1.8 MICROCOPY RESOLUTION TEST CHART. N1 N-1 Pl[PFAVl. 'AW~RQ A

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1 "AD-6AiS 778 RESISTANCE CURVES FOR CRACK GROWTH UNDER LANE-STRESS i/i CONDITIONS IN AN EL..(U) NORTHWESTERN UNIV EVANSTON IL STRUCTURAL MECHANICS LAB J D ACHENBACH ET AL. JAN 85 UNCLASSIFIED NU-SML-TR-85-1,4-76-C-0063 N. F/6 20/11 NL Iso llioll

2 * LA 1.8 MICROCOY RESOLUTION TEST CHART N1 N-1 l[favl. 'AW~RQ A

3 RESISTANCE CURVES FOR CRACK GROWTH UNDER LANE-STRESS CONDITIONS IN AN ELASTIC ERFECTLY-LASTIC MATERIAL Lf J.D. Achenbach and Z.L. Li Department of Avil Engineering Northwestern University Evanston, IL Office of Naval Research N C-0063 L4i January NU-SML-TR-No Approved for public release; distribution unlimited,0o "-" % t ' m. "% -. ", "-. "-"- ", "*,,, ',. ' -%. -" -".. -", './ -. '".." -... o " %

4 // / 7 Abstract )A recently developed solution for the plastic strain, (x,t), on y the crack line is used in conjunction with a critical strain criterion tconstruct curves for K(a) versus a, where a is the increase in crack length. Resistance curves have been computed for various values of the critical plastic strain. They show a monotonic increase of K(a) with increase in crack length, to a constant steady-state value. / D TIC Co Y INSECTED

5 1_-VI 1. Introduction For small-scale yielding, the condition for continued crack advance may be written as K = KR(a), (1.1) where K on the left-hand side is regarded as the "applied" K, and a in K(a) is the increase in crack length. The curve of KR(a) versus a is called the resistance curve or R-curve. Under plane stress conditions, KR(a) tends to be a monotonically increasing function of a, at least for many metals. Following the notation of Ref.[l,p.20], we will use KC to denote the value of K for initiation of crack propagation, i.e., C = K R(O). In this paper we consider a general geometry such as an edge crack in a thin sheet, and we construct resistance curves. These curves are based on a critical strain criterion for crack propagation, which stipulates that crack growth will proceed when the plastic strain on the crack line maintains f a critical strain level e at a distance xf ahead of the crack tip. The y critical strain criterion was proposed by McClintock and Irwin [2] and it has been used frequently for the Mode-Ill case. Rice [3,p.281] presented an expression for the plastic shear strain on the crack line for Mode-Ill crack propagation in an elastic perfectly-plastic material. It was also shown in Ref.[3] that the fracture criterion of critical plastic shear strain leads to an integral equation for the plastic zone size required for quasi-static crack extension. 4,

6 U 2 The construction of curves for KR(a)/KIC presented in this paper for plane stress, follows in general outline the method of Ref.[3]. The construction is based on an expression for the crack-line strain which was recently presented by Achenbach and Dunayevsky [4] and Achenbach and Li [5]. As shown in some detail in Ref.[5], in the plastic loading zone the stresses and strains can be expanded in powers of the distance, y, to the crack line. Substitution of the expansions into the equilibrium equations, the yield condition and the constitutive equations, yields a system of simple ordinary differential equations for the coefficients of the expansions. This system is solvable if it is assumed that the stress a is uniform on the crack line. y By matching the relevant stress components and particle velocities to the dominant terms of appropriate elastic fields at the elastic-plastic boundary, a complete solution was obtained for the plastic strain, E, in the plane of y the crack. The solution depends on position on the crack-line and time, and applies from the propagating crack tip up to the moving elastic-plastic boundary. It is shown in this paper that the critical strain criterion yields an integral equation for x (a), where x (a) is the extent of the plastic p p zone on the crack line as a function of the increase in crack length. The integral equation has been solved numerically for various values of Ef /(y)b, where (cy)b is the crack-line strain at the elastic-plastic boundary. A relation between K and x (a) subsequently p

7 S p 3 yields resistance curves KR(a). The geometry is shown in Fig. 1. The x 3 -axis of a stationary coordinate system is parallel to the crack front, and x 1 points in the direction of crack growth. The position of the crack tip is defined by x I = a(t). A moving coordinate system, x,y,z, is centered at the crack tip, with its axes parallel to the xlx 2 and x 3 axes. 2-.

