also from the existing variations including couple stresses. The solution of the static problem, for a simply supported beam loaded by a force at the middle of the beam, is defined and the first (least) eigen-frequency is found. The present model is proved to be stiffer.

A variational approach to the decomposition of unstable viscous fluids and its consistent numerical approximation

Abstract: The mixing and de-mixing properties of highly viscous fluids can be described as a diffusive system driven by an external velocity field. In order to analyze the micro-morphological evolution with a diffusion theory of heterogeneous mixtures we apply a Cahn-Hilliard phase-field model, which is here extended by a convective term. As the model is based on an energetic variational formulation, we state the underlying energy minimizing principles. For consistent finite element analysis we provide a piecewise smooth and globally C 1 -continuous approximation by means of rational B-spline basis functions. Numerical examples exhibit the physical properties of the model. Simulations of phase decomposition and coarsening controlled by diffusion and by convection show the robustness and the versatility of our approach.

FEM analysis of Cosserat plates and shells based on some constitutive relations

Abstract: We carry out further study of the constitutive laws for Cosserat plates developed by Altenbach and Eremeyev. In particular, we examine the problem of choice of the micropolar material coefficients. Requiring that the constitutive matrix is positive definite, we establish some bounds on values of micropolar constants. The constitutive relations for Cosserat plates have been implemented into formulation of shell finite elements developed within the framework of the statically and kinematically exact, nonlinear six-parameter shell theory. By the linear parametric-sensitivity analysis we study the influence of the micropolar material constants on the response of shell structure with orthogonal intersections of branches. Such structures can naturally be analyzed using the six-parameter shell theory. Having established the most influential coefficients, we show how these values affect the behavior of the structure in the nonlinear range of deformations.

On restoring of the pre‐stressed state in elastic bodies

Abstract: This research is devoted to the development of theoretical foundations for identification of an essentially inhomogeneous prestressed state by analyzing the gain-frequency characteristic of boundary points of the body. The proposed scheme to the reconstruction of inhomogeneous prestresses is constructed on the iterative processes. It includes the finite element solution of the direct problem and the regularizing procedure to solve the Fredholm integral equation of the first kind in the inverse problem. In the series of one-dimensional model examples it was shown that this scheme is effective.

Quasistatic delamination models for Kirchhoff‐Love plates

Abstract: A quasistatic rate-independent brittle delamination problem and also an adhesive unilateral contact problem is considered on a prescribed normally-positioned surface in a plate with a finite thickness. By letting the thickness of the plate go to zero, two quasistatic rate-independent crack models with prescribed path for Kirchhoff-Love plates are obtained as limit of these quasistatic processes.

Dynamic stress and electric field concentration in a functionally graded piezoelectric solid with a circular hole

Abstract: This work addresses the evaluation of the stress and electric field concentrations around a circular hole in a functionally graded piezoelectric plane subjected to anti-plane elastic SH-wave and in-plane time-harmonic electric load. All material parameters vary exponentially along a line of arbitrary orientation in the plane of the piezoelectric material under consideration. The computational tool is a non-hypersingular traction based boundary integral equation method (BIEM). The kernel functions used in the BIEM are exact fundamental solutions that have been derived in previous work by the authors. Numerical solutions for the stress and electric field concentration factors (SCF and EFCF, respectively) around the perimeter of the hole are obtained. The simulation demonstrates the efficiency of the computational approach and its potential to reveal in an adequate way the dynamic stress and electric field distribution around the hole. Presented are results showing their dependence on various system parameters as e.g. the electro-mechanical coupling, the type of the dynamic load and its characteristics, the wave-hole and wave-material interaction and the magnitude and direction of the material inhomogeneity.

Wave propagation in an elastically supported string with point-wise defects: gap-band and pass-band effects

Abstract: Wave excitation, propagation and diffraction phenomena as well as related resonance trap-mode, gap-band and pass-band effects occurring in one-dimensional waveguides with obstacles are considered. The waveguide is a spring-supported string with pointwise changed cross-section and/or spring force. The analysis is based on semi-analytical solution in terms of the waveguide's Green functions with unknown coefficients. The latter are obtained from a linear algebraic system whose eigenvalues are spectral points of the problem considered. One of the purposes is to examine the relation between the spectral point allocation in the complex frequency plane and the resonance wave phenomena observed. It is shown that pass-band phenomena are controlled by natural frequencies approaching to the real axis.

