MECHANICS AND PHYSICS OF ENERGY DENSITY

ENGINEERING APPLICATION OF FRACTURE MECHANICS Editor-in-ChieI George C. Sih

1. G.c. Sih and L. Faria (eds.), Fracture mechanics methodology: Evaluation of structural components integrity. 1984. ISBN 90-247-2941-6.

2. E.B. Gdoutos, Problems of mixed mode crack propagation. 1984. ISBN 90-247-3055-4.

3. A. Carpinteri and A.R. Ingraffea (eds.), Fracture mechanisms of con­crete: Material characterization and testing. 1984. ISBN 90-247-2959-9.

4. G.c. Sih and A. DiTommaso (eds.), Fracture mechanics of concrete: Structural application and numerical calculation. 1984. ISBN 90-247-2960-2.

5. A. Carpinteri, Mechanical damage and crack growth in concrete: Plastic collapse to brittle fracture. 1986. ISBN 90-247-3233-6.

6. J.W. Provan (ed.), Probabilistic fracture mechanics and reliability. 1987. ISBN 90-247-3334-0.

7. A.A. Baker and R. Jones (eds.), Bonded repair of aircraft structures. 1987. ISBN 90-247-3606-4.

8. J.T. Pindera and M.-J. Pindera, Isodyne stress analysis. 1989. ISBN 0-7923-0269-9.

9. G.c. Sih and E. B. Gdoutos (eds.), Mechanics and physics of energy density. 1992. ISBN 0-7923-0604-X.

10. B.E. Gdoutos, Fracture mechanics criteria and applications. 1990. ISBN 0-7923-0605-8.

11. G.c. Sih, Mechanics of fracture initiation and propagation. 1991. ISBN 0-7923-0877-8.

Mechanics and Physics of Energy Density Characterization of material/structure behavior with and without damage

George C. Sih Institute of Fracture and Solid Mechanics Lehigh University Bethlehem, Pennsylvania, USA

and

Emmanuel E. Gdoutos Department of Civil Engineering Democritus University of 7hrace X anthi, Greece

Kluwer Academic Publishers Dordrecht / Boston / London

Library of Congress Cataloging-in-Publication Data

Mechanics and physIcs of energy density characterIzatIon of material/structure behavior with and without damage / [edIted by] George C. Sih and Emmanuel E. Gdoutos.

p. ern. -- (EngIneering applIcatIon of fracture mechanics 9) Proceedings of a symposIum held July 17-20. 1989, at the

Democrltus University of Thrace. Includes bIblIographIcal references

1. Fracture mechanlcs--Congresses. 2. StraIns and stresses--Congresses. I. Sih, G. C. (George C.) II. Gdoutos, E. E., 1948-

III. SerIes. TA409.M399 1990 620.1' 126--dc20 90-41854

ISBN-13: 978-94-010-7373-8 DOl: 10.1007/978-94-009-1954-9

e-ISBN-13: 978-94-009-1954-9

Published by Kluwer Academic Publishers, P.O. Box 17,3300 AA Dordrecht, The Netherlands.

Kluwer Academic Publishers incorporates the publishing programmes of D. Reidel, Martinus Nijhoff, Dr W. Junk and MTP Press.

Sold and distributed in the U.S.A. and Canada by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A.

In all other countries, sold and distributed by Kluwer Academic Publishers Group, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

printed on acid:free paper

All Rights Reserved © 1992 Kluwer Academic Publishers Dordrecht. The Netherlands. Softcover reprint of the hardcover I st edition 1992 No part of the material protected by this copyright notice may be reproduced or utilized in any fonn or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Contents

