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About this book

This book presents a liber amicorum dedicated to Wolfgang H. Müller, and highlights recent advances in Prof. Müller’s major fields of research: continuum mechanics, generalized mechanics, thermodynamics, mechanochemistry, and geomechanics.

Over 50 of Prof. Müller’s friends and colleagues contributed to this book, which commemorates his 60th birthday and was published in recognition of his outstanding contributions.

Table of Contents


Chapter 1. Magnetorheological Elastomer’s Material Modeling and Parameter Determination by Using the Energy-based Method

Functionalized materials provide tailored properties to design smart structures. For example, by adding polarized particles in a polymer, a composite material is generated, which couples deformation with electromagnetism. This magnetorheological elastomer (MRE) is a particle reinforced polymer matrix. Such a composite material deforms under an externally applied magnetic field so materials response is steered without contact. In order to achieve a simulation of an engineering design with MRE, we need an appropriate constitutive (material) equation modeling the deformation behavior accurately under different magnetic fields. We aim at determining the parameters in such a material equation out of experiments by using an inverse analysis. Although the material equation is nonlinear in deformation, its material parameters are mostly in such a way that we acquire a linear regression problem by using the energy-based method. Hence, the obtained parameters are unique and the method is fast allowing us to try out various material models.We present the method for determining the material parameters out of experimental data obtained by a standard rotational rheometer. The proposed material equation with its determined parameters can be used in a computation, for example by the finite element method.

Bilen Emek Abali, Hua Yang

Chapter 2. On the Size Effects in Indentation Testing of Elastic Functionally-graded Materials

The size effect in the small-scale indentation testing is studied for a functionally-graded material (FGM) whose shear elastic modulus varies according to the exponential law. Under the simplifying assumption of zero Poisson’s ratio, the asymptotic model of the indentation stiffness for an axisymmetric frictionless indenter is developed in the case when the contact radius is small compared to the inhomogeneity characteristic size. The so-called sample size effect is considered on the example of a simply supported FGM plate indented at the center of its top surface. A certain range of applicability of the first-order asymptotic models has been established by comparison with the approximate analytical solution available in the literature.

Ivan Argatov

Chapter 3. The Effect of Mechanical Load-induced Intraosseous Pressure Gradients on Bone Remodeling

It is well established that changes in bone blood and interstitial fluid flows are associated with changes in the bone remodeling process. These flows in bone are a result not only of trans-cortical pressure gradients produced by vascular and hydro-static pressure, but also of mechanical loadings. Mechanical load-induced intraosseous pressure gradients may result in some fluid stimuli effects which, in turn, may enable bone cells to detect external mechanical signals. In this paper, the exploitation of a 2D continuum model based on classical poroelasticity is presented within a variational framework. The investigation is aimed at describing how mechanical actions can affect the remodeling process of a bone tissue. The focus is on the introduction of a physically motivated strain energy contribution aimed to take into account the presence of saturating fluid in the interconnected pores of bone tissue. The interaction with a bio-resorbable organic ceramic material like those used in bone graft implants is also considered in presented model. Numerical results are provided in a relevant exemplary case.

Emilio Barchiesi, Ivan Giorgio, Faris Alzahrani, Tasawar Hayat

Chapter 4. Mechanical and Thermodynamic Materials Properties Derived by Semi-empirical Atomic Potentials with Special Focus on Ag, Cu, and the Binary Alloy Ag-Cu

The following contribution deals with the relationship between atomic interaction potentials and macrospcopic materials properties typically required in engineering disciplines such as mechanical engineering or thermodynamics. Special focus is exemplarily turned to the so-called Nearest Neighbor Embedded-Atom-Method, which has proved to reliably calculate various materials properties especially for FCC lattice configurations. An energy expression for binary alloys is derived and linked to the elastic constants as well as to the phase diagram construction. The obtained equations are applied to the binary brazing alloy Ag-Cu, and the results are compared to experimental data. Finally the theory is extended to lattice dynamics/vibrations in order to calculate temperature-depend materials quantities such as the heat capacity.