8 4 2. Crack-Line Strain For monotonic loading, expressions for the strain rate y(xlt), on the crack line ahead of a propagating crack tip, are given in Refs. [ 4] and [5] as E. 2&(t) x (t) B A(t)+B 2 p (t) C 1 (t)+c 2 kp (t) y (la(t 9n I[x 1 -a(t)] 2, (2.1) ka1 x-a (t) x (t) p where x 1 defines the point of observation in the stationary coordinate system, while a(t) defines the position of the crack tip and x (t) is the size of the plas- tic zone along the crack line. Also, k is the yield stress in shear, E is Young's modulus, and the constants B 1, B C and C have been derived in Ref.[ 4] as B I - 8(l+)[(K + 5) + 2(K + 1)V-2] -4 (2.2) B2.i(l+V)[(K + 5) + (K + )/22.3) C I - l(l+v)[-(k + 5) + (K + 1)V2] + 2 (2.4) C2.l(l+)[-(K + 5) + 2(K + i)v2],(2.5) where v is oisson's ratio and K is defined as (3-v)/(l+v). In the stationary coordinate system, (2.1) can be integrated to yield cy(xlt) - (Ey) + E (xlt), (2.6) yly B where (ey)b is the elastic strain at the elastic - plastic boundary: k (e k(2-v) ' (2.7) yb E 2.7)

9 5 e(xlt) is the plastic strain t Ey(Xt) = f [y(xl,s)ds (2.8) y t p The lower limit t, which is the time that the elastic-plastic boundary reaches position x I and the plastic strain starts to accumulate, follows from a(t p) + x p(tp) = x (2.9) By integration by parts of the terms multiplying B 2 and C 2, E(xlt) can be separated into two components: where t E(xl 1 t) = se[ x l t)] +, (xls)ds, (2.10) t E s 1 C d- -1) (2.11) k y [EYx 1 st)] = B 2 ( - I ) 2 2 g(xlt) = x (t)/[xl-a(t)], (2.12) and E= ((s) f[x-a(s),x (s) ] (2.13) kyl 1 p 1 C 1 f[xl-a(s),x p(s)] = x 1 -a(s) {2ing(xl's) + BI-B 2 E(xls) + 1(XlS)], 2 (2.14)

10 6 Here we have used (2.9) at the lower limits of integration of Eq.(2.8). The lower limit t in (2.10) is defined as t = tf for tf > t p, where tf is the time that crack propagation starts. When tp > tf, the lower limit is defined as t = t p We will now assume that crack propagation is governed by the criterion of a critical plastic strain at a fixed micro-structural distance xf ahead of the crack tip, i.e., at x = a(t) + xf* The condition for stable crack propagation then is f where E y t E = E [xp(t)/xf] + f [a(t) + xf~si ds, (2.15) t is the critical value of the plastic strain, and the right-hand side follows from (2.10). Equation (2.15) is an equation for x p(t). For the case tf < t, the integral in (2.15) vanishes when the upper limit is taken as t = tf, and the following equation for xp(tf) is obtained B 2 1 E f] (t ) 1 = 0 (2.16) xf p (tf) [ C 2 - B 2 - E I p f 2 2 xf A convenient form of (2.15) can be obtained by replacing the independent variable t by crack length a. Thus we consider the strain as well as the position of the elastic-plastic boundary as functions of a. After introducing the normalizations a a/xf, xp(a) = Xp [t(a)]/xf, (2.17a,b) we can write....

