ABSBHACT A THEOKLTICAL STUDY OF DISLOCATIOR EEJECTS .ON Th3 STATIC AND DYQAMIC MODULI OF CRYSTALS by David H. Y. Yen The relation between the static and dynamic moduli of a perfect crystal is derived from the laws of thermody- namics. However, if a crystal contains impurities, this relation is much more complicated. In this paper, the ef- fects due to dislocations are studied. The nonlinear stress-dislocation strain law derived by Granato and Lucke to account for a strain amplitude de- pendent internal friction is used to define the change of effective static modulus. The stress-dislocation strain law depends on the distribution function of dislocation loop lengths. A different distribution function is suggest- ed and a different stress-dislocation strain law derived. Numerical results of the changes of static and dynamic mo- duli are obtained by using both stress-dislocation laws. The results are also compared and discussed. A THEORETICAL STUDY OF DISLCCATION EFFECTS ON THE STATIC AND DYNAMIC MODULI OF CRYSTALS By David H. Y. Yen A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Applied Mechanics 1961 ii ACKNOWLEDGEMENT The writer is greatly indebted to Dr. T. Triffet for his kind guidance throughout the preparation of this thesis. He also wishes to express his sincere thanks for the help given to him by the Division of Engineering Research. iii CONTENTS LIST OF FIGURES LIST OF TABLES LIST OF SYMBOLS CHAPTER I INTRODUCTION CHAPTER II THEORY CHAPTER III NUMERICAL RESULTS CHAPTER IV DISCUSSION AND CONCLUSIONS BIBLIOGRAPHY Page iv vi 23 34 38 Figure Figure Figure Figure Figure Figure Figure Figure iv LIST OF FIGURES A BOWED-OUT DI SLOCATION A QUALITATIVE SKETCH OF THE STRESS- DISLOCATION STRAIN RELATIONSHIP STRESS-DISLOCATION STRAIN CURVE, FOBNIUL-A I, r =5 STRESS-DISLOCATION STRAIN CURVE, STRESS-DISLOCATION STRAIN CURVE, FORMULA I, r=50 STRESS-DISLOCATION STRAIN CURVE, STRESS-DISLOCATION STRAIN CURVE, FORMULA II, 3/ =10 STRESS-DISLOCATION STRAIN CURVE, Page 21 24 25 26 27 28 29 Table Table ‘Table ' Table II III Iv‘ LIST OF TABLES VALUES OF ( 3n,0'.)ON THE anew, CURVES STRAIN AMPLITUDE INDEPENDENT CHANGES or STATIC MODULUS CHANGE_OF STATIC MODULUS IN THE EARLY NONLINEAR RANGE CHANGE OF DYNAMIC MODULUS IN THE EARLY NONLENEAR RANGE Page 30 31 32 33 LIST OF TABLES Page Table I VALUES OF ( &~,¢r,.)oN THE anew. CURVES 30 Table II STRAIN AMPLITUDE INDEPENDENT CHANGES OF STATIC MODULUS 31 .Table III CHANCE OF STATIC NCDULUS IN THE EARLY NONLINEAR RANGE 32 ' Table Iv' CHANGE OF DYNAMIC MCDULUS IN THE EARLY NCNLTNEAR RANGE 33 vi LIST OF SYMBOLS A: Effective mass per unit length of dislocation B: .Damping force per unit length of disleation C: ,Effective tension in dislocation G: _ True shear modulus . M.Unit matrix 13:..Network length of dislocation; average value of LN'S LC: Length of dislocation segment separated by impurities; average value of LC's N(L)dL: Distribution function of dislocation loops Q: _§3fi/TL:}Q R4C. Q': Slope of the6=tdacurve T: Temperature Z: DistanCe between an impurity and dislocation axis a : tomic spacing; Burger's vector of dislocation c-' : Elastic contant matrix c : Specific heat capacity Cottrell's force a5 L U r: Ratio of internal friction to change of dynamic modulus s;~: Elastic compliance matrix J R: wave number v: Velocity of stress wave propagation oh. ‘9 : Changes of strains 2;,2/ with temperature I ”In/Le I C: fat 3 vii LIST OF SYMBOLS CONTINUED Difference in atomic radii of solute atom and solvent atom divided by the atomic radius of the.solvent atom Elastic strain flat 3",”: Total strain :48: 8N3, 0.. 5-H u’ .JI 9%,"3 Dislocation strain Strain at which the stress-dislocation strain relation starts to be nonlinear Stress Component Stress at which the stress-dislocation strain relation starts to be nonlinear Poisson's ratio Frequency Dislocation density «‘7‘ - 40L. Characteristic length of dislocation determined by Dislocation displacement CHAPTER I INTRODUCTION In the study of theory of elasticity, the general- ized Hooke's Law is postulated after the concepts of stress and strain are introduced and formalized. It is well known that this generalized Hooke's Law, being the starting point of traditional elasticity, has the form of a 6 by 6 square matrix for a completely anisotropic medium, i.e., the con- stants CU between stress and strain components in the fol- lowing system of equations (9 6‘: 2m; e,- (;=.,2.