LIBRARY Michigan University This is to certify that the dissertation entitled SOME NON—L I NEAR ASPECTS OF CRACK—T I P FIELDS IN FINITE ELASTICITY presented by Jae-Sung Yang has been accepted towards fulfillment of the requirements for Ph.D. degreein MeChaniCS (cm—4t v "‘ Major professor Dam November 12, 1985 Mclli.,...Aa‘—_..: . - F‘ ”L . , . . 0‘27" RETURNING MATERIALS: ‘EV1ESI_J Place in book drop to remove this checkout from w your record. FINES will ' be charged if book is returned after the date stamped below. SOME NON-LINEAR ASPECTS OF CRACK-TIP FIELDS IN FINITE ELASTICITY by Jae-Sung Yang A DISSERTATION Submitted to Michigan State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in Mechanics Department of Metallurgy, Mechanics and Materials Science 1985 ABSTRACT SOME NON-LINEAR ASPECTS OF CRACK-TIP FIELDS IN FINITE ELASTICITY by Jae-Sung Yang This dissertation has two parts, each concerned with some non-linear aspect of crack-tip fields in finite elasticity. The first part of the current dissertation is concerned with the deformation field near the tip of a crack in an incompressible solid which is loaded in Mode 1. The material model used here exhibits strain softening in shear at large deformations, which allows for a loss of ellipticity of the governing differential equations. As a result, one expects zones of localized shear to emanate from each crack-tip and extend into the interior of the body. A local asymptotic analysis near the crack-tip is carried out and both elliptic and non-elliptic solutions to the governing equations are found. Neither of these solutions, alone, satisfy the boundary conditions and so, neither provides a complete solution to the problem. However, a composite deformation field consisting of a "patching together“ of these two separate solutions can be constructed. There are certain lines of strain discontinuity emanating from each crack-tip, corresponding to the boundaries between these different solutions, modelling a narrow zone of localized shear. Jae-Sung Yang In the second part we are concerned with the derivation of simple explicit expressions for J, the energy release rate associated with quasi-static crack growth in nonlinearly elastic solids. The J-integral is the central theoretical quantity behind nonlinear fracture mechanics for rate-independent materials under monotonic loading; can also be regarded as a measure of the intensity of the crack tip singularity fields. The estimation of J for elastic-plastic crack problems is usually achieved by interpolation between the values of J corresponding to linearly-elastic and pure power-law materials. These interpolation schemes involve coefficients determined through a finite element analysis; moreover, different interpolation schemes are required for different constitutive descriptions. It is observed in this dissertation that one can avoid such interpolations by using a certain analytical estimation procedure. A center-cracked strip in Mode III is used to illustrate this. The results for J obtained by this scheme are appropriate for general constitutive relations and are very accurate under certain conditions. Consequently, the need to interpolate between particular materials can be avoided. ACKNOWLEDGEMENTS I would like to express my deepest gratitude to my adviser, Professor Rohan Abeyaratne for his help and support during all stages of this investigation. It has been my pleasure to learn from him and to have a research Opportunity with him. No graduate student could have a more understanding, patient and generous adviser. Grateful thanks are extended to Professor C.0. Horgan for his generous help and assistance throughout my graduate studies and to the other members of the guidance committee, Professors N. Altiero and D.H. Yen. I also owe my sincere thanks to my parents, parents-in-law, and lovely wife for their encouragements and supports, without which my study could have not been carried out. Thanks are also due to Mrs. Arlene Klingbiel for typing the manuscripts. While preparing this dissertation, I held teaching assistantships awarded by the Department of Metallurgy, Mechanics and Materials Science and Research Assistantships supported jointly by the U.S. National Science Foundation under Grants MEA 83-19616, MEA 78-26071 and the 0.5. Army Research Office under Grant DAAG 29-83-K-0145. The support of these institutions is gratefully acknowledged. ii TABLE OF CONTENTS LIST OF FIGURES O O O O O O O O O O O O O O 0 INTRODUCTION 0 O O O O O O O O O O O O O C O O O O O O O O O 0 PART I: CHAPTER 1. CHAPTER 2. CHAPTER 3. CHAPTER 4. CHAPTER 5. CHAPTER 6. PART II. CHAPTER 7. CHAPTER 8. CHAPTER 9. CHAPTER 10. CHAPTER 11. REFERENCES FIGURES LDCALIZED SHEAR DISCONTINUITIES NEAR THE TIP OF A MODE I CRACK INTRODUCTION . . . . . . . . . . . PRELIMINARIES PERTAINING TO FINITE FORMULATION OF THE MODE I CRACK PRO ASYMPTOTIC SOLUTIONS TO THE FIELD E 4.1 Non-elliptic Solution in H . 4.2 Elliptic solution in E3 . . . MATCHING . . . . . . . . . . . . . SUMMARY OF RESULTS . . . . . . . ESTIMATION OF ENERGY RELEASE RATES: TO INTERPOLATION INTRODUCTION . . . . . . . . . . . PRELIMINARIES PERTAINING TO FINITE FORMULATION 9.1 The Crack Problem . . . . . . 9.2 The Energy Release Rate . . . SOLUTION . . . . . . . . . . . . . DISCUSSION . . . . . . . . . . . . iii PLANE-STRAIN . BLEM . . . . . QUATIDNS AN ALTERNATIVE ANTI-PLANE STRAIN Page iv 15 18 25 28 32 35 39 44 46 48 53 56 59 FIGURE PART I: PART II: LIST OF FIGURES LDCALIZED SHEAR DISCONTINUITIES NEAR THE TIP OF A MODE I CRACK Response curve in simple shear for the piecewise power‘Iawmateria‘ooooooooooooooooo Geometry of the global crack problem . . . . . . . . Domains of validity of the solutions to the differential equations . . . . . . . . . . . . . . . Sketch of u(e) vs. 6 as defined by (4.24) with o O on R. (2.3) Since the body is assumed to be incompressible, the deformation must be locally volume-preserving so that J(x) - 1 on R. (2.4) The right and left Cauchy-Green tensors E and G are defined respectively by c = FTF, g = ffT. (2.5) 5 “A Let t be the Cauchy (true) stress tensor field on R* accompanying the deformation at hand. The equilibrium equations are 7 div 1 = g, I = rT on R*, (2.6) A IN when body forces are presumed to be absent. The Piola (nominal) stress tensor corresponding to r is given, in view of (2.4), by p(x) = 1(y(x))(FT(x))'1 on R. (2.7) Equations (2.2), (2.4), (2.6), and (2.7) lead to the equilibrium equations in the reference configurations div p(x) = o, oFT = FoT on R. (2.8) Furthermore, the nominal and true surface tractions are given by s = 0N “A on S, t = tn on 5*, (2.9) where S and 5* are surfaces in R and R*, 5* = y(S), while N and n are unit normals to S and 5*, respectively. It follows then that s = O on S if and only if t = O on 5*, (2.10) which is a useful result since it allows the boundary condition on a traction free surface 5* to be specified on its undeformed image S. He now turn to the constitutive law and suppose that the body is homogeneous, incompressible, and elastic, and that it possesses an elastic potential H a N(F). The nominal stresses are then given by aim , (2.11) where p(x) is a scalar field arising because of the incompressibility constraint. In the case where the material is isotropic, H depends on F in a special manner, viz. u a "(11912): (2.12) where I1, I2 are the principal scalar invariants of G: 1 = tr e 1 . 