Strain energy density criteria for dynamic fracture and dynamic crack branching

Theoretical and Applied Fracture Mechanics 5 (1986) 117-123 North-Holland

117

STRAIN ENERGY DENSITY CRITERIA FOR DYNAMIC FRACTURE AND DYNAMIC CRACK BRANCHING M. RAMULU and A.S. KOBAYASHI Department of Mechanical Engineering, University of Washington, Seattle, WA 98195, U.S.A.

Dynamic extension of Sih's fracture criterion based on strain energy density factor, r~.(dW/d V ), is used to analyze dynamic crack propagation and branching, influence of the nonsingular components, which are known as the higher order terms (HOT) in the crack tip stress field, on the strain energy density distribution at a critical distance surrounding the crack tip moving at constant crack velocity is examined. This r,,(dW/dV) fracture criterion is then used to analyze available dynamic photoelastic results of crack branching and of engineering materials.

1. Introduction

The mechanics associated with the curved crack paths or bifurcation events received continuous attention in the literature. Comprehensive review on crack curving and branching were published by the authors [1] and Dally [2]. Literature on crack branching during rapid fracture discusses its correlation with either the dynamic crack tip stress field, initiation of secondary cracks, or with stress wave loading. These studies deal primarily with the stress intensity factor, the strain energy release rate at the instant of branching and the branching crack velocity. The crack branching angle as well as the crack curving angle can also be determined by a crack directional stability criteria under combined tension and shear loadings, i.e., for a mixed mode stress field. This crack stability criterion is governed by either the maximum circumferential stress at the crack tip [3] or at a specified distance from the crack tip [4-6], the maximum energy. release rate [7,8], and the minimum strain energy density factor surrounding the crack tip [9,10] or at a specified distance from the crack tip [11]. Sih [12] extended the minimum strain energy density factor criteria to rapidly propagating cracks for predicting the crack branching angle with a singular stress field. Rossmanith [13] and the author [14-16] have shown that the nonsingular component in the stress acting parallel to the crack path has considerable influence on the directional stability of a moving crack and incorporated this component

for estimating the curving and bifurcation angles. The additional higher order terms in the crack tip stresses, which are used to account for the spread of advanced cracks, are the cause of incipient branching in brittle materials, and enlarge the fracture process zone which results in roughening of the fracture surface. The presence of unsuccessful multiple cracks or crack interaction with the stress waves [17-19] under mode I loading are often associated with the slight unsynunetry in the fringe patterns of the dynamic photoelastic experiments of rapidly propagating cracks. This unsymmetry in the fringe pattern was modeled by Rossmanith [17] and the authors [19] by adding the higher order terms of mode I and mode II stress components to the mode I singular crack tip stress field. This inevitable involvement of the higher order terms forms the basis of the dynamic fracture analysis in this paper and the prediction of crack-growth direction of a constant crack velocity using the strain energy density factor, rc(dW/dV ), criterion.

2. Dynamic mixed mode crack tip stress fields

The crack tip dynamic state of stresses under mixed mode conditions was given by Nishioka and Atluri [20] in terms of the local rectangular (x, y) and polar (r, 0) coordinates. The three rectangular stress components under mode I and mode II conditions are given as:

0167-8442/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

118

M. Ramulu, A.S. Kobayashi / Strain energy densi(v criteria...

) i,,/2-,~cos ("-)02 (l+2S(-~2)r("/2-"cos (n~-10,-2h(n)~ 3 1 = E A,. B,(c______Q)n(n + 1)

%.

