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2022 | Book

Theoretical Analyses, Computations, and Experiments of Multiscale Materials

A Tribute to Francesco dell’Isola

Editors: Prof. Dr. Ivan Giorgio, Prof. Dr. Luca Placidi, Dr. Emilio Barchiesi, Prof. Dr. Bilen Emek Abali, Prof. Dr. Holm Altenbach

Publisher: Springer International Publishing

Book Series : Advanced Structured Materials


About this book

This book is devoted to the 60th birthday of the Prof. Francesco dell’Isola, who is known for his long-term contribution in the field of multiscale materials. It contains several contributions from researchers in the field, covering theoretical analyses, computational aspects and experiments.

Table of Contents

33. Correction to: Theoretical Analyses, Computations, and Experiments of Multiscale Materials


Ivan Giorgio, Luca Placidi, Emilio Barchiesi, Bilen Emek Abali, Holm Altenbach

Theoretical Analysis

1. A Different Catch for Poisson

Poisson’s ratio, similar to other material parameters of isotropic elasticity, is determined via experiments corresponding to small strains. Yet at small-strain linear elasticity, Poisson’s ratio has a dual nature; although commonly understood as a geometrical parameter, Poisson’s ratio is also a material parameter. From a geometrical perspective only, the concept of Poisson’s ratio has been extended to large deformations by Beatty and Stalnaker. Here, through a variational analysis, we firstly propose an alternative relationship between the Poisson ratio and stretches at finite deformations such that the nature of Poisson’s ratio as a material parameter is retained. In doing so, we introduce relationships between the Poisson ratio and stretches at large deformations different than those established by Beatty and Stalnaker. We show that all the nonlinear definitions of Poisson’s ratio coincide at the reference configuration and thus, material and geometrical descriptions too coincide, at small-strains linear elasticity. Secondly, we employ this variational approach to bring in the notion of nonlinear Poisson’s ratio in peridynamics, for the first time. In particular, we focus on bond-based peridynamics. The nonlinear Poisson’s ratio of bond-based peridynamics coincides with 1/3 for two-dimensional and 1/4 for three-dimensional problems, at the reference configuration.

A. Derya Bakiler, Ali Javili
2. Nonlinear Deformation of a Clamped-Edge Strip-Like Nano-Film

Nonlinear deformations of a clamped-edge nano-film are considered on the basis of an extension of von Kármán’s theory of elastic plate, taken into account the Gurtin–Murdoch surface elasticity and Kirchhoff’s hypothesis. Unlike most of the previous related theories, surface tension, usually omitted in the conjugation condition of Young–Laplace law in the transverse direction, is incorporated in the two-dimensional motion and constitutive relations for membrane forces together with quadratic terms equal to the von Kármán-type strains. The influence of the linear and nonlinear terms of surface tension is illustrated in the cases of nonlinear bending, post-critical compressive buckling, and free transverse vibration of a striplike nano-film with clamped edges.

Anatolii Bochkarev
3. Closed-form Analytic Solutions of the Problem of a Hollow Sphere Made of Second Gradient Plastic Porous Material and Subjected to Hydrostatic Loading

Gologanu, Leblond, Perrin, and Devaux (GLPD) developed a constitutive model for ductile fracture for porous metals based on generalized continuum mechanics assumptions. The model predicted accurately ductile fracture process in porous metallic structures under several complex loads. The GLDP model’s performances over its competitors has attracted the attention of several authors who explored further capabilities of the model. The aim of this paper is to provide analytical solutions for the problem of a porous hollow sphere subjected to hydrostatic loadings, the matrix of the hollow sphere obeying the GLPD model. The exact solution for the expressions of the stress and the generalized stress the GLPD model involved are illustrated for the case where the matrix material does not contain any voids. The results show that the singularities obtained in the stress distribution with the local Gurson model are smoothed out, as expected with any generalized continuum models. The paper also presents some elements of the analytical solution for the case where the matrix is porous and obeys the full GLPD model at the initial time when the porosity is fixed. These analytical solutions can serve as benchmark solutions to assess numerical implementations of any second gradient constitutive model.