Thomas Böhme

Chapter 5. Mechanical Response Change in Fine Grain Concrete Under High Strain and Stress Rates

Experimental results on assessing the effects of strain and stress rates on the behavior of fine-grain concretes are presented. Specimens of fine-grain and fiberreinforced concretes were dynamically tested using the Kolsky method and its modification, the “Brazilian test”. As a result of the experiments, values of the Dynamic Increase Factor (DIF) were determined for both the materials studied. Their curves as a function of strain and stress rates were constructed. The experimental data is compared with the theoretically obtained values of DIF as a function of strain rate available in the literature

Francesco dell’Isola, Anatoly M. Bragov, Leonid A. Igumnov, Bilen Emek Abali, Andrey K. Lomunov, Dmitry A. Lamzin, Alexander Yu. Konstantinov

Chapter 6. Estimating Fatigue Related Damage in Alloys under Block-type Non-symmetrical Low-cycle Loading

Processes of plastic deformation and damage accumulation in polycrystalline structural alloys are investigated under block-type, nonstationary, non-symmetric cyclic loading. In the framework of damage mechanics, a mathematical model is proposed that effectively describes elastoplastic deformation and fatigue related damage accumulation processes under low-cycle loading. This model can be subsumed under three main parts: the relations defining elastoplastic behavior of the material; the equations describing damage accumulation kinetics; the strength criterion of the damaged material. For validating the model, we perform a numerical analysis and a comparison with the data from full-scale experiments.We demonstrate that the proposed model qualitatively and quantitatively describes the main effects of plastic deformation and damage accumulation processes in structural alloys under complex loading scenarios. Moreover, fatigue related lifetime of the structure is accurately captured by this model as well.

Francesco dell’Isola, Ivan A. Volkov, Leonid A. Igumnov, Simon R. Eugster, Svetlana Yu. Litvinchuk, Dmitri A. Kazakov, Vasilii A. Gorohov, Bilen Emek Abali

Chapter 7. On Non-holonomic Boundary Conditions within the Nonlinear Cosserat Continuum

Within the framework of the nonlinear micropolar elastic continuum we discuss non-holonomic kinematic boundary conditions. By non-holonomic boundary conditions we mean linear relations between virtual displacements and virtual rotations given on the boundary. Such boundary conditions can be used for modelling of complex material interactions in the vicinity of the boundaries and interfaces.

Victor A. Eremeyev

Chapter 8. Nonlinear Localized Waves of Deformation in the Class of Metamaterials as Set as the Mass-in-mass Chain

A well-known mathematical model representing a chain of oscillators consisting of elastic elements and masses, each containing an internal oscillator and describing the class of acoustic metamaterials “mass-in-mass”, is generalized by taking into account the nonlinearity of the external and (or) internal elastic elements. As a result of analysis of the long-wavelength approximation of the obtained system, it is shown that spatially localized nonlinear deformation waves (solitons) can be formed in a metamaterial, under dynamic influence on it. The dependencies connecting the parameters of a localized wave are determined: amplitude, velocity and width with inertial and elastic characteristics of the metamaterial.

Vladimir I. Erofeev, Daniil A. Kolesov, Alexey O. Malkhanov

Chapter 9. Modelling of a Hydrogen Saturated Layer Within the Micropolar Approach

This paper is concerned with modeling the strongly inhomogeneous hydrogen distribution over a sample by means of the micropolar continuum approach. The presence of micro-cracks covering the lateral surface of the sample is modeled by means of a distributed couple stress prescribed as a boundary condition. The applied couple stress produces a longitudinal displacement in return, which quickly fades away from the surface. The tensile displacement increases the intergranular space in the vicinity of the sample boundary and initiates hydrogen absorption from the environment. A comparison between widths of the surface layer that were experimentally determined and the ones that were analytically obtained allows estimating a value of one of the non-classical elastic parameters.