1 (tr e)2 - tr(Gz)] (2 13) 1 ~’2 '2": ~ ~ 0 . Suppose now that the domain R occupied by the undeformed body is a right cylinder with generators parallel to the x3-axis. Let D be the open region of the (x1, x2)-plane occupied by the interior of the middle cross section of this cylinder at x3 = 0. Suppose further that the deformation (2.1) is a plane deformation so that .Ya ‘ xa‘+ “u(xlt x2): YB = x3 0" R0 (2014) Throughout this problem, a comma followed by a subscript indicates differentiation with respect to the appropriate coordinate and Latin subscripts take the values 1, 2, 3, while Greek subscripts take the values 1, 2. Repeated subscripts are summed over the proper range. It follows from (2.2) and (2.14) that Fad = “1,8! Fo3 = Fae = 0. F33 = 1. (2.15) The nominal stresses are now given by aim _1 913(5) °oa ' 3F;- " P FBo’ c’33 ‘ '31;— ' P: (2°16) If we assume that the elastic potential H is such that . “ . o (2.17) 3' «3 3:3“ for every F such that (2.15) holds, then we further have 030,3 dag . o. (2.18) If we now define I by 10 I = FQBFQB (2.19) we have, because of (2.5), (2.13), and (2.15) that I . 11 - 1 a 12 - 1. (2.20) In the case when the material is isotropic, we have from (2.12), (2.20) that, in plane deformations, * w . H(1+1, 1+1) (2.21) so that if we define the plane strain elastic potential H(I) By * , u(x) = u(1+1, 1+1), I > 2, (2.22) we have that "(F) - H(I) where I = FQBFGB. It follows from this that A 3HIE) g 2F H'(I). (2.23) a as 08 From (2.5), (2.16), (2.23), we conclude that I dug: 2H'(I)FaB - pFaB. (2.24) The deformation (2.14) is a simple shear in the x1-direction if it has the form U1 3 kxz, uz = 0, (2.25) where k is the amount of shear in the x1, xz-plane. Then from (2.19), one gets I . 2 + k2 (2.26) and the stress of primary interest, 112, is found from (2.7), (2.15), (2.24), (2.26) to be 112 a ;(k) - 2kH'(2+k2), 0 < k < o; (2.27) ;(k) denotes the response function _o_f_ the material EMM. Its secant, tangent, and infinitesimal shear moduli are given by 11 us(k) s-IIAL, ut(k) =-E§§51, no = ut(0), (2.23) respectively. The secant modulus is assumed to be positive, so that from (2.27), (2.28), u5(k) =.2u'(1) > o. (2.29) The in-plane behavior of an incompressible material, in plane strain, is essentially governed by its response to simple shear. In particular, agprlane deformation of an incompressible body can always be decomposed locally into a rigid-body rotation followed, or preceded, by a simple shear [11]. The amount of this ”effective local shear", ke, is k, = (1-2)1/2, 1 a tr FTF . (2.30) If A, 1'1 are the principal stretches of the deformation, one can equivalently write k, = (1 - x-ll. (2.31) Therefore from (2.24), (2.29), (2.30), the in-plane response of an isotropic, incompressible, elastic material in ggy_plane deformation can be written as ea, . u,(ke)F,B - p(§)rgi, k, = (1-2)1/2. (2.32) From (2.24), (2.8), (2.19), (2.4), (2.3) and (2.1), one may obtain the displacement equations of equilibrium for plane deformations of a homogeneous, isotropic, incompressible hyperelastic material. They are -1 8a a F 0 on D, det f = 1 on D, (2.33) coev6(5) "7.86“ ”’3 where Fag = ua,3 + 5o3 and 12 azu a = ' ' ca8Y5(E) m 2H (”GCYGBG + 4H (I)FGBFY5° (2.34) It has been shown, (Abeyaratne [11]), that (in the presence of (2.29)) this system of partial differential equation is elliptic at a solution (ua,p) and at a point (x1,x2) if and only if 2H“(1)(I-2) + H'(I) > O. (2.35) A physical interpretation of this ellipticity condition may be obtained in terms of the concept of the local amount of shear. Recall the definition of the shear stress reSponse function ;(k): in.) = 2kH'(2+k2), m < ., (2.35) Differentiating (2.36) with reSpect to k and observing that the secant modulus is assumed to be positive leads to ;'(k) = 2H'(2+k2){2k2 53-22%)- + 1}. (2.37) H'(2+k ) He therefore find that (2.35) is equivalent to ;'(ke) > o, ke - (I-2)1/2, (2.38) from which we conclude that the system of partial differential equations (2.33) is elliptic at a solution 2, p and at a point x if and only if the tangent modulus evaluated at the effective local shear ke is positive. Thus, in particular, if the material at hand has a monotonically increasing response curve in shear, it follows that ellipticity prevails throughout. In this study, we are concerned wfith the case in which the material behavior allows for ellipticity to be lost at severe levels of deforma- k tion. He suppose that the shear response function T(k) is linear for 13 small amounts of shear and has a power-law form for large amounts of shear. Specifically, we take uok for 0 < k < k0 1(k) : 2 1 (2.39) k n- a uoko(-E—) for ko < k < , o where no, kc, and n, (Po > 0, k0 > 0, and O < n < 1/2), are material constants. A sketch of ;(k) vs. k is shown in Figure 1. Note that when 0 < n < 1/2, as is assumed here, ellipticity is lost whenever the effective local shear ke exceeds kg. The loss of ellipticity of the governing partial differential equations leads to the possible occurrence of elastostatic fields which are less smooth than previously assumed. Therefore, we now have to relax the smoothness assumptions made previously in order to account for such "weak solutions.“ Particular interest lies in the case wherein the field quantities possess the degree of smoothness assumed previously everywhere except on one or more curves on 0. Accordingly, it is assumed that, although 2 is continuous in 0, there is a smooth curve S in D such that (i) p and u are respectively once and twice continuously differentiable in 0-5, (ii) p and V2 suffer finite jumps across S. Under these new conditions, the field equations discussed previously are to hold in 0-5. In addition, it is required that the nominal traction s be continuous across S; this in turn implies that the true traction t is continuous across 5*. Therefore, from (2.9), we get [SJN = O on S, [:]n = O on 5*, (2.40) where [.] indicates the jump across the appr0priate curve. A curve 5 (or its image 5*) carrying jump discontinuities in F, p, and a while 14 preserving continuity of displacement and traction is called an eguilibrium shock. CHAPTER 3 FORMULATION Q§_THE MODE I_CRACK PROBLEM Let D, the undeformed cross-section of the cylindrical body, be the exterior of the straight-line segment L (Figure 2), L . I 5| xz = O, -b < x1 < b}; (3.1) L represents a traction-free crack of length 2b, aa2(x1, or) . o, -o < x1 < b. (3.2)1 The body is subjected at infinity to a uni-axial stress in a direction normal to the crack 022 + 0., 012 + 0, “11 + 0, 021 + O as x12 + xzz + a. (3.3) The problem to be considered is the following: For the material characterized by (2.32), (2.39), we seek a suitably smooth deformation field y(x), a nominal stress field p(x) and a pressure field p(x), all on D, satisfying the field equations (2.2)-(2.4), (2.30), (2.8), (2.32), (2.39) and the traction-free and prescribed-load boundary conditions (3.2), (3.3). As mentioned previously, it is possible for the governing differential equations here to lose ellipticity.2 This suggests that we seek the solution to the aforementioned boundary-value problem in a class of functions which are less smooth than that in which one would have otherwise sought the solution. Accordingly, we merely require that A y(x) be continuous and have piecewise continuous first and second A partial derivatives on 0. Furthermore, y(x) is to be bounded near the crack-tip. Note that the preceding smoothness requirements admit ’I:2 As mentioned previously, Greek subscripts take the values 1,2 and repeated subscripts are summed. 