Roy Burson, Koffi Enakoutsa
4. Quantum Dynamics Effects on Amplitude-Frequency Response of Superharmonic Resonance of Second-Order of Electrostatically Actuated NEMS Circular Plates

This work deals with the effects of Casimir and/or van der Waals forces (quantum dynamics phenomena) on the amplitude-frequency response of the superharmonic resonance of second-order of axisymmetric vibrations of electrostatically actuated nanoelectromechanical systems (NEMS) clamped circular plates. Electrostatic actuation consists of alternating current (AC) voltage of magnitude to produce hard excitations and of frequency near one fourth the natural frequency of the clamped circular plate. The intermolecular forces Casimir and van derWaals, damping force, and electrostatic force are the forces acting on the NEMS plate. Six Reduced Order Models (ROMs) with one and up to 6 modes of vibration are used. The ROM with one mode of vibration is solved using the Method of Multiple Scales (MMS) in which the hard excitations are modeled using first-order and second-order models of hard excitations electrostatic force. Also, Taylor polynomials up to 25th degree are used to approximate the electrostatic, Casimir and van der Waals forces in the ROM with one mode of vibration. MMS predicts the amplitude-frequency response (bifurcation diagram) of the resonance. The other ROMs, using from two to six modes of vibration are solved using two methods, namely continuation and bifurcation using AUTO software package to predict the amplitude-frequency response, and numerical integration using Matlab to predict time responses of the NEMS plate. The amplitude-frequency response predicts a softening effect, and the existence of three branches, two stable and one unstable. A saddle-node bifurcation point of amplitude of 0.24 of the gap, and end points of amplitudes of 0.66 and 0.75 of the gap of unstable and stable branches, respectively, are predicted. The increase of Casimir and/or van der Waals forces shifts the branches, bifurcation points, and endpoints to lower frequencies.

Dumitru I. Caruntu, Julio S. Beatriz
5. Propagation of Chaos for a Stochastic Particle System Modelling Epidemics

We consider a simple stochastic N-particle system, already studied by the same authors in Ciallella et al. (2021b), representing different populations of agents. Each agent has a label describing his state of health. We show rigorously that, in the limit N → ∞ propagation of chaos holds, leading to a set of kinetic equations which are a spatially inhomogeneous version of the classical SIR model. We improve a similar result obtained in Ciallella et al (2021b) by using here a different coupling technique, which makes the analysis simpler, more natural and transparent.

Alessandro Ciallella, Mario Pulvirenti, Sergio Simonella
6. On the Constitutive Assumptions for a Continuum Model of Scintillating Crystals

For inorganic scintillating crystals, we show how the evolution equation for the charge carriers densities can be obtained by modeling the crystal as a continuum with structure. The resulting equation is a Reaction-Diffusion-Drift one. We deal with various hypotheses on the reaction/recombination term which are induced by two different choices for the associated entropy, namely those based either on the Gibbs-Boltzmann or Fermi–Dirac statistics.

Fabrizio Daví
7. Strong Ellipticity Within the Strain Gradient Elasticity: Elastic Bar Case

In this note we discuss the strong ellipticity condition within the nonlinear strain gradient elasticity. Considering a one-dimensional case, i.e. an elastic bar loaded by a tensile force, we analyze the correspondence of violation of the strong ellipticity condition and compare the results with classic nonlinear elasticity. The correspondence of ellipticity loss to the non-uniqueness of solutions are discussed in more detail.

Victor A. Eremeyev, Emanuele Reccia
8. Two Thermodynamic Laws in Phenomenological Mechanics of Continuum: Postulates or Definitions?

The role of the fourth and fifth postulates of continuum mechanics, also known as the first and second laws of thermodynamics, in the axiomatization of phenomenological theory is discussed. It is shown that, in contrast to the statistical and molecular approaches, the internal energy and entropy of an individual (liquid) volume can be fully determined by specifying their source, flow across the boundary, and production. Thus, two thermodynamic postulates serve as definitions. Energy conjugate pairs of quantities of different physical nature and the possibility of extending the table of postulates are discussed.