Series on engineering application of fracture mechanics IX

Foreword Xln

Editors' preface XV

Contributing authors XIX

Photographs XXI

1. Synchronization of thermal and mechanical disturbances in uniaxial specimens / G.c. Sih 1 1.1 Introductory remarks 1 1.2 System inhomogeneity and continuity 2 1.3 Simultaneity of displacement and temperature change 6 1.4 Isoenergy density theory 9 1.5 Axisymmetric deformation 15 1.6 Nonequilibrium response of cylindrical bar specimen

in tension 16 1.7 Conclusions 33 References 33

2. Thoughts on energy density, fracture and thermal emission / R. Jones, M. Heller and L. Molent 35 2.1 Introduction 35 2.2 The F -111 wing pivot fitting 35 2.3 Damage assessment of an F/A-18 stabilator 39 2.4 The finite element model 44 2.5 Thermoelastic evaluation of damage 47 2.6 Stress fields from temperature measurements 52 2.7 Conclusions 56 References 56

V

VI Contents

3. Effects of fillers on fracture performance of thermoplastics: strain energy density criterion / T. Vu-Khanh, B. Fisa and 1. Denault 59 3.1 Introduction 59 3.2 Experimental consideration 60 3.3 Fracture analysis 61 3.4 Strain energy density criterion 64 3.5 Conclusions 71 References 72

4. Strain energy density criterion applied to characterize damage in metal alloys / V.S. Ivanova 75 4.1 Introduction 75 4.2 Strain energy density criterion 75 4.3 Thermal/mechanical interaction in solids 77 4.4 Damage characterization 79 4.5 Transition of micro- to macrodamage 82 4.6 Concluding remarks 84 References 84

5. Local and global instability in fracture mechanics / A. Carpinteri 87 5.1 Introduction 87 5.2 Strain energy density fracture criterion 88 5.3 Strain-hardening materials 89 5.4 Strain-softening materials 94 5.5 Size effects on strength and ductility 103 References 107

6. A strain-rate dependent model for crack growth / G.E. Papakaliatakis, E.E. Gdoutos and E. Tzanaki 109 6.1 Introduction 109 6.2 Description of the method 110 6.3 Specimen geometry and material properties 112 6.4 Stress analysis 113 6.5 Crack growth initiation 118 6.6 Concluding remarks 119 References 119

7. Extrusion of metal bars through different shape die: damage evaluation by energy density theory / J. Lovas 121 7.1 Introduction 121 7.2 Yielding/fracture initiation in plastic deformation 122 7.3 Nonlinear behavior of extruded metal 124 7.4 Analysis of failure initiation sites 129 7.5 Conclusions 135 References 136

Contents VII

8. Failure of a plate containing a partially bonded rigid fiber inclusion / E.E. Gdoutos and M.A. Kattis 137 8.1 Introduction 137 8.2 A partially bonded rigid elliptical inclusion in an infinite

plate 138 8.3 Local stress distribution and stress intensity factors 140 8.4 Failure initiation from the crack tip or the fiber end 144 References 147

9. Crack growth in rate sensitive solids / O.A. Bukharin and L.V. Nikitin 149 9.1 Introductory remarks 149 9.2 Sih criterion 149 9.3 Linear viscoelastic solid 150 9.4 Crack growth in uniformly applied stress field 151 9.5 Conclusions 153 References 153

10. Strain energy density criterion applied to mixed-mode cracking dominated by in-plane shear / K.-F. Fischer 155 10.1 Preliminary remarks 155 10.2 Sih's strain energy density criterion 157 10.3 Mixed-mode cracking dominated by in-plane shear 159 10.4 Discussions 163 References 164

11. Group-averaging methods for generating constitutive equations / M.M. Smith and G.F. Smith 167 11.1 Introduction 167 11.2 Generation of scalar-valued invariants 167 11.3 Generation of tensor-valued invariant functions 170 11.4 Applications 173 References 178

12. A dislocation theory based on volume-to-surface ratio: fracture behavior of metals / C.W. Lung, L.Y. Xiong and S. Liu 179 12.1 Introduction 179 12.2 Super-dislocation model 179 12.3 Plastic zone size 183 12.4 Dislocation distribution in plastic zone 184 12.5 Crack in semi-infinite medium 186 12.6 Relation of volume/surface ratio to plate ligament 187 12.7 Specimens with different volume/surface ratio 188 12.8 Conclusions 191 References 193