.=1

~

+ S:tr ": "cos(

?/

+

"

n

2

2) s i n ( -n~ - l ) O 2 _ 2S, r~~/2-'' sin(~n _ 1 )O, + ( l + SS~-7---h(n)r~"/2-" ( 1 + 2 S ( - S 2 2 ) r ~ "/z-l, s i n ( 2 - 1 ) 0 1

B,,(c) +

L.# A l l , , .=1 ~

- 2h(fi)r~ "/2-'' s i n ( 2 - 1 ) 0 2

n(n + 1) - ( l + S2)r("/2-1' s i n ( ~// - l)O, + 2 h ( h ) ~ " / 2 - " s i n ( ~// - l)02 2

2Slr~./2_l)cos( 2 _ 1)01

(1 +$2S22) h(~)r~.,/2_,, c o s ( ~n _ 1)02 (1)

where C2

C 2

St = 1

h(n)=

S~ = 1 - - c~'

C?' 2S1S2 1+S22 1 +$2 2

1 + S2

S,(C) = - D(C) '

232

a,,(C) = - D(C)'

D(C) = 4 S , S z - (1 + Sd) 2,

fornodd, for n even,

h(-h)=h(n+l),

r 2 = x 2 + y 2,

r~ = x 2 + S2y 2,

r2 = x 2 + S2y 2,

tan O, = S, tan O,

tan 02 = $2 tan O,

r and 0 are the polar coordinates with the origin at the moving crack tip, C, C 1 and C2 are the crack velocity, dilatational and distortional wave velocities, respectively. The general solution expressed in eq. (1) yields the singular stresses when n = 1, i.e., A n = K I and A m = K . , which are stress intensity factors of mode I and mode II, respectively. The constant stress, o0x, is related to the higher order term (HOT) for n = 2.

6B.(C)(S 2 OOx "~-

21~ --

tiates when r~(dW/dV) reaches a critical value of [rc(dW/dV)]c. The elastic strain energy density, W, per unit volume, V, of the material is dW

dV-- 2E

S2 I ) AI2,

All 2 = 0

L

x+O'~v+O'z2:)

l+vV(ax.,+o,.,.+o. ._.)2+2o:,,](3)

(2)

for plane strain condition

o==u(o~+avy )

(4)

3. Dynamic strain energy density factor criterion

and thus The strain energy density factor, rc (dW/dV), criterion developed by Sih [9] is based on the local density of the strain energy field at a core radius rc from the crack tip. It assumes that fracture ini-

dW dr-

l+v[ 2E ( 1 - v ) ( ° L + O ' ~ " )

-2v(oxxo,,v ) + 2o~,,].

(5)

M. Ramulu, A.S. Kobayashi /$trai~ energy density criteria...

where E and v are the modulus of elasticity and Poisson's ratio, respectively. As discussed earlier, for n = 2 the first nonsingular term exists only in q~x, stress component of mode I loading and is zero in all other components of mode I and mode II crack tip stress fields. These and other higher order terms for n >/2, on the strain density factor surrounding the crack tip are believed to influence the spreading of advanced cracks which in turn leads to crack curving and branching. By denoting the energy density function ( d W / d V ) in the presence of higher order terms as ( d W / d V ) , , the intensity of the strain energy density factor for the state of plane strain at a critical distance, re, can be written as: rc(dW/dV) .. The subscript index, n, indicates the number of the terms considered in determining the strain energy density factor. By using eqs. (1)-(5) and evaluating for n = 1, the strain energy density factor S became

S = [arrAnt + 2a,2AnA,n +

a22A2n].

(6)

The coefficients a u depend upon C, C t, C2 and Poisson's ratio v. The fracture criterion based on the minimum strain energy density factor notes that the crack will extend to the location of the minimum strain energy density factor, [rc(dW/dV)]mi n. In the presence of higher order terms, strain energy density factor, criterion can be written as

119

approximated by setting C--0.01C2 in eqs. (1) and (7) where the results agreed well with that of [12,13]. Crack velocities of C / C 2 = 0.01-0.8 were chosen to generate the dynamic strain energy density factor in the vicinity of the crack tip. The representative static and dynamic strain energy density factor distributions in the vicinity of the crack tip at a distance rc = 1.3 m m are shown in Fig. 1 for Ai2 < 0, AI2 = 0, and AI2 > 0 under mode I loading. Figure 2 shows the angular variation of static and dynamic (rcdW/dV) with the third higher order term A~3 which is compressive in nature, in addition to remote stress under pure mode I loading (i.e., K n = 0). Figure 2(a) shows the strain energy density factor distribution for A 12 = - 0 . 2 5 and AI3 = --0.3 for v = 0.36 and Fig. 2(b) represents the distribution of strain energy density factor for A~2--0.25 and At3 = - 0 . 3 . The minimum value of strain energy density factor is almost constant within + 10 ° for At2 < 0 at lower crack velocities. However, the minimum strain energy a

15-

rc = 1 3 Azl = I 0 C/Cz 08

A~z:'0.25 n =2

I0-

~ 0

d r~ - ~ d0

"=0

at

0--0c.

