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Thermomechanics of Solid Materials with Application to the Gurson-Tvergaard Material Model

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Thermomechanics of Solid Materials with Application to the Gurson-Tvergaard Material Model VTT PUBLICATIONS 312 F19800095 Thermomechanics of solid materials with application to the Gurson-Tvergaard material model Kari Santaoja VlT Manufacturing Technology . TECHNICAL RESEARCH CENTRE OF FINLAND ESP00 1997 312 Kari Santaoja Thermomechanics of solid materials with application to the Gurson-Tvergaard material model TECHNICAL RESEARCH CENTRE OF FINLAND ESP00 1997 ISBN 951-38-5060-9 (soft back ed.) ISSN 123.54621 (soft back ed.) ISBN 95 1-38-5061-7 (URL: http://www.inf.vtt.fi/pdf/) ISSN 14554849 (URL: http://www.inf.vtt.fi/pdf/) Copyright 0 Valtion teknillinen tutkimuskeskus (VTT) 1997 JULKAISIJA - UTGIVARE - PUBLISHER Valtion teknillinen tutkimuskeskus (VTT), Vuorimiehentie 5, PL 2000, 02044 VTT puh. vaihde (09) 4561, faksi (09) 456 4374 Statens tekniska forskningscentral (VTT), Bergsmansvagen 5, PB 2000,02044 VTT tel. vaxel(O9) 4561, fax (09) 456 4374 Technical Research Centre of Finland (VTT), Vuorimiehentie 5, P.O.Box 2000, FIN42044 VTT, Finland phone internat. + 358 9 4561, fax + 358 9 456 4374 VIT Valmistustekniikka, Ydinvoimalaitosten materiaalitekniikka, Kemistintie 3, PL 1704,02044 VTT puh. vaihde (09) 4561, faksi (09) 456 7002 VTT Tillverkningsteknik, Material och strukturell integritet, Kemistvagen 3, PB 1’704,02044 VTT tel. vaxel(O9) 4561, fax (09) 456 7002 VTT Manufacturing Technology, Maaterials and Structural Integrity, Kemistintie 3, P.O.Box 1704, FIN42044 VTT, Finland phone internat. + 358 9 4561, fax + 358 9 456 7002 Technical editing Kerttu Tirronen VTT OFFSETPAINO. ESP00 1997 Santaoja, Kari. Thermornechanics of solid materials with application to the Gurson-Tvergaard material model. Espoo 1997. Technical Research Centre of Finland, UTPublications 312. 162 p. + app. 14 p. UDC 536.7 Keywords thermomechanical analysis, thermodynamics, plasticity, porous medium, Gurson- Tvergaard model ABSTRACT The elastic-plastic material model for porous material proposed by Gurson and Tvergaard is evaluated. First a general description is given of constitutive equations for solid materials by thermomechanics with internal variables. The role and definition of internal variables are briefly discussed and the following definition is given: The independent variables present (possibly hidden) in the basic laws for thermomechanics are called controllable variables, The other independent variables are called internal variables. An internal variable is shown always to be a state variable. This work shows that if the specific dissipation function is a homogeneous function of degree one in the fluxes, a description for a time-independent process is obtained. When damage to materials is evaluated, usually a scalar-valued or tensorial variable called damage is introduced in the set of internal variables, A problem arises when determining the relationship between physically observable weakening of the material and the value for damage. Here a more feasible approach is used. Instead of damage, the void volume fraction is inserted into the set of internal variables, This allows use of an analytical equation for description of the mechanical weakening of the material. An extension to the material model proposed by Gurson and modified by Tvergaard is derived. The derivation is based on results obtained by thermomechanics and damage mechanics. The main difference between the original Gurson-Tvergaard material model and the extended one lies in the definition of the internal variable 'equivalent tensile flow stress in the matrix 3 material' denoted by oM. Using classical plasticity theory, Tvergaard elegantly derived an evolution equation foroM.This is not necessary in the present model, since damage mechanics gives an analytical equation between the stress tensor u and oM. Investigation of the Clausius-Duhem inequality shows that in compression, states occur which are not allowed. 4 PREFACE The present publication was prepared for the RAKE project (Structural Analyses for Nuclear Power Plant Components). The main objective of the RAKE project is to create, evaluate and apply effective and reliable structural analysis methods for the safety and availability assessment of nuclear power plant applications. In particular, they are applied on pressure vessels and piping. There are three target research areas: to develop fracture assessment tools, to assess component behaviour under realistic loading cases, and to verify the methods using large scale experiments. This work forms part of the first item on the above list. The RAKE project belongs to the Finnish research programme on the 'Structural Integrity of Nuclear Power Plants' (RATU2). The RATUZ programme is set for the period 1995- 1998. The author would like to thank professor Martti Mikkola - Helsinki University of Technology (Espoo, Finland) - for his review and valuable comments and the extensive work he put into this report. The comments given by professor V. N. Kukudianov - the Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow - are appreciated. I am grateful to professor Nguyen Quoc Son - Ecole Polytechnique (91 128 Palaiseau Cedex, France) - for our fruitful discussions during his visit to Helsinki (Finland), on September 30 - October 4. 