Ksenia Frolova, Elena Vilchevskaya, Vladimir Polyanskiy, Ekaterina Alekseeva

Chapter 10. Types of Physical Nonlinearity in the Theory of Constitutive Relations and the Generalized Poynting Effect

The certain class of constitutive relations are considered that connect the symmetric stress tensor and the symmetric strain tensor by means of isotropic potential tensor nonlinear functions in three-dimensional space. The various definitions of tensor nonlinearity are given as well as their equivalence is shown. From the perspective of mathematical theory about the tensor nonlinear functions, an interpretation of the Poynting effect is given, which is well known in experimental mechanics. It is demonstrated that such an effect is not necessarily the consequence of tensor nonlinearity in constitutive relations; instead, it is effected by the quadratic dependence on invariants in certain material functions. Therefore, in the physically linear case for a small strain, this dependence is absent. Concerning this “order of smallness,” the Poynting effect is investigated and a possibility is discussed for simulating such an effect by means of the tensor linear constitutive relations.

Dimitri V. Georgievskii

Chapter 11. Eigenstresses in a Nonlinearly Elastic Sphere with Distributed Dislocations

The problem of the eigenstresses due to distributed edge and screw dislocations in a hollow nonlinearly elastic sphere is considered. The dislocation density is given by an arbitrary spherically symmetric tensor field. For a general isotropic elastic material, the problem is reduced to a one-dimensional nonlinear boundary value problem. By replacing the unknown functions, the boundary value problem with nonlinear boundary conditions is transformed to a problem with linear ones. Numerical solutions are constructed for specific models of compressible and incompressible materials. The analysis of the influence of dislocations on a stress state of an elastic sphere at large deformations is carried out.

Evgeniya V. Goloveshkina, Leonid M. Zubov

Chapter 12. Fundamental Solution for the Generalized Plane Stress of a Nanoplate

The fundamental solution for the generalized plane-stress problem of an infinite, isotropic elastic plate subjected to a point force is presented taking into account surface stresses in the plate faces. Constitutive equations is derived using the stress-strain relations for the bulk material and Gurtin–Murdoch’s linearized surface elasticity equations for the surfaces of the plate supposing that the residual surface stress is negligibly small compared with the surface elasticity parameters. The complex relations (Green functions) for the stresses and displacements in the explicit form are evaluated using Goursat–Kolosov complex potentials and Muskhelishvili representations. It is shown that in the case of the generalized plane stress, the fundamental solution depends on the thickness of the plate that is the size effect intrinsic to the nanoobjects.

Mikhail A. Grekov

Chapter 13. Isotropic Linear Viscoelastic Reduced Cosserat Medium: an Acoustic Metamaterial and a First Step to Model Geomedium

The reduced Cosserat medium is a continuum whose body points possess rotational degrees of freedom, and there is a reaction to the rotation of a body point relatively to the background of centres of mass, but no stresses are caused by the gradient of micro-rotation. This theory is useful for modelling rocks and soils containing heterogeneities, a geomedium with blocky structure, certain composites with inclusions as well as seismic metamaterials. In this work we consider the influence of viscosity in the linear isotropic reduced Cosserat medium on the propagation of shear waves. We find that viscosity may change drastically the wave propagation. In some cases, the material behaves as a double negative acoustic metamaterial for shear waves, i.e. there is a decreasing part of the dispersion curve for a certain band of frequencies. We also observe that the attenuation in such a continuum does not necessarily increases with frequency, as it happens in the classical viscoelastic medium. It may have one maximum at a certain frequency, or have maximum and minimum. Similar phenomena are observed in the range of seismic frequencies for a geomedium (Sato et al, 2012). The theory considered in this work is only the first step to model the geomedium since it does not take into account previous stress state and existing couplings between pressure and shear-rotational waves due to rock anisotropy caused by gravity.

