Further study on strain singularity behavior of moving cracks in elastic-viscoplastic materials

Theoretical and Applied Fracture Mechanics 14 (1990) 233 242 Elsevier

233

Further study on strain singularity behavior of moving cracks in elastic viscoplastic materials Y.C. Gao ftarbin Shipbuilding Engineering Institute, Harbin, P.R. Chma

The time dependent crack tip stress field is investigated for an elastic viscoplastic material whose properties are adjustable in accordance with the artificial viscosity. Two types of strain singularities are obtained; they depend on whether the viscosity coefficient is high or low and possess a common limit at a critical value of the viscosity. The power law singular behavior prevails for high viscosity while the logarithm singular behavior applies to low viscosity. The present findings are offered in an attempt to clarify the ways with which the crack tip singular behavior depends on the constitutive coefficients.

1. Introduction Character of the crack tip singular behavior could take different forms depending on the nonlinearity a n d / o r dissipative nature of the material. A variety of mathematical solutions could thus be conceived though their modelling of the physical phenomenon should not vary appreciably. One of the early works on moving cracks in nonlinear materials was concerned with the anti-plane shear of a quasi-static crack in a perfectly plastic material [1]. For the corresponding quasi-static problem for cracks undergoing in-plane extension, the solutions were given in [2,3]. The discrepancy was attributed to certain inherent assumptions in the quasi-static formulation [4]. While the inclusion of dynamic effects [5 7] overcame the deficiency of the quasi-static assumption, additional questions were raised in connection with plastic shocks. A special solution of the work in [5] was presented in [8] but it contained no singularities. Recognizing that the constitutive relation could influence the crack tip singular behavior, an elastic viscoplastic material was introduced [9] for the dynamic moving crack problem. The logarithm singular solution did not exist when the viscosity effects were large. It is the objective of this work to show that the strain singularity changes to a power law behavior as the artificial viscosity increases beyond a critical value at which both solutions possess a common limit. These findings are discussed in connection with the crack tip stress behavior deduced from other higher order theories [10-13] in mechanics for nonlinear and dissipative materials.

2. Eiastic-viscoplastic material model The elastic viscoplastic material model [9] considered is made of three elements as shown in Fig. 1. They are the elastic, plastic and viscous element. Let the total, elastic and plastic strain be denoted,

r/

13"

O"

F(~p) Fig. 1. Schematic of elastic viscoplastic model. 0167-8442/90/$03.50 c,~;1990

Elsevier Science Publishers B.V.

234

Y.C. Gao / Strain singulariO, behavior of mooing {'racks in elastic-viscoplastic materials

respectively, by e, % and ep such that {1)

6. : ~e -F Ep

and kp = X{}o

G = o/E,

(2)

with X0 being the flow factor and E the Young's modulus. The dot represents differentiation with respect to time t. If o, o,, and % represent the total stress, viscous and plastic stress in that order, then the following relations are assumed O=Ov+ %

3i

in which av = "9~p,

Op = F ( e p ) .

The viscosity coefficient is '0 and from the above relations that

4}

F(Ep) iS a

function depending on the plastic strain. It can be deduced

d ~=T+Xo

5)

with X defined as X =

1

--

F(ap)

>~ 0

(6)

and Ep can be written as o

Ep:E

7)

E"

In three dimensions, the expressions for the uniaxial model may be written in tensorial form = C: 6 + XS.

(8)

In eq. (8), S is the deviator of o and C is the fourth order compliance tensor. The dual scalar product is used. If o* and %* stand, respectively, for the effective stress and plastic strain, then a

*

(3S

"

S ) 1/2,

Ep =

.

~Ep ~ p ) | ,,2

f(2..

dr,

(9)

The rate of effective plastic strain is

.. = - ~ 2 Ep

[o._

t o}

It is now more pertinent to investigate the singular behavior of the crack tip strain a n d / o r stress field by taking dynamic effects into account.

3. Asymptotic behavior

In what follows, asymptotic expressions of the strain rate or stress will be obtained for the plane extension of a crack where inertia effects are included. The resulting singular behavior is obtained by enforcing the free traction condition on the crack surfaces in the limit as the distance to the crack tip becomes vanishingly small.

Y.C. Gao / Strain singulariO' behavior of moving cracks in elastic viscoplastic materials 3.1. General

235

equations

Consider first a perfect plastic and incompressible material under plane strain such that

F(e~) =

%~,

v = 0.5

(II)

and

e:: : 0

(12)

with the z-axis corresponding to the longitudinal dimension of the body. The Poisson's ratio is v and o> in the first of eqs. (11) is the yield strength. Under these