D. R. Axelrad · W. Muschik (Eds.) Recent Developments in Micromechanics Proceedings of the Mini-Symposium on Micromechanics at the CSME Mechanical Engineering Forum 1990 June 3-9,1990, University of Toronto, Canada Springer-Verlag Berlin Heidelberg NewY ork London Paris Tokyo Hong Kong Barcelona Prof. Dr. D. R. Axelrad Dept. of Mechanical Engineering McGill University Montreal, QC H3A 2K6 Canada Prof. Dr. rer. nat. W. Muschik Institut mrTheoretische Physik PN7-1 Technische Universitat Berlin 1000 Berlin 12 FRG ISBN-13: 978-3-642-84334-1 e-ISBN-13 978-3-642-84332-7 001: 10.1007/978-3-642-84332-7 This work is subject to copyright. All rights are reserved, whether the whole orpart of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broad casting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September9, 1965, in its current version and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin, Heidelberg 1991 Softcover reprint of the hardcover 1s t edition 1991 The use of registered names, trademarks,etc. in this publication does not imply,even in the absence ofa specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. 61/3020-543210 -Printed on acid-free paper Preface This volume contains the lectures presented at the mini-symposium on "Micromechanics" held in conjunction with the CSME Mechanical Engineer ing Forum 1990 between the 3rd and 8th June, 1990 at the University of Toronto, Canada. The expressed purpose of this symposium was to discuss some recent developments in the Micromechanics of Materials and how ad vances in this field now relate to the solution of practical engineer ing problems. Due to the time limit set for this section of the Engineer ing Forum as well as the restriction on the number of papers to be pre sented, it was not possible to cover a much wider range of topics. How ever, an attempt was made to include the most important advances asso ciated with the progress made in micromechanics in its application to material science and engineering over the past decade. Thus, the topics are concerned with: the fundamental aspects of the thermodynamics of structured solids (part I), - the micromechanical behaviour of alloys (part II), - the modelling of the material behaviour on the basis of continuum theory (part III), and finally the important new approach to the characterization of various mate rials and their responses to external agencies by the use of proba bilistic micromechanics (part IV). We would like to take this opportunity to thank the Chairman of the Organizing Committee, Prof. F.P.J. Rimrott, and the President of the CSME, Prof. T.S. Sankar, for their permission to prepare this publica tion coincident with the Proceedings of the CSME Engineering Forum 1990. We would also like to express our thanks to the members of the Local Or ganizing Committee for the support given by them in arranging the vari ous seminars, and the staff of Springer Verlag, in particular Mr. A. von Hagen, for the efficient production of this volume. Montreal, Berlin D.R. Axelrad and W. Muschik October, 1990 List of Contributors D. R. AXELRAD P. MAZILU S.R. BOHN R.E. MEYER W. FRYDRYCHOWICZ I. MfiLLER Z.L. GONG W. MUSCHIK R.K.T. HSIEH R. PERALTA-FABI T.R. HSU J.W. PROVAN M.R. KAMAL G.C. SIH J.A. KONIG E. STEIN R. MAHNKEN K.W. WILMANSKI G.A. MAUGIN G. ZHANG Contents Part I. Thermodynamics of Materials: 1. Non-Equilibrium Thermal/Mechanical Behaviour by G.C. Sih . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 2. Internal Variables in Non-Equilibrium Thermodynamics by W. Muschik . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3. Evaporation in Porous Media by R.E. Meyer . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35 Part II. Micromechanics of Alloys: 1. Micro-Macro and Meso Mechanics of Alloys with Shape Memory by Ingo Miiller . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2. Nonlinear Localized Effects in Micromechanics by G.A. Maugin .................. . 63 3. Micromechanics of Conductive Polymeric Materials by R.K. T. Hsieh . . . . . . . . . . . . . . . . . . . . . . . 79 4. Large Elasto-Plastic Deformation of Two-Phase Alloys - Structural Continuous Model by K.W. Wilmanski. . . . . . . . . . . . . . . . . . . . . . . .. 84 Part III. Micro-Continuum Mechanics: 1. Micromechanical Modelling and Computation of Shakedown with Nonlinear Kinematic Hardening - Including Examples for 2D-Problems by E. Stein, G. Zhang, R. Mahnken and J.A. Konig. . . 99 2. New Variational Principles in Heterogeneous Elasticity Derived by Lagrange's Multiplier Method by P. Mazilu . . . . . . . . . . . . . . . . . . . . . . . . . . 115 3. A Constitutive Model for Cyclic Inelastic Deformation of Solids by T.R. Hsu and Z.L. Gong. . . . . . . . . . . . . . . . . 127 VIII Part IV. Stochastic and Experimental Micromechanics: 1. Stochastic Flow Dynamics of Polymer Melts by D.R. Axelrad, M.R. Kamal and W. Frydrychowicz. . . .. 147 2. Microstructure and the Fatigue Reliability of Metals by J.W. Provan and S.R. Bohn . . . . . . . . . . . . . . . . .. 162 3. The Measurement of Microdeformations in Structured Solids by D.R. Axelrad. . . . . . . . . . . . . . . . . . . . . . . . . .. 172 4. Micro-Rheology of Multi-Phase System: An Experimental Assessment by R. Peralta-Fabi . . . . . . . . . . . . . . . . . . . . . . . . .. 