Dimitri Georgievskii
9. On an Extended Family of Quasi-equivalent Models of the Gradient Elasticity Theory

There are investigated the formulations of boundary value problems in the Mindlin-Tupin gradient theory characterized by a higher differential order of equilibrium equations and a varied spectrum of boundary value problems, formulated both on a piecewise smooth surface and on the edges of this surface. We consider the possibility of simplifying boundary value problems by eliminating boundary conditions at the edges by introducing an extended spectrum of gradient applied models in the class of equivalent models having the same potential energy density. For this purpose, we investigate the variational statements of boundary value problems, which establish admissible kinematic connections on the surface in the form of linear combinations of the displacement vector and the first derivatives of displacements (both normal and tangential). Classes of gradient models obtained by introducing kinematic constraints on the surface, in which there are no boundary conditions at the edges, are indicated. These include models built by introducing kinematic constraints on the displacement vector and some special classes of models in which the kinematic constraints on the surface are set to the derivatives of displacements.

Sergey Lurie, Petr Belov, Yury Solyaev
10. Continuum Models via Granular Micromechanics

The utilization of continuum modelling in describing complex material systems is historically widespread because of the computational feasibility of such an approach. Nevertheless, continuum models must be informed about the microstructure to have a satisfactory predictive capacity. Knowledge of micro-structural features is fraught with challenges due to a litany of factors, including random structure, contact conditions, and contact constitutive laws. In this scenario, the granular micromechanics approach (GMA) is away to conveniently provide a linkage of the grain-scale behavior to the collective behavior of grains. In this short review, we describe some salient features of GMA and discuss outlook of GMA based continuum models.

Anil Misra, Luca Placidi, Emilio Barchiesi
11. Some Variational Principles in the Three-Dimensional Micropolar Theories of Solids and Thin Solids

In this work, we formulated the variational principles of Lagrange, Castigliano, the generalized Reissner-type variational principles (GRTVP), aswell as the principle of virtual work and the principle of complementary virtual work of threedimensional micropolar mechanics (MM) of solids of some rheologies in the case of potentiality, as well as nonpotentiality of stress and couple stress tensors. Proceeding from them and applying the new parametrization (NP) of the domains of single-layer and multilayer thin bodies, the variational principles corresponding to the theories of single-layer and multilayer thin bodies are formulated. In particular, the generalized Reissner-type operator of three-dimensional MM of solids is constructed, on the basis of which the generalized Reissner-type operators of three-dimensional MM of solid single-layer and multilayer thin bodies with one small size are constructed. From the latter Reissner-type operators, in turn, the GRTVP of three-dimensional MM of solid single-layer and multilayer thin bodies with one small size are derived under the NP of the domains of these bodies. It should be noted that the advantage of the NP is that it is experimentally more accessible than other parameterizations used in the scientific literature. Further, using the method of orthogonal polynomials, from the above-mentioned GRTVP, the GRTVP of MM of solid single-layer and multilayer thin bodies with one small size under the NP of the domains of these bodies in moments with respect to the system of Legendre polynomials are derived. Moreover, in the case of the theory of multilayer thin bodies, the representation of the generalized Reissner-type operator is given and the generalized Reissner-type variational principle is formulated, both in the case of complete contact of adjacent layers of a multilayer structure, and in the presence of zones of weakened adhesion. In addition, the description of obtaining of dual Reissner-type operators and the GRTVP, as well as of Lagrangian and Castiglianian and variational principles of Lagrange and Castigliano is given. The interface (interphase boundary) is described by a surface of zero thickness.

Mikhail Nikabadze, Armine Ulukhanyan
12. Asymptotic Comparison of the Strain-Gradient and Micromorphic Models when Loading Forces Are Widely Spread

In this paper we reconciliate different homogenization results which describe the effective behavior of a heterogeneous material either by a strain-gradient model either by a micromorphic one. Indeed we prove that the solutions of both models are asymptotically very close when considering a loading with increasing wavelength. This result is obtained using the Fourier analysis on the tensor spaces and applies to a large class of micromorphic models. However, we provide an example of a micromorphic model that does not belong to this class and thus cannot be approximated by a strain-gradient model.