----é) (I—l) '3! take up 36 independent valuesm However, depending upon the existence of a strain-energy function and the symmetry pro- perties of a specific material, this number of independent elastic constants is. greatly reduced. In the simplest case for a material whiCh possesses complete isotropy, there are only two independent elastic constants. Because in engineering design the values of these elastic constants are at least equally as important as the mathematical theory of the strength of materials itself, the measurement of elastic constants of materials has had a long history. Results of such measurements on various materials by previous researchers have been reviewed and summarized in )1,2 the papers by Hearmon (1946:1956 and by Huntington (1958).3 According to the ways these measurements are made, the elastic constants have the names of static and dynamic constants. In the former case the modllus is obtained by directly measuring the stresses and strains, while in the latter case measurements may be made, for example, by using resonance techniques to obtain the velocity of stress wave propagation through the material; therefrom the elastic con- stants are calculated. Static methods give the isothermal co stants, in the sense that the temperature is kept constant during the measurement; while dynamic methods give the adiabatic cons- tants, implying that during the measurement heat neither flows in nor flows out. For a perfect crystal, these two constants are related by the following equatiOA - '7 . . ' " M I -‘— [bo'.J -15....41 '2: 0‘ fi—J‘—- (1-2) j Sfafic dynamm I“ LP where (5’j] , the elastic compliances matrix is defined as the inverse of the elastic constants matrix, i.e., aim-5,, = 1 (1-5) Equation (1-3) can therefore be written as I ”I / _'_‘ C\.""1/‘ /. w —Uw -—-— ———..%—,—-—— (1-4) nyfl‘C Glann:C )‘vl'D Herecfi;aj are the changes of strainseqtu with temperature T, ‘f is the density, and czp the Specific heat capacity. It is seen from equation (1-4) that the staticaand dynamic constants are directly related by the thermodynamic proper- ties of the material. For a crystd. which is other than perfect, it is eXpected that the imperfections will affect the relationship 3 between static and dynamic constants in equation (led). While the meaning of the expression "imperfections in cry- stals" has many implications, namely the following six pri- mary types of imperfections: (a) phonons (b) electrons and holes (0) excitons (d) vacant lattice sites and interstitial atoms (e) foreign atoms in either interstitial or substitu- tional positions (f) dislocations and the following three transient imperfections (g) light quanta (h) changed radiations (i) unchanged radiations, it is only the effects of dislocations and those which inter- act with dislocations during their motions under the action of stresses that will be studied in what follows. In 1941, Read4 first suggested that dislocation mo- tion under applied stress might contribute to the observed strain and give rise to that portion of the internal friction in metals which cannot be explained by other mechanisms. There have been a great number of theoretical as tell as experimental investigations since Head's suggestion on the effects of impurities, and of cold work and annealing of the Specimens. Among them, Koehler (1952) made a theoretical study of the influence of dislocatiOns and impurities on the damping and the elastic conSEants of metal single crys- tals by using the idea that the motion of a dislocation un- der an oscillating stress can be considered analogous to the motion of a damped vibrating string. By the method of suc- cessive approximations KOehler solved the differential equa- tions of motion and was thereby able to express the internal frition loss and change of elastic modulus as a function of frequency. He also derived eXpressions for the strain- amplitude dependent internal friction loss and change of elastic modulus by using the idea that the dislocation line would break