TECHNICAL UNIVERSITY OF LIBEREC FACULTY OF MECHANICAL ENGINEERING Doctoral Dissertation 2010 Ing. Hoang Sy TUAN TECHNICAL UNIVERSITY OF LIBEREC FACULTY OF MECHANICAL ENGINEERING Ing. Hoang Sy Tuan ELASTIC AND VISCOELASTIC BEHAVIOUR OF COMPOSITES WITH ELASTOMERIC MATRIX ELASTICKÉ A VISKOELASTICKÉ CHOVÁNÍ KOMPOZITŮ S ELASTOMERICKOU MATRICÍ Doctoral Dissertation submitted to Department of Engineering Mechanics, in partial fulfillment of the requirements for the degree of doctor of Philosophy in Mechanical Engineering Supervisor: doc. Bohdana Marvalová, CSc.
Liberec – 2010 Acknowledgments The work presented in this dissertation has been possible thanks to several individuals whose contributions I would like to acknowledge. I first wish to express my deep gratitude and many thanks to my supervisor and her family, Associate Professor Bohdana Marvalová for all her guidance, support, advice and encouragement throughout my study in more than three years. The support which I have received from her is immeasurable. For this, I will never forget her.
I would like to thank Dr. Plešek, Prof. Holeček, Dr. Šimůnková and Prof.
Holý for their useful lectures. Specially, I am very grateful to Dr. Petríková for the financial support to perform experiments during my work. I wish to acknowledge the important role of my family in my life, especially my parents.
They always encourage and remind me to work hard to achieve my goals. They also teach me to realize the value of education and knowledge. I also thank all my friends who have shared difficulties in real life. Their warm friendship is the reason I do not feel lonely.
Finally, I would like to thank all the members of Department of Engineering Mechanics for their kindness, friendliness and essential support during my work. I wish to thank Faculty of Mechanical Engineering of Technical University of Liberec and the Czech government for their provided fellowship. This work was supported by the subvention from Ministry of Education of the Czech Republic through the Contract Code MSM 4674788501. v Prohlášení Byl jsem seznámen s tím, že na mou doktorskou práci se plně vztahuje zákon č.
121/2000 o právu autorském, zejména §60 (školní dílo) a §35 (o nevýdělečném užití díla k vnitřní potřebě školy). Beru na vědomí, že Technická univerzita v Librci má právo na uzavření licenční smlouvy o užití mé práce a prohlašuji, že souhlasím s případným využitím mé práce (prodej, zapůjčení apod. Jsem si vědom toho, že užít své doktorské práce či poskytnout licenci k jejímu využití mohu jen se souhlasem Technické univerzity v Liberci, která má právo ode mne požadovat přiměřený příspěvek na úhradu nákladů, vynaložených univerzitou na vytvoření díla (až do jejich skutečné výše). vlastnoruční podpis Místopřísežné prohlášení Místopřísežně prohlašuji, že jsem doktorskou práci vypracoval samostatně s použitím uvedené literatury, pod vedením školitelkou.
vi ABSTRACT The viscous behavior of the fiber-reinforced composite materials with rubber-like matrix is modeled in the continuum mechanics framework by the Helmholtz free energy function and the evolution equations of the internal variables. The decomposition of the free energy function and the chosen viscoelastic model are bases for formulation and description of the viscous characteristics of these anisotropic materials. Numerical simulations to predict the response of these materials in finite strains are performed. The dissertation focused on experimental evaluating the purely elastic and viscoelastic material parameters of proposed models via some standard experiments on relaxation, such as simple tension, pure shear and biaxial tensile tests.
Both the isotropic and anisotropic materials were tested. Several numerical examples were implemented in FEM software COMSOL Multiphysics and compared with the experimental results. The applications of the model were enlarged to predict other viscoelastic phenomena i. creep and influence of loading velocities on stresses.
The influence of the directions of reinforcing fibers was also examined. The viscoelastic model was applied to a practical example that is an air-spring with two fiber reinforcements undergoing an internal pressure. An extension of nonlinear theory for rubber-like anisotropic composites was applied to magneto-sensitive (MS) elastomers under an external magnetic field. The constitutive equations of both magnetic and mechanical fields were presented.
Some numerical computations of a coupling of magnetic and mechanical problems were illustrated in order to describe a nonlinear characteristic of MS elastomer. Key words: Composites, rubber-like matrix, fiber-reinforced, viscoelasticity, magneto-sensitive elastomers, experimental, FEM. vii ABSTRAKT Viskózní chování kompozitních materiálů s pryžovou matricí vyztuženou kordy je modelováno v rámci mechaniky kontinua pomocí Helmholtzovy funkce volné energie a vývojových rovnic pro vnitřní proměnné. Rozklad funkce volné energie a zvolený viskoelastický model jsou základem pro formulaci a popis viskózní vlastnosti těchto anizotropních materiálů.