2. The corresponding problem for a material which does not lose ellipticity has been considered by Stephenson [16]. 15 16 the possibility of finite jump discontinuities in f and 3 across curves in D. In the event that such a curve S exists, equilibrium considerations require that the nominal traction s be continuous across it. Such a curve S across which the displacements and tractions are continuous but the displacement gradients are discontinuous is referred to as an “equilibrium shock” (or 'shock'). Finally, let (r,6) be local polar coordinates in the undeformed configuration as shown in Figure 2. Then x1 - b = r cose, x2 = r sine, r > 0, -n < 0 < n. (3.4) On using (2.28) and (2.32), the equilibrium equations (2.8)1 may be shown to imply u (k )-u (k ) 3k 3k ‘32., t e s e __J; 1 __J: 3r u5(ke)Hr + ke grr 3r +:2'gre ae 1’ (395) u (k )-u (k ) 3k 3k 3 t e s e e l e 3%: "5(ke)He + ke gre 37+:2'988 WJ’ where we have set RY 3! 8y 3’ 2 0‘ .__e_._e Hrg'fi'tk'ydt’ I"6331—6" ya’grr ar ar ’ (3.6) 9rd ar 89 ’ 988 as as ’ and 2 a 1 (3.7) ke grr“:2'966 '3' Similarly, (2.3), (2.4) yields J ,_,_1_ (3’1 3,2 _ 3’2 33'1 __, 1 (3.8) r Gr 30 Gr 30 ’ and the boundary conditions (3.2) on the crack surfaces give 17 By 31! 3y 31! 1 2 a 2 1 _ "5(ke) 3'6" "P 3F" 0' "s("e) ae ' '9 a—r‘ " 0 (3'9) on 6 = i n, 0 < r < 2b. 0n using (3.6), (3.8), one can show that the boundary conditions (3.9) are equivalent to 9669rr‘r2. gre’oo P'us(ke)/9rr 0" 9'1 I. 0 < r < 2b. (3.10) For the material characterized by (2.32), (2.39) under consideration here, the equilibrium equations (3.5), in view of (2.28), reduce to aplar a "oHra ap/ae = "OHS (3.11) at points in D where 0 < ke < kc and to 331 0 e2".2 ’10 2n-3 3"e 2 3"e or T k 2n-2 Hr + (ll-UR 2n-2 ke [29" 37- + :2 9re '59—]. o o 2n-2 k u 3k 3k a no e o 2n-3 e 2 e 3% ' k 2n-2 He + ("'1) k2n-2 ke [29rd '5?" 1' :2" 988 88‘]: o o (3.12) at points where ke > k0. CHAPTER 4 ASYMPTOTIC SOLUTIONS IQ_THE FIELD EQUATIONS In this chapter we proceed to calculate asymptotic expressions for the deformation field near the right crack-tip. He will determine two solutions of the governing differential equations, each valid on some sub-domain of the crack-tip zone. In the next chapter, these solutions will be combined in a suitable manner in order to generate a solution of the complete (asymptotic) boundary-value problem. One of the deformation fields determined is a non-elliptic one, and satisfies (3.8), (3.12) in a zone H ahead of the crack-tip, (see Figure 3). The other is elliptic, and is determined from (3.8), (3.11) in zones E+, E' adjacent to the crack-faces. The regions E+, E‘, and H are described by: E+ = { (r,e)| 0 < e < n, 0 < r < r0}, H = { (r,e)|-eo < e < 00, 0 < r < r0}, (4.1) E- = { (r,9)|-n < e < 0, 0 < r < r0}, where 00 (O,n) is an angle to be determined. 4.1 Non-elliptic solution on H He assume that the deformation field in H admits an asymptotic representation of the form §a(x1.x2) ~ rmuaw) as r + 0. (4.2) 18 19 where ua(e), (uaua$0), are smooth functions defined on some range -eo 0. Equations (4.2), (4.4) and the incompressibility condition (3.8) yield u1 J2 - uluz a 0 on ‘90 < e < 00, (4.5) which may be integrated to give uGI a aau(e) on - 80 < 0 < 60. (4.6) Here a1, a2 are constants, (aaaa t O), and u (f 0) is a smooth function on -60 < e < 60. In view of the nature of the prescribed loading (3.3), and the geometric symmetry present in the current problem, one expects the deformation field to possess the following symmetries: 3:7 As mentioned previously, the deformation field must remain bounded at the crack-tip. Despite (4.4), the deformation (4.2) is bounded when the crack-tip is approached from within the (eventuET) hyperbolic domain (see Chapter 6). 20 y1(r.6) = yum-6). yzine) = -yz(r.-6) for 0 O) and 8(6) through mu(e)=p(e) cosu(e), 0(e)=o(e)sint(e) on -°o < 8 < 80. (4.18) Eliminating u(e) from (4.18) gives 5(e)/p(e) = {0811(9)} tan 4(6). (4.19) On the other hand (4.11), (4.17) and (4.18) provide (2n41)(3(e)/o(e))tant(e) + {1(0) + m + 2(m-1)(n-1)} = 0. (4.20) Equation (4.19) can now be used to eliminate 5/9 in (4.20) which leads to I {n/(I-n)+cos 28} + {wo+c052¢} = 0 on -80 < 8 < 00, (4.21) where we have set "0 a 1 - m(1-2n)/(1-n) > 1. (4.22) The symmetry condition (4.10), in view of (4.18) requires that ((0) a 1/2. (4.23) Integrating (4.21) and using (4.23) gives 0(8) implicitly as 22 8 = -(m -.%) + +11Tm IIEnn wltanTIEwltan(0 --%)]. (4-24) where 1/2 W1= ((Wo+1)/(Wo-1)) > 0. (4.25) and wb is given by (4.22). Equation (4.24) cannot be inverted explicitly to furnish 0 = 0(8). A typical graph of 8 vs. 0 as described by (4.24) is shown in Figure 4. It indicates that 0(6) is multi-valued in general but that a single-valued branch, which satisfies 0(0) 2 1/2, may be chosen. This corresponds to the solid portion of the curve in Figure 4. From here on, it is this single-valued branch that we consider. The graph indicates that 0(8) increases monotonically on ('90: 00) from the value 0b=1/2 c05'1(-n/1-n) to u-0o and is antisymmetric about 8 = 0, 0 = 0/2. Furthermore, we find from (4.24) that 8 = - tan'1(/1-2n) + l-m 1'2" w tan'1{w ll-Zn} 0 < 6 <1 0 + I-m 1-2n 1 1 ’ o ‘ (4.26) We now proceed to find u(8) in terms of 0(8). Eliminating p(8) from (4.18) gives 0 J - mu0 tan0 = O on -80 < 8 < 00, (4.27) so that (4.21) can be written in the form {wo + c0520}G + mu0In/(1-n) + c0520}tan0=0. (4.28) Integrating (4.28) yields the following expression for u(e): u(8) = a cos0(8){c0520(8) + wo}"(1""')/2 , (4.29) with 0(8) being the single-valued inverse of (4.24) on -80 < 8 < 80, In equation (4.29), a is a positive constant and wb is given by (4.22). 23 For later purposes it will turn out that we need an explicit expression for u(8) at small values of 6. This can be readily obtained by using a Taylor expansion of u(e) about 8 = O. From (4.10), (4.18), (4.19), (4.23), we have u(0) = 0, 0(0) = 9(0), (4.30) 0(0) = 5(0). (4.31) {7(0) = ;(0)-p(0) (1(0))2. (4.32) while (4.21), (4.23) provide 0(0) a -m. (4.33) On the other hand (4.18)1, (4.29) give p(e) a mm{c0520(0) + wo}'(1'm)/2 on -80 < 8 < 80, (4.34) so that _ 2n-1) -(1-m)/2 9(0) - am 1-" } » (4.35) 3(0) = 0. 3(0) = W 0(0)- (4.36) Therefore, (4.30)-(4.36) indicate that 3 "(9) = p(OILe - m{2(1-g%:m(4n-3)}_§_J + 0(65) as 8 + 0. (4.37) At this stage the displacement field in H has been found to be y1 = 0(r'"), y2 = rmu(8) + o(r"') as r + O, (4.38) where u(e) is given by (4.24) and (4.29). We now proceed to obtain a better estimate on y1. Accordingly, we suppose that YI 3 P£V(9) t DIP“). y2 = rmu(6) + o(rm) as r + 0, (4.39) _ 24 n which 2 (>m) is an unknown exponent and v(8) is an unknown smooth function. Symmetry (4.7), requires that v(8) = v(-8). The incompressibility condition (3.8), on using (4.39), leads to r”""""2 (zvu - mvu) + o (r‘i‘m'z) = I, A (4.40) so that necessarily, t < 24m. (4-41) One must consider the two cases 2 < 2-m and z = 2-m separately. First, if z < Z-m, (4.40) implies that xvi - mvu = O on -80 < 8 < 00, (4.42) which may be integrated to give v(8) as v(e) = clu(e)I£/m. (4.43) where c is a constant. Since t/n < 1 and u(0) . 