RUPTURE PHENOMENA IN VISCOELASTIC MATERIALS Thesis by Wolfgang Gustav Knauss In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1963 To my patient wife ACKNOWLEDGEMENTS It is a pleasure to express my appreciation to Professor M. L. Williams for suggesting this interesting problem area and for his valuable comments, especially during the latter stages of this work. I wish to thank Dr. R. F. Landel of the California Institute of Technology Jet Propulsion Laboratory for making his laboratory facilities so readily available for the determination of the material properties. The experimental material was supplied by Drs. R. B. Kruse and T. A. Neely of the Thiokol Chemical Corpora- tion, Huntsville , Alabama. My thanks are also due to Mrs, R. Lees and Mr. K. Hebert of the California Institute of Technology Computing Center for their continuing advice and assistance during the programming of the numerical calculations for an IBM 7090 computer. I am indebted to Mrs. Elizabeth Fox and Miss Helen Burrus for their excellent work in typing this manuscript and to Mrs. Betty Wood for her assistance in lettering the figures. Two National Aeronautics and Space Administration Fellowships during the course of this research are also grate- fully acknowledged. Part of this research was supported by the National Aeronautics and Space Administration Research Grant NsG-172-60. ABSTRACT A failure theory for high polymers is developed from the hypothesis that weak regions exist in the material. Defects nucle- ate in these regions through bond rupture until the defects reach a size which is critical for the applied boundary loading. This critical condition is based on energy balance considerations, By considering the relaxation of the polymer chain in terms of the phenomenological stress-strain behavior and the rupture of chemical bonds in terms of an Arrhenius type rate law, the theory is able to accommodate an arbitrary stress or strain history, and shows reasonably good agreement with experiments which cover a large range of conditions. In addition the stress analysis of a special crack geometry is presented. The geometry consists of a thin infinite strip containing a semi-infinite crack, For a uniform separation of the infinite boundaries an infinitesimal elasticity solution is obtained with the help of the Fourier transform and Wiener-Hopf techniques The effect of large strains on the stresses near the crack tip is studied experimentally and a surprising correlation with the infinitesimal elasticity solution is found, TABLE OF CONTENTS PAR T TITLE PAGE I THE RUPTURE OF POLYMERIC MATERIALS 1 A. ME CHANICAL BEHAVIOR OF VISCOELASTIC 8 MATERIALS 1.1 Representation of Stress -Strain Behavior 1.2 Time - Tempe rature Equivalence 1.3 Effect of Finite Strain 1.4 Isothermal vs. Adiabatic Rupture Properties B. THE PHENOMENOLOGICAL DESCRIPTION OF THE RUPTURE PROCESS 1.5 The Molecular Structure of Polymers a. The polymer chain b. Effect of chain slipping c. The chemical bond d. Atmospheric effects 1.6 Microscopic Aspects a. The nature of polymer defects b. The effect of volume size upon rupture properties 1.7 Macroscopic Considerations a. Fracture markings b. Effect of the multiaxial stress state c. Energetics of crack growth C. THE RUPTURE MODEL 1.8 Defect Growth 1.9 A Condition of Critical Loading TABLE OF CONTENTS (Cont'd) PART TITLE PAGE D, APPLICATION TO RUPTURE PREDICTION 1.10 The Constant Strain Rate Test 1. 11 Rupture Initiation Time 1,12 Crack Propagation at Constant Velocity a. Analytical considerations b. Experimental analysis 1.13 Suggested Applications of the Failure Theory a. Crack acceleration b. Cumulative damage in the constant strain rate test c. The dual strain rate test d. Stress cycling 1. 14 Filled Polymer Systems E. EXAMINATION OF THE FRACTURE MODEL F. CONCLUSION I1 STRESS ANALYSIS OF A SPECIAL CRACK GEOMETRY A. CLASSICAL ELASTICITY SOLUTION 2. 1 Statement of the Mathematical Problem 2.2 Solution 2.3 Evaluation of Stress and Displacement Integrals 2.4 Numerical Evaluation B. EXPERIMENTAL ANALYSIS OF THE STRESSES IN A SLIT RUBBER SHEET 2.5 Large Deformation Stress Analysis TABLE OF CONTENTS (Cont'd) PAR T TITLE PAGE 2.6 Experimental Procedure 114 2.7 Numerical Evaluation 115 C, CONCLUSION 120 REFERENCES 122 FIGURES 129 APPENDIX A 157 APPENDIX B 161 - 1 - INTRODUCTION In the past, investigators have tended to explain material failure by establishing the criticality of various stress or strain combinations, or by the energy instability concept pioneered by Griffith. Since in their usual application these criteria are rate and thus time independent they have been of restricted usefulness for materials possessing strongly time dependent mechanical properties. Excepting the analysis of