away from the impurity atoms at large strain amplitudes. Koehler's vibrating string model was further deve- lOped by Granato and Lucke (1956)6, who solved the differ- ential equations of motion in a much more general way, making it possible to consider the dependence of the internal fric- tion loss and change of BlaSLlC modulus on the loop length for all frequencies. The vibrating-string model, as develOped by Granato and Lhcke, leads to two types of loss and change of modulus. The first type is a dynamic one due to the damping of the vibrating dislocation segments, and is strain-amplitude in- dependent. The second type is due to the Tact that during the loading and unloading parts of the stress cycle, points on the stress-dislocation strain diagram do not follow the same path, thus giving rise to a hysteresis loop. For low frequencies, the kilocycle range, the stress-dislocation strain relationship is independent of frequency, and so are the internal friction loss and change of elastic modulus. As this paper is not primarily concerned with the internal friction and change of elastfl: modulus in various materials as functions of frequency, amplitude, and the de- gree of impurity, but rather with the static and dynamic elastic constants in a dislocation-containing crystal, a review of the various theories on the internal friction and change of elastic constants will not given here. However, it is interesting to note that in order to account for a strain-amplitude dependent internal friction loss and change of elastic modulus, a hysteresis loop in the stress-dislo- cation strain diagram was suggested in the Granato and Lucke deve10pment by extending the idea of breakaway first used by Koehler. It was found that both the internal friction and the change of dynamic constant are directly proportional to the area of the hysteresis loop on the stress-dislocation strain diagram, with the two proportionality constants being of the same order of magnitude. The static modulus of a crystal is obtained from the stress-strain curve under static loading. Theoretically, the load is applied to the Specimen in a time of length infinity. Under this type of loading one can not expect to have the same stress-dislocation strain relation as obtained before, as judged from a theoretical point of view, since the impu- rities will not pin the dislocations in the way suggested, but follow the applied stress in a diffusion process. However, as the diffusion process is an extremely slow one, while in actual practice the static loading and unloading is always accomplished in a finite length of time, it is reasonable to assume that diffusion of impurities will not occur in actual cases. Under this assumption, the string model and idea of breakaway can still be applied. The stress-dislocation strain loop is dependent on the magnitude of the maximum stress. The path of a point on the loop is non-linear during the increasing-stress por- tion of the cycle due to successive breakaways of the dis- location line from impurities. But the path during decreas- ing stress is linear since the entire bowed-out dislocations come back to their original position as single loops. This linear portion occurring during unloading will be used to define the effective static modulus, Which is dependent on the maximum stress as mentioned above. In the following chapter, both the effective static modulus and the dynamic modulus as a function of the maxi- mum stress, and,hence a function of the hysteresis loop, will be studied and compared numerically. Furthermore, as the hysteresis loop in the stress-dislocation strain diagram also depends on the distribution function of dislocation loops (i.e., the number of loops for a given length C as a function of () under each stress, it is expected that a different distribution of dislocation lines from the one used by Granato and Lucke will affect the hysteresis 100p and hence the relationships between tne static and dynamic modulus. In the Granato and Lucke theory, the initial dis- tribution of loop lengths Lc(the lengths determined by im- purity atoms) is an exponential function, while in the final stage the lengths have a delta function distribution with