Pierre Seppecher, Lukáš Jakabčin
13. Quasiconvexity and Rank-One Convexity in Cosserat Elasticity Theory

The quasiconvexity and rank-one convexity conditions of conventional nonlinear elasticity theory are extended to nonlinear Cosserat elasticity. These furnish necessary conditions for energy minimizers in conservative boundary-value problems.

Milad Shirani, David J. Steigmann
14. Models of Viscoelastic Materials: A Review on Historical Development and Formulation

A classification of different mathematical models of viscoelastic materials is presented. The review covers the classical models of viscoelasticity with integer order derivatives, as well as models with fractional derivatives and fractional operators. This paper provides a detailed historical background of the basic viscoelastic models with their mechanical schemes and mathematical formulations. A comparative analysis of contribution of Western and Russian scientists to the development of linear viscoelasticity is carried out. The paper fully tracks the recent theories on the topic of linear and nonlinear viscoelasticity.

Marina V. Shitikova, Anastasiya I. Krusser
15. Invariance Aspects of F = FeFi Representations in Coupled-Field Problems

In multi-field problems, the free energy contributions of different fields are coupled. For the kinematic of deformation, explicit assumptions on the independent fields are necessary, and the intermediate configurations of composed deformations are not uniquely determined. Here, a thermodynamic free energy potential with two primary variables is analyzed. The two energy representations ψ(c; C) and ψ(c; Ce), based on the concentration c, the right Cauchy-Green tensor C, and its elastic part Ce, are studied in full detail. The considerations are specified to the chemomechanical coupling of a regular solution model and two types of elastic energy potentials, but all derivations can be generalized to arbitrary material models.

Marek Werner, Kerstin Weinberg


16. Strain-Gradient Modeling and Computation of 3-D Printed Metamaterials for Verifying Constitutive Parameters Determined by Asymptotic Homogenization

Metamaterials exhibit significantly different mechanical deformation than in classical “first-order” theory. One possible modeling approach is to use a “straingradient” theory by incorporating also higher gradients of displacements into the formulation. This procedure clearly brings in additional constitutive parameters. In this study, a numerical framework is presented by applying strain-gradient theory to 3-D printed structures with an infill ratio used frequently in additive manufacturing for weight reduction. This choice causes metamaterials; the additional constitutive parameters in the strain-gradient model are determined by an asymptotic homogenization. In order to demonstrate the reliability of this methodology, we verify the accuracy by computations using the finite element method.

Gokhan Aydin, M. Erden Yildizdag, Bilen Emek Abali
17. On Boundary Layers Observed in Some 1D Second-Gradient Theories

Boundary layers are regions into a material domain where gradients localize. They often arise in non-local theories such as second gradient ones, which introduce various internal length scales. This work aims at exploring the properties linked to boundary layers for a few typologies of one-dimensional continua moving in plane. More particularly, three cases are explored: the deflection of an extensible Euler–Bernoulli beam, and the axial deformation of a pantographic beam with nonlinear first gradient and second gradient effects. It is concluded that the size of boundary layers depends on the internal lengths and, when nonlinearities are considered, on the external load.

Emilio Barchiesi, Alessandro Ciallella, Ivan Giorgio
18. Design and Parametric Enhancement of a Flexible Planar TEG—Numerical Study

A thermoelectric generator (TEG) can recover waste energy and convert it into electricity. Therefore, a TEG is a sustainable and reliable device that can be applied in several fields. The current work aims to go further in the design and enhancement of a flexible planar TEG made with bismuth telluride. For this purpose, ANSYS® software was used. An initial design of the generator was created, and a sensitivity analysis was carried out to understand which parameters most impact its performance. Setting a satisfactory mesh, numerical simulations of the TEG were performed in its operating conditions to obtain the corresponding output properties. The device was enhanced for an industrial application through a numerical model based on the parametric enhancement of the most relevant design parameters, namely the height and width of the semiconductor components and the number of thermocouples. A theoretical model was used for comparison of the numerical model, and both models were validated with experimental data from literature. The numerical simulations produced more complete results, such as contours of temperature and voltage in the TEG and led to an optimal design capable of reaching a maximum of 0.10 W and an output voltage of 9.973 V when exposed to a temperature gradient of 130 ℃. Additionally, the generator fit within the desired area of this specific industrial application and exhibited an efficiency of approximately 5%. This research study is expected to encourage TEG manufacturing in industrial applications.