Collinear unequal crack series in magnetoelectroelastic materials: Anti‐plane case

Abstract: The purpose of the present work is to study collinear unequal crack series in magnetoelectroelastic materials, which are of more practical significance than the widely studied collinear equal cracks Fracture analysis is performed by the combined methods of distributed generalized dislocations, Green’s function and singular integral equations (SIEs) The theoretical derivation is validated by the degeneration of SIEs in special cases Parametric studies on the numerical results of mechanical strain energy release rate (MSERR) yield the following conclusions (a) As two unequal cracks approach each other, interference occurs in crack tip fields, resulting in an obvious increase of MSERR The ratio between the crack space and the length of the shorter crack is a main factor affecting the interference (b) Negative or positive electric loading may shield or enhance crack growth, respectively In the case of positive magnetic permeability, the effect of magnetic loading is analogous to that of electric loading; however, if magnetic permeability is negative, the result is completely opposite (c) Positive values of magnetic permeability are found to be more reasonable than the negative values

Inverse scattering problem for a rigid scatterer or a cavity in elastodynamics

Abstract: We consider the inverse scattering problem for the shape determination of a rigid scatterer or a cavity in a homogeneous and isotropic elastic medium. Both problems are formulated in R 2 for time-harmonic fields and longitudinal or transversal incident plane waves. Representation of the scattered field as a single- or a double-layer potential, equivalently, leads to a system of two nonlinear integral equations for the density and the parametrization of the boundary. A detailed numerical implementation is presented for computing the corresponding solutions of both systems and numerical reconstructions are given to show the effectiveness of the method. c

Exact solution to the antiplane problem of doubly periodic conducting rigid line inclusions of unequal size in piezoelectric materials

Abstract: The problem of a doubly periodic array of conducting rigid line inclusions in piezoelectric materials under far-field antiplane mechanical load and inplane electric load is investigated, where the fundamental cell contains four rigid line inclusions of unequal size. An exact solution to the problem is presented by employing the conformal mapping technique and the elliptical function theory. The closed form formulae for the stress and electrical displacement intensity factors at the rigid line inclusion tip and the effective electroelastic moduli of such composites are derived. Many new and existing solutions can be regarded as the special or degenerated cases. Numerical examples are provided to show the interesting electroelastic interaction phenomenon induced by multiple rigid line inclusions. The present work is helpful in understanding the optimization mechanism of some naturally occurring composites and in designing novel materials by direct engineering of their microstructure.

Traffic flow with unobservant drivers

Abstract: We discuss mathematical properties of a coupled macroscopic second-order traffic model with different pressure laws on the connected roads. Of particular interest is the zero pressure limit and the arising solutions to the coupled equations. This model can be used for prediction of traffic accidents on roads.

Completely passive natural convection

Abstract: We show that a unique, nontrivial, natural convection state exists under the Boussinesq approximation and completely passive boundary conditions.

Heat transfer in a micropolar fluid flow past a permeable continuous moving surface

Abstract: A boundary layer analysis is presented for the fluid flow and heat transfer characteristics of an incompressible micropolar fluid flowing over a permeable continuous moving surface with uniform surface heat flux conditions. Two cases are considered, one corresponding to a plane surface moving in parallel with a free stream and the other, a surface moving in the opposite direction to the free stream. The resulting system of non-linear ordinary differential equations is solved numerically using fourth-order Runge-Kutta method with shooting techniques. Numerical results are obtained for velocity, angular velocity, and temperature distributions, as well as the friction factor, local Nusselt number and wall couple stress for prescribed values of the velocity ratio parameter, suction/injection parameter, and micropolar parameter. The obtained results are presented graphically and in tabular form and the physical aspects of the problem are discussed for both parallel and reverse moving plates relative to the free stream direction.

On the dynamical theory of thermoelastic simple shells

Abstract: In this paper we present a mathematical study of the equations of motion for orthotropic thermoelastic simple shells. We use a direct approach to the mechanics of thin shells, in which the shell-like body is modeled as a deformable surface endowed with a triad of orthonormal vectors connected with each material point. The thermal effects are described by introducing two temperature fields which represent the temperature on the two major faces of the three-dimensional shell. In the framework of the linear theory, we establish first the uniqueness of solution to the associated boundary–initial–value problem. Then we prove the properties of reciprocity, we give a variational characterization, and we investigate the continuous dependence of solutions on the external data. Finally, we present an existence result for the weak solutions to the equations of motion of thermoelastic simple shells.

Dynamics of mechanical systems with nonlinear nonholonomic constraints – I The history of solving the problem of a material realization of a nonlinear nonholonomic constraint

Abstract: The paper brings forth a detailed analysis of the solution of the problem of the material realization of a nonlinear nonholonomic constraint (NNC). The existing models of the NNC are shown that can be classified into two groups: the first group comprises correctly realized physical models, while the second group contains the so-called “quasinonlinear” nonholonomic constraints which in fact represent mathematical models. The correctness of the cited models is considered in detail, and the essential nature of such constraints, the basic of which is holonomic, is shown. The second group of models, i.e., the “quasinonlinear” NC (nonholonomic constraints) in fact represents the given program of motion, while the additional force, which carries out the program, has the analytical form of the reaction of the NNC. That is why are presented the models of the NNC which possess a clear physical sense, on the basis of which certain statements on the method of variation and the reaction of the NNC can be given. With regard to the clear physical sense and the nature of the models cited, the NNC that come out of them are used quite normally in the analysis of motion of such a system. The cited models, together with standard models oh nonholonomic Mechanics (sphere, disk, blade) make a group of basic nonholonomic constraints which can be classified, according to the three criteria, into certain types. Finally, it is shown that the cited model can be used for the construction of “nonholonomic chains”, both open and closed ones.