Ln, the network length, equal to a constant. There remains the contradiction that in the initial distribution of LC there are lengths greater than Ln. For this reason, an alternate assumption is made about the distribution of loop lengths after the breakaway from impurities occurs, and the consequent results are studied and compared. CHAPTER II THEORY In this chapter, a brief summary of the vibrating string model of dislocation movement under stress, as deve- lOped by Granato and Lucke, will be given in Section 1. Section 2 covers the derivations of the functional depend- encies of the distribution of dislocation 100p lengths on stress Suggested by Granato and Lficke, as well as those suggested by the author of this paper. In Section 3, the stress-dislocation strain laws as consequences of the deri- vations of Section 2will be given. The definition of sta- tic effective modulus and its relation to the dynamic modu- lus as a result of the stress-dislocation strain hysteresrs loop will be given and discussed in Section 4. Section I The Vibrating String Model It is known that a crystal contains dislocations in the form of a three-dimensional network. If the crystal contains a large enough concentration of impurity atoms, which interact with the dislocation lines through the so- called Cottrell mechanism, there are two characteristic dislocation lengths in the model. They are : The network length Ln’ and the Length LC which is caused by the pinning action of the 100ps by the impurities. In general, both Ln and LC have distribution functions which again are functions of the applied load and other work conditions. When a shearing stress d‘is applied to the crystal, two kinds of strain will occur: The elastic strain £2! , which would be the only strain we could have if the crystal were "perfect"; and a dislocation Strain Edu due to the re- coverable motion of the dislocations under the acti0n of stress 0‘ , i.e., E '3 Eel+ 8.1:. (2-1) The shearing stress er and shearing strains are related by the wave equation, a. ‘4 a f a - v“ ‘ “L— = 0 (2-2) ax‘ at‘ Where x is the direction of stress amplitude, and [p the density. We also have the Hooke's Law r __L. 5"" = ‘6;— (= ‘44 0') (2—3) where C: is the true shear modulus. The dislocation strain, £,fi, , caused by a loop of length L in a cube of edge L is given by 8d“:- if; (2_4) where f is the average diSplacement of a dislocation of length l, and a the Burger's vector. lO } is given by: .. I I z -—;/ Wyn/y (2-5) with y being the coordinate in the dislocation line as shown in Figure l. “’7 Fig. l A bowed-out dislocation Now, if .A. is the total length of moveable dis- location line in a unit cube, then 6w: = i110 = 7—) é'wdy (2-6) 0 The equation of the vibrating string model for the diSplaoement of the dislocation under stress is a i a5 a j A ‘ 5 9‘4 ‘ C y? ‘ or (2-7) tr where 3‘ 3'(x,7,t) 11 effective mass per unit length = 71/701 damping force per unit length tension in the string = a C: az/'1T(I-U) , Poisson's ratio "\JOtUtD II II II I! and the boundary conditions as shown in Figure l are ,|§(‘o0otl - O [3C)L, L, t) = o (2-8) To summarize, we have the following system of two simultaneous partial differential equations: d5..- 5’ a,” _ Afa i -d' .1» ‘3 w __ “'7‘ am , > , (2-9) \ L. A ‘1 _‘)l~ - A :15 V 555-- I :73 ._.. a , are x “A K I ) (f‘ V) (— = (a - - (2-10) and ‘3‘ _ 4;}. 4aF‘ ~'.‘ I (. (Mung t. 2.; an») —— ..... A..- ‘- } FA Z an“) A ((0:.-_,-:2)‘+;':-Jd)1,'"’z (2 11) 330 n _ ,. I 8 _". .1 /z ;_ + "’ Jig.— Where d z t-Q-"J'= (2n 1! A l K A) I é" .Qn a)"; v.1“ with Z M0 “1593': 9'36! “W ‘ I (2-12) .2 V H [ll/0.,az—u‘l/It4éwdjlj 12 44-41 _.-_-.‘_ft’:é_‘_“.°7_ 2. n [(AIf—Jj‘mwdflj viw) = v00- (2-13) if only the first term of the above series for f is used, i.e. .. -'.‘a 9 - 4““ 5' ’7 3 -— ---_.__. an -— _... "14’ I [(u),‘- v‘j‘flwd/‘J (2-10)’ in which ./ = A; A $.85i9f gil "19 a f 0 w’c’- ’~ A . Using the notations D =<#/d and fl = Q/Jb, the change of dynamic modulus is given by AC: ____ deg _ A.1l__z[ (I— L27“] (2 14) L31 Va 7’ (1,- 11))”4 2711)). 2 1 v1 as L:=.Z = - . This gives the dependence of the 9'). 