J Mech Behav Mater 2016; 25(3-4): 77–81
David J. Unger* Path-dependent J-integral evaluations around an elliptical hole for large deformation theory
DOI 10.1515/jmbm-2016-0008 where W is the strain-energy density, x and y are Cartesian coordinates, and Γ is the path of integration, on which ds Abstract: An exact expression is obtained for a path- is the differential arc length, u is the displacement, and T dependent J-integral for finite strains of an elliptical is the traction. For cases where the integral becomes path- hole subject to remote tensile tractions under the Tresca dependent, its evaluation around Γ is no longer zero as deformation theory for a thin plate composed of non-work indicated in Eq. (1). hardening material. Possible applications include an ana- In [6, 7], an explicit analytical solution for an elliptic lytical resistance curve for the initial stage of crack propa- hole in a non-work hardening material was derived. In [8], gation due to crack tip blunting. the analogous problem was solved for large deformations. Keywords: nonlinear crack problem; path-dependent A pair of superposed coordinates was defined in [8], where integral; R-curve; resistance curve; Tresca yield condition. the uppercase letters represent the initial undeformed state of the material and the lowercase letters represent the deformed configuration, as indicated in Figure 1. The natural or logarithmic strain rate was determined 1 Introduction there as
Many finite element schemes of simulated crack problems vv ε log ==00, (2) have demonstrated the path dependence of the J-integral αα ρα++FR() Fv()ατ+ 0 [1] as a result of non-proportional loading. This type of behavior may occur because of finite deformations of where an elastic-plastic material [2] or through the use of flow ab22 theories of plasticity with small strains in elastic-plastic F()α = , (cab22os αα+ 22sin)3/2 analyses [3, 4]. In general [5], any nonlinear elastic mate- (3) uv==τρ, Ru+ . rial for finite deformations will exhibit path-dependent 00 0 J-integrals for crack problems. In Eqs. (2) and (3), R is the distance along a normal However, no exact mathematical evaluation of a or slip line from the elliptical hole to a particular point path-dependent integral has been derived to date from in the undeformed configuration (Figure 1). The angle α which one may draw more general conclusions about measures the inclination of an arbitrary slip line orthonor- how parameters affect these integrals. Here, exact mal to the elliptical boundary in both the (X, Y) and (x, expressions are derived for J-integrals of finite deforma- y) systems. The magnitude of displacement u is uniform tions of a nonlinear elastic material satisfying the Tresca 0 for all slip lines and acts in the direction of the slip line. yield condition. A traction-free elliptical hole subject The parameter v is the constant rate of displacement with to remote tensile tractions for a non-strain hardening 0 respect to the loading parameter τ. material is used to represent a crack under plane stress The nonlinear strain energy density may be deter- loading conditions. mined from the following integral, assuming the material The closed J-integral, as defined in [1] for a plane is nonlinear elastic. problem, is ττvd τ ∂ ==ετ log 0 u Wk22∫∫αα dk JΓ =⋅Wdyd-0T s= , (1) RF++()ατv ∫ ∂x 00 0 Γ (4) vu00τ =+2lkkn1=+2ln1, RF++()ααRF()
*Corresponding author: David J. Unger, Department of Mechanical where k is the yield strength in pure shear. No other and Civil Engineering, University of Evansville, 1800 Lincoln Avenue, strain component contributes to W. The coordinate Y in Evansville, IN 47722, USA, e-mail: [email protected] differential form in the initial configuration notation [8] is 78 D.J. Unger: Path-dependent J-integral evaluations around an elliptical hole for large deformation theory
Y, y A much more challenging integral to evaluate analyti-