11 7 a f s [x(a)] + f f [a + l-n,x p(n)]dn (2.18) a where f[a n,x (n)] a [2 n + B - B 2 +C_ (2.19) p a---i1 2 3 C2 = Xp (n)/(a r) (2.20) The lower limit a in Eq.(2.18) is defined by and a =0, when a + 1 < x (0) (2.21) -p a + x (a) = a + 1, when x (0) < a + 1, (2.22) where x (0) = x p(t f)xf Equation (2.21) corresponds to tf > t, while (2.22) corresponds to tf < t p Equation (2.18) is a Volterra integral equation for x (a), which can be solved by a step-by-step procedure. If x (a), is known for 0 < a < a I < xp(0) - 1, then for a = a 1 + A < x (0) - 1, Eq.(2.18) yields [(a 1+A) - f f[a 1 +A+l-nx (n)]dn (2.23) 0 where A is small. The integral in (2.21) can be approximated by a+a a 1 f al ++l-rxp(n)]dn f f[a 1 +A+l-n,x r(n)]dn + f(a 1 +A+l-n,x p(a)]a (2.24) 0 0 The integral on the right-hand-side of (2.24) is known. Substitution of (2.24) in (2.23) yields a cubic equation for x (a 1 +A), which can be solved. This value for xp(al+a) is used as the starting point for an iteration procedure whereby subsequent values of x (a 1 +A) are substituted in the integral in '....

12 Eq.(2.21) to yield improved values of x (a 1 -+A), until a desired accuracy has been achieved. For the first step in this procedure we use x (A) x (0) + x'(o)a, where x (0) follows from (2.16), and p p p p x'(0) = dx (a)/da at a = 0, can be obtained from (2.18). p p For the case when x (0) - 1 < a, defined by (2.22), the lower limit of p the integral is a function of the upper limit, a, and of the unknown function x p(a). Again, if x p(a) is known for a < a then for a = a 1 + A, Eq. (2.22) * - * -- * yields a + x (a) = a + A + 1, where x (a) is known since a < a Hence we can solve a = a (ala). Using an analogous method as before, Eq.(2.18) can subsequently be solved for x (a +A). p In the actual computations the following values of the relevant parameters have been considered: f _pf S= /(cy ) = 2, 6,and 10 (2.25) y y y B The oisson's ratio was taken as v = 0.3. For a = 0, the value of x (0) follows directly by computation of the real root of Eq.(2.16), since x (0) = x (tf)/xf. The numerical results show a subsequent increase of x (a) with a, where x (a) approaches an asymptotic value for large a. p 'p For quasi-static steady-state crack extension the following relation was derived by Achenbach and Li [5]: k I 1 3 e (x) Zn B 1 Zn () - S [(-) - 1i]} (2.26) p p p The critical strain criterion then yields the relation - [n [O./ -1] = 0 (2.27) KYp 1 np 3 p

13 9 Equation (2.27) can be solved for x to yield a result which is independent of crack-tip speed and loading state. This result is just the asymptotic value of x (a) as a increases. p A convenient form of the resistance curve shows the ratio K IK versus a dimensionless crack length. Here K I is the present stress intensity factor, and KC is the value of the stress intensity factor which is required to satisfy the fracture criterion for a stationary crack. In Ref.[5, Eq.57] an expression was presented which relates x (t) to KI/k. In the present notation this expression states 2/ x (a) - p 9 r x f [K (a)/k] 2 (2.28) It follows that K(a)/KIc = C(a)/Kic = [x p(a)/x p(0)] (2.29) The quantity xf which enters in a, can be eliminated by equating the result for x (0) obtained from (2.16) to x (0) as obtained from (2.28). By using the resulting expression for xf in the definition of a, as given by (2.17a), we find - a 9ra x p(0) a = - = - - (2.30) xf 2/2 [K i/k] f For the three values of c given by Eq.(2.25), curves for K(a)/KIc y versus a are shown in Fig. 2. It is noted that the resistance curves show a monotonic increase with increase in crack length, to a stable phase of crack propagation where KR(a)/KIC assumes the steady-state value. -- ",. J-