Elena F. Grekova, Rafael Abreu

Chapter 14. Numerical Analysis of Free Vibrations of Piezoelectric Cylinders

Various approaches to solving the linear electroelasticity problems of finite-length cylinders with the discrete-continuous (spline-collocation) and discrete (finite-element) methods are considered. An axisymmetric problem on free vibrations of hollow finite-length cylinders made of piezoelectric materials is solved within the framework of the 3D electroelasticity theory. The lateral surfaces of the cylinders are free of external actions and covered by short-circuited electrodes. The cylinder material is radially polarized. The cylinders with a clamped end are analyzed numerically using two approaches. Practical agreement of the results obtained testifies that the solution is accurate.

Alexander Ya. Grigorenko, Igor A. Loza, Sergiy N. Yaremchenko

Chapter 15. Qualitative Investigations of Experiments Performed on 3D-FDM-printed Pantographic Structures Made out of PLA

Additive manufacturing methods, commonly known as 3D printing, enable the design and manufacturing of complex and sophisticated material fabrics with a special substructure resulting in extraordinary macroscopic deformation behavior. Such a man-made structure is also referred to as a metamaterial. So called pantographic structures, which can be described as metamaterials with a substructure that is composed of two orthogonal arrays of beams connected by internal cylinders, were manufactured using fused deposition modeling technique. In order to further understand the peculiarity of its deformation behaviors, a plane sheet was also printed to be used as a comparison. Different types of experiments were performed and evaluated qualitatively by the means of digital image correlation being able to localize the initial area of out-of-plane movements in shearing tests for both specimen. Results of quasi-static standard tension and shearing tests indicate a resilient material behavior during high elastic deformations resulting in a high resistance against total failure of the structure. Furthermore, cyclic long-term tests show a viscoelastic deformation behavior of the thermoplastic material. PSs show linear as well as non-linear elastic deformation response in all experiments except the cyclic tension test.

Arion Juritza, Hua Yang, Gregor Ganzosch

Chapter 16. Calculation of Stress Intensity Factors for an Arbitrary Oriented Penny-shaped Crack Under Inner Pressure in an Orthotropic Electroelastic Material

The electroelasticity problem for an arbitrarily oriented disc-like crack under internal pressure in an orthotropic electroelastic material was considered. Generalizing of the Willis approach for an elastic material, using the Fourier transform of the Green’s function for an infinite anisotropic electroelastic space, the problem of electroelasticity is reduced to finding unknowns of the jumps of displacements and electric potential through the surface of a circular crack. Quadrature Gauss formulas were used to calculate one-dimensional integrals. Testing the approach in the particular case of the problem for which an exact solution is known confirms the effctiveness of the used approach. The distribution of stress intensity factors (SIF) along the boundary of a disc-shaped crack (under internal pressure) in a piezoelectric orthotropic material under various orientations of a crack was studied. A significant effect of the crack orientation on the SIF distributions was established.

Vitaly S. Kirilyuk, Olga I. Levchuk, Holm Altenbach

Chapter 17. On the Quasi-Static Approximation to the Initial Traction Boundary Problem of Linear Elastodynamics

Conditions are investigated that are suffcient for the quasi-static approximation to be valid in the initial traction boundary value problem of linear elastodynamics on a bounded three-dimensional spatial region. The approach consists of two main steps both of which involve trace inequalities to derive explicit estimates. The first establishes continuous dependence of the solution upon the inertia. The second treats continuous dependence of the inertia upon surface traction and body-force. Circumstances when the approximation is not valid are briefly discussed.

Robin J. Knops

Chapter 18. Delamination Buckling in Composite Plates: an Analytical Approach to Predict Delamination Growth

An analytical modelling approach is presented which is capable of determining the post-buckling responses as well as the onset of delamination growth of multi-layered composite plates with an embedded circular delamination. In order to overcome current drawbacks of analytical models regarding embedded delaminations, the model employs a problem description in cylindrical coordinates and a novel geometric representation of delamination growth in conjunction with a Rayleigh-Ritz formulation and the so-called crack-tip element analysis. The modelling approach is applied to study the compressive response of composite plates with thin-film delaminations loaded under radial compressive strain. Post-buckling responses and the onset of delamination growth are determined for several layups. The results are in very good agreement with finite element simulations while requiring low computational cost.