B ′ Slip line cally occurs along path AFB of Figure 1, whose form for a =R+u ρ 0 constant but arbitrary value of R is C ′ Γ′ Uniform (1) displacement u0 α J = J Deformed contour R=const of elliptical hole B π/2 Γ u C R =+2lkRn1 0 +Fd()ααcos α, (9) b O X, x ∫ ( ) E F E′ F ′ -/π 2 RF+ ()α Elliptical hole D a A where F(α) is given in Eq. (3). Initial attempts at evaluating this integral directly, using a symbolic mathematical com- D ′ puter program, failed to produce meaningful results. Nev- A′ ertheless, under the specified sequence of transformation
Figure 1: Path of integration in both the initial coordinates (X, Y) of variables to follow, an exact expression can be found. and in the deformed state (x, y). Making the following substitutions in Eq. (9), where several functions are identified as standard Jacobian ellip- tic functions [9], =+αα+ αα dY sin dR ((RF ))cos d . (5) αα=→am(\zm)ddz= n dz
2 The J-integral (1) is now divided into two distinct parts coscα==os(am(zm\))cn zm, =1-(/ba), sinsαα==in(am(zm\))sn zF, ()=(/ba23) nd z, (10)
(1)(2) ∂u J ==WdYJ,-T⋅ dS. (6) ΓΓ∫∫∂X ΓΓ one obtains
Km( ) As in the case of small deformation theory [7], the u J =+2lk n1 0 ∂ ∂ ∫ + 23 traction T and u/ X vectors remain orthogonal to one -Km( ) Rb(/az) nd another during deformation and, consequently, they 23 (2) ×+((Rb/)az nd ) dn zz cn dz, (11) provide no contribution to JΓ . Because of this, the only nontrivial component of JΓ reduces to that of the first inte- where the modulus of the elliptic functions is m. The com- (1) gral of (6), i.e. JΓ . plete elliptic integral of the first kind that is found in the Now there are two distinct paths to be analyzed sepa- limits of integration in Eq. (11) is defined by (1) π/2 rately for JΓ , as shown in Figure 1. One particular class dθ K()m = . is along the Y-axis, i.e. BC (α = π/2) and DA (α = -π/2), such ∫ 2 (12) 0 1-msin θ that One should be aware that a common alternative repre- R2 u 2 (1) 0 sentation for parameter m is k as in [10]. If this notation is J π =+2lkdn1 R. α=± ∫ 2 (7) 2 R Ra+ /b employed, then it also carries over to the related Jacobian 1 elliptic functions modulus in (10). This integral has the elementary solution An additional transform of z to φ in Eq. (11) of the form
2 φ= cn zd z (1) 2k ab++()uR02 ∫ J π = bu ln α=± b 0 2 ++ 2 ab()uR01 1 -1 =→cos(dn zz)dn =cosm φ, (13) bu m ++2 + 0 ()abR2 ln1 2 ab+ R2 will simplify this integral to bu -1 22 2 0 cos(ba/-b ) u -lab++R1 n1. (8) 0 () 2 + J =+4lk n1 abR1 ∫0 23 +Rb(/am) sec φ ×+((Rb23/)amsec)φφcosmd φ, (14) Provided one keeps the paths BC and DA of equal length, the two contributions from Eq. (8) cancel each where an additional factor of 2 ahead of this integral other over the closed path Γ and consequently do not accounts for the symmetry of the integrand with respect (1) contribute to the path dependence of JΓ . to the X-axis that was used. D.J. Unger: Path-dependent J-integral evaluations around an elliptical hole for large deformation theory 79
Further, use of the transformation where tn represents a root of either the numerator or denominator of the argument of the log function. Subse- φ= -1 λ (1/)m cos, (15) quently, using standard properties of logarithms, one can expand the integrand of Eq. (20) into 12 distinct integrals, reduces the integral in Eq. (14) to all of which have a linear dependence of t in the argument 1 3 4k ua0 λ of the log function. Hardy [13] provides criteria from which J =+ln1 22∫ 32 one can predict that the integral in Eq. (20) should contain ab- ba/ Raλ +b only elementary transcendental functions and rational ()Raλλ32+bd × . (16) functions in its evaluation. This surprising prediction λλ22 1- proved true. Although both the numerator and denominator of the The following and the last of the transformations (see argument of the log function in Eq. (18) are of sixth order, [11]) reduces the algebraic function outside of the argu- exact roots of these reduced polynomials can be readily ment of the log function in Eq. (16) to a rational function found algebraically. The exact evaluation of Eq. (20) was of t while leaving the argument of the log a rational func- obtained in this fashion using Mathematica® 1.0 (Wolfram tion, albeit of a higher order Research Champaign, IL, USA); however, its length pre- 2t cludes the entire solution being reproduced here. λ= . (17) + 2 1 t Representative behavior using the entire solution is plotted in Figure 2 for a specific value of displacement Consequently, upon substitution of λ in Eq. (16), one u /a = 0.02. Note the symbol J appearing in the figure, as a finds that 0 0 normalizing parameter, is the value of J obtained in [7] for