VIII Contents

13. The effect of microcracks on energy density / N. Laws 195 13.1 Introduction 195 13.2 Microcracked solid with given crack density 196 13.3 Microcrack nucleation 198 References 201

14. Convex energy functions for two-sided solution bounds in elastomechanics / A.A. Liolios 203 14.1 Introduction 203 14.2 General problem in elastostatics 203 14.3 Convexity of strain energy and Hilbert space: elastic system 205 14.4 Global solution bounds 206 14.5 Local solution bounds 207 14.6 Concluding remarks 208 References 209

Series on engineering application of fracture mechanics

Fracture mechanics technology has received considerable attention in recent years and has advanced to the stage where it can be employed in engineering design to prevent against the brittle fracture of high-strength materials and highly constrained structures. While research continued in an attempt to extend the basic concept to the lower strength and higher toughness materials, the technology advanced rapidly to establish material specifications, design rules, quality control and inspection standards, code requirements, and regulations of safe operation. Among these are the fracture toughness testing procedures of the American Society of Testing Materials (ASTM), the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Codes for the design of nuclear reactor components, etc. Step-by-step fracture detection and prevention procedures are also being developed by the industry, government and university to guide and regulate the design of engineering products. This involves the interaction of individuals from the different sectors of the society that often presents a problem in com­munication. The transfer of new research findings to the users is now becoming a slow, tedious and costly process.

One of the practical objectives of this series on Engineering Application of Fracture Mechanics is to provide a vehicle for presenting the experience of real situations by those who have been involved in applying the basic knowledge of fracture mechanics in practice. It is time that the subject should be presented in a systematic way to the practising engineers as well as to the students in universities at least to all those who are likely to bear a responsibility for safe and economic design. Even though the current theory of linear elastic fracture mechanics (LEFM) is limited to brittle fracture behavior, it has already provided a remarkable improvement over the conventional methods not accounting for initial defects that are inevitably present in all materials and

G. C. Sih and E. E. Gdoutos (eds): Mechanics and Physics afEnergy Density, IX-XI. © 1992 KIUI'veI' Academic Publishers.

x Series on engineering application of fracture mechanics

structures. The potential of the fracture mechanics technology, however, has not been fully recognized. There remains much to be done in constructing a quantitative theory of material damage that can reliably translate small specimen data to the design of large size structural components. The work of the physical metallurgists and the fracture mechanicians should also be brought together by reconciling the details of the material microstructure with the assumed continua of the computational methods. It is with the aim of developing a wider appreciation of the fracture mechanics technology applied to the design of engineering structures such as aircrafts, ships, bridges, pavements, pressure vessels, off-shore structures, pipelines, etc. that this series is being developed.

Undoubtedly, the successful application of any technology must rely on the soundness of the underlying basic concepts and mathematical models and how they reconcile with each other. This goal has been accomplished to a large extent by the book series on Mechanics of Fracture started in 1972. The seven published volumes offer a wealth of information on the effects of defects or cracks in cylindrical bars, thin and thick plates, shells, composites and solids in three dimensions. Both static and dynamic loads are considered. Each volume contains an introductory chapter that illustrates how the strain energy criterion can be used to analyze the combined influence of defect size, component geometry and size, loading, material properties, etc. The criterion is particularly effective for treating mixed mode fracture where the crack propagates in a nonself similar fashion. One of the major difficulties that continuously perplex the practitioners in fracture mechanics is the selection of an appropriate fracture criterion without which no reliable prediction of failure could be made. This requires much discernment, judgement and experience. General conclusion based on the agreement of theory and experiment for a limited number of physical phenomena should be avoided.

Looking into the future the rapid advancement of modern technology will require more sophisticated concepts in design. The micro-chips used widely in electronics and advanced composites developed for aerospace applications are just some of the more well-known examples. The more efficient use of materials in previously unexperienced environments is no doubt needed. Fracture mechanics should be extended beyond the range of LEFM. To be better understood is the entire process of material damage that includes crack initiation, slow growth and eventual termination by fast crack propagation. Material behavior characterized from the uniaxial tensile tests must be related to more complicated stress states. These difficulties should be overcome by unifying metallurgical and fracture mechanics studies, particularly in assessing the results with consistency.