(7)

The half crack branching angle, 0~, at which [r~dW/dV]~ possesses stationary values, can be obtained from eq. (7). As a check on the accuracy of eqs. (5) and (7), the dynamic r~(dW/dV) should coincide with its static value in [12], when the crack velocity C ~ 0 for n = 1.

b

~

All : 1.0 n :I

15" 0.8

~o.~,5-

C

g •~

A~ :

15

1.0

AI~= 0 2 5 08

n

: 2

4. Numerical results In order to study the variation of the strain energy density factor with angular orientation, the theoretical ( d W / d V ) values were computed for each 0.25 degrees, using a Poisson's ratio, v = 0.36, Youngs Modulus, E = 3.74 GPa, critical radius rc = 1.3 mm, A n = K I = 1 M P a f m , and K u / K l = 0 value. Static strain energy density factor was

-180 -140 -I00 - 6 0

-20

20

60

I00

140

180

Angle, 0c

Fig. 1. Angular distribution of strain energy density factor

r~.(dW/dV) in presence of first nonsingular stress component 0o. ~ •

120

M. Ramulu, A.S. Kobayashi / Strain energy density criteria...

~

u :056 Au : 10 Am: -0.25 A13:-0 30

15-

C/Cz 08

10-

a

20"

DILATATIONAL 15-

C/----~22

rc

0.8

I0-

:

] 3

A~t = 1.0

k.

~5-

~0 : 036 qx~ : I0 A[2:0 25 AI~:-0 30

1508

q~ I

co

n =3

0

-

~

~ 15-

5-

I0-

o4

5"

-180 -140 -I00

-20 20 60 I00 140 180 Annl~ R Fig. 2. Angular distribution of strain energy density factor r~(dW/dV) in presence of higher order terms under pure mode 1 loading.

b

20-

DISTORTIONAL

rc =13 A~l = I0 C/C 2 04

0.8

-60

density factor for A~2 > 0 at lower crack velocities yields minimum r c ( d W / d V ) at greater than + 30 ° when crack velocity C/C2>~0.5. This is not surprising since the compressive remote stress stabilizes the crack path and the tensile stress enhances the directional instability of a propagating crack. Figure 3 shows the angular distribution of dilatational and distortional strain energy density factors for the strain energy density factors in Fig. 1 for A l l = 1.0. It can be observed that the energy is constant between + 13 degrees for lower crack velocities and the maximum magnitudes of the distortional energy and corresponding dilatational energy are minimum at C/C2 <~0.5. The minimum or maximum spread of dilatational and distortional energy is constant within the + 13 degrees angular region, respectively. A similar observation was made when we added A~3 term to the stress field. Figure 4 shows the branching angle versus crack velocity corresponding to a singular stress field with varying Poisson's ratio. The minimum strain energy density occurred at angle 0 = 0 for crack velocities C / C t ~<0.46; it becomes non-zero for C/C1 > 0.46. Note that the branching angle increased with increasing crack velocities. However, at higher velocities, the difference in branching angle is very small regardless of the difference in

-180 -140 -I00 - 6 0

-20 20 Angle,e

60

I00

140

180

Fig. 3. Angular distribution dilatational and distortional strain energy density factors under singular stress field.

magnitudes of Poisson's ratio. Table 1 shows the predicted crack branching angles using only the singular stress field with critical crack velocities of various materials represented by Poisson's ratio. These results suggest that the branching angle is very sensitive to the Poisson's ratio J,. Note that the branching angle of +11 to +23 degrees occurred in these materials within the crack velocity range of C / C 2 = 0.47-0.64, which is higher than the experimentally observed crack velocities [1] irrespective of the +-80-

+60

rc : 1 3 AII: I 0

j

0 2 5 ~

+4'

0 2 9 %

+-2C

J

0''

O0

t,

o12'o14

p

J

'o16'o:8

'

I.b

Crock velocity, C/Ca

Fig. 4. Effect of Poisson's ratio u on crack branching angle under mode 1 singular field condition.