2 3 1 small displacements and strains, i.e. 2kb 1 8(Rt0 ) J =+aa22-b 1 + 22∫ − 222 ab- bbtt(1+ ) J00=4.ku (21) 13++tD238( )3tt34++t 6 × 0 For the Tresca yield condition, the yield strength in ln23346 dt, (18) 13++tR8( )3tt++t 0 tension σ is equal to twice the yield strength in shear k. 0 The crack opening displacement δ can be defined as the where various parameters appearing in Eq. (18) are change in separation between points C and C′ plus D and defined by D′ of Figure 1, which is equal to 2u0. Therefore, Eq. (21) can
1/3 be rewritten as aR RD==, (/aU b21),/3 00b2 0
UR00==ρ +u . (19) 1.0
Symbolic mathematical computer programs can gen- erally integrate a logarithm of the linear function of the integration variable multiplied by a rational function of the same variable outside the argument. Similarly, stand- 0
u /a=0.02 0.5 J ard integral tables such as [12] typically list entries having 0 / J the same mathematical construct. By factoring the numerator and denominator of the argument of the log function in Eq. (18), a general repre- sentation of the integral assumes the following form 1.0 0.0 2 1 3 1.0 21kb 8(Rt0 ) J =+1 + 22 ∫ tt22(1+ )2 ab- aa22-b 0.5 - 0.5 bb R/a b/a 6 (20) ∏(-ttn ) ×lnn=1 dt, 0.0 12 ∏(-ttn ) Figure 2: Normalized J-integral as a function of elliptical axes n=7 aspect ratio and normalized distance from the internal boundary. 80 D.J. Unger: Path-dependent J-integral evaluations around an elliptical hole for large deformation theory
1/3 =σδ 22 J00, (22) ba--+ ab au ==0 cd, 2 . (24) ba+ --ab22 b which is reminiscent of the form of the crack tip opening displacement δt, defined in [1], for the Barenblatt-Dugale model of crack tip plasticity. In Figure 3, one finds relationship in Eq. (23) plotted
A special and important case (R = 0) of the complete versus the aspect ratio b/a for various values of u0/a. One analytical solution will now be given explicitly. Its length observes that as the elliptical hole geometry approaches is considerably shorter than the general result, making a line crack (b→0) the J-integral evaluation approaches it possible to present it here. This formula applies to the zero for all values of displacement. evaluation of the integral in Eq. (20) along a path follow- In Figure 4, the normalized J-integral from the com- ing one half of the elliptical hole, i.e. as AFB approaches plete solution is plotted as a function of the normalized DEC in Figure 1. This path produces the smallest value of distance from the elliptical hole (R/a) for the range of per- the J-integral possible for a fixed aspect ratio b/a and dis- missible aspect ratios (0 < b/a < 1) at a fixed value of dis- placement u0/a. It assumes the mathematical form placement u0/a = 0.02. As b approaches zero in Figure 4, the inside envelope of curves approaches the locus of the J ub =+0 asymptotic expansion of the complete solution, i.e. (/bu0 )ln12 J 0 R=0 a 2 u0 21bd 2-1 - J ∼4kR ln 1+→ as b 0. (25) + 1-dctan-3/π 4 2 1+d R au0 1-(/ba) -3tan(-1 --aa+ 22bb)/ The limit of J in the asymptotic expansion of Eq. (25) (23) as R goes to zero is zero. 1/32 1/32 -1 1(+ -1) d ++1(-1) dctan 2/3 1(+ -1) d 1/3 2/32 -1 1(+ -1) d 2 Discussion + 1-(-1) dctan, 2/32 1-(-1) d There exist no finite element analyses for large deforma- tion crack or notch problems involving the Tresca yield where the two parameters c and d found in Eq. (23) are condition. Should there be any in the future, the solution defined by obtained here can serve as a benchmark for comparison. However, if one compares how the value of the nor- malized J-integral changes with distance from the notch 1.0 boundary in Figure 4 to the elastic-perfectly plastic data 0.001 (n = 0) plotted in Figure 9 of [2], the two loci qualitatively 0.01 resemble one another despite several differences that 0.8 exist between the two analyses. 0.1 For example, both analyses indicate that for finite deformations, the J-integral tends toward zero as the paths u0/a 0.6 0 J