Anton Köllner, Fabian Forsbach, Christina Völlmecke

Chapter 19. Dynamical Vector Fields on Pantographic Sheet: Experimental Observations

In this work, we will present and discuss some experimental observations of the dynamical displacement vector field on a pantographic sheet. We will sketch the experimental setup and we will qualitatively describe the observed behavior for a set of relevant frequencies.

Marco Laudato, Fabio Di Cosmo, Rafał Drobnicki, Peter Göransson

Chapter 20. Numerical Solution of the Tri-harmonic Kirchhoff Plate Equation Resulting from a Strain Gradient Theory

A second gradient continuum theory is formulated based on second gradients of displacements. For a reduction of additional material parameters, the modified strain gradient model is used and a partial dierential equation of rank six is developed using the Kirchhoff plate assumptions. The solutions of the governing tri-harmonic plate bending equation incoorperate size-effects. Balance equations are presented and higher-order stress-strain relations are derived. In order to account for second gradients of displacements, which manifest themselves in the higher-order terms of a strain energy density, a C1–continuous displacement field is preferable. So-called Hermite finite element formulations allow for merging gradients between elements and are used to achieve global C1–continuity of the solution. Element stiffness matrices as well as the global stiffness matrix are developed for a lexicographical order of nodes and for equidistantly distributed elements. The convergence, the C1–continuity, and the size effect are demonstrated.

Christian Liebold, Belal M. Dawwas

Chapter 21. Implications of the Lagrange Identity in Thermoelasticity of Dipolar Bodies

This paper is concerned with the mixed initial-boundary value problem in the context of the theory of thermoelasticity of dipolar bodies. We prove a uniqueness theorem and some continuous dependence theorems without recourse to any energy conservation law, or to any boundedness assumptions on the thermoelastic coefficients. This was possible due to the use of Lagrange’s identity. Because of the flexibility of this identity, we also avoid the use of positive definiteness assumptions on the thermoelastic coefficients.

Marin Marin, Andreas Öchsner, Sorin Vlase

Chapter 22. Theory and Computation of Nonlinear Damage Accumulation for Lifetime Prediction

Nonlinear damage accumulation is modelled for the lifetime prediction in order to capture the loading sequence effect, which is the influence of the chronological order of the loading values on the lifetime. The prediction results from the solution of the damage evolution equation, which is defined according to the theory of continuum damage mechanics and applied together with a cohesive zone model for structural adhesive joints. The damage model consists of a creep and fatigue damage part, both taking into account the influence of the mean stress and the load multiaxiality on the predicted time to rupture. The analytical investigation of the model shows the meaning of the model parameters and propose their identification by means of tests with static and constant amplitude loading. In order to capture the loading sequence effect by nonlinear damage accumulation, the fatigue damage part is enhanced with a factor, which influences the predicted lifetime due to variable amplitude loading in the case of pure fatigue damage, while the prediction for constant amplitude loading is unaffected. The influences of the enhancement on the predicted lifetime and the damage evolution are discussed. The comparison of lifetimes with numerical predictions proves the validity of the proposed approach.

Anton Matzenmiller, Ulrich Kroll

Chapter 23. A Non-equilibrium Approach Concerning Thermostatics of Schottky Systems

Non-equilibrium processes in Schottky systems generate by projection and relaxation reversible accompanying processes which serve as thermostatic approximations of different accuracy. The compatibility of the accompanying processes with the non-equilibrium ones according to the embedding theorem is shortly discussed.

Wolfgang Muschik

Chapter 24. On the Temperature Gradient in the Standard Troposphere

The temperature gradient in the upper troposphere is –0.65 $$ \frac{\text{K}}{{ 1 0 0 {\text{m}}}} $$ . We suggest that the thermal diffusion factor of air in the gravitational field determines this gradient.

