Altenbach / Müller / Abali Higher Gradient Materials and Related Generalized Continua

1. Auflage 2019
ISBN: 978-3-030-30406-5
Verlag: Springer International Publishing
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 120, 246 Seiten

Reihe: Advanced Structured Materials

This book discusses recent findings and advanced theories presented at two workshops at TU Berlin in 2017 and 2018. It underlines several advantages of generalized continuum models compared to the classical Cauchy continuum, which although widely used in engineering practice, has a number of limitations, such as: •     The structural size is very small. •     The microstructure is complex. •     The effects are localized. As such, the development of generalized continuum models is helpful and results in a better description of the behavior of structures or materials. At the same time, there are more and more experimental studies supporting the new models because the number of material parameters is higher.


Prof. Dr.-Ing. habil. Dr. h.c.mult Holm Altenbach is a member of the International Association of Applied Mathematics and Mechanics, and the International Research Center on Mathematics and Mechanics of Complex Systems (M&MoCS), Italy. He has held positions at the Otto von Guericke University Magdeburg and at the Martin Luther University Halle-Wittenberg, both in Germany. He graduated from Leningrad Polytechnic Institute in 1985 (diploma in Dynamics and Strength of Machines). He defended his Ph.D. in 1983 and was awarded his Doctor of Technical Sciences in 1987, both at the same institute. He is currently a Full Professor of Engineering Mechanics at the Otto von Guericke University Magdeburg, Faculty of Mechanical Engineering, Institute of Mechanics (since 2011), and has been acting as Director of the Institute of Mechanics since 2015. His areas of scientific interest are general theory of elastic and inelastic plates and shells, creep and damage mechanics, strength theories, and nano- and micromechanics. He is author/co-author/editor of 60 books (textbooks/monographs/proceedings), approximately 380 scientific papers (among them 250 peer-reviewed) and 500 scientific lectures. He is Managing Editor (2004 to 2014) and Editor-in-Chief (2005 - to date) of the Journal of Applied Mathematics and Mechanics (ZAMM) - the oldest journal in Mechanics in Germany (founded by Richard von Mises in 1921). He has been Advisory Editor of the journal 'Continuum Mechanics and Thermodynamics' since 2011, Associate Editor of the journal 'Mechanics of Composites' (Riga) since 2014, Doctor of Technical Sciences and Co-Editor of the Springer Series 'Advanced Structured Materials' since 2010. He was awarded the 1992 Krupp Award (Alexander von Humboldt Foundation); 2000 Best Paper of the Year-Journal of Strain Analysis for Engineering Design; 2003 Gold Medal of the Faculty of Mechanical Engineering, Politechnika Lubelska, Lublin, Poland; 2004 Semko Medal of the National Technical University Kharkov, Ukraine; 2007 Doctor Honoris Causa, National Technical University Kharkov, Ukraine; 2011 Fellow of the Japanese Society for the Promotion of Science; 2014 Doctor Honoris Causa, University Constanta, Romania; 2016 Doctor Honoris Causa, Vekua Institute, Tbilisi, Georgia; 2018 Alexander von Humboldt Award (Poland).  Bilen Emek Abali has studied and worked on various continents, and is currently a Postdoctoral Associate at the Technische Universitaet Berlin, in Germany. He has lectured on various topics, including mechanics, composite materials, numerical methods, and multiphysics simulations at different universities. Dr. Abali's research focuses on thermodynamical derivation of governing equations and their computation in engineering systems, especially in multiphysics applications. With distinguished scientists in several countries, he has been working on analytical solutions for verifying computations of heterogeneous materials in solids; developing and validating novel numerical solution strategies for multiphysics, including fluid structure interaction and coupled electromagneto-thermomechanical systems; investigating further theoretical methods in describing metamaterials with inner substructure; studying fatigue-related damage in metal alloys; applying mechanochemistry for a theoretical description of stresses in batteries; and also developing inverse analysis methods to characterize soft matter. For all computations, he utilizes and develops open-source packages and makes all codes publicly available in order to encourage scientific exchange.

Autoren/Hrsg.

Altenbach, Holm

Müller, Wolfgang H.