/ 1.0
J b/a 1 1.0 b/a 0 0.4 0.8
0.6 0 J / J u /a=0.02 0.2 0.4 0 10.0 0.2
0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 b/a R/a
Figure 3: Normalized J-integral along the elliptical hole boundary Figure 4: Envelope of normalized J-Integral values versus normalized versus axes aspect ratio and normalized displacement. distance from elliptical boundary at a fixed displacement. D.J. Unger: Path-dependent J-integral evaluations around an elliptical hole for large deformation theory 81
of integration approach the notch boundary. McMeek- M is taken as 2 [16] or higher [15]. The flow stress σflow may ing [2] attributes this effect due to crack tip blunting. In also be substituted into Eq. (26) in place of σ0 in order to contrast, in an elastic-perfectly plastic analysis performed incorporate effects due to strain hardening [15]. in [3] for a sharp crack, a normalized J-integral in their Figure 3 approaches a non-zero value of J as the path of integration approaches the crack tip. No blunting of the References crack tip was incorporated into this particular analysis. Both analyses [2] and [3], however, use a flow theory of [1] Rice, JR. J. Appl. Mech. 1968, 35, 379–386. plasticity under the von Mises yield condition. [2] McMeeking, RM. J. Mech. Phys. Solids 1977, 25, 357–381. Lastly, it should be pointed out that for a value of [3] Carka, D, Landis, CM. J. Appl. Mech. 2011, 78, 011006. b/a equal to zero, the analytical solution employed here [4] Carka, D, McMeeking, RM, Landis, CM. J. Appl. Mech. 2012, 79, 044502. degenerates [6, 7]. However, one may approach a line [5] Chen, HK, Shield, RT. Z. Agnew. Math. Phys 1977, 28, 1–22. crack to any degree of accuracy desired as long as b [6] Unger, DJ. Theor. Appl. Fract. Mech. 2005, 44, 82–94. remains finite. [7] Unger, DJ. J. Elasticity 2008, 92, 217–226. For the case b = 0, a statically admissible solution is [8] Unger, DJ. J. Elasticity 2010, 99, 117–130. proposed in [6]; however, it requires the formation of a [9] Abramowitz, M, Stegun, IA. Handbook of Mathematical Func- stress discontinuity. A stress discontinuity is also encoun- tions with Formulas, Graphs and Mathematical Tables. National Bureau of Standards, Series 55, U.S. Printing Office: Washing- tered in the plane stress perfectly plastic analysis [14] ton, DC, 1964. using the von Mises yield condition for a semi-infinite line [10] Byrd, PF, Friedman, MD. Handbook of Elliptic Integrals for Engi- crack. neers and Scientists, 2nd ed., Springer-Verlag: New York, 1971. If this J-integral is interpreted as an energy dissipa- [11] Zwillinger, D. Handbook of Integration. Jones and Bartlett: tion rate instead of an energy release rate, then a theoreti- Boston, 1992, 145–147. [12] Gradshteyn, IS, Ryzhik, IM. Table of Integrals, Series, and cal resistance curve (R-curve) is obtained in closed form Products. Academic Press: Orlando, 1980. for the initial stage of crack extension due to crack tip [13] Hardy, GH. The Integration of Functions of a Single Variable, 2nd blunting, e.g. in Eq. (25) as b→0 with u0 = Δa. Previously edn., Dover Phoenix Editions: Dover, Mineola, New York, 2005, [15], modeling of this region was limited to relationships 60–61. having the form [14] Hutchinson, JW. J. Mech. Phys. Solids 1968, 16, 337–347. [15] ASTM Committee E 1820-1. Standard Test Method for Measure-
J =∆Maσ0 , (26) ment of Fracture Toughness. Annual Book of ASTM Standards, Vol. 03.01. ASTM, West Conshohocken, PA, 2002, 1031–1076. which is an offshoot of Eq. (22), where the change of crack [16] Hertzberg, RW. Deformation and Fracture Mechanics of Engi- length due to blunting Δa is approximated by δ/2 [16] and neering Materials, 4th ed., Wiley: Hoboken, NJ, 1996, 366.