Ingo Müller, Wolf Weiss

Chapter 25. A Brief History of Mechanical Stress and the Method of Experimental Micromechanics with the Raman Microprobe

The Raman microprobe is a unique tool for experimental stress analyses at the microscopic level, capable to overcome the drawbacks of other probes including low spatial resolution and lack of tensorial stress deconvolution. The presence of elastic stress in the lattice of crystalline materials results in shifts in frequency of Raman bands, which in turn obey individual dependencies, related to the specific molecular vibrations that they represent. More importantly, the observed frequency shifts depend on the reciprocal orientation between different stress components and the vibrating lattice. We present here some basic algorithms for stress assessments with the Raman micro-probe and validate them with some examples of practical engineering applications. We also introduce some details for three-dimensional measurement procedures using a confocal microprobe through a characterization of the probe itself. The high spatial resolution, the complete contactless nature, and the possibility to deconvolute the six independent components of the stress tensor from the Raman spectrum are key features, which place the Raman method at the frontiers of modern micromechanics.

Giuseppe Pezzotti

Chapter 26. Analytical Solutions of 2-dimensional Second Gradient Linear Elasticity for Continua with Cubic-D4 Microstructure

We consider in this paper analytical solutions for some remarkable cases and for a linear anisotropic D4 second gradient elastic model. The purpose is that of constitutive parameter identification. In general, analytical solutions are considered less important than in the past due to fast numerical tools and to the fact that they are generally very difficult to achieve. However, they still play very important roles such as for numerical comparison and for dealing with pathological mechanical systems.

Luca Placidi, Giuseppe Rosi, Emilio Barchiesi

Chapter 27. Gradient Theory of Adhesion and Tabor Parameter

There are two basic approaches to mathematical description of media having microstructure, or more general, one or more intrinsic characteristic lengths. The first approach is to consider the underlying structure explicitly. The opposite possibility is to try to simulate the medium still as a homogeneous one but having either additional degrees of freedom or being characterized by some intrinsic characteristic length. This second way is the way used in a wide spectrum of micropolar and gradient approaches.In the present paper, the philosophy of the gradient approaches is applied to the problem of adhesion. Adhesive forces have some range of action which naturally introduces a length parameter into the system. Relative role which plays this characteristic length in adhesive behavior is governed by a parameter introduced 1977 by Tabor. However, the interaction range is only one possible characteristic length in adhesion. Other lengths may be connected with spatial heterogeneity of the surfaces at the micro scale. We discuss the possibility of describing adhesion with a finite characteristic length in the framework of a continuum theory introducing proper gradient expansion. The result is both encouraging and disappointing: The gradient theory provides a description of adhesion with finite length which is much simpler compared with known explicit approaches. However, it remains phenomenological, so that proper physical interpretation of the characteristic length is not straightforward.

Valentin L. Popov

Chapter 28. Cavity Flow of a Micropolar Fluid - a Parameter Study

This paper presents a parameter study of the flow of a micropolar fluid. The underlying equations and the choice of boundary conditions are discussed. Two flow situations are considered: Couette flow as a reference problem and the liddriven cavity problem. The governing equations are specialized for the case of twodimensional flow and discussed in dimensionless form. Several dimensionless parameters common in the theory of micropolar fluids are identified and their impact on the solutions is analyzed using the finite element method.

Wilhelm Rickert, Sebastian Glane

Chapter 29. Graded Insulation to Improve High Pressure Resistance in Deepwater Flowlines: a Closed Form Analytical Elastic Solution

In this paper, an insulated pipe system comprising the inner pipe, insulation and outer jacket is investigated in the context of elasticity theory with the view to establish whether introducing stiffness gradient in the insulation would improve performance of the pipe under external pressure. Closed form analytical solutions are derived for stresses and displacements in the pipe system. Comparative analysis of pipes with homogeneous and graded insulation is performed and beneficial effect of graded insulation on stresses and displacements in the pipe is established.