2 change of dynamic modulus on frequency. Section II Distribution of Loop Length as a Function of Stress It can be shown that for frequencies in the kilo- cycle range, the expression e-—£J, ['(wf- «0‘2" + (a 00‘] "'1 is essentially ‘ ‘ = - ‘C. as u: << w, and ()0 a o , A» if therefore, 12 fl "’ V ('00) == V°["' :Ldfii __-.__ I, _ (2'13) if only the first term of the above series for f is used, i.e. é —-fi°r . F9 e‘d° 11,, 5" T [(a.»,t.e*/‘+:e}:}3] <2-10>' in which u, efé. A ”8519‘. 9-‘2 7’15: f 0 IT’C' / ’ A . Using the notations D =0 60‘] y, L . . I L A is essentially ‘ . = -- t as u) << w, and a —. a , A: if C. o therefore, 13 J .TFTA_ .'”9 573.; SM, L g 2. (2-10)" The force exerted by the dislocation line at any instant is 3‘ '2 “INT-v) (2-15) where a} 40.;L ¢ := pf“: 'LLCIC” :2 Therefore, ' 4arLL+El In... a L (T. , 4.)...“ = (2-15) We By Cottrell's theory,7 breakaway occurs when A”, = ,5 2 4—95-94 (2-17) where e’ is the difference in atomic radii divided by the atomic radius of the solvent atom, and Z the distance between the impurity atom and the dislocation line. So breakaway occurs when .; (2-18) Granato and Lfioke assumed that the initial distri- bution of 100p lengths is exponential, i.e. 1 Le. Q dL L ‘ L LL where Lc here means the average value of the Lc's. When N;LU)==- the strain becomes very large, the loop lengths were assumed to have a delta function distribution. For intermediate 14 strains,they assumedthat the distribution function, denoted by N'(()d£, consists of two parts, i.e., (1) an exponential part due to 100ps in networks which have not yet broken away; and (2) a delta function part due to the already bro- ken away part, L 1 J11 e—Ao 1: (irtL,Lfl) dl OéZ‘ i ”QM: (2-19) té(L'1~)MdL ,{e(3 ~ “ d: ‘l U" {:L(* I , q , I N(()dL - (2-22.) F , i 4 / r r, ) -10."- “’L. -L l— ([4 9 _ 5 “ [it "— -uJ 4w db ,. .1 b (“(14'4'0)’. ’0 ‘ . \Hz.l({)d( 3'- ’] offi/ (1.! f A, P 'i. v L" M .51 4 bN .. Substituting (2-22) into (2-25) and gration, one obtains ,. -‘i ‘ “i"“l‘i-"I-"’ n+9. é,d,5 "‘fJJo 3.“. o. q -3. .> (‘7‘ .r s—(I+~:)+:. ' r x v I O -:l ‘r‘f Z {-(HZ); ' ) ii 9‘ V. .t - >‘-:- + + k 3! c l- "? 4» 5’3 4.. z. j w J (2-28) 20 In the early stages 3‘37; >2 «9:9 '3 0 , so a first simplification gives 1 x ’ Q‘ ,Lds=jWAl.NLr+(fl7f 2—+ 53- ‘ (249) +3 3 + 3") , 'J' For cases where q is much larger than r, a further simplication is given by éndu == 0 Ll't -, —-n “ j r 3’9 0-3 (2-30) ’5 - F or i 5 3.,~ (Formula II) Laid/‘5 =- £21 ‘-+- "‘35“:- ‘3- J“ Some numerical results of the stress-dislocation strain laws, equations (2-27) and (2-30) will be given in the next chapter. Section IV The Dependence of Changes in the Static and Dynamic Modulus on the Stress Amplitude It is readily seen that both the stress-dislocation strain laws as given by equations (2-27) and (2-30) (refer- red to as Formula I and Formula II in the following analysis) have the following prOperties. -— ————-y— ————— —— . v i ‘, gm' 5 ‘!'-__.._ 21 (34:)? I l f ,/ ”I (65“:1 "0/ ’ I II ’ l (1 .~ I I A | LM 1 ; o" .0 5' Fig. 2 A qualitative snetch of the stress— dislocation strain relationship (1) When af is small .:.w; is linearly proportional to g” , the prOportionality constant being Q for Formula II. For Formula I, this linearity holds until q = r. (2) c4555 increases rapidly (and nonlinearly) as 6' increases. However, when. f‘beoomes sufficiently large, the curve approaches the asymptote c.ns a le+ KIEZSZir+ti~J or en” '= Q.Lrli<'respectively for the two theories. (3) When the maximum stress CT, (the point B) is rea- ched, and the quarter cycle of decreasing stress starts, the path will then be a straight line BO, determined by 6} . The slepe of this straight line will be used to define the change of the effective static modulus. (4) The area OABO has been shown by Granato and Lucke to be proportional to the amplitude - dependent internal friction loss and the change of dynamic modulus. They also showed that the ratio of the internal frictial to the change of dynamic modulus, called r by them, is a constant. 22 As the internal friction is given by the quotient of the area OABO divided by twice the area of the triangle OBB', the change of dynamic modulus can be obtained once r is XIIOWIIO - ‘ —‘.——‘rA”‘-"—“~"*v—‘ fee—.