M. Ramulu, A.S. Kobayashi / Strai'n energy density criteria... Table 1 Crack branching velocity and initial crack branching angle for various materials

0.22 0.23 0.24 0.25 0.26 0.27 0,28 0.29 0,30 0,31 0,32 0,33 0,34 0,35 0,36 0.37 0.38 0.40

Ci/C2

C/C2

0c

1.669 1.688 1.709 1.732 1.756 1.782 1.809 1.838 1.870 1.905 1.944 1.985 2.031 2.082 2.138 2.201 2.273 2.449

0.470 0.480 0.490 0.500 0.510 0.520 0.540 0.550 0.560 0.555 0.570 0.575 0.590 0.600 0.605 0.610 0.620 0.636

17 16 15 15 15 15 22 23 23 11 17 13 19 19 16 13 14 13

+-80-

re

121

= 1.3

u : 0.33

Au= 1.0 +-60

Remarks

/ +_40

AIz:075, AI3=0160" ~

oo:o.,o,:1i / '°°:°"T / / i

+_20-

Glass (heavy flint)

I

/

0

oo

o2

r

04

material used. The discrepancy between the experimental observation and theory can be partly attributed to the finite geometry and boundary effects which influences all singular and higher order terms in the crack tip stress field. In the following, some representative values of the higher order terms (HOT) are used to show the difference in the predicted branching angles with HOT and without HOT in the stress field. Figure 5 shows the branching angle versus the crack velocities for ~ = 0.33, in the presence of higher order terms under the pure mode I loading condition. The addition of positive higher order terms enhanced the branching angle while negative higher order terms suppressed branching. The crack is directionally stable even at higher crack velocities of C/C2 = 0.6 in the presence of negative HOT terms. Also, at higher crack velocities, the resulting branching angle is not influenced by the higher order terms, with differences of nearly 8 degrees. Quite often the dynamic photoelastic fringe patterns associated with a propagating curved crack exhibits a slight unsymmetry in isochromatics under mode I loading. This unsymmetry in isochromatics is modeled by adding the mode II higher order terms, without the model II singular term. Since the applied remote load is only mode I loading, it is appropriate to disregard the singular

o'8

C/C 2

Crock velocity,

Homalite-100, aluminum Polycarbonate

_A12:_075, A13:O60 AI2:-050,A[3:O60 -- Ax2=-0.23, A13:0

Am:0.50,A,3=O60 ~-7 _/

-

,o

Fig. 5. Effect of mode 1 higher order terms (HOT) on crack branching angle:

term and add only A l l 3 t e r m as was demonstrated previously in [18,19]. Figure 6(a) shows the angular distribution of the strain energy density factor for crack velocities of C / C 2 = 0.01-0.8 and A n = -0.25, Al3 = 0.3 and A n 3 = - 0 . 3 for ~,=0.36. The rc(dW/dV)3 distribution is slightly unsymmetric at lower crack velocity and increased at higher crack velocities. For a crack velocity of 0.4 6 C/C2 <~0.7, the minimum strain energy density factor is located at about 20 ° but at C / C 2 = 0.8, the minimum strain energy density factor occurs asymmetrically at about 55 and 60 °. Reversing the sign of AH3 from

a

rc : 1 3

AI~ l.O Am:=0.25

C/Cz

A~ =0.30 AI,,:'O.30

O.e

IO-

~'~

b .~ 15-

~

A /

\

/

~

rc :13 10 Alz:025 A~3=0.30

All:

08 /t~

0 ~

-180

140

I00

60

20

20

Angle,0

6O

I00

)z~O 180

Fig. 6. Angular distribution of strain energy density factor,

rc(dW/dV) in presence of mode I and mode II higher order terms.