This series is therefore offered to emphasize the applications of fracture mechanics technology that could be employed to assure the safe behavior of

Series on engineering application of fracture mechanics XI

engineering products and structures. Unexpected failures mayor may not be critical in themselves but they can often be annoying, time-wasting and discrediting of the technical community.

Bethlehem, Pennsylvania

G.c. SIH

Editor-in-Chief

Foreword

This volume on the application of energy density in mechanics and physics is concerned with the field of material/structure technology. Theoretical and experimental contributions were presented as lectures at the Democritus University of Thrace, Xanthi, Greece, during the period of July 17 to 20, 1989. They included individuals from Australia, Canada, German Democratic Republic, Greece, Hungary, Italy, Poland, People's Republic of China, USA und USSR. Most rewarding are the exchange of new ideas and informal discussions that were made possible only by a limited size audience. Discussions on the development of new materials and their applications were particularly relevant to the on-going research of the European Common Community.

The co-organizers of this symposium are indebted to those who have spent many hours of their valuable time in preparing the manuscript. Financial support from

Ministry of Industry, Research and Technology Democritus University of Thrace Lehigh University Prefecture of Xanthi Municipality of Xanthi Commercial Bank of Greece National Bank of Greece Cooperative Cigarette Company (SEKAP) Cooperative Meat Company (SEPEK) Cooperative Milk Company

are also gratefully acknowledged. The following members of the local organizing committee are credited with operating the visual equipment and organizing social activities:

G. C. Sih and E. E. Gdoutos (eds): Mechanics and Physics of Energy Density, XIII-XIV. © 1992 Klllwer Academic Publishers.

XIV

Dr. E.E. Gdoutos Dr. D.A. Zacharopoulos Dr. G. Papakaliatakis Mrs. Z. Adamidou Mr. A. Betzakoglou Ms. C. Gemenetzidou Mr. N. Prassos

X ant hi, Greece July 1989

Foreword

G.c. SIR E.E. GDOUTOS Co-Organizers

Editors' Preface

The motivation to publish this volume stems from the central role that energy density plays in the formulation of continuum mechanics theories and characterization of material/structure behavior. So many developments in mechanics and physics depended on the positive definiteness of volume energy density and its scalar invariant property. Examples are the Kirchhoff uniqueness proof in the linear theory of elasticity and determination of constitutive relations based on the theoretic invariant approach. Proposed by E. Beltrami as early as 1903 was that the limiting capacity of a solid could be related to elastic energy stored in a unit volume of material. This unique character of the volume energy density function was also recognized in the 1960's by the late L. Gillemot as a useful parameter for characterizing the plastic behavior of metals and weldments.

The versatility of this quantity has been expanded over the years to account for failure by yielding and fracture which included initiation, stable growth and onset of global instability. It matters not of material type nor of the source of energy whether chemical, electrical or otherwise. Disruptions in material/ structure behavior can, in general, be associated with the fluc­tuations of energy in a unit volume and/or surface of material. What is not commonly recognized is the relation between surface and volume energy density even though it has been explicitly expressed in the crystal nucleation theory of Gibb where the energy proportional to the nucleus volume exchanges directly with that to the nucleus surface area. Further clarification on how energy density could be applied to formulate theories and to explain physical phenomena is one of the main objectives of the fourteen chapters in this volume.

In their work, O.A. Bukharin and L.V. Nikitin derived the volume energy density for a linear viscoelastic solid and showed that the threshold stresses corresponding to the initiation of stable and unstable crack growth can be

G. C. Sill and E. E. Gdolltos (eds): Mechanics and Physics oj Energy Density. XV-XVIII. © 1992 Kluwer Academic Publishers.