0, 6(0) ¢ 0, it follows that 5(0) is bounded only if t a 0. Thus one finds that v(8) = constant, 8 a O. (4.44) This corresponds to a rigid-body translation in the x1-direction. Consequently, it is the second possibility, 2 = 2-m, (4-45) that is relevant. In this case, (4.40) implies that (2-m)vu-mvu = 1 on (-60, 00). The solution v(8) of this equation, which is bounded at 8 = 0 is 6 me) = - finall‘z’w'“ I [u(m'z’mdo for o < e < 60. (4.46) O with symmetry, v(8) = v(-8), defining v(8) on (-80, 0). From (4.46), (4.35)-(4.37), we find that for small angles 8, v(8) can be approximated to be 25 v(e) = m [1+ W2] + 0(6“). (4.47) In summary, we have found that the deformation field near the crack—tip in region H is given, asymptotically, by (4.39). Here the exponent m is yet to be determined. The function u(e) is known (except for the amplitude a) and is given by (4.24), (4.29). The exponent t = 2-m and the function v(8) is given by (4.46). The region H is defined by (4.1), (4.26). 4.2. Elliptic solution on E1 Next, in E+, we take y(x) to be of the form .Ya ~ ba + r"u(,(8) as r + 0. (4.48) Since we have ellipticity here ke < kc, and this requires 8 > 1. Furthermore, not both u1, uz can vanish identically. Incompressibility (3.8) together with (4.48) provides erV'Z (ulfiz-uzul) + 0(r2V'2) = 1 as r + 0. (4.49) Since 2v-2 > O, (4.49) requires that v = 1. Thus UIOZ - uzdl = 1 on O < 8 < n. (4.50) On the other hand the equilibrium equations (3.11), on using (3.6), (4.48) and v=1, lead to a (II +u ). (4.51) aplar ~ noua(ua+ua)/r, 3p/88 ~ "0 a a a Eliminating the pressure p from (4.51) yields d 3-5 (uaua) + ZlimuOI = 0, which may be integrated4 to give 4. The constant of integration can be shown to be zero by using (4.51), (4.53) and the boundary condition (3.IO)3. 26 u1(u1+ul) + u2(uz+u2) = O on O < 8 < n. (4.52) On the other hand, differentiating (4.50) with respect to 8 gives u1(u2+u2) - u2(u1+ul) = 0 on O < 8 < n. (4.53) Since u12 +-u22 ¢ 0, the pair of equations (4.52), (4.53) require that ul+u1 = O , u2+u2 = O on 0 < 8 < n. (4.54) Next, on using (4.48) with val and (3.6), the boundary conditions (3.10)1, (3.10)2, on the crack-faces give uBuBGaJ, = 1, [Julia - 0 on 8 a n. (4.55) Thus on solving (4.54), subject to (4.50), (4.55), one finds u1(8) ao cose - (bo/(a§+ b§)) sine, (4.56) 2 2 . u2(6) bo cose + (ao/(ao+bo)) Slne, for 0 < 8 < n, where a0, b0 are unknown constants, aoz + b0? t 0. In order to calculate the leading terms for the stress components in E*, it turns out that we need an expression for the deformation to an order higher than in the foregoing. Therefore, we now assume (ya ~ ba + rua(8) + rkfa(8) as r + 0, (4.57) where k > 1 is a constant and fa(8) are smooth functions on [0,0]. The higher order consideration (4.57) yields, through the incompressibility condition (3.8), ulfz - uzfl + k(Ozf1 - Olfz) = O on 0 < 8 < u, (4.58) while the equilibrium equations (3.11), together with (3.6), give ap/ar ~ unrk-Z ua(fa+k2fa), ap/ae ~ nork-l ua(fa+k2fa). (4.59) 27 This indicates that the pressure field is of the form p(x1, x2)~p0+r9f3(8) as r + 0, where q k -1 > 0 and (k-1)f3 = uoua(fa+k2fa), i3 uoOa(fa+k2fa) on 0 < e < n. (4.60) The boundary conditions (3.10)1 and (3.10)2, in view of (3.6), (4.55), (4.57), yield uaug(uaf3+kugfa) = O on 6 = n, (4.61) uafa + kuafa = 0 on 8 = 1:, while the third of (3.10), together with (4.59)1, requires that uyuyuauau3(f3+k2f5) + 2k(k-l)faua = 0 on 8 : n. (4.62) In summary, we have found that the deformation field near the crack-tip in region E+ is given, asymptotically, by (4.57). Here the exponent k > 1 is yet to be determined. The functions u1(8), u2(8) are given by (4.56) while f1(8), f2(8) are to be determined from (4.58), (4.60)-(4.62). Symmetry (4.7), gives the deformation in E'. CHAPTER 5 MATCHING In the previous section we found two solutions to the governing differential equations, each valid on some subdomain of the crack-tip zone. Figure 3 shows the domain of validity of each solution. In this chapter, we will combine these deformation fields in a suitable manner in order to obtain a solution of the complete (asymptotic) boundary-value problem. It is sufficient to restrict attention to the upper half-plane. One can show from (4.24), (4.26), (4.29), (4.39), (4.45), (4.46), (4.56), (4.57) that5 ya? does not match continuously onto yafl across the line 8 = 80. Thus we are led to seek two curves 5*, S' lying in H across which the deformation is continuous, (see Figure 5). These curves are defined in a neighborhood of the right crack-tip by St: 8 = i 8(r), 0 < r < r0, (5.1) where we suppose that 8(r) 8 Ar5 + o(r5) as r + 0. (5.2) Here A > 0 and s > 0 are constants. Note that if s = 0 the curves 5* make angles tA with the x1-axis at the crack-tip. In this case one must have 0 < A < 80. If s > 0, the curves 51 are tangent to the x1-axis at the crack-tip. The crack-tip zone may now be divided into three zones 2", E ‘andK as follows (see Figure 5): 5. The superscripts E’and H will be used in this section to denote quantities associated with the elliptic and non-elliptic solutions, respectively. 28 29 E“ = {(r,8)| 8(r) < 6 < w, 0 < r < r0}, ’1 = {(r.6)| -9(P) < 9 < 9(r). 0 < P < r0}, (5.3) £,- a {(r,8)| -n < 8 < -8(r), O < r < r0}. The deformation field near the crack-tip is now taken to be H (5.4) fix 0" K: E ya on 2*, 8'. Yo.’ where yafl and yaE were found in Sections 3.1 and 3.2 respectively. It is clear from the preceding analysis that (5.4) does in fact satisfy all requirements, provided that this deformation field and the corresponding traction field are continuous across S+ and S', i.e. ya = ya on S , (5.5) s E = sH on S+. (5.6) a (I First we match y1 across the shock and find through (5.2), (4.39), (4.45). (4.47). (4.57). (4-55). (5.5). that I 2-m b1 + aor (2-m)p(0) r as r + 0, (5.7) and so, recalling that m < 0, b1 8 0, do 8 O. (5.8) Similarly, matching y2 across the shock leads to 02 + bor + rkf2(0) ~ Ap(0)r5+m as r + 0. (5.9) Observe from (4.57) that b2 denotes the crack-opening displacement. Thus we expect b2 > 0 and so (5.9) gives b2 2 Ap(0), s a -m > 0. (5-10) He may also match the deformation y1 to a higher order to find, through (5.2), (4.39), (4.45), (4.47), (4.57), (4.56), (5.5) and (5.8), that 3D -1 2- I-m k 1 m -bo Ar + r f1(0) ~ (z-n)o(0) r as r + 0. (5.11) Assuming f1(0)¢0, this gives k = 1-m, f1(0) . Abo'l. (5.12) Next we compute expressions for the Piola stress components in E+ and H. First, we calculate the pressure field using (3.12), (3.11), (3.6), (4.39), (4.45), (4.56), (4.57), (5.8) to get, in H and ET, respectively pH(r,8) = 0(r2(m-1)(O-2)), ' k-I pE(r,e) ~ po + udEE:T>I-b;lsin8(f1 + szl) + (5.13) 2 bocose(f2 + k f2)} , where p0 is a constant. He now calculate the stress fields through (2.2), (2.30), (2.28), (2.32), (2.39) to get, jg_H, a), = 0(r(m-1)(Zn-3)), H (5.14) 012 = 0(r(m‘1)(2"'3)). ogl = uokoz'zn G28) [mucose - Gsin8]r(m-1)(Zn-1) + o(r(m-1)(20-1)), egg 2 uokoz'zn G(g)1[musin6 + Ocose]r(m-1)(Zfl-1) + o(r(m-1)(20-1)), and _i_9_ E"’, E . . O11I {u°(kf1c°se ‘ f15199) ' Po(kfzsin8+fzcos8)}pk-l+ 0(rk'1). E - ° - 012 a (pobo-uobo 1)+[uo(kf151n8 + flcosfl) + po(kf2cose fzsine) + - " 2 " 2 k-1 k-I (bouo/(k-1)){-bolsin6(f1 + k f1) + bocos8(f2 + k f2)}]r + o(r ), E -1 k-l 021 = uobO - pob0 + 0(r ), (5-15) 31 Gig = {uo(kfzsin8 + fzcose) - po(kf1cose-f15in8)}rk'1 + 0(rk'1). The traction-free condition 012 a 022 a 0 on the crack requires that Po iuobo'2 (5.16) as well as that (4.61), (4.52) hold. In calculating the Piola tractions, note that the unit normal vector N to 5* has components N1 = -A(1+s)rs + o(r5), N2 = 1 + 0(r25). (5.17) He may now match the nominal tractions across 5*. 