,,Acoustic and Mechanical Properties of Viscoelastic, LinearElastic, and Nonlinear Elastic Composites

Von der Fakultat fur Maschinenwesen der Rheinisch-Westfalischen TechnischenHochschule Aachen zur Erlangung des akademischen Grades eines Doktors der

Ingenieurwissenschaften genehmigte Dissertation

vorgelegt von

Heiko Topol

Berichter: Universitatsprofessor Dr.-Ing. Dieter WeichertProfessor Dr. Sc. Vladyslav V. Danishevskyy

Tag der mundlichen Prufung: 29. Oktober 2012

Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfugbar.

Acknowledgements

Iam very grateful to work under Professor Dr. Sc. Igor V. Andrianov.Because of his personality, his patience, numerous helpful discussionswith him, his advices, the great opportunity to learn from him, andthe time he always took for me (when I sent him e-mails late in thenight, I often received his answer in the same night), research under his

guidance has always been a pleasure for me. Professor Dr. Sc. Andrianovs greatsupport contributed significantly to my achievements.

I would like to thank Univ.-Professor Dr.-Ing. Dieter Weichert for his numerousadvices, the great opportunity to carry out research at his Institute of General Mech-anics, and the chances to gain very valuable experiences in science and life, not onlyin Aachen, but also in the Peoples Republic of China. He encouraged me and sup-ported me and my work continuously.

I would also like to thank Professor Dr. Sc. Vladyslav V. Danishevskyy, an avidsupporter of my work. I sought his advice on countless occasions. Despite his du-ties, he always found time to help me and to answer my questions. He had a greatinfluence on my research. I feel very honored by Professor Dr. Sc. Vladyslav V.Danishevskyy for taking the way from Dnipropetrovsk in Ukraine to Aachen totake part in the defense of my doctoral thesis.

I am grateful to Professor Dr. Lorenz Singheiser, who accepted the position asthe chairman of the examination board of my defense.

I also want to thank the following colleagues for a good cooperation, their help indifferent situations, and many pleasant moments during my time at the Institute ofGeneral Mechanics: Dr.-Ing. Min Chen, M. Sc. Bei Zhou, Dr.-Ing. Russell Todres,Dr. rer. nat. Michael Ban, apl. Professor Dr.-Ing. Marcus Stoffel, Mrs. Julia Blu-menthal, Dipl.-Ing. Jeong Hun Yi, Dr. Lele Zhang (Beijing Jiaotong University).

Thanks a lot to the persons, especially to my friends, who often remind me thatthere is also a life outside research.

Abstract: This works deals with the analysis of the acoustic and mechanical prop-erties of composite structures. For the investigation of the acoustic properties, linearelasticity, geometric, physical, and structural nonlinearity, and viscoelasticity aretaken into account. Imperfect bonding between components is also considered. Dif-ferent approximation techniques, such as the asymptotic expansion method and theplane-wave expansion method, are used to obtain the macroscopic acoustic proper-ties of the structure and to determine the frequency bands (passing and stoppingbands) of the composite. Stress distribution in the viscoelastic matrix of a fiber-reinforced composite after loading of the fibers is also investigated.

Zusammenfassung: Die vorliegende Arbeit behandelt die Untersuchung der akus-tischen und mechanischen Eingenschaften von Verbundwerkstoffen. Bei der Un-tersuchung der akustischen Eigenschaften werden lineare Elastizitat, geometrische,physikalische, und strukturelle Nichtlinearitaten und Viskoelastizitat berucksichtigt.Imperfektes Bindungsverhalten zwischen den Komponenten wird ebenfalls betrachtet.Zur Untersuchung der makroskopischen, akustischen Eigenschaften und der Frequenz-bander (Pa- und Stoppbander) des Verbundwerkstoffs werden verschiedene Approx-imationstechniken angewandt, wie z.B. die Anwendung asymptotischer Folgen undder Plane Wave Expansion-Methode. Weiterhin wird die Spannungsverteilung inder viskoelastischen Matrix in faserverstarkten Verbundwerkstoffen nach Belastungder Fasern untersucht.