Jsou uvedeny numerické simulace pro přepověď odezvy těchto materiálů na konečné deformace. Disertační práce se zaměřuje na experimentální určení elastických a viskoelastických materiálových parametrů navrhovaných modelů pomocí některých standardních testů jako je tahová zkouška, čistý smyk a dvouosé tahové zkoušky kvazistatické i relaxační. Byly testovány izotropní a anizotropní (kompozitní) materiály. Několik numerických případů je implementováno do MKP prostředí COMSOL Multiphysics a srovnáno s experimentálními výsledky.
Aplikace modelu byly rozšířeny o numerickou předpověď dalších viskoelastických jevů jako je tečení a vliv rychlosti zatížení na napětí. Vliv směru vlákenné výztuže byl rovněž zkoumán. Viskoelastický model byl aplikován na numerickou simulaci vnitřním přetlakem zatížené vzduchové válcové pružiny, jejíž pryžový plášť je vyztužen dvěma skupinami kordů. Nelineární materiálový anizotropní model byl rozšířen na případ magneto-sensitivních elastomerů mechanicky zatížených ve vnějším magnetickém poli.
Numerické simulace odezvy tělesa s magneto-mechanickou vazbou jsou uvedeny s cílem popsat nelineární vlastnosti magneto-sensitivních elastomerů. Klíčová slova: Kompozity, pryžová matrice, kordová výztuž, viscoelasticita, magneto-sensitivní elastomery, experiment, MKP. viii Table of Contents List of figures xiii List of tables xvii Notation and symbols xviii Chapter 1. Introduction 1 Outline of the dissertation ………………………………………………….
Overview of literature 5 Chapter 3. Free energy functions and constitutive relations for equilibrium and viscoelastic response of rubber-like composite 11 3. Free energy functions ………………………………………………………. Volumetric free energy function ……………………………………….
Isotropic (isochoric) free energy functions ……………………………. Anisotropic (isochoric) free energy functions ………………………… 13 3. Equilibrium stress responses in plane-stress deformations …………………. Isotropic rubber-like materials ……………………………………….
Composites reinforced by two families of fibers ……………………. Non-equilibrium stress responses …………………………………………… 16 3. Solution of evolution equations for overstresses ……………………. Solution of evolution equations for inelastic strains ………………….
Experiments and material parameter identification 19 4. Experimental equipments and specimens …………………………………… 19 4. Universal testing machine TIRAtest 2810 ……………………………. Biaxial testing equipment …………………………………………….
System Q-400 digital cameras Dantec Dynamics ……………………. Experiments and identification methods ……………………………………. Isotropic composite materials …………………………………………. Pure shear test ………………………………………………………… 27 4.
Biaxial tensile test ……………………………………………………. Fiber-reinforced composite materials ………………………………………. Numerical simulations of viscoelastic composites 35 5. FEM implementation in COMSOL Multiphysics ………………………….
Isotropic (hyperelastic) rubber-like materials ………………………………. Equilibrium stress-strain responses …………………………………… 36 5. Viscoelastic behavior of isotropic materials …………………………. Effect of loading velocities …………………………………….
One-step and multi-step relaxations …………………………. Prediction of creep process …………………………………… 42 5. Fiber-reinforced composites ………………………………………………… 43 5. Equilibrium response of fiber-reinforced composite in pure shear …….
Elastic response of a rectangular fiber-reinforced composite plate with a hole …………………………………………………………………. Viscoelastic response of fiber-reinforced composite …………………. Viscoelastic behavior of an air-spring ………………………………. Equilibrium responses of an air-spring tube ………………………….
Viscous responses of an air-spring tube ………………………………. Viscous responses of internal stress-like and strain-like variables …………. Magneto-sensitive elastomer materials 55 6. The free energy function in terms of invariants ……………………….
Numerical simulations of MS elastomers ………………………………. FEM solutions of MS isotropic materials ……………………………. A plane strain compression of isotropic material block ……… 61 6. A simple shear strain of isotropic material plate …………….
A pure shear deformation of isotropic material tube …………. FEM solutions of MS anisotropic materials …………………………. Compression of anisotropic material block …………………. Simple shear of anisotropic materials ………………………… 71 6.
Conclusions, discussions and future perspectives 75 Literatures 77 Publications 83 Appendix A. Overview of continuum mechanics A-1 A. Finite strain kinematics ……………………………………………………. Motions of continuum bodies ………………………………………….
Cauchy stress tensor and equilibrium equation ………………………. Alternative stress tensors ……………………………………………… A-8 A. Conjugate pairs of stress and strain tensors …………………………… A-9 A. Conservation of mass ………………………………………………….