Maria Carolina Fernandes, Cristina Furtado, Daniela Campanhã, Sónia Isabel Silva Pinto
19. Implementation and Comparison of Non-Newtonian Viscosity Models in Hemodynamic Simulations of Patient Coronary Arteries

Realistic numerical simulations of blood flow in patient-specific coronary arteries constitute a challenge in the study of hemodynamics. Several blood viscous models are available; yet their direct comparison has not been carried out. This work innovates by programming and implementing six viscosity models (Carreau, Carreau-Yasuda, Casson, Cross, generalized power-law and power law) as user-defined functions in ANSYS® Fluent to compare the major hemodynamic parameters, the time-averaged wall shear, the oscillatory shear index and the relative residence time, and to evaluate atherosusceptibility in coronary arteries. The study used the left coronary arteries of an apparently healthy patient and an unhealthy patient, with 40% stenosis in the left anterior descending. Flow simulations comprised two sets (steady-state and pulsatile flows), each based on Newtonian and non-Newtonian fluid, i.e., a total of four cases. Results indicate that the Casson model originates larger areas of atherosusceptibility and the generalized power-law models returns the most healthy results. The non-Newtonian pulsatile cases show less critical areas than the remaining studied regimes.

Maria Carolina Fernandes, Luísa Costa Sousa, Catarina Ferreira de Castro, José Manuel Laginha Mestre da Palma, Carlos Conceição António, Sónia Isabel Silva Pinto
20. Bending/Tension of Plate Reinforced by a System of Parallel Fiber

We present a 3-D to 2-D dimension reduction procedure as applied to the periodicity cell problem (PCP) of the homogenization theory for plates reinforced with a unidirectional system of fibers. The original 3-D PCP is reduced to several 2-D problems. The reduction procedures are not trivial, in one case we encounter the incompatibility condition, which makes impossible to transform the 3-D elasticity problem to the 2-D elasticity problem (only the transformation to 2-D thermoelasticity problem is possible). Numerical analysis of 2-D periodicity cell problems demonstrates new phenomena: the boundary layers on the top and bottom surfaces of the plate and, as a result, the wrinkling of the top and bottom surfaces of the plate. Note that these phenomena never occur in uniform plates or plates made of uniform layers.

Alexander G. Kolpakov, Sergei I. Rakin
21. Semi-automatic Method of Stent Development for Hemodynamic Simulations in Patient Coronary Arteries with Disease

Atherosclerosis contributes to the development cardiovascular diseases, the leading cause of death in the world. Complications arising from atherosclerosis, such as stenosis (an abnormal narrowing of a blood vessel, which can lead to its clogging) exacerbate the risk of cardiovascular disorders. Besides aging, sedentary lifestyle, unhealthy diet and tobacco consumption are among the risk factors which increase the likelihood of developing atherosclerosis overtime. One way to prevent stenosis development, due to atherosclerosis, is by inserting stents inside the relevant blood vessels. Stents are small metal tubes which can be expanded to keep the passageway open and improve blood flow, essentially resulting in an arterioplasty. The achievement of an accurate stent, such as accurate mesh, length and other features, specific for a patient coronary artery with atherosclerosis, is still a challenge in clinical practice. Therefore, after developing numerical based coding solution that simulate hemodynamic conditions as close as possible to reality, the goal of the present work is to develop a semi-automatic method to create a stent in the stenotic location of patient-specific coronary arteries. As far aswe know, no authors have been able to quickly and effectively place the stent in a model of a patient’s artery, which is subject to the complex geometry of the coronary such as curvature, tortuosity, etc. The impact of stent length was considered, in order to verify which is the ideal, for a patient case, avoiding restenosis occurrence. After hemodynamic simulations in the model artery with stenosis and in the model artery with stent, it is observed that strong atherosusceptible regions just after the stenosis are eliminated after stent insertion.