0n using (2.9), (5.14)-(5.17) we find that the continuity of $1 a OlgNg requires 2 {ubil(0) + pokf2(0) +-E§:¥(?2(0) + k2f2(0))}rk-1 + o(rk-1) - 0. (5.18) This implies that . bzu ” ubf1(0) + pokf2(0) +-—§:§(r2(0) + 12r2(o)) = 0. (5.19) Similarly, on matching the traction component 52 = angg, we get nok02-2n p(o)2n-1 p(m-1)(2n-1) ~ -(hobo-pooo-1)A(1-n)r-m’+ (uof2(0) - pokf1(O))r‘m, (5.20) from (2.9), (5.14), (5.15), (5.17), (5.10), which provides m = 1 - 1/2n, (5.21) u0i2(0)‘-pokf1(0) a A(1-m)(uobo-pob;1) + uokg'2“of331 . (5.22) CHAPTER 6 SUMMARY inRESULTS In summary, we have investigated the near-tip displacement and stress fields under plane strain mode I conditions for the piecewise power-law material characterized by (2.39), (2.32). He found two shock lines 5*, S' emanating from the crack-tip (Figure 5) and described asymptotically by 8 ~ o,r1(0)r-1+1/2n. (5.1) The curves 5*, 5’ separate the crack-tip zone into regions I; T. E; ' (adjacent to the crack-faces) and K (ahead of the crack), defined by (5.3), (5.1), (6.1). Note that 5+ and S'are tangential to the xl-axis at the crack-tip. The deformation and stress fields vary smoothly within E T, E 'and K ; the defamation and the tractions are continuous across S+ and S‘. The deformation gradient and stress are discontinuous across these curves. The deformation field in the elliptic zone 5 + is given by ’1 fl "”4“” + rmnffle)’ onET. (5.2) y2 - b2 + rbocose + r1/2“f2(6). Here b2 = bof1(0)p(0) > 0 and b0 > O are constants with p(O) given by (4.35), (5.21). Further, the smooth functions f1, f2 (and f3) are given by fi(9) = koZ‘Z" 9(012"'1 91(6), i = 1, 2 and 3, (5.3) where according to (4.58), (4.60)-(4.62), (5.19), (5.22), (4.56), (5.8), (5.12), (5.16) and (5.21), 91, gz (and 93) are to be found from the linear boundary-value problem 32 33 bocose 91 + bo‘lsine 92 + k(bosin8 91-b0'1cose 92): on (O,n), uOI-bo‘lsin8(gl+k291) + boc058(g2+kzgz)} = (k-1)g3 on (0, n), "ofbo‘lcose(91+k291) + bosine(92+k292)} = -93 on (0.“). 92 + k{k+2(k-1)bo'4}92 = O on 8 = O and 8 a n, (6.4) 52 - kbo291 = 1 on 6 = 0. 92 - kbo‘zgl = 0 on e . n, where k = 1/2n. He have been unable to determine a closed form solution of this linear boundary-value problem. On the other hand, for a given value of b0, it may be solved numerically for each value of n. The leading order homogeneous deformation characterized by (6.2) describes a state of uni-axial tension in the yz-direction. The principal stretches of this deformation are be, 00-1, and in view of ellipticity, are restricted by Ibo-bo'll < k0. The Piola stress components in ET are given by (5.15) with po=uobo‘2, k = 1/2n. In the hyperbolic region K , the defamation field is given by yl ~ r1+1/va(e), ll)( (5 5) ya 2 rl'I/Z"U(9). ’ . with u(8) given by u(8) . acos0(8){cos20(8) + (1-2n+2n2)/(2n(1-n))}'1/4", 8 a -(0-n/2) + tan‘1{(1-2n)‘1tan(0-w/2)}, (6.6) and v(8) given by (4.46) with m = 1-1/2n. 0n using the fact that the shock-lines S2 are tangential to the x1-axis at the origin, and u(0)=0, it may be easily verified that the deformation (6.5) is bounded in K . The Piola stress components in K are given by (5.14) with m = 1-1/2n. Finally we note that the asymptotic expressions for the various field quantities determined here are known completely in terms of two unknown constants a and b0. Necessarily, an asymptotic solution 34 procedure cannot provide the value of both of these constants since they must depend on the global problem. It may however be possible to determine the value of one of them through a higher order calculation. PART II ESTIMATION OF ENERGY RELEASE RATES: AN ALTERNATIVE TO INTERPOLATION CHAPTER 7 INTRODUCTION Fracture mechanics is focused in two principal directions, the development of phenomenological explanations of crack extension, and the description of the micromechanical process of material separation on the microscale. In the first, emphasis is placed on predicting crack extension behavior, usually in terms of a single parameter which characterizes the near-tip stress field. Linear elastic fracture mechanics is a case in point, where predictions are made in terms of the elastic stress-intensity factor K, which serves to characterize the influence of applied loads and geometry on the near-tip field under elastic (or even small-scale yielding) conditions. Hence, the analytical problem in elastic fracture mechanics is to determine the stress intensity-factor. Rice showed that in the elastic range, the value of the so-called J aintegral is proportional to the square of the stress intensity-factor [17]. The J - integral continues to be path independent even for nonlinearly elastic materials. It is the central theoretical quantity behind nonlinear fracture mechanics for rate - independent materials under monotonic loading, since it can be regarded, from a physical point of view, as a measure of the intensity of the crack-tip singularity fields. Also, it represents the energy release rate (see, for example, the recent survey article, Hutchinson [18]). If a crack in a deformed solid is to propagate, it requires energy and the “energy release rate“ represents the rate at which energy is 35 36 made available to the crack for this purpose. Experiments carried out by Begley and Landes [19] have demonstrated the potential that this parameter has for use as a fracture initiation criterion. The importance of the energy release rate as a fracture initiation criterion has prompted many studies to calculate its value under nonlinear conditions. These have primarily consisted of numerical and experimental investigations, e.g. [20-23], as well as analytical approximations based on interpolation, e.g. [24,25]. The only exact solution appears to be that due to Amazigo [26] for a pure power - law material under conditions of anti - plane shear. In his paper [24], Shih proposed some approximate (but accurate) formulas for estimating the relation between the path - independent integral, J, the applied stress, the load point displacement, and the constitutive parameters for cracked bodies of strain hardening elastic -plastic materials. The formula makes use of results from the elastic and fully plastic solutions so as to interpolate from the small - scale yielding range to the fully plastic range. However, this interpolation scheme involves coefficients that are determined through a finite element analysis. Moreover, different constitutive descriptions (e.g. a Ramberg - Osgood material as against power - law material) require different schemes. Recently, Abeyaratne [27] suggested an analytical estimation procedure for calculating the value of the energy release rate associated with Mode I and Mode III crack problems for an infinite body composed of a nonlinearly elastic material. The results are accurate even up to moderately large loads. This scheme does not suffer from the 37 drawbacks mentioned above and adapts an approximation that is a natural extension of the small-scale yielding concept. The small-scale yielding approximation for the energy release rate uses the argument that when the applied loads are small, the nonlinearities in the problem are confined to a small region near the crack tips and that the linear elastic solution provides a good approximation elsewhere. The present scheme, essentially, replaces the linear elastic solution in the preceding argument by a better approximation. Consider, for example, a Mode 111 crack problem under remotely applied uniform shear. In the absence of the crack, the body is in a homogeneous state of (finite) anti-plane shear. If one were to assume that the presence of the crack causes only a small perturbation