Momentum balance principles ………………………………………. Balance of mechanical energy ………………………………………… A-11 A. Balance of energy in continuum thermodynamics ……………………. Anisotropic viscoelastic models A-13 B.
The decomposition of the free energy function ………………………. Constitutive equations of stress responses ……………………………. Transversely isotropic materials ……………………………… A-15 B. Two fiber-reinforced composite materials …………………….
Evolution equations with internal variables …………………………………. Internal stress-like variables …………………………………………… A-18 B. Internal strain-like variables …………………………………………… A-19 Appendix C. Implementation of user material models in Comsol Multiphysics A-23 C.
Definition of modules for viscoelastic problems ……………………………. Definition of constitutive equations ………………………………………… A-23 C. Redefine free energy functions ……………………………………………… A-24 C. Formulation of evolution equations ………………………………………… A-25 C.
Define a function in Comsol ………………………………………………… A-25 C. Integration of volume ………………………………………………………. Some m-function and script files in Matlab A-27 D. Some functions of analyzing data …………………………………………… A-27 D.
Main program of analyzing data ………………………………………. Estimation of purely elastic coefficients ……………………………………. The main program of evaluation of purely elastic parameters ………. Estimation of viscoelastic coefficients ……………………………………….
Main program of evaluation of viscoelastic material parameters ……. A-31 xii List of Figures Figure 1.1 Some applications of FREs …………………………………………….1 Universal testing machine TIRAtest 2810 ………………………….2 The main window application of TIRAtest software …………………… 19 Figure 4.3 Biaxial testing equipment ……………………………………………….5 Main window of Istra 4D software ……………………………………… 20 Figure 4.6 Shapes of specimens for simple tensile, pure shear and biaxial tests .7 Mullins effect in cyclic tests …………………………………………….8 Strain evaluation by the image correlation ……………………………… 22 Figure 4.9 A standard Maxwell element ……………………………………….10 Controlled displacement in the simple tensile test ……………………….11 Applied force in the simple tensile test …………………………….12 Estimation of elastic coefficients of the isotropic material by a simple tensile test with different models ……………………………………….13 Estimation of viscoelastic coefficients of the isotropic material by a simple tensile test with neo-Hookean and Ogden models ………….14 Controlled displacement in the pure shear test ………………….15 Applied force in the pure shear test …………………………………….16 Principal stretches in the pure shear test ……………………………….17 Estimation of elastic coefficients of the isotropic material by a pure shear test with different models …………………………………….18 Estimation of viscoelastic coefficients of the isotropic material by a pure shear test with neo-Hookean and Ogden models ………….19 Estimation of elastic coefficients of the isotropic material by a biaxial tensile test with different models ……………………………………….20 Estimation of elastic coefficients of the composite reinforced with different fiber orientations by a pure shear test ………………………….21 Estimation of viscoelastic coefficients of the composite reinforced with different fiber orientations by a pure shear test ………………………….1 Geometries and FE models in simple tension and pure shear deformations …………………………………………………………….2 First principal stress versus stretch of simple tension deformation …….3 First principal stress versus stretch of pure shear deformation …….4 Stress distribution of simple tension and pure shear ……………….5 Stress-strain response at different loading velocities …………………… 40 Figure 5.6 Deformation of isotropic materials in a simple tension at different extensions ……………………………………………………………….7 The second Piola-Kirchhoff stresses and overstresses versus time at different stretches ……………………………………………………….8 Stress and deformation of simple tension in relaxation ………………… 41 Figure 5.9 Displacement and Cauchy stress of simple tension in multistep relaxations ………………………………………………………….10 Controlled force and displacement of simple tension in a creep ……….11 Geometry and FE model of fiber-reinforced composite in pure shear deformation …………………………………………………………….12 Deformation and stress distribution of composite with different fiber angles ……………………………………………………………….13 Equilibrium Cauchy stress with different fiber directions in pure shear deformation …………………………………………………………….14 Deformation and stress distribution of a composite plate with a hole ….15 Cauchy stress versus the first principal stretch of a composite with a hole 46 Figure 5.16 Displacement and Piola-Kirchhoff stress of the anisotropic composite with fiber angles by 300 in relaxations ………………….17 The components of overstresses in the pure shear deformation ………… 47 Figure 5.18 A sketch scheme of geometry and FE model of an air-spring tube subjected an internal static pressure …………………………………….19 Deformation and stress in the tube inflated by an internal pressure 0,9MPa the angle of fibers =300, 350 and 400 …………………….20 “Inversion phenomenon” happens to a tube reinforced by fibers with =400 ……………………………………………………………….21 The internal pressure and the longitudinal stretch with different fiber angles ……………………………………………………………….22 External force acts on the tube in a creep ……………………………….23 Deformation and stress of the tube at different time instants in a creep .