M. Ramulu, A.S. Kobayashi / Strain energy density criteria...

122

Table 2 Effect of mode Ii higher order terms on crack extension angle At1 = 1.0, AI2 = 0.25, AI3 = 0.3 and AH3 = 0.3 a Poisson's ratio

Crack velocity

Crack extension angle

Remarks

C/C2

O~

0.25

0.50 0.54 0.58 0.62 0.66 0.72

2.50 11.75 33.25 + 44.25 + 51.00 _+57.75

Crack kinking Crack kinking Crack kinking Branching Branching Branching

0.29

0.56 0.60 0.64 0.70

5.25 27.00 ± 40.75 ± 52.00

Crack kinking Crack kinking Branching Branching

0.33

0.58 0.64 0.70

3.0 30.25 ± 47.00

Crack kinking Crack kinking Branching

a Crack extension angles are negative for A m > 0.

negative to positive in Fig. 6(b) simply reverses the peaks in the strain energy density factor distribution. At lower crack velocities the positive higher order terms of mode II stress field enhances crack kinking and unsymmetric crack branching could occur at higher crack velocities. Therefore, for An3 < 0, a positive kinking angle is obtained while for All 3 > 0, a negative kinking angle results. This effect is similar to the presence of positive and negative mode II stress intensity factors, respectively. The series of experimentation shows that branching could occur even in the presence of K n at higher crack velocities.

Qualitative results of crack branching angles are presented in the following for Poisson's ratio of u=0.25, 0.29 and 0.33 for All = 1.0, AI2 = - - 0 . 2 5 , Ai3 = 0.3 a n d higher order terms involving mode II nonsingular terms of A m = 0, An2 = 0 and All 3 = 0.3. Table 2 shows the effect of HOT on crack branching angles for varying crack velocities. At lower crack velocities of C / C 2 ~ 0.6, the mode II higher order terms contributed to crack kinking and branched asymmetrically at higher velocities. This observation is consistent with the present results in that the HOT effect is negligible at higher crack velocities. An enhanced branching angle also is clearly observed at higher velocities irrespective of the number of higher order terms present. Positive higher order terms enhanced magnitudes of the branching angle at all crack velocities. However, the crack branching angle was always 0 degree when C / C 2 <~0.4 for all compressive or negative higher order terms. Representative experimental crack branching data associated with running cracks in Homalite100, polycarbonate, steel, aluminum [21] and glass were evaluated and presented in Table 3. There is good agreement between measured and predicted angles using higher order terms, whereas significant differences are observed when HOT terms were not used.

5. Conclusions (1) The minimum value of rc(dW/dV ) strongly depends on the first nonsingular stress, OOx, for a given Poisson's ratio under any mode of deforma-

Table 3 Measured and calculated values of branch angle Material

Crack Velocity

re (mm)

C/C 2 Steel Aluminum Homalite-100

Polycarbonate

Measured branch

Predicted angle 2 0~

angle 20c

Without HOT

With HOT

Remarks

0.44 0.14 0.10

1.0 1.3 1.3

66 88 52

0 0 0

64 84 52

0.46

1.3

50

0

47

0.38 0.45

1.3 0.75

28 34

0 0

0 0

Pressurized pipe Pressurized pipe Wl-rdcb specimen Biaxially loaded Centrally notched specimen SEN specimen SEN specimen

M. Ramulu, A.S. Kobayashi / Strain energy density criteria...

tion. It also depends strongly on the crack velocity and varies systematically with the nonsingular stresses. (2) For small values of C / C 2 <~0.4, the fracture angle 6c, is found to be 0 for all negative higher order terms of mode I loading and 0c * 0 for all positive values of HOT, irrespective of crack velocity. Negative higher order terms suppresses the crack extension angle 6c whereas positive HOT always enhances the 8c, regardless of the Poisson's ratio. (3) Depending on the test conditions and the shape of the fractured specimens, theoretically predicted angle could deviate from experimentally measured fracture angles when the second order term is neglected.