creep deformations, this deficiency has been of little practical concern because the usual engineering metals possess only limited rate dependence. But with the increasing incorporation of viscoelastic materials such as the organic glasses and rubbers in structural design, as well as the use of metals at higher and higher temperatures, an ex- tension of the presently existing knowledge to rate dependent failure processes is believed timely . Depending upon the temperature and the rate of load appli- cation, viscoelastic materials can exhibit such different behavior as brittle rupture response or viscous flow and non-recoverable permanent set. In this regard they constitute a class of materials that form a transition between the essentially rigid engineering metals and the highly deformable liquids. One would therefore expect their failure characteristics to reflect certain features of one or the other limit states. In the following development we shall use the conceptual similarity to metallic behavior in explaining certain aspects of failure. As an important corollary, and because such a similarity -2- should not be one-sided, one would also expect that the study of rupture phenomena in viscoelastic materials might also aid in elucidating some, possibly yet undetermined, phenomena in metal fracture. Although a metallurgist and a physical chemist may be sceptical of an association of dislocations and chemo -molecular bonds it is, nevertheless, an accepted fact that the high mobility both of dislocations in crystals and of molecular configurations in amorphous polymers are at the origin of fracture behavior in the respective materials. It appears therefore that the study of rupture phenomena in viscoelastic materials may not only consist of an important extension of existing fracture knowledge -per -se, but also provide an important source of new ideas for understand- ing rupture in the engineering metals. Since the analysis of viscoelastic rupture includes the analysis of stresses and strains in the fracturing solid, a part of this dissertation is devoted to the stress analysis of a test geornetry employed here in crack propagation studies. Inasmuch as this stress analysis can apply also to materials which are not viscoelastic it has been separated from the rupture analysis in a second part of this dissertation, Part I is thus concerned with the rupture analysis of viscoelastic materials. Following the presentation of back- ground information, a rupture model is deduced which is subse- quently used to predict rupture in some selected test situations. Part I1 contains the stress analysis for the stretching of a thin sheet containing a semi-infinite crack, Although the *dxlacrroa8 acrr-es aq30 laays xaqqnx -e uo dpnqs I-eluarxx? -xadxa UE UJ pauru-exa uaq s? XOpWyaq ur-exqs - ssaxls x-eaug-uou pu-e suo?7emxojap a8z-e~30 a~uanp?ay;L od$;zqsqa TEF?~TSSE-~30 dxoaq ayl uo pas-eq s? srsd.~-eu-e~~3qaxoay~ aqs73a333 uoyernxojap a8x-e~30 uo;~-exap~suo~ayl saxpbax d~-[-enw.eI ?x-ed uy payeax3 uoqenlIs -[-e3rsdyd -£- -4- PART I THE RUPTURE OF POLYMERIC MATERIALS Initially the exploration of rupture in polymeric materials was related to the strength characteristics of natural and synthetic textile fibers; but recently this problem has obtained new impetus from developments in solid propellant rocket technology where the fracture threshold and its control is of major production and design importance. Solid propellant materials consist of a physical mixture of oxidizer granules and a rubber matrix filling the interstitial spaces between the oxidizer particles. Their characteristic behavior arises from both the viscoelastic nature of the rubber and the relaxation effects associated with the restricted motion of the oxidizer particles. More specifically, the rupture behavior is complicated by the irregular separation of the rubber from the hard filler grains prior to gross fracture. In order to avoid such effects of the interaction between filler and rubber matrix, it seems prudent for an initial investigation of viscoelastic rupture to consider only a one phase system such as a pure rubber. Building upon this foundation one then hopes to extend the re- sults to a two phase system, and eventually to solid propellant fuels, by judicious extrapolation and continuing research. This dissertation will therefore be concerned with the time dependent failure of viscoelastic materials which can be considered as macroscopically amorphous and homogeneous. -5 - The most extensive study of rate dependent rupture behavior has been made by Rivlin, Thomas, and Greensmith (1- 8). These authors have investigated the tear behavior of rubbers by extending the Griffith concept so as to include a rate dependent