Roberta Sburlati, Maria Kashtalyan

Chapter 30. On Brake Pad Shim Characterization: a Homogenization Approach and Finite Element Analysis

Brake squeal is a typical problem of “Noise, Vibration, Harshness” (NVH) phenomena in the automotive world leading to potential customer complaints. This high frequency noise in the audible frequency range of approximately 1 kHz to 15 kHz is induced by self excitation resulting from the frictional contact between brake pad and disk. A typical industrial countermeasure to address this problem is the mounting of thin composite structures consisting of elastomer and steel layers, so called shims, on the pad backplates. They are applied to increase the damping and to influence the vibration shapes.The computational modeling of shims using Finite Elements is still a complex task and shows significant potential for improvement. To avoid problems resulting from element sizes of the partially very thin layers a classical homogenization theory from literature is considered. This homogenization approach maps shim properties in an improved manner which contributes to substantially smaller model sizes as well as less simulation effort and time. Therefore, analytical approaches for constrained layer damping structures are introduced and corresponding theoretical results are presented. To validate these theoretical results, experimental investigations are carried out on shims bonded to structures, especially steel plates and brake pads.

Dominik Schmid, Nils Gräbner, Utz von Wagner

Chapter 31. Teaching Mechanics: Inequalities in Statically Indeterminate Static Friction Problems

In this contribution, we provide a mathematical treatment of the inequality systems arising from two toy examples of statically indeterminate, static friction problems with multiple contact planes. Both examples illustrate general phenomena that are not covered by introductory textbooks. The first one illustrates that a certain inequality might restrict the solution set for a certain choice of a parameter, while being not restrictive for another choice of the parameter. The example also illustrates why it is not suffcient to investigate only the limit case of impending motion (H = μ 0 N ) for more complex problems.The second example illustrates that the positiveness of contact forces can also be a limiting condition. It treats a problem where equilibrium might be lost by slipping or by losing contact, where it is a-priori unclear which condition is decisive. Finally, we investigate the intriguing question how this knowledge can be exploited by a purely hypothetical evil professor—not to upset anyone, we give him a common German surname as a working name, say professor Müller—to set up test problems that are solved wrongly by the majority of students.

Patrick Schneider, Reinhold Kienzler

Chapter 32. Initial Damage of Composite Materials

In composite materials, damage occurs even without excessive loading, because of the discrepancy between material parameters of adhered materials. This damage (damage of nonconformity) is expressed as a violation of the continuity at the interface. A mathematical model of the interaction of inclusion particles and the composite matrix, based on taking into account nonlocal interactions of contacting materials particles is proposed. The quantifying parameter of damage is determined from the condition of stationarity of the energy of the elastic deformations of the discrepancy. The distribution of the nonconformity damage found in this way is proposed to be used to determine the initial distribution of the scattered damage of a homogeneous material that models composite material. Such scattered damage can be used to solve evolution equations in problems of fracture mechanics.

Vladimir S. Shorkin, Victoria Yu. Presnetsova, Vadim M. Presniakov, Sergey N. Romashin, Larisa Yu. Frolenkova, Svetlana I. Yakushina

Chapter 33. How the Properties of Pantographic Elementary Lattices Determine the Properties of Pantographic Metamaterials

In this paper we describe a three-scales homogenization process which we use to determine a macroscopic model for pantographic metamaterials. The smallest scale refers to the length at which the considered deformable mechanical system can be modeled as a Cauchy’s continuum. Of course, at this scale, its geometry is rather complex. The meso-scale refers to a length at which the system can be modeled as a Hencky-type discrete system constituted by masses interconnected by extensional and rotational springs. At macro-scale the model to be used is a generalized plate whose deformation energy depends on geodesic curvature. While the direct identification from the smallest scale to the macro-scale seems rather diffcult, the identification from smallest scale to meso-scale can be successfully obtained. The geometrical properties, along with Young and Poisson coeffcients of the used isotropic material, at the smallest scale determine the extensional and rotational stiffnesses to be used at the meso-scale. On the other hand, the Piola-type identification process allows us to determine the stiffnesses of the macroscopic generalized plate model, via a simple asymptotic expansion. We have observed that this process is valid in both cases when the smallest scale is of the order of microns and when the smallest scale is of the order of tenth of millimeters. Some experimental and numerical results supporting this statement are exhibited.