‘fil‘, I .3? "2;; 23 CHAPTER III NUMERICAL RESULTS In order to plot the stress-dislocation strain curves given by Formula (I) and Formula (II), the following numeri- I cal'values were chosen for the parameters appeared in, the equations. They apply to a 99.999% pure copper crystal, 5 and are essentially the same as those used by Koehler in his investigation. = 2.55 A = 2.55' 10"8 cm a. A = 1.4 . 1016a .-.-. 3.5x 105.3 cal/cm} LC: “LI—2" 1030 = 2.12:: 10"5 am f.= 8.93 gram/cm3 V = 3.49 G = 4.53:21011 dynes/cm2 A = 2.5111'10'14 gram/cm c = 3.90 x 10"4 gram cm/se02 Q = -éi‘f}é:__3i = L; “0‘” "“z/dyne 7? = 4 «10"6 dynes f" = 4": 1,: = 1.8 X lO7 dynes/cm’2 Three different values of 3’, i.e., 2; = 5, 3’ = 10 and 5’ r. 50 are used; and six curves are plotted up to the range of strains which is several times 2,, a, being defined as the point where the stress-dislocation strain curves start to be nonlinear. On curves where no abrupt changes from "linear" to "nonlinear" can be found, the point (6., t...) {i}! . Cm . , . _. . . 66/1]? in) 5‘fr’655 * DIS/OCCvf/On 5750/0 Curt/e «Carma 1’0 (1) 3’: 5’ —6 éxro .— —6 5W0 *— '6 Wm r. —-6 5X/0 — 2X/56_ -6 ~70 — /, l L J l l 0 xiv/06 arm’s 3x/06 42/06 5x/c 25 6dr: Cry 57‘r'853— D/‘s/oco7/fon jfi-oin Curve I cm FarmU/a (I) b’:/0 -6 r ’- . 7— air/0 1‘2‘.‘4m~ m 1" 7‘ I I -6 ZA’IO _ /‘51/3 ’— Mm H -6 $3 01 x 0 .6 6 f I I 0 oJx/g ””06 gyro 21/0 dyan/c a 6 d3: 0% .4 31/0 r ngo -6 ZA’ lo /'§K/3 /X/o {y‘SIIO 25 m r. (,5 0 QJKQ /X/o6 L/y. FOI’MU /0 I1=IO afiuwo (I) 57""853 " Dis/06077010 jfi'afn Curve EXIoé dwwgé i m .‘ 'f?’_— ——-‘— v 44* ‘. "H J. 26 \S'frE‘SS'D/‘S/Ow‘flbn .57’r0/h CUM/E? AEOqu/Q (I) D’= 5o .. I '7] I. 2100 - ? k 3 -@ [pr/o ” _/ Mm" L— *6 0.9/0 t // _'__.-—"’/’ “-1...." “.I P J 1 l 1 : L) 3 5,?106 /\’/¢‘6 /.5')4/(/6 dg/flgf/g " Cir. 27 E30735 0%,” 574*855 -— D/S/OCof/on JfI’O/n Carve FOrmu/O (.27) {=5 -7 5100 r- - -7 (KW-0? gx/o __ -7 Ljno #- —7 /x/0 - 051/0 O ,x/Of 21/05- 34735 41/05 5X/03" 5/.4_ é 28 6005 Cmcm 6%I'955'D/IS/OCO‘flb/7 54/”01'n Curt/E acormu/o ('1) f: /o -7 [37/0 .- 1 z I -7 /X/0 - --7 @970 — / 4' 1 J 1 l 5 J : 4 r 5 O afAr/of /,Y/05 45’005 4I/0 (Eur/0 dynir/Irz #1 C) . 7 50w“ 5%,» 579955 - 0 [5/0 coffon 5770 In, Cum/6 fOrm u/a {177 : O 45 r 5 1,5X/0 - .8 IX/O — / f — 8 QSX/O .— //'A // _/ / r " I I i l l L Jl r, 0. 57/0 ‘1' 7/ ,y m 4’ /. {Jr/c 4’ 2.470 4’ 2. [Avg “Ll 0/7”?" 2 cm 30 is defined as the point at which the nonlinear effect con- tributes 1 % to the total strain. Values of’(£~,cv)are given in Table I. Table I Values of(£..a',)on the 2d,, - a’ Curves Formula 3 K I r” dynfjg’,’ u, 92;“ I 5 ' , 3.6 . 106 8.6 x 10‘7 I I 10 1.8 .106 2.5 e 10‘7 I , 50 . 1.1 x 106 1.5 x 10’7 II E 5 2.6 . 105 5.4 . 10'Q II J" 10 1.5 . 105 f 1.8 . 10"8 II I 50 1.9 x 104 i 2.5 x 10"9 It is seen from the above table that the ('3N‘,r;) points fall in ranges of different order. ' For cases in which 5:3. is smaller than r; , there is no amplitude-dependent change of dynamic modulus because the hysteresis loop has zero area. However, there is a change of static modulus determined by the s10pe of the straight portion and the true elastic modulus. This change is amplitude independent and has a constant value until";m is equal to f} .' Table II gives this amplitude independ- ent changes ofstatic modulus. 31 Table II Strain Amplitude Independent Changes of Static Modulus Formula I K' I ( A G/G)static I 5 ‘y - 0.098 'I ‘ 10 I. - 0.060 I - 50 ‘ - 03056 H I 5 - 0.056 II E 10 ; - 0.056 I Results in the above Table Were obtained as follows: For a certain stress, the total strain is given by E+o+ao = as] + 84:; =- («é—+6.“)? where Q' is the slope of the straight portion of the hystere- sis 100p I . 0' (.1 (J = a + A = ‘ — I --._I , q tfofal (+016? (-91) _ -_E:'.T.‘i’__ ._ .. u a’ (3 1) ‘ usfoh‘c UT [4’ ‘3 Q, . ' (.5 1 Taking or = 4.53 x In" “7" an} I "J 9M”, and u? = ‘3 = "J "0 /""7.""‘ one obtains (———““' — - or Jfaffc — 0‘ o 53 as given in rows 3 to 6. Results in the first two rows were obtained in a similar way. The change of static modulus for strains greater than 4%” ,‘but still in the early nonlinear range, is given 32 in Table III. Points chosen are those with strains equal to 22..., $¢~ , and 4.... , the calculations technique is si- milar to that described above. 