XVI Editors' Preface

obtained in a straightforward manner bypassing much of the complexities required in previous works on the same subject M,M. Smith and G.F. Smith made use of the scalar property of the strain energy density function for generating the invariants that describe the material symmetry. Developed are procedures for deriving constitutive relations. Use is also made of the strain energy density by A.A. Liolios to establish upper and lower bounds in the numerical analysis of problems in elastostatics and elastodynamics.

A dislocation model was proposed by W. Lung, L.Y. Xiong and S. Liu to obtain the fracture behavior of macro crack under the influence of plasticity. Influence of the ligament from the crack to the plate edge on plasticity can be related to the change in volume with surface, a quantity that interconnects the surface and volume energy density. Microscopic damage and critical incre­ment of crack growth prior to macroinstability for thermoplastics were successfully explained by T. Vu-Khanh, B. Fisa and J. Denault They applied the strain energy density criterion and showed that the addition of fillers led to a reduction in the critical strain energy density. This can be interpreted as an enhancement of the resistance of thermoplastics to fracture. V.S. Ivanova modelled the microdamage and macrodamage of metal alloys by considering the critical distortional and dilatational energy density. The former was identified with the latent heat of melting and latter with the internal energy density calculated from the heat capacity at constant pressure. Analytical results are related to damage zone sizes at the microscopic and macroscopic scale level; they compared well with experiments. As the density of micro­cracks in a unit volume of material changes, the corresponding energy density would be affected as discussed by N. Laws. The response for open and closed cracks can be different and the influence on the strain energy density function must be determined accordingly.

The energy density criterion is employed by 1. Lovas to evaluate the damage in aluminum and stainless steel bars that are extruded through dies of different shapes. Sites of potential failure by fracture are predicted from the maximum of minimum strain energy density. An analytical/experimental procedure was used and the predictions agreed with the tests. Presented by K.F. Fischer is how mixed mode fracture could be characterized using the criterion of strain energy density. The direction of crack initiation is assumed to coincide with the location of minimum energy density factor or sm\ while the onset of crack growth is assumed to occur when smill reaches some critical value. Results are displayed in terms of plots of Mode I and II stress intensity factors and discussed in connection with other failure criteria. Critical loads to initiate crack growth and fracture trajectories for cracks at interface of inclusion and matrix are obtained by E.E. Gdoutos and M.A. Kattis. They made consistent use of the strain energy density criterion. The influence on change in the local material behavior on subcritical crack growth was examined by G.E. Papakaliatakis, E.E. Gdoutos and E. Tzanaki. Each

Editors' Preface XVII

material element along the prospective path of crack growth can follow a different elastic-plastic stress/strain response and its behavior can be affected by the previous loading step, Crack growth is assumed to occur in segments corresponding to a constant ratio of the strain energy density factor and growth segment Strain rate and strain rate history dependency led to crack growth data that deviated appreciably from those found by the classical theory of plasticity,

As the failure behavior of small specimens or structures can differ significantly from the larger ones, A. Carpinteri applied the strain energy density criterion to obtain the scaling relations that include the combined influence of specimen size, material and loading steps. Crack growth resistance curves are found for concrete-like materials that undergo hardening and softening. For a number of years, R. Jones, M. Heller and L. Molent at the Aeronautical Research Laboratory in Melbourne, Australia, have effectively applied the strain energy density theory to the design of reinforcement for repairing aircraft skin structures damaged by stress corrosion cracking. Summarized in their Chapter are some of the more recent works that illustrate how the temperature field can be measured and used to determine the energy density field of flawed structures.