Luís Matias, Catarina Ferreira de Castro, Carlos Conceição António, Luísa Costa Sousa, Sónia Isabel Silva Pinto
22. The Efficient Trabecular Bone Remodeling Numerical Tool Enabling Multiple Load Case Simulation

Paper concerns a simulation of the trabecular bone remodeling process taking into account its real geometric form. The efficient trabecular bone remodeling numerical tool enabling multiple load case simulation is presented. The observation proposed by Julius Wolff—called the Wolff’s law—can be described as a structural adaptation of the bone to the external forces. Thus the trabecular bone remodeling process numerical simulation has to include the very important aspect of external load, namely the variable loads. For simulation purposes it means, that the numerical tool must be able to simulate multiple load case and the geometric form of the bone must correspond to these loads. Technically the numerical system is .Net C# project designed with Inversion of Control paradigm design pattern that provides pluggable and extensible platform.

Jan Polak, Michał Nowak
23. Modeling the Magnetic Relaxation Behavior of Micropolar Ferrofluids by Means of Homogenization

In this paper a simple particle population homogenization approach is used in order to estimate the magnetic relaxation time of a ferrofluid by means of a microscopic analysis. At a macroscopic level the ferrofluid is modeled as a micropolar fluid with rotational degrees of freedom. The governing equations for these degrees of freedom are the spin balance and the magnetic relaxation equation. They are solved analytically for a simple unidirectional magnetic setup. On a microscopic level the ferrofluid is considered to consist of rigid spherical permanent magnets suspended in a non-magnetic carrier fluid. Due to both, the friction of the micro magnets with the carrier fluid and their own inertia, the alignment of the magnets with an applied external field is retarded. By neglecting thermal effects and therefore the Brownian motion, it is possible to reduce the equations of motion to a nonlinear pendulum equation, which is readily solved using numerical methods for ordinary differential equations. By averaging over all possible initial configurations of the micro magnets, a pseudo homogenization is obtained, which can then be compared to the macroscopic solution. From this comparison the relaxation time at a continuum level can be estimated.

Wilhelm Rickert, Max Winkelmann, Wolfgang H. Müller
24. Numerical Homogenisation of Gradient Materials

In this contribution, we present a numerical homogenization procedure for gradient materials. In particular, we investigate the prototypical example of fiber reinforced materials and demonstrate, that even at a typical microscale we obtain second order continuum. In particular, we make use of immersed technologies to embed the fibers within the matrix material. Introducing a novel approach IGA2 using spline based approximations on both, the micro- and the macroscale, allows for the numerical homogenisation of complex microstructures with anisotropic second gradient contributions of the fibers as shown in a representative example.

F. Schmidt, M. Krüger, C. Hesch
25. Modeling the Slow Crack Growth of an Edge Crack Within the Cohesive Zone Model Approach

The problem of quasi-static growth of the mode I edge crack in a viscoelastic material is solved. The stages of analysis are indicated, namely (i) choice of rheological model, (ii) choice of crack model and fracture criterion, (iii) solution of the elastic problem of crack mechanics within the chosen model approach, (iv) constructing the viscoelastic solution of the problem and equations of crack propagation. The last two of these stages were given particular attention. The proposed algorithm for determining the crack opening in the framework of the cohesive zone model approach is based on the methodology developed in previous papers of the authors. A regularized singular integral equation is used to obtain an elastic crack opening. When modeling the quasi-static crack growth, the hypothesis of the independence of the cohesive law on the rate of slow crack propagation is assumed to be true. To construct numerical solutions, the exponential kernel of the slow crack growth equations is utilized. The smoothed triangular traction–separation law with the hardening segment ensures smooth crack closure. Auxiliary solutions of the problem that are obtained at each step and should be used to describe hereditary viscoelastic behavior are illustrated. The dependence of crack length on time is obtained for some numerical values of model parameters. The proposed methodology for modeling slow crack growth has demonstrated its effectiveness by fast convergence of solutions at each iteration of the algorithm.