2 Table III Change of Static Modulus in the Early Nonlinear Range ( A G/G)static at 2.4. = Ebrmula; 3’ ON ' .22” I )9” l 4.9” a” I § 5 ' - 0.098 - 0.182: - 0.226i - 0.262 8.6»10'7 I i 10 - 0.060 - 0.112? - 0.146: - 0.177- 2.5. 10‘1 1.. . 50 ‘ - 0.056 - 0.115§ - 0.155) - 0.188 1.51 10" II 5 - 0.056 - 0.169: - 0.213; - 0.244‘ 3.41-10”3 II . 10 i - 0.0565 - 0.197g - 0.246i - 0.290; 1.8-rlO-8 11 50 g - 0.056 - 0.194% - 0.251 - 0.300; 2,5. lo'q The change of dynamic modulus,as eXplained in Chapter II, is given in Table IV at ;,Ma, =.;.~, 5.e~ and‘4ey The results were obtained by dividing the area of the hys- teresis 100p by ten times a.m%,.'ffia, . In this way, a factor of re = 5 ( r being the ratio 0f the internal fric- tion to the change of dynamic modulus) has been used. 33 Table IV Change of Dynamic Modulus in the Early .' Nonlinear Range ' ( 4 G/G)dynamic at a...=~ Formula; 3’ ‘gl ‘45“ I 5a,. 1 4,“ c». I § 5 E 0 f - 0.029: - 0.057 ; 0.041? 8.6 10 _ I i 10 g 0 - 0.051i - 0.043l - 0.050‘ 2.5 10 I 50 E 0 g - 0.028! - 0.045% - 0.055: 1.5 10 II .5 0 E - 0.019 - 0.056; - 0.040 3.4 10 III “10 o z - 0.029 - 0.039! - 0.045 1.8 10 II _ 50 0 - 0.029, - 0.044 - 0.053 2.5 10 Results given in Table I through IV will be discu- ssed in the next chapter. 34 CHAPTER IV DISCUSJION AND COLCLUSIORS Results obtained in Chapter III will be discussed first. It is seen from Table I that the two theories pre- dict different orders of magnitude of stress. Whereas the Granato and Lucke theory predicts that the stress-disloca- tion strain relation starts to be nonlinear at stresses of n? é: d9 jémz, the theory suggested is this paper the order 10 gives results of the order [0 5 ‘J""e‘/,,,...tl for low values (low concentration of impurities) and of even lower order for high 6’ values. Also, results obtained by the present theory are seen to be more sensitive to the value assigned to r than those obtained by the Granato and Lucke' theory. It must be left to experimenms to determine which theory gives the best results. The numerical results given in .Table II and III are left uncompared, since no exPerimental results are available. However, as one sees that some of the parameters involved in the equations are not well established, the values assigned to them are also questionable. In fact, the stress-dislocation strain law depends on two parameters, namely Q and {7 . ‘Q in turn depends on J , Lc , and f" on f. and Lw_ . Therefore, a series of static measurements, with some of these parameters properly controlled, might yield information about their exact values. For instance, one might use neutron irradiation to produce interstitial 35 atoms and lattice vacancies, which would pin the dislocations and thus reduce the average value of L2; . As an example of the dependence of the results on the parameters, one can see that if a value for Q one tenth that previously assigned is used, results in Tables II and III will also be reduced by approximately a factor of 10. Results given in Table IV are too large as compared with experimental results? since the latter are in the order of lO'J to 10-5 . The results given in Table IV are inde- pendent of Q when only the relation between change of dyna- mic modulus and stress amplitude is considered. However, these results depend on r , the ratio of internal friction and change of dynamic modulus, and the choice of r" = 5 in the present case is rather arbitrary. Granato and Lucke concluded that r is of the order of unity in a detailed analysis, but experimental results indicate that r ranges from the order of unity to the order of ten.5 The purer a specimen is, the greater the value r takes. The significance of the suggested theory of the dis- tribution of dislocation lengths will now be discussed. As indicated in Chapter II, the broken-away portion of the 100p lengths has an exponential distribution 64/“ for I 51‘ °° This means that for network lengths not less than ii, the probability of finding two adjoining 100ps, separated by an impurity on a particular network 100p, with a sum equal to or greater than J: , is one. Obviously this is a very good approximation for low 3’ values. This is because 36 r' is equal to the ratio of the average value of L” to