14 10 Acknowledgment This paper was written in the ccurseof research sponsored by the Office of Naval Research (Contract No. N C References [1] J. W. Hutchinson, Nonlinear Fracture Mechanics, Department of Solid Mechanics, The Technical University of Denmark, [2] F. A. McClintock and G. R. Irwin, lasticity aspects of fracture mechanics. In Fracture Toughness Testing and its Applications, ASTM ST 381, American Society of Testing Materials, hiladelphia (1965). [3] J. R. Rice, Mathematical analysis in the mechanics of fracture. In Fracture: An Advanced Treatise (Edited by H. Liebowitz) Vol. II, pp , Academic ress, New York (1968). [4] J. D. Achenbach and V. Dunayevsky, Crack growth under plane stress conditions in an elastic perfectly-plastic material. J. Mech. hys. Solids 32, (1984). [5] J. D. Achenbach and Z. L. Li, lane stress crack-line fields for crack growth in an elastic perfectly-plastic material. J. Engineering Fracture Mech., in press.

15 1 X 2 yj X 1 *. _ X a t) Xp(t) Fig. 1. Geometry of propagating crack; a(t) is increase in crack length; x = x (t) defines the elastic-plastic p boundary. - 9 U' O...

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17 *: Unclassified SECURITY CLASSIFICATION OF THIS AGE (*%en Data.Entered)_ 13 BEFORE COMLETING FORM READ [NSTRUCTIONS REORT DOCUMENTA:TION AGE BFR OEiGFR r I. REORT NUMBER 2. GOVT ACCESSION NO. 3. RECIIENT'S CATALOG NUMBER L" ". NU-SML-85-1 " 4. TITLE (amd Subtitle) S. TYE OF REORT & ERIOD COVERED Resistance Curves for Crack Growth under Interim lane-stress Conditions in an Elastic erfectly-lastic Material 6. ERFORMING 01G. REORT NUMBER 7. AUTHOR(s) S. CONTRACT OR GRANT NUMBER(s) J. D. Achenbach and Z. L. Li NOOO C ERFORMING ORGANIZATION NAME AND ADDRESS 10. ROGRAM ELEMENT. ROJECT. TASK AREA & WORK UNIT NUMBERS Northwestern University, Evanston, IL CONTROLLING OFFICE NAME AND ADDRESS 12. REORT DATE Office of Naval Research January 1985 Structural Mechanics rogram 13. NUMBER OF AGES Department of the Navy, Arlington, VA I4. MONITORING AGENCY NAME & ADDRESS(I1 dlfferent from Controlling Office) IS. SECURITY CLASS. (of thli report) Unclassified ISa. DECL ASSI FI CATION/ DOWN GRADI NG SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, It different from Repot) IS. SULEMENTARY NOTES to be published in Engineering Fracture Mechanics 19. KEY WORDS (Continue on reverse aide it necesasy and identify by block number) crack growth Mode I plane stress perfect plasticity strain field 20. ABSTRACT (Continue on rovers& aide It neceeuui and Identify by block nmb*er) A recently developed solution for the plastic strain, e(x,t), on the y crack line is used in conjunction with a critical strain criterion to construct curves for KR(a) versus a, where a is the incr'ase in crack length. Resistance curves have been computed for various values of the critical plastic strain. They show a monotonic increase of K (a) with increase in crack length, to a constant steady-state value. DD I F 2 N! EDITION O I NOV 65 IS OBSOLETE Unclassified SECURITY CL.ASSIFICATION OF THIS MAGE (When Data Entered)

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A STUDY OF SEQUENTIAL PROCEDURES FOR ESTIMATING. J. Carroll* Md Robert Smith. University of North Carolina at Chapel Hill.

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