Emilio Turco

Chapter 34. Metallic Interconnection Technologies for High Power Vertical Cavity Surface Emitting Lasers Modules

Highly reliable power VCSEL (Vertical Cavity Surface Emitting Lasers) array systems with an optimized optical output require a plan parallel assembly for a homogeneous radiation and an advanced packaging design to ensure good heat dissipation and an overall reliable performance. The aim of this study is to evaluate if metallic interconnection technologies like soldering and silver sintering can meet these requirements. Therefore, GaAs dies with VCSEL arrays of more than 2000 single lasers were mounted on substrates by soldering using AuSn20 and SnAg3 solder as well as by applying pressure assisted silver sintering. The samples were analyzed using ultrasonic microscopy (C-SAM), X-ray microscopy and 3D laser profilometry. Cross-sections of selected samples were made and analyzed using light- and scanning electron microscopy (SEM). Soldered and silver sintered samples were subjected to thermal cycling between -55°C and +125°C to validate the reliability of the metallic interconnects. Furthermore, it was tested, if it is possible to assemble a DCB onto a micro channel water cooler made of copper by pressure assisted silver sintering in order to enable an advanced heat transfer of the high power VCSEL module.

Constanze Weber, Lena Goullon, Matthias Hutter, Martin Schneider-Ramelow

Chapter 35. Coupled Thermal and Electrochemical Diffusion in Solid State Battery Systems

In the light of today’s extensive research on rechargeable batteries a electro-chemically diffusion model for a temperature sensitive multi-phase solid is presented. The derivation of the model is based on the framework of the Thermodynamics of Irreversible Processes with the assumption of a local equilibrium. The physical effects which are accounted for are: the flux of ions including chemical reactions, the heat flux, the electrical current, and their coupling resulting, e.g. in phase decomposition, thermal diffusion, and thermoelectric effects. Two numerical examples illustrate typical applications and demonstrate the versatility of the coupled multi-physics model.

Marek Werner, Kerstin Weinberg

Chapter 36. Nonclassical Bending Behavior of Thin Strips of Photochromic Liquid Crystal Elastomers Under Light Illuminations

Photochromic liquid crystal elastomers (LCEs) bend when irradiated by light of suitable wavelength. However, due to the rotation of the liquid crystal director, rather large shear strains are inevitably produced and some basic assumptions of the classical simple beam theory of Euler-Bernoulli fail to be satisfied. In this work, we use the first-order shear deformation beam theory of Timoshenko to model the unusual quasi-soft bending behavior of soft LCEs under light illuminations. The results show that in addition to the large shear strain, the effect of initial effective length ratio makes a great difference to the deflections due to the rotation of director. This represents the first direct verification that Euler-Bernoulli beam theory fails to deal with such nonclassical bending of soft LCEs, while Timoshenko beam model can work suffciently well, which also gives a possible method to measure the effective opto bending moment experimentally.

Yang Zhang, Yongzhong Huo

Chapter 37. A Simple Qualitative Model for the Pressure-induced Expansion andWall-stress Response of Fluid-filled Biological Channels

This work investigates the effects of a pressure increase in deformable fluid-filled biochannels, such as arteries and veins. Simple qualitative expressions are developed relating pressure-induced changes to the biochannel expansion, volumetric flow rate, and biochannel wall stress. Such relations are necessary for a rapid analysis in potential applications such as post-traumatic stress, hemorrhagic strokes, atherosclerotic plaque buildup, etc. The relations are based on the development of functions that correct classical pressurized thin-tube expressions for hoop stress for finite deformations.

Tarek I. Zohdi
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