that of L, . On the other hand, the Granato and Lucke theory will be a good approximation for high 3’ values,since then LN is much larger than L,,and in the initial distribution lengths greater than L” can be neglected. In the present investi- gation, numerical results are compared for the cases I = 5, 3' = 10, and 5': 50; but just how good an approximation each theory gives is still in question. Several other points should at least be mentioned. Throughout this paper, the terms "elaStic constant" and "elastic modulus" have been used interchangeably; because in the present case only one shearing stress and shearing strain are used, the latter means the elastic shear modulus G and the former means c44, with G = c44. However, in general, though they are directly related, these two do not equal to each other. If normal stress is used instead of shearing stress, i.e., the longitudinal wave instead of the transverse wave, an orientation factor taking account of the orientation re- lations between the direction of propagation of the longi- tudinal wave and the slip plane and slip direction has to be introduced. The effect of this orientation factor will not be discussed here, but can be found in Reference 6 in the Bibliography. The use of the vibrating string model to account for the internal friction and change of elastic modulus in crystals due to dislocations is but one of several existing 37 theories in the study of dislocation damping in crystals. However, as indicated by Niblett and Wilks8 recently (1960), the vibrating string model generally gives a fairly satis- factory account of both the frequency dependent and ampli- tude dependent internal friction loss and change of dynamic modulus. It was also indicated that this theory enables one to take account of the effects on internal friction and change of dynamic modulus due to temperature, cold work or annealing, and impurities. The strain amplitude dependent part of the theory consists in the use of a nonlinear stress- -dislocation strain law. This nonlinear relationship can be used to define the change of static modulus, which can be measured eXperimentally. Therefore, a combination of static and dynamic measurements furnishes a method of check- ing the current theories. Furthermore, the stress-disloca— tion strain law depends on several parameters such as 11., the dislocation density, and.£c, the average length of dislocations between impurities, in different ways. By pr0perly changing and controlling such parameters, and making both static and dynamic measurements, more informa- tion abdut the internal structure of a crystal can be ob- tained. 38 BIBLIOGRAPHY l. Hearmon, R. F. S. "Elastic Constants of Anisotropic Materials", Reviews of Modern Physics, Vol. 18, 1946, p. 409 2. Hearmon, R. F. S. : "Elastic Constants of AnisotrOpic haterials--II", AdVances in Physics, Vol. 5, 1956, p.323 3. Huntington, H. B. ' "The Elastic Constants of Crystals", Solid State Physics, Vol. 7, Academic Press Inc., New York, 1958, p. 215 4. Read, T. A. "Internal Friction of Single Crystals of COpper and Zinc", Transactions of the American Institute of Mining and Metallurgical Engineering, Vol. 143, 1941, p. 30 5. Koehler, J. S. "The Influence of Dislocations and Impurities on the Damping and the Elastic Constants of Metal Single Crystals”, Imperfections in Nearly Perfect Crystals, John Wiley and Sons, Inc., New York, 1952, p. 197 6. Granato, A. and Lucke, K. "Theory of Mechanical Damping due to Dislocations" Journal of Applied Physics, Vol. 27, 1956.11.553: 39 also, "Application of Dislocation Theory to Internal Friction phenomena at High Frequencies” Journal of Applied Physics, Vol. 27, 1956, p.789. These two papers are based on a Ph.D. thesis by A. Granato. This thesis is available from Univer- sity Microfilms, Ann Arbor, Michigan, Publication No. 131712. Cottrell, A. H. "Effect of Solute Atoms on the Behavior of Dis- locations", Report-of a Conference on the Strength of Solids, University of Bristol, England, Physi— cal Society, London, 1948, p.30 Niblett, D. H. and Wilks, J. "Dislocation Damping in Metals", Advances in Physics, Vol. 9, 1960. p.1 440.13”. .. 1 -. 14. .- «411‘ BBBBB W”. 1711 MW yygflylflfiytl at 8