Abali, Bilen Emek

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;List of Contributors;14
4;1 A Computational Approach for Determination of Parameters in Generalized Mechanics;18
4.1;1.1 Introduction;18
4.2;1.2 Homogenization Between Micro- and Macroscales;21
4.3;1.3 Determination of Parameters;23
4.4;1.4 Computation of One Specific Case;25
4.5;1.5 Algorithm for All Deformation Cases;26
4.6;1.6 Examples;27
4.7;1.7 Conclusion;30
4.8;References;31
5;2 Extensible Beam Models in Large Deformation Under Distributed Loading: a Numerical Study on Multiplicity of Solutions;36
5.1;2.1 Introduction;37
5.2;2.2 The Model;38
5.2.1;2.2.1 Kinematics and Deformation Energy;38
5.2.2;2.2.2 Lagrange Multipliers Method;39
5.3;2.3 Numerical Simulations;40
5.3.1;2.3.1 Numerical Methods;40
5.3.2;2.3.2 The Number of Equilibrium Configurations when the Load Increases;42
5.3.3;2.3.3 Equilibrium Configurations;43
5.3.4;2.3.4 Parametric Study on the Extensional Stiffness;46
5.4;2.4 Conclusions;49
5.5;Appendix;53
5.6;References;54
6;3 On the Characterization of the Nonlinear Reduced Micromorphic Continuum with the Local Material Symmetry Group;59
6.1;3.1 Introduction;59
6.2;3.2 Micromorphic Continua;60
6.3;3.3 Local Material Symmetry Group;62
6.4;3.4 Relaxed Micromorphic Medium as a Micromorphic Subfluid;63
6.5;3.5 Conclusions;68
6.6;References;68
7;4 Structural Modeling of Nonlinear Localized Strain Waves in Generalized Continua;71
7.1;4.1 Introduction;71
7.2;4.2 Principles of Structural Modeling;72
7.3;4.3 One-dimensional Model of a Nonlinear Gradient-elastic Continuum.;76
7.4;4.4 Nonlinear Strain Waves;78
7.5;4.5 Conclusions;82
7.6;References;83
8;5 A Diffusion Model for Stimulus Propagation in Remodeling Bone Tissues;85
8.1;5.1 Introduction;86
8.2;5.2 Accepted Assumptions and Main Variables;88
8.3;5.3 Poromechanical Formulation;91
8.4;5.4 Evolutionary Equations for Bone Remodeling;93
8.5;5.5 Stimulus ModelingWithout Time Delay and Diffusion Phenomena;95
8.6;5.6 An Improved Version of Stimulus Modeling;96
8.7;5.7 Numerical Simulations;98
8.8;5.8 Conclusions;103
8.9;References;105
9;6 A C1 Incompatible Mode Element Formulation for Strain Gradient Elasticity;111
9.1;6.1 Introduction;111
9.2;6.2 From Local Balance of Momentum to Minimization of the Elastic Potential;113
9.3;6.3 Ciarlets Elastic Energy;115
9.3.1;6.3.1 Strain Gradient Extension;115
9.3.2;6.3.2 Stress-strain Relations;116
9.3.3;6.3.3 Material parameters;117
9.4;6.4 Element Formulation;117
9.4.1;6.4.1 Overview;117
9.4.2;6.4.2 Incompatible Mode Element Formulation;118
9.4.3;6.4.3 Numerical Integration;121
9.4.4;6.4.4 Testing of the Implementation;122
9.5;6.5 Single Force Indentation Simulations;123
9.5.1;6.5.1 The Boundary Conditions;123
9.5.2;6.5.2 Meshing;124
9.5.3;6.5.3 Results;124
9.5.3.1;6.5.3.1 Transition Behavior as ? = 0. . .?;124
9.5.3.2;6.5.3.2 Singularity;124
9.5.4;6.5.4 A Comment on Pseudorigid Bodies;126
9.6;6.6 Sharp Corner Simulations;127
9.6.1;6.6.1 The Boundary Conditions;127
9.6.2;6.6.2 Meshing;128
9.6.3;6.6.3 Results;128
9.6.3.1;6.6.3.1 Transition Behavior as ???;128
9.6.3.2;6.6.3.2 Convergence on Mesh Refinement;130
9.7;6.7 Improvement of the Element Formulation;132
9.8;6.8 Convergence Study;132
9.9;6.9 Conclusion;134
9.10;References;135
10;7 A Comparison of Boundary Element Method and Finite Element Method Dynamic Solutions for Poroelastic Column;137
10.1;7.1 Introduction;137
10.2;7.2 Mathematical Model;139
10.2.1;7.2.1 usi–p-formulation in Laplace Domain;140
10.2.2;7.2.2 usi–p-formulation in Time Domain;141
10.3;7.3 Boundary Integral Equation and Boundary Element Methodology;141
10.4;7.4 Laplace Transform Inversion;143
10.5;7.5 Numerical Example;143
10.6;7.6 Conclusion;146
10.7;Appendix;147
10.8;References;149
11;8 From Generalized Theories of Media with Fields of Defects to Closed Variational Models of the Coupled Gradient Thermoelasticity and Thermal Conductivity;151