of this homogeneous state, one can then carry out an analysis based on the theory of small deformations superposed on a large deformation. The results based on such an approximation would clearly be invalid in the vicinity of the crack-tips, but one would expect it to provide a reasonable approximation at points distant from the crack. Since our interest here lies églgly_in estimating the energy release rate, and since this can be written in the form of a path independent line integral J taken over a contour that is far from the crack-tips over most of its length, one expects such an approximation to be suitable for our purposes, at least when confined to moderate levels of loading. The analytical scheme in [27] is appropriate for general constitutive relations and is not restricted to a particular model. Consequently, one does not need to go through an elaborate calculation each time the material changes. This important observation was not made 38 in [27] and it is this point that we wish to make in this study. He use a center - cracked strip in Mode III to illustrate this. CHAPTER 8 PRELIMINARIES PERTAINING TO FINITE ANTI-PLANE STRAIN Suppose that an isotropic, homogeneous, incompressible elastic body occupies the region R in the undeformed state, and consider the deformation y = y (x) = x + u (x) on R, (8.1) where i is the position vector in R, y the position vector in the deformation image R* of R, and u is the displacement vector. This mapping is one-to one. The deformation gradient tensor F is defined by f (5) = V y (f); (8.2) the incompressibility of the material requires that J = det F (x) = 1 on R, (8.3) for every admissible deformation, so that the deformation is locally volume-preserving. let 9= 5 5“ <84) be the left Cauchy - Green tensor for the deformation, and set 11 = Tr g, 12 - 1/2 [(Tr g)? - Tr (82)], (8.5) so that 11, 12 are two of the three principal invariants of G. Here a superscript T stands for transpose and the third invariant is 13 a det E . 02, and is equal to 1 by the incompressibility condition (8.3). At the undeformed state I1 = 12 a 3, and 11 > 3, 12 > 3 for all deformations. The mechanical response of an isotropic, incompressible, hyperelastic material is governed by its strain energy function per unit 39 4O undeformed volume H, which depends only on the strain invariants 11, 12: H a H (11, 12). If I is the Cauchy stress tensor (force per unit deformed surface area), the stress - deformation relation is aw 8H 5=2gq§+3g(m1-§)g-p1 on) where 1 is the unit tensor and the scalar p is an arbitrary pressure whose presence is necessary to accomodate the incompressibility constraint (8.3). The Piola stress a (force per unit undeformed area) is then related to T according to 3 ‘ I‘ET)’1- (8.7) From (8.6), (8.7), and (8.4) it follows that 8 3H T -1 g -- 21%;. +350, 1 - (1)131 - p(E) . (8.8) The differential equations of equilibrium may be expressed either in tenms of 3, regarded as a function on R*, or in terms of g, regarded as a function on R. In the absence of body forces, these equations are div 5 = O on R*, (8.9) div 3 = O on R. The tensor I is symmetric (32:1); 2 in general is not. If S* is a traction free portion of the boundary of the region R* occupied by the deformed body, T satisfies 1 n = 0 on 5* (8.10) 41 where n is a unit normal vector on S*. If S is that portion of the boundary of R which is carried onto 5* by the deformation (8.1), as a consequence of (8.7), (8.10), 3 satisfies 3 N = O on S (3-11) where N is a unit normal vector on S. Suppose now that R is a cylindrical region, and choose rectangular cartesian coordinates x1, x2, x3 with the x3 4 axis parallel to the generator of R. The deformation (8.1) is an anti- lane shear if it is of the form y1 = X1. y2 = X2. y3 = X3 + U(X1.X2)- (8.12) It is convenient to think of the 'out-of—plane' displacement u as a function on the plane cross-section D of R for which x3 = 0. Here x1, yi are the components of x, y, respectively, in the given frame. In the present problem we consider a class of incompressible materials for which H is independent of the second invariant H = H(11), (3-13) so as to ensure that the governing field equations have a solution of anti-plane shear type (Knowles [13]). He assume that H(I1) is twice continuously differentiable for I1 > 3 and that it vanishes in the undeformed state, H(3) a 0. If (8.13) holds, the two forms (8.6), (8.8) of the constitutive law take the simpler forms T 8 2H'(I1)G - p l , “ “ “ (8.14) o = 2H'(I1)F - p(FT)‘1. where the prime indicates differentiation with respect to the argument of N. For the deformation (8.12), the matrices of components of the tensors F and G are readily found from (8.2), (8.4) to be 1 O O 1 0 u,1 F = O 1 D , G = O 1 0,2 , (8,15) u,1 u,2 1 u,1 u,2 1 + |vu|2 IV'UI2 = U.au.o- . (8.16) and Here a subscript preceded by a comma indicates partial differentiation with respect to the correSponding x - coordinate; and Greek subscripts have the range 1, 2, while Latin subscripts take the values 1, 2, 3; repeated subscripts are summed over the appropriate range. From the first of (8.15), one confirms that the incompressibility condition (8.3) is automatically satisfied, while the first of (8.5) and the second of (8.15) furnish 11 = 12 a 3 + |Vu|2. (8.17) From (8.14), (8.15) we find the Cauchy and Piola stress components in terms of u, p. Through the equilibrium equations for the Piola stresses, one can determine p which when substituted back into the expressions for the stresses gives (e.g. Knowles [14]) 130 3 T03 a 036 a 2H.(Il)usas “63 ‘ [2"'(11) + do(X3+U) + 9110.8. res = °a8 = ' [do(x3+U) + d115a8. (8.18) T33 = 2H'(Il) |Vu|2 - do(X3+u) - d1, 033 = - [do(x3+u) + d1]. Here 6GB is the Kronecker delta, and do, d1 are constants. The differential equations of equilibrium require that [2H'(3+|Vu|2)u,o],o = do on 0 (8.19) The constants do and d1 must be determined from boundary conditions in a particular anti-plane shear problem. The deformation (8.12) is a simple shear in the xg-direction if u 43 has the form u(x1,xz) 8 k xz, (8.20) where k is the amount of shear in the X3, x1 - plane. Then from (8.17), one gets 11 = 12 -- 3 + k2 (8.21) and the stress of primary interest, 132, is found from the first of (8.18), (8.20), (8.21), to be r32 . t(k) . 2ku'(3+k2). (8.22) The graph of 1(k) vs. k is henceforth to be referred to as the response curve jp_simple shear. Then the corresponding secant, tangent, and infinitesimal shear moduli (all assumed to be positive) are defined by 115(k) ails-fl, ut(k) $3391, no = ut(0). (8.23) Finally, returning to a general anti-plane deformation (8.12) and comparing (8.17) with (8.21), leads one to define the ”effective shear“ as ke(§) = |vu(§)| on D. ' (8.24) CHAPTER 9 FORMULATION 9.1 The crack problem. Consider now an infinite strip of width 2b, whose cross section 0 contains a crack of length 28. Let (X1,X2,X3) be rectangular Cartesian coordinates chosen such that the x3-axis is parallel to the edges of the crack with the origin lying on the crack, midway between its edges (see Figure 6). Suppose that the strip is subjected at infinity to a simple shear parallel to the edges of the crack, and that both the crack and the long sides at x1 = tb, remain traction-free in the deformed configuration. Thus by (8.11), on the plane surface of the crack, the Piola stresses must satisfy 012 = 022 a 023 8 O for x2 = Oi, - a < x1 < a. (9.1) Reference to (8.17) shows that the first conditon in (9.1) is automatically satisfied, while the second conditon requires that do = d1 = O. (9.2) Because of (9.2), one can reduce, with the aid of (8.17), the Cauchy and Piola stress components given in (8.18) to the following forms: T3a’3 103 a 036 = dog 8 2H'(3+|Vu|2)u,o on D, coo . coo . 