Mikhailo Selivanov, Lidiia Nazarenko, Holm Altenbach
26. An Insight into Computational Challenges in Damage Mechanics: Analysis of a Softening Hooke’s Spring

While many efforts are being currently spent to forge reliable damage laws based on the physics of the materials to be studied, damage modeling is still addressed numerically too naively in many situations. This article highlights some topical conceptual aspects that have been up to now dealt with too superficially by comparing the performances of different numerical algorithms in solving Karush–Kuhn–Tucker conditions for a simple linearly softening Hooke’s spring. It is concluded that even such a primitive model, because of the multiplicity of solutions satisfying simultaneously equilibrium, damage law and irreversibility conditions, actually requires well-established numerical algorithms to face unexpected challenges. A comparison between different numerical strategies, beyond highlighting critical behaviors of traditional algorithms, permitted to observe an appealing robustness shown by an iterative strategy based on the fixed-point theorem. As a closure remark, evidences collected within this contribution naturally lead to the following question, which is left open for future studies: Is it possible to envisage the formulation of a criterion– possibly an energetic one, like that distinguishing stable and unstable solutions in elasticity—to establish which solution should be considered as valid in a given situation?

Salvatore Sessa, Emilio Barchiesi, Luca Placidi, Massimo Paradiso, Emilio Turco, Nahiene Hamila
27. Thermodynamic Compatibility of the HystereticPoly Uniaxial Material Implemented in OpenSees

The fulfillment of the Drucker postulate applied to a phenomenological hysteretic constitutive model is hereby investigated. Such a material is defined in terms of analytical functions so that it is capable of determining the response and its tangent operator in closed form and does not require any iterative algorithm. Hence, the constitutive model is very appealing for several applications, including structural analysis and homogenization techniques. Within this context, the thermodynamic compatibility implied by the Drucker’s postulate aims to ensure that the model does not provide responses associated with negative values of the dissipated energy, this in order to fulfill the 2nd law of thermodynamics. In particular, the research is focused on two peculiar phenomena associated with non-consisten energy dissipations: the negative softening and the hysteretic crossing paths. It is shown that the thermodynamic compatibility may be violated because of negative softening although it is possible to determine a displacement range for which the material preserves its physical significance. On the contrary, it is proved that the analytical formulation of the investigated model avoids the crossing path phenomenon thus ensuring the fulfillment of the Drucker’s postulate.

Salvatore Sessa, Nicoló Vaiana, Massimo Paradiso, Luciano Rosati
28. Studying the Higher-Order Inertia in the Second-Order Theory of Elasticity for Modeling Metamaterials

Modeling the materials with a complex microstructure, such as metamaterials, is challenging especially in the dynamic regime. Higher-gradient models have been widely used for modeling the mechanical behavior of metamaterials. In dynamic loading problems, the inertia plays an important role. Including higher-order inertia in the model could possibly improve the accuracy of the model close to the eigenfrequencies of the structure. Such inertial terms have been presented in theory but they are not understood experimentally, therefore it has not been possible to quantify their value. Herein, we consider a macro-scale model for a pantographic structure and simulate a dynamic loading on it.We run the simulation for a range of frequencies of loading and for a number of arbitrary values for a higher-order inertial term that we have added to the model. The results show a clear relation between the value considered for the inertial term and the eigenfrequency of the structure that we get from the model. This result sheds light on finding an algorithm for determining the higher-order inertial terms experimentally in further studies.

Navid Shekarchizadeh, Alberto Maria Bersani
29. Structural Analysis of Doubly-Curved Shells with General Boundary Conditions

The paper focuses on a bi-dimensional (2D) formulation for the dynamic and static analysis of arbitrary shaped laminated doubly-curved shells enforced with general boundary conditions via the Generalized Differential Quadrature (GDQ). Following the Equivalent Single Layer approach, a 2D theory based on a miscellaneous assessment of the displacement field variable is provided, accounting for different higher order theories. The geometry of the structure is described with a set of principal coordinates. The fundamental equations are derived from the Hamiltonian principle, together with the natural boundary conditions. Unconventional constraints are assessed by means of in-plane and out-of-plane sets of linear elastic springs distributed along the shell edges. The accuracy of the formulation is outlined by means of a series of validating examples. Doubly-curved shells of variable thickness and different curvatures enforced with non-conventional boundary conditions are investigated. In particular, mode frequencies and shapes, as well as the static three-dimensional deflection of the structure, have been calculated employing different kinematic assumptions. The results have been successfully compared to predictions by high-computationally demanding Finite Element simulations. The methodology outlined in this chapter well predicts with a reduced computational effort both the static and the dynamic response of generally anisotropic laminated structures embedding all the effects that are usually depicted by 3D formulations.