1996. I am grateful to Mrs Hilkka Hanninen and to Mr Tuomo Hokkanen for their drawings. The RATU2 research programme has been funded mainly by the Ministry of Trade and Industry (KTM), the Finnish Centre for Radiation and Nuclear Safety (STUK), lmatran Voima (IVO), Teollisuuden Voirna Oy (TVO) and the Technical Research Centre of Finland (VTT). This task was funded by KTM and VTT. Their financial support is greatly appreciated. 6 CONTENTS ABSTRACT .... .... 3 PREFACE ....... .... 5 LIST OF SYMBOLS 10 1 INTRODUCTION 13 2 THERMOMECHANICAL PRELIMINARIES . 17 3 CONTINUUM MECHANICS ..... 27 3.1 Law of conservation of mass ... ... 27 3.2 Law of balance of momentum ............. 27 3.3 Law of balance of moment of momentum ... 29 4 THERMODYNAMICS .. ... 33 4.1 General remarks .... ... 33 4.2 Thermostatics .... ... 37 4.3 Thermodynamics .... ... 40 5 THERMOMECHANICS .............. ... 43 5.1 Major dialects of thermodynamics .... ...... 41 5.2 Thermodynamics with internal variables . ........ 46 5.3 Variables describing the modelled process ...... 51 5.4 Law of caloric equation of state and the axiom of local accompanying state .... ... 55 5.5 First law of thermodynamics ........ ... 58 5.5.1 Heat equation .......... ... 63 5.6 Second law of thermodynamics .... ... 65 7 5.7 Clausius-Duhem inequality .............................. 68 5.8 Principle of maximal rate of entropy production ............... 72 5.8.1 Normality rule in a general case ....................... 72 5.8.2 Normality rule for the specific complementary dissipation function (pc .................. 79 5.8.3 Normality rule for thermoplastic material behaviour ......... 82 6 FOURIER'S LAW OF HEAT CONDUCTION ..................... 91 7 GENERAL FORMULATION FOR THE THEORY OF THERMOPLASTICITY ....................... 93 7.1 Kelvin-Voigt type of material models ....................... 96 7.2 Maxwell type of material models ........................... 98 7.3 State equations and normality rule for Maxwell type of material models ................................. 101 7.4 Maxwell type of material models with elastic strain tensor ............................... 104 8 MATERIAL MODELS FOR ISOTROPIC AND KINEMATIC HARDENING 109 8.1 Rates of the internal forces 'p' and 'p2 ..................... 109 8.2 Consistency condition and the multiplier ..................111 8.3 Clausius-Duhem inequality .............................. 113 8.4 Particular material models ............................... 113 8.4.1 Duhamel-Neumann form of Hooke's law ................113 8.4.2 Special models for plastic flow ........................ 115 9 GURSON-TVERGAARD MATERIAL MODEL .................... 121 10 EXTENSION FOR THE GURSON-TVERGAARD MATERIAL MODEL . 133 10.1 Specific complementary Helmholtz free energy itc ............ 134 10.2 Effective stress tensor 0 ............................. 139 8 10.3 Yield function F and evolution equations ... , 141 10.4 Clausius-Duhem inequality ................ .. ... 144 10.5 Plasticity multiplier ................................ 146 11 DISCUSSION AND CONCLUSIONS ... 141 REFERENCES ... ...157 APPENDICES A Double-dot product of a skew-symmetric third--order tensor and a symmetric second-order tensor B Legendre transformation C Legendre partial transformation D Divergence of the dot product of a second-order tensor and a vector E Stress power per unit volume F Partial derivatives of the second deviatoric invariant JJa- p’) 9 LIST OF SYMBOLS surface area of a subsystem in the initial configuration deviatoric part of the internal force B deviatoric part of the plastic strain tensor 8 yield function void volume fraction scalar-valued internal force function showing the inhomogeneity of the material fourth-order identity tensor For every second-order tensor A holds, I A = A : I = A kinetic energy function showing the inhomogeneity of the material outward unit normal power input heat input rate material parameters in the Gurson-Tvergaard material model heat flux vector heat source per unit mass entropy fourth-order compliance tensor for Hookean material compliance tensor for Hookean material with spherical voids S deviatoric stress tensor S specific entropy specific entropy rate specific entropy production rate specific entropy production rate (thermal part) specific entropy production rate (mechanical part) absolute temperature time surface traction vector 10 U internal energy
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