Perhaps, one of the most significant post World War II developments in the field of material/structure behavior is the recognition that design requirements based on classical continuum mechanics solutions alone were not sufficient. Additional failure criteria must be supplemented to account for the criticality of defects or discontinuities. The impetus were thus provided to put the 1921 concept of A.A. Griffith into practice. A quantity that became the focus of attention in fracture mechanics research was the energy required to create a unit area of fracture surface, the equivalent of which are many only for the idealized situation of no energy dissipation where the material can recover completely. The restrictive character of linear elastic fracture mechanics gave information only on the terminal condition which does not provide the safety measures required in many of the high performance structures. Needed is the assessment of how and where defects would initiate in addition to their subsequent growth behavior. To this end, the volume energy density criterion were advanced to address the complete process of defect initiation, subsequent growth and onset of instability. Much of the work can be found in the introductory chapters of the seven volumes of the series on Mechanics of Fracture and the references therein. It represented a contrast to the surface energy density quantity associated with energy release rate, stress intensity factor, crack opening displacement, etc. Failure analyses in fracture mechanics have thus followed two separate schools of thought: one on surface energy density and the other on volume energy density. This trend has prevailed up to this date. Discussion on energy density would be incomplete without touching on the reconciliation of the two approaches. This can be made possible only by

XVIII Editor's Preface

revoking a fundamental assumption in continuum mechanics. That is by not letting the size of the continuum element to vanish in the limit. Because it is the finite rate change of volume with surface that links the surface and volume energy density. The ways with which this can be done is summarized in the first chapter of this volume. It hinges on interlacing the mechanical and thermal effects so that the complete hierarchy of the material damage process can be assessed quantitatively without ambiguity. Deformation and temperature changes in uniaxial specimens, once synchronized, are found to be highly nonhomogeneous. The corresponding thermal/mechanical properties are transitory in character. Derived are nonequilibrium results that correlate well with real time data. Unless displacement and/or temperature data are quantified by the time and size over which measurements are made, they would be vulnerable to miscue.

To conclude, the editors wish to accredit the authors for their consulted efforts that made the material coherent as a volume. The knowledge gained in this exposition hopefully would simulate others to further the use of energy density in other fields of physics and engineering. Completion of the manuscripts depended much on the assistance of Mrs. Barbara DeLazaro and Constance Weaver whose efforts are acknowledged.

X anthi, Greece July 1989

G.c. SIR E.E. GDOUTOS

Volume Editors

Contributing authors

O.A. Bukharin USSR Academy of Sciences, Moscow, USSR

A. Carpinteri Politecnico di Torino, Torino, Italy

J. Denault National Research Council Canada, Boucherville, Quebec, Canada

B. Fisa National Research Council Canada, Boucherville, Quebec, Canada

K.-F. Fischer Ingenieurhochschule Zwickau, Germany

E.E. Gdoutos Democritus University of Thrace, Xanthi, Greece

M. Heller Aeronautical Research Laboratory, Melbourne, Australia

V.S. Ivanova USSR Academy of Sciences, Moscow, USSR

R. Jones Aeronautical Research Laboratory, Melbourne, Australia

M.A. Kattis Democritus University of Thrace, Xanthi, Greece

xx Contributing authors

N. Laws University of Pittsburgh, Pittsburgh, Pennsylvania

A.A. Liolios Democritus University of Thrace, Xanthi, Greece

S. Liu Institute of Metal Research Academia Sinica, Shenyang, China

J. Lovas Technical University of Budapest, Budapest, Hungary

C.W. Lung Institute of Metal Research Academia Sinica, Shenyang, China

L. Molent Aeronautical Research Laboratory, Melbourne, Australia

L.V. Nikitin USSR Academy of Sciences, Moscow, USSR

G.E. Papakaliatakis Democritus University of Thrace, Xanthi, Greece

G.c. Sih Lehigh University, Bethlehem, Pennsylvania

G.F. Smith Lehigh University, Bethlehem, Pennsylvania

M.M. Smith Lehigh University, Bethlehem, Pennsylvania

E. Tzanaki Democritus University of Thrace, Xanthi, Greece

T. Vu-Khanh National Research Council Canada, Boucherville, Quebec, Canada

L.Y. Xiong Institute of Metal Research Academia Sinica, Shenyang, China

At the banquet: speakers, organizers, publishers and guests,

Committee of the ladies,

Greetings from the Mayor of XanthL K Benis. at Town HaR

Outing to the Isiand of Thassos,

,'\t the footsteps of Xenia HoteL

ChHtting \vith the Rector E. Galousis of Democritus University Thrac{\