Francesco Tornabene, Matteo Viscoti, Rossana Dimitri


30. Characterisation of Mechanical Properties of Wood: Size Effect

In the areas such as cultural heritage, there are various non-standard sample sizes and shapes to characterise the mechanical properties of materials. In addition, there is a need for minimum intervention, which leads to minimisation of samples in sizes and numbers. The aim of this study is to investigate the size effect on material properties of hard wood, as an orthotropic material, in terms of stiffness properties. The compression tests in combination with image analysis technique were used to find the stiffness of the oak samples in the radial direction of wood. Small clear specimens made from oak (Quercus robur L.) were tested. The specimens were divided into two groups: A and B. The testing volumes of group A and B were 10 × 10 × 10 mm3 and 25 × 25 × 25 mm3 respectively. A total number of 8 samples from group A and 9 samples from group B were tested. The results show that the average difference between the two sizes of the samples in term of stiffness in radial direction is 5.5%, with slightly higher values for smaller cubic specimens. More experiments in all orthogonal directions of wood are needed to confirm the results of this study.

Reza Afshar
31. Covering a Surface with Pre-stressed Ribbons: From Theory to Nano-Structures Fabrication

The paper deals with the fabrication of nano-shells from pre-stressed nanoplates release. Due to geometrical and technological restrictions we have to cover a given surface with three-dimensional thin ribbons. We discuss the key role of the geodesic curvature in the design of such shell-ribbons. We show that including small strains but large rotations we are able to control the metric tensor of both un-deformed (or planar) and deformed (or shell) ribbons by an appropriate choice of the width and thickness of the ribbons. Moreover, the strain tensor is controlled by the difference between the curvature of the planar (un-deformed) ribbon and the geodesic curvature of the supporting curve of the shell (deformed) ribbon. Under suitable constitutive assumptions, we deduce the field equations, the boundary conditions and the design equations. The former relate the pre-stress in the planar layer to the final geometry of the desired shell-ribbon. A fine tuning of the composition, geometry and of the pre-stress of the plate-ribon is necessary to design and fabricate the shell-ribbon. We design and fabricate a partial cover of the sphere with constant latitude ribbons starting from planar multi-layer semiconductor materials grown by molecular beam epitaxy. The details of fabrication method and its limitations are discussed in detail.

Alexandre Danescu, Philippe Regreny, Pierre Cremillieu, Jean-Louis Leclercq, Ioan R. Ionescu
32. Experimental and Theoretical Investigations of Auxetic Sheet Metal

This paper deals with the mechanical and thermal behavior of aluminium sheets with a rectangular perforation structure exhibiting an auxetic behaviour. The negative Poisson’s ratio is basically achieved by the rigid rotation mechanism of the squares between the perforations. In this work uniaxial quasistatic tensile tests are carried out to characterize the perforation pattern. During the mechanical tests, the samples are observed simultaneously with a optical camera, used for Digital Image Correlation (DIC) to determine the local deformation fields, and passive thermography to visualize the heat evolution in the sample due to plastic deformation process. This led to the in situ thermomechanical characterization of the component and moreover to the determination of the effective material parameters such as Young’s modulus, Poisson’s ratio and thermoelastic constant. The results were then compared and validated by means of FEM simulations. Furthermore, the thermographic images were optimized in this work by using optical images to segment the sample from the background. These improved images were used to extract the temperature change due to plastic deformation to determine the yield stress.

Arash Gordanshekan, Tobias Heib, Wolfgang Ripplinger, Hans-Georg Herrmann, Stefan Diebels
Theoretical Analyses, Computations, and Experiments of Multiscale Materials
Prof. Dr. Ivan Giorgio
Prof. Dr. Luca Placidi
Dr. Emilio Barchiesi
Prof. Dr. Bilen Emek Abali
Prof. Dr. Holm Altenbach
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