Contents

List of Figures v

Nomenclature vii

1. Introduction 1

1.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Objectives and structure of the thesis . . . . . . . . . . . . . . . . . . 4

1.3. Analysis of periodic structures . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1. Floquet-Bloch approach . . . . . . . . . . . . . . . . . . . . . 6

1.3.2. Asymptotic homogenization method . . . . . . . . . . . . . . 6

1.3.3. Plane-wave expansion method . . . . . . . . . . . . . . . . . . 7

1.3.4. Comparison of the asymptotic homogenization method withthe plane-wave expansion method . . . . . . . . . . . . . . . . 8

2. Mechanical models of composite materials under consideration 9

2.1. Constitutive equations of the theory of linear elasticity . . . . . . . . 9

2.1.1. Elastic constants . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.2. Decomposition of the stress and the strain tensor . . . . . . . 12

2.1.3. Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.3.1. Propagation of anti-plane shear waves . . . . . . . . 13

2.1.3.2. Longitudinal vibration of a rod . . . . . . . . . . . . 13

2.1.3.3. Heterogeneous material . . . . . . . . . . . . . . . . 14

2.1.4. Different characteristics in bonding between components . . . 15

2.1.4.1. Perfect bonding . . . . . . . . . . . . . . . . . . . . . 16

2.1.4.2. Thin interphase simulating bonding condition . . . . 16

2.1.4.3. Boundary condition at an interface describing bonding 17

2.1.4.4. Different bonding conditions for pressure and tension 18

2.2. Constitutive equations of the theory of nonlinear elasticity . . . . . . 19

2.2.1. Physical and geometrical nonlinearity . . . . . . . . . . . . . . 20

2.2.2. Structural nonlinearity . . . . . . . . . . . . . . . . . . . . . . 22

2.3. Constitutive equations in viscoelasticity . . . . . . . . . . . . . . . . . 24

2.3.1. Models for viscoelastic behavior . . . . . . . . . . . . . . . . . 25

2.3.2. Correspondence principle . . . . . . . . . . . . . . . . . . . . . 26

2.3.2.1. Initial and final value theorem . . . . . . . . . . . . . 28

2.3.3. Wave propagation in viscoelastic solids . . . . . . . . . . . . . 29

ii Contents

3. Wave propagation in linear elastic material 31

3.1. Longitudinal waves in a layered composite . . . . . . . . . . . . . . . 313.1.1. Exact solution of the frequency band structure by the applic-

ation of the Floquet-Bloch approach . . . . . . . . . . . . . . 323.1.2. Short-wave solution by the plane-wave expansions method . . 343.1.3. Long-wave asymptotic approach by the higher-order homo-

genization method . . . . . . . . . . . . . . . . . . . . . . . . 393.1.4. The effects of imperfect interface - symmetric behavior . . . . 46

3.1.4.1. Exact solution: application of the Floquet-Bloch ap-proach . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.1.4.2. Short-wave solution by the plane-wave expansionsmethod . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.1.4.3. Long-wave asymptotic approach by the higher-orderhomogenization method . . . . . . . . . . . . . . . . 50

3.1.5. The effects of imperfect interface - non-symmetric behavior . . 553.1.5.1. Long-wave asymptotic approach by the higher-order

homogenization method . . . . . . . . . . . . . . . . 563.2. Anti-plane shear waves in fibrous composites . . . . . . . . . . . . . . 57

3.2.1. Governing relations . . . . . . . . . . . . . . . . . . . . . . . . 573.2.2. Short-wave solution by the plane-wave expansions method . . 583.2.3. Long-wave asymptotic approach by the higher-order homo-

genization method . . . . . . . . . . . . . . . . . . . . . . . . 603.2.4. The effect of imperfect interface . . . . . . . . . . . . . . . . . 62

3.2.4.1. Short-wave solution by the application of the plane-waveexpansion method . . . . . . . . . . . . . . . . . . . 62

3.2.4.2. Long-wave asymptotic approach for small inclusions . 633.2.4.3. Long-wave asymptotic approach for large inclusions . 683.2.4.4. Inclusion of arbitrary size . . . . . . . . . . . . . . . 69

3.2.5. The effect of disordering . . . . . . . . . . . . . . . . . . . . . 713.2.5.1. Perfect bonding between components . . . . . . . . . 723.2.5.2. Imperfect bonding . . . . . . . . . . . . . . . . . . . 73

3.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4. Wave propagation in nonlinear elastic material 77

4.1. Longitudinal waves in a layered composite . . . . . . . . . . . . . . . 774.1.1. Long-wave asymptotic approach by the higher-order homo-

genization method . . . . . . . . . . . . . . . . . . . . . . . . 784.2. The effect of imperfect interface . . . . . . . . . . . . . . . . . . . . . 82

4.2.1. Long-wave asymptotic approach by the higher-order homo-genization method . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5. Viscoelastic composite materials 87

5.1. Longitudinal waves in a viscoelastic layered composite . . . . . . . . . 875.1.1. Exact solution of the harmonic band structure by the Flo-

quet-Bloch approach . . . . . . . . . . . . . . . . . . . . . . . 885.1.2. Short-wave solution by the plane-wave expansions method . . 90

Contents iii

5.1.3. Long-wave asymptotic approach by the higher-order homo-genization method . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2. Load-transfer in fiber-reinforced composites . . . . . . .