188 Closure 204 Part I Thermodynamics of Materials Non-Equilibrium Thermal/Mechanical Behaviour G. C. SIH Institute of Fracture and Solid Mechanics. Lehigh University Bethlehem. Pennsylvania 18015 USA Summary A solid undergoes nonequilibrium changes when disturbed. a phenomenon that occurs at both the microscopic and macroscopic scale level. These changes depend on the rate at which the process takes place and cannot be described in terms of physical parameters representing the system as a whole. Their description involves synchronizing the thermal fluctuation with mechanical motion of the material at each location and time. The concept of an isoenergy density space is introduced such that the ener gy state in a volume element can be characterized by the time history of only one stress and one strain component regardless of whether the element is in a uniaxial or multiaxial state of stress. A unique correspondence between the uniaxial and multiaxial stress state can thus be established without loss in generality. Unlike the classical approaches in continuum mechanics. the response of an isoenergy density element requires the simul taneous specification of element size, time and temperature. Mutual inter action of size/time/temperature cannot be ignored for nonequilibrium states because temperature averaged over a macroelement within a given time inter val can differ qualitatively and quantitatively from that over a microele ment and different time interval. The same applies to mechanical motion. As an example, data for the nonequilibrium thermal/mechanical response of an aluminum cylindrical bar specimen stretched at a displacement rate of 1.27 x 10-4 cm/sec will be presented. A two-phase medium is considered such that the conditions at the solid/air interface are allowed to chanqe with time rather than specified or ignored as in the classical treatment of equilibrium processes. Thermal/mechanical disturbances are found to pre vail in the surrounding of the specimen; they diminished at distances more than one-half the specimen length. The response is transitory in character for the entire load history and depends on the initial thermal environment next to the specimen as well as the rate of the applied external disturb ance. Introduction A physical system is said to be in a state of nonequilibrium when its con stituents change continuously with time in a nonhomogeneous fashion. That is. the thermal/mechanical properties cannot be described by parameters rep resenting the system as a whole. Traditional approaches have relied on ~ta­ ~tiC6 and kinetiC6 for developing molecular models where conditions of ir reversibility at both the microscopic and macroscopic scale level are im posed. While numerical data have been obtained for coefficients in the transport equations, they cannot be readily used in engineering application. 4 Theories of nonequilibrium ~heromadynamiCb [1-3] are equally unsatisfying for they do not consider the mutual interaction of temperature change with the motion of mass elements. Thermal and mechanical effects are left un synchronized. Inhomogeneity being characteristic of the nonequilibrium process cannot be adequately described by applying the same equation of state or constitutive relation to all mass elements in a system. Moreover, the linear relation assumed for the entropy exchanged between the system and surrounding and that gene4ated in the system is also disturbing because it excludes the possibility of interaction between the heat exchanged and generated. These limitations can be overcome only if thermal change and mechanical motion are considered intrinsically as one of the same operation. As a fundamental departure from the classical field theories, the isoenergy density theory [4,5] takes into account the change of element size that no longer shrinks to zero. The elements can expand and/or contract depending on the local rotation and deformation without restrictions in magnitude and finiteness. They are known as the isoenergy density elements because their orientation and size are determined at each location and time increment ac cording to the load history from the condition that the same energy is transmitted across all the mutually orthogonal surfaces. Unique correspond ence of energy state between uniaxial and multiaxial stress/strain state is thus made possible without loss in generality. A brief summary of the iso energy density theory will be given for the sake of continuity. Isoenergy Density Theory The change of volume with surface denoted by &V/&A or V is assumed to van ish in the limit when the element size is shrunk indefinitely. This assump tion will no longer be made in the isoenergy density theory [4,5]. For an orthogonal coordinate system in three dimensions, the three components Vj (j = 1,2,3) establish the relations the components Sj (j = 1,2,3) of the surface energy density vector and the vol ume energy density W as S. = V. • W, j = 1,2,3 (1) J J A system of rectangular Cartesian coordinates si (i 1,2,3) is used to 10- cate the surfaces of energy transmission. I~aene4gy Efeme~. An isoenergy surface possesses the property that the transmission of energy on such a surface is directional independent. If si (i = 1,2,3) are the current state coordinates, then the condition