[7]

[8]

[9]

[10]

[11]

[12]

[13]

Acknowledgment The work reported here was obtained under O N R Contract No. N00014-76-C-0000 NR 064478. The authors wish to acknowledge the support and encouragement of Dr. Y. Rajapakse, ONR, during the course of this investigation.

[14]

[15]

[16]

References [1] M. Ramulu and A.S. Kobayashi, "Mechanics of crack curving and branching-a dynamic fracture analysis", lnternat. J. Fracture 27 (3-4), 187-201 (1985). [2] J.W. Dally, "Dynamic photoelastic studies of fracture", Exp. Mech. 19, 349-367 (1979). [3] F. Erdogan and G.C. Sih, "On the crack extension in plates under plane loading and transverse shear", J. Basic Engrg. 88, Set. D., 519-527 (1963). [4] J.G. Williams and P.D. Ewing. "Fracture under complex stress-The angled crack problem", lnternat. J. Fracture Mech. 8, 441-446 (1982). [5] I. Finnie and A. Saith, "A note on the angled crack problem and the directional stability of crack", lnternat J. Fracture 9, 484-486 (1979). [6] J. Eftis, N. Subramanian, and H. Liebowitz, "Crack

[17]

[18]

[19]

[20]

[21]

123

boarder stress and displacement equations revisited", Eng. Fracture Mech 9, 189-210 (1979) K. Palaniswamy and E.G. Knauss, "Propagation of a crack under general in-plane tension", lnternat. J. Fracture Mech. 8, 114-17 (1982). M.A. Hussain, S.L. Pu and J. Underwood, "Strain energy release rate for a crack under combined mode I and mode II", A S T M STP 560, 2-28 (1974). G.C. Sih, "Some basic problems in fracture mechanics and new concepts", Engrg. Fracture Mech. 5, 365-377 (1983). G.C. Sih, "Strain energy density factor applied to mixed mode crack problems", Internat. J. Fracture Mech. 10, 305-321 (1974). G.C. Sih and M.E. Kipp, "Fracture under complex stress-The angled crack problem", Discussion, Internat. J. Fracture 10, 261-265 (1984). G.C. Sih, "Dynamic crack problems: Strain energy density fracture theory", in: G.C. Sih, ed., Elastodynamic Crack Problems, Vol. 4, Noordhoff, Leyden, 17-37, (1977). H.P. Rossmanith, "Crack Propagation and Branching", Proc. Syrnp. on Absorbed Spec. Energy~Strain Energy Density, G.C. Sih, E. Czoboly,and G. Gillemot eds., Akademial Kiado, Budapest, 283-294 (1982). M. Ramulu and A.S. Kobayashi, "Dynamic crack curving-A photoelastic evaluation", Exp. Mech. 23, 1-9 (1983). M. Ramulu, A.S. Kobayashi and B.S.-J. Kang, "Dynamic Crack Branching-A Photoelastic Evaluation, Fracture Mech. ASTM. STP 833, 130-142 (1984). M. Ramulu and A.S. Kobayashi, "Strain energy density fracture criterion in elastodynamic mixed mode crack propagation", Engrg, Fracture Mech. 18, 1087-1098 (1983). H.P. Rossmanith, "How mixed is dynamic mixed-mode crack propagation?-A dynamic photoelastic study". J. Mech. Phys. Solids 31,251-260 (1983). H.P. Rossmanith and A. Shukla, "Dynamic photoelastic investigation of interaction of stress waves with running cracks", Exp. Mech., 21,415-422 (1981). M. Ramulu, A.S. Kobayashi and D.B. Barker. "Analysis of dynamic mixed mode isochromatics", Exp. Mech. 25, 344-353 (1985). T. Nishioka and S.N. Athiri, "Path independent integrals, energy release rates, and general solutions of near-tip fields in mixed mode dynamic fracture mechanics". Engrg. Fracture Mech. 17, 1-22 (1983). M. Ramulu, A.S. Kobayashi and B.S.-J. Kang, "Dynamic crack curving and branching in line-pipe", A S M E J. Pressure Vessel Technology 104, 317-322 (1982).