11.1;8.1 Introduction;151
11.2;8.2 Kinematics of Gradient Continuous Media and Gradient Media with Fields of Defects;154
11.3;8.3 Variational Statement of Generalized Gradient Media with Fields of Defects;156
11.4;8.4 Mathematical Statement for Generalized Gradient Dilatation Model;158
11.5;8.5 Identification of Generalized Stress Factors of the Model;159
11.6;8.6 Particular Model: Gradient Thermoelasticity;163
11.7;8.7 Particular Model: Gradient Thermal Conductivity Model;164
11.8;8.8 Conclusion;167
11.9;References;168
12;9 Mathematical Modeling of Elastic Thin Bodies with one Small Size;171
12.1;9.1 Introduction;172
12.2;9.2 On Parametrization of a Thin Body Domain With one Small Size with an Arbitrary Base Surface;174
12.3;9.3 Presentations of the Equations of Motion, Heat Influx and Constitutive Relations of Micropolar Theory;184
12.3.1;9.3.1 Presentations of the Equations of Micropolar Theory;184
12.3.2;9.3.2 Representation of the Equation of Heat Influx in Micropolar Mechanics of a Deformable Thin Solids;185
12.3.3;9.3.3 Representations of Hooke’s Law and Fourier’s Heat Conduction Law;185
12.4;9.4 Some Recurrence Relations of the System of Legendre Polynomials on the Segment [?1, 1];187
12.4.1;9.4.1 Main Recurrence Relations;187
12.4.2;9.4.2 Additional Recurrence Relations;188
12.5;9.5 Moments of Some Expressions Regarding the Legendre Polynomial System;188
12.5.1;9.5.1 Moments of Some Expressions Regarding the Legendre Polynomial System;189
12.6;9.6 Different Representations of the System of Motion Equations in Moments;192
12.6.1;9.6.1 Presentations of the System of Motion Equations in Moments with Respect to Systems of Legendre Polynomials;193
12.7;9.7 Representations of Constitutive Relations in Moments;197
12.8;9.8 On Boundary and Initial Conditions in Micropolar Mechanics of a Deformable Thin Body;199
12.8.1;9.8.1 The Boundary Conditions on the Front Surface;199
12.8.2;9.8.2 Boundary Conditions in Moments in the Theory of Thin Bodies;201
12.8.3;9.8.3 Kinematic Boundary Conditions in Moments;202
12.8.4;9.8.4 Physical Boundary Conditions in Moments;203
12.8.5;9.8.5 Boundary Conditions of Heat Content in Moments;205
12.8.5.1;9.8.5.1 Boundary Conditions of the First Kind in Moments;205
12.8.5.2;9.8.5.2 Boundary Conditions of the Second Kind in Moments;206
12.8.5.3;9.8.5.3 Boundary Conditions of the Third Kind in Moments;206
12.8.6;9.8.6 Initial Conditions in Moments;206
12.9;9.9 Problem Statements in Moments of Micropolar Thermomechanics of a Deformable Thin Body;207
12.9.1;9.9.1 Statement of the Coupled Dynamic Problem in Moments of (r,N) Approximation;207
12.9.2;9.9.2 Statement of a Non-stationary Temperature Problem in Moments of (r,N) Approximation;208
12.9.3;9.9.3 Statement of the Uncoupled Dynamic Problem in Moments of the (r,N) Approximation;209
12.10;References;210
13;10 Application of Eigenvalue Problems Under the Study of Wave Velocity in Some Media;216
13.1;10.1 Kinematic and Dynamic Conditions on the Strong Discontinuity Surface in Micropolar Mechanics;216
13.2;10.2 Equations for Determining the Wave Velocities in an Infinite Micropolar Solid;219
13.3;10.3 Classical Materials with Anisotropy Symbols {1,5} and {5,1};223
13.4;10.4 Classical Material with an Anisotropy Symbol {1, 2, 3} (Cubic Symmetry);228
13.5;10.5 Classical Material with an Anisotropy Symbol {1,1,2,2} (Transversal Isotropy);229
13.6;10.6 Micropolar Material with a Center of Symmetry and the Anisotropy Symbol {1,5,3};232
13.7;10.7 Conclusion;233
13.8;References;234
14;11 Theoretical Estimation of the Strength of Thin-film Coatings;236
14.1;11.1 Introduction;237
14.2;11.2 Theoretical Position;238
14.3;11.3 Basic Assumptions;241
14.4;11.4 Comparison of Calculation Results with Known Data;244
14.5;11.5 Conclusion;244
14.6;References;245