0 on 0, (9.3) 133 . zu'(3+IVu|2)|Vu|2, 033 = 0 on 0. A Recalling that r a 1(k) represents the response function in simple shear of the current material, in view of (8.23) one can now write 44 45 115018) = 2H'(3+k:), (9.4) 8 I 2 2 u 2 ”t(ke) 2w (3+ke) + 4 ke w (3+ke) where the first invariant I1 is now given, from (8.17), (8.24), by 2 I1 8 3 + ke . (9.5) Here ke designates the effective amount of the shear |Vu|. The constitutive law of the current incompressible material (8.14)1 yields the components of the Cauchy (true) stresses accompanying an anti-plane deformation, through (9.3), (9.4), in terms of the out-of—plane displacement as T301' T013 ’ Ps(ke)uoa. TUB = 0, T33 3 "5(ke)k: o (9.6) Consequently, the boundary value problem governing the out-ofeplane displacement u(x1,x2) consists of the equilibrium equation (8.9)1 which, in view of (9.6), is [us(ke)u,1],1 + [us(ke)u,2],2 = O on D (9.7) with the traction free boundary conditions on crack and side surfaces T32 = u5(ke)u,2 8 O at xz = 01, - a < x1 < a, (9.8) 131 a us(ke)u,1 = O at X1 = ib, - a < X2 < a, In addition to the free-surface conditions (9.8), one imposes the requirement that, at infinity, the displacement field should approach that of a simple shear parallel to the crack surface and perpendicular to the cross-section D, u ~ kuxz as xi + xg + w, - b < x1 < b, (9.9) 46 where kw is the remotely applied shear. The differential equation (9.7), the boundary conditions (9.8), (9.9) and the further requirement that u be suitably smooth on D and bounded near the crack tips, comprise a nonlinear boundary value problem for the out-of—plane displacement u(x1,x2). 9.2. The enepgy release rate. There is associated with the anti-plane Shear problem formulated above, a path independent integral of the type first discovered by Eshelby [28] and later exploited by Rice [17] in connection wfith crack problems. The path independent integral associated with finite anti-plane shear is J a JP [H(3+k§) n1 - t3anau,1]ds. (9.11) Here 1 is any simple closed curve which encloses the right crack tip in its interior but does not include the left one and no are the components of the unit outward normal to I while 5 is arc length along 1. Now let P(a) denote the total potential energy of the body and loading system currently under consideration. Let P(a+Aa) denote the total potential energy of a system which is identical to the previous one, except that the crack in this body has length 2(a+Aa). Then the enepgy release rate G associated with the first body is defined by _lim P(a+Aa) - P(a) G ' Aa+o Aa (9.12) . .22 - aa' Rice [17] has shown that J = G, provided the path of integration 1 is as 47 described above. Therefore, the value of the path-independent integral J, (9.11), taken around the crack tip can be viewed as representing the energy release rate. Our purpose in this problem is to calculate the value of G (or equivalently J) associated with the particular boundary value problem (9.7)-(9.9). CHAPTER 10 29811191 In the absence of the crack, the deformation of the body is one of simple shear and the associated displacement field is u . k.x2 on D. If we assume that the presence of the crack causes only a small disturbance of this crack-free equilibrium state, we would have u(x1,x2) a kaxz + O(x1,x2), where [Val << 1 on O. (10.1) While such a hypothesis is undoubtedly invalid in the vicinity of the crack, it seems reasonable to assume that it would be true at points distant from the crack tips. Since our interest here lies ipl_el_y in calculating the value of the path-independent integral J, and since for this purpose we may consider a path that is essentially far removed from the crack tip (provided the crack length 2a is not too close to the strip width 2b), the error introduced by such an assumption is probably small. Thus motivated, we now assume that the displacement field is of the form (10.1). Then from (10.1), we get A A u,1 a u,1, u,2 = k» + u,2 (10.2) so that 2 2 2 ~ ~ 2 e 2 ke = |Vu| - k“ + 2 k.P:2 + |Vu| , |Vu| = u,au,a. (10.3) Next, we calculate the Cauchy stresses to leading order with the aid of (9.4), (9.6), (10.3) as 48 49 T31 us(ke)u.1 2R'(3+kiazkafi,2)fi,1 2(fi'(3+k§) + 'lI-(3+ki) (2195.916,1 (10.4) 29' ($49,296,1 us(k.,)U.1 ~ 2 where us(ka) - 2H'(3+k~) is the secant modulus at infinity. Similarly, we get 132 = bs(ke)u.2 ZH'(3+kE) k” + 4w'(3+ki) kflfi,2+ 2fi'(3+kf,)fi,2 (10.5) To. + “t(k..)“'2 where "t(k.) - ZR'(3+kE) + 4N"(3+ki)k3 is the tangent modulus at infinity. We have set Is. 2 I“ 8 2H (3+k”)k~, so that r. is the remotely applied shear stress. Similarly we can linearize the nonlinear boundary value problem (9.7)-(9.9) to get us(kalu.11 + ut(k~)U.22 = 0 on D. -T a 691:0 0nx1=tb,lle<°°, N u + O as |x2| + a , |x1| < b. 50 Correspondingly, we can calculate the linearized version of J as 2 follows. First, H(3+ke) in the integrand of (9.11) is linearized by H(3+k§) w(3+ki +2k u,2+ke 2), = w(3+k2 ) + N' (3+ki )(2k u,2+ke 2) +— 2w'(3+ki )( 2K + (1,2 + k2 2 (1007) fi(3+ki) + t¢O,2 + R'(3+ki)fi,§ +‘%’"t(k.)332 , and T36060.1 by r3°nau,1 = {ZR'(3+kE)O,1n1 + (r.,.,+ut(lr,,)h,zlnzla,1 , (10.8) so that the J integral (9.11) is now linearized to yield . ~2 ~~ ~2 J £ 1/2{ut(k~)u,2n1-20t(ka)u,Zu,1n2-us(ka)u,1n1}ds. (10.9) Our present task therefore is reduced to solving the 112322 boundary value problem (10.6) for O(x1,x2) and to then evaluate the integral (10.9). ‘ The linear boundary-value problem (10.6) may be further simplified by the scaling £1 = x1. 52 - mxz, m2 - “5(ko)/ut(kw) and by setting U(51,E2) = u(x1,x2) + kdmzxz. (IO-10) This leads to A A “91 = U919 “all a U911: u,2 = mU,2 - kednz, u,22 = mZU,22, (10.11) 51 so that the boundary-value problem (10.6) yields V2U(E1,Ez) . 0 on 0, U,2 a O on 62 = 01, |61| < a, 0,1 = 0 on 51 a to, |£2| < a, (10.12) U + (kam)€2 on 52 * °. [Ell < b. The mathematical problem (10.12) is identical to the linear elasticity problem governing the Mode III center-cracked strip problem. The solution to this problem is well-known and we find ([31]) K III ”S” U ~ 2 m gosh-g- as r + O, |8| < 1!. (10.13) where (r,8) are polar coordinates in the (61.62) plane such that r case = 51 - a, r sine-£2 and km = c“ A‘aég. tan (13%)]1/2. (10.14) On using (10.11), (10.13) we may calculate y KIII sin-; "9 a " ( )7 7 . K O 111 “’5 '2' 2 "'2 "‘ '“"‘("")7"us k” In 7—2" " "oom Thus we may now evaluate the J integral (10.9) (by choosing the contour to be an ellipse of major axis a and minor axis ma in the (x1,x2)- plane): 52 8 1! Sin 3 KIII '2' 2 r cos 8 .. 1,, var-v.0 1W .. 2..) ——-—... cos a (k)(u:ltnm '2' -2k¢m)r9-—-se— KIII “"2” K111 “52 2"tam”.- us(k¢)7m an) '7—_(s"'u (k )7m - kamz) r sin 8}d8, v/ll (k~)u (kg) K =1} . my: .8442:- sin2 g-)cos 8 + 2 sin gcos ‘3‘ sin 8}d8, : ‘2' 'lus(k~)"t(k~ )(TTIIK-I—IT-F’ which in view of (10.14), gives the desired result 2 ‘1' co 1 2b J = 2. [USOCI non...) ir a [Tr'a' tan gg]. (10.16) (10.17) Here us(ka.) and ut(k..) are the values of the secant and tangent moduli far from the crack and r... is the remotely applied shear stress. CHAPTER 11 DISCUSSION Equation (10.17) is a simple formula for the value of J in terms of constitutive, geometric and load parameters. He now specialize it to certain special material. In the case of a linear material, k/ko a r/to, we have from (8.23) that "5(ko) = ut(k.) ' To/ko (ll-1) and so (10.17) yields (the “small-scale yielding" result) 1 1.. 2 2b «a J/tokoa = 2- (7), IEE tan (5)]. (11.2) On the other hand, in the case of a pure power-law material described by k/ko = u(t/to)", we have 1' o (I-n)/n n k“ ’ (aka) (11.3) 85(k.) = ‘1' o k (1-n)/n’ n(ako) " ut(k.,) = so that (10.17) yields (the "fully plastic solution“) ‘1' o + J/‘tokoa - 55(7): "/n‘ r [E—g- tan (3%)]. (11.4) If one were now interested in a, say, Ramberg;0sgood material, 53 54 k/ko = r/ro + u(t/to)" (11.5) the traditional procedure for determining J has been to interpolate between (11.2) and (11.4). [The need for such a procedure stems from the fact that (11.2) and (11.4) are usually found directly, by exploiting certain special features of those material, rather than from a more general result such as (10.17): no such specialized methods have been available to study more general materials such as (11.5)]. The interpolation formula proposed by Shih [24] for the Ramberg-Osgood material is J] k (1,)21f1(a/b.1) (1%)",1 f1(a/b.n) (116) 100a 3 —- ___.._._._.... . o (I lib) o (I-a/b)n where the function f1(a/b, n) is determined numerically through a finite element analysis (see Table 3 in [24]). Improvements to (11.6) based on the notion of an effective crack length, have also been pr0posed in [24]. The advantage of the present procedure is that the result (10.17) is not restricted to a particular shear response function. For a Ramberg-Osgood material one obtains, from (10.17) and (11.5), the expression (in the 'elastic-plastic range") T T J/tokoa a-%(;:)2[1+o(n+1)(1:)"'1mflzo(( :) 20-2 (11.7) 11/2_ 2b tan (% Figure 7 shows the variation of the energy release rate J/rokoa with the applied load talro (in the case n a 3, a . 3/7 and for a/b a 1/8 and 55 1/4). The solid curve correSponds to the formula (11.7) derived here while the dashed curve describes the result according to the interpolation formula (11.6). The agreement is seen to be good. LIST OF REFERENCES 10. 11. 12. 13. REFERENCES Knowles, J. K. and Sternberg, E., “An Asymptotic Finite-deformation Analysis of the Elastostatic Field Near the Tip of 8 Crack, ' J. of Elasticity, Vol. 3,1973 p. 67. Knowles, J.K. and Sternberg, E., "Finite-Deformation Analysis of the Elastostatic Field Near the Tip of a Crack: Reconsideration and Higher-Order Results,” g, pf Elasticity, Vol. 4, 1974, p. 201. Knowles, J.K. and Sternberg, E., 'On the Ellipticity of the Equations of Nonlinear Elastostatics for a Special Material,"gyup§ Elasticity, Vol. 5, 1975, p. 341. Knowles, J.K. and Sternberg, E., 'On the Failure of Ellipticity of the Equations for Finite Elastostatic Plane Strain,“ Arch. Rat. Mech. Ana.,_Vol. 63, 1977, p. 321. Knowles, J.K. and Sternberg, E., “On the Failure of Ellipticity and the Emergence of Oiscontinuous Gradients in Plane Finite Elastostatics,“ gyhpj Elasticity, Vol. 9, 1979, p. 2. Abeyaratne, R .C., “Oiscontinuous Deformation Gradients in Plane Finite Elastostatics of Incompressible Materials,“ J. of Elasticity,_Vol. 10, 1980, p. 255. Hutchinson, J. H. and Neale, K. R., “Finite Strain J2 Deformation Theory in Finite Elasticity,‘ Martinus Nijhoff, The Hague, 1982, p. 237. Rudnicki, J. H. and Rice, J. R., “Conditions for the Localization of Deformation in Pressure-Sensitive Dilatant Material, ' J. Mech. Phys. Solids, Vol. 23, 1975, p. 371. Abeyaratne, R. and Triantafyllidis, N., 'An Investigation of Localization in a Porous Elastic Material Using Homogenization Theory,“ gyhpI_Applied Mechanics, Vol. 51, 1984, p. 481. Knowles, J.K. and Sternberg, E., “Oiscontinuous Deformation Gradients Near the Tip of a Crack in Finite Anti-Plane Shear: An Example,‘ Lpfi Elasticity, Vol. 10, 1980, p. 81. Knowles, J.K. and Sternberg, E., "Anti-Plane Shear Fields with Oiscontinuous Deformation Gradients Near the Tip of a Crack in Finite Elastostatics," gyhpj Elasticity, Vol. 11, 1981, p. 129. Abeyaratne, R .C., ”Oiscontinuous Deformation Gradients in the Finite Twisting of an Incompressible Elastic Tube,“ J. of Elasticity,,Vol. 11, 1981, p. 43. Abeyaratne, R .C., "Oiscontinuous Deformation Gradients Away from the Tip of a Crack in Anti-Plane Shear,“ J. of Elasticity,,Vol.11, 1981, p. 373. 56 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 57 Fowler, G.F., ”Finite Plane and Anti-Plane Elastostatic Fields with Oiscontinuous Deformation Gradients Near the Tip of a Crack,” g, p: Elasticity, Vol. 14, 1984, p. 287. Hutchinson, J.R., ”Plastic Stress and Strain Fields at a Crack-Tip,” gg_Mech. Phys. Solids, Vol. 16, 1968, p. 337. Stephenson, R.A., ”The Equilibrium Field Near the Tip of a Crack for Finite Plane strain of Incompressible elastic materials,” g, p: Elasticity, Vol. 12, 1982. p. 65. Rice, J.R., ”A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks.” ASME Journal 55 Applied Mechanics, Vol. 35, 1968, pp. 379-386. Hutchinson, J.R., ”Fundamentals of the Pheonomenological Theory of Nonlinear Fracture Mechanics,” ASME Journal pj_Applied Mechanics, Vol. 50, 1983, pp. 1042-1051. Begley, J.A., and Landers, J.D., ”The J Integral as a Fracture Criterion,” in: Fracture Toughness, ASTM STP 514, 1972, pp. 1-23. Ranaweera, M.P. and Leckie, F.A., ”J Integral for Some Crack and Geometries,” International Journal prFracture, Vol. 18, 1982, pp. 3'18. ' McMeeking, R.M., ”Path Dependence of the J Integral and the Role of J as a Parameter Characterizing the Near Tip Field,” in: Flaw Growth and Fracture, ASTM STP 631, 1977, pp. 28-41. Griffis, C.A., and Yoder, G.R., ”Initial Crack Extension in Two Intermediate Strength Aluminum Alloys,” ASME Journal_p[ gpgineeripg.Materials and Technology, Vol. 98, 1976, pp. 152-158. Clarke, G.A., Andrews, H.R., Paris, P.C., and Schmidt, D.H., ”Single Specimen Tests for J1c Determination,” in: Mechanics pi Crack Growth, ASTM STP 590, 1976, pp. 27-42. Shih, C.F., ”J Integral Estimation for Strain Hardening Materials in Antiplane Shear Using Fully Plastic Solutions,” in: Mechanics 'pI_Crack Growth, ASTM STP 590, 1976, pp. 3-22. Shih, C.F., and Hutchinson, J.R., ”Fully Plastic Solutions and Large Scale Yielding Estimates for Plane Stress Crack Problems,” ASME Journal p[_Engineering_Material and Technology, Vol. 98, 1976, pp. 2893295? Amazigo, J.C., ”Fully Plastic Crack in an Infinite Body Under Anti-plane Shear,” International Journal pj_Solids and Structures, V01. 10, 1974, pp. 1663‘16150 Abeyaratne, R., ”On the Estimation of Energy Release Rates,” ASME Journal 2: Applied Mechanics, Vol. 50, 1983, pp. 19-23. 28. 29. 30. 31. 58 Eshelby, J.D., ”The Continuum Theory of Lattice Defects,” Solid State Physics, Vol. 3, Academic Press, 1956. Knowles, J.K., ”On Finite Anti-Plane Shear for Incompressible Elastic Materials,” g, Austria]. Math. Soc. 19 (Series B), 1976, pp. 400-415. Knowles, J.K., ”The Finite Anti-Plane Shear Field Near the Tip of 3 Crack for a Class of Incompressible Elastic Solids,” International Journal 2: Fracture, Vol. 13, 1977, pp. 611-639. Hutchinson, J.R., ”Nonlinear Fracture Mechanics, Technical University of Denmark, 1979. FIGURES r(k)# O< n < l/2 To ———————— l I | ' l-Zn . #Okao/k) | #0“ : l I I l | | 1 > o k, k Figure 1. Response curve in simple shear for the piecewise power-law material. REGION 0 Figure 2. Geometry of the global crack problem. ll‘P(9) 7"? \ l’v'w" ‘. ”/2 -. W0 :12" ‘0‘ (":1 4. die \ L x 1 f— 8 8 O 9 Figure 4. Sketch of u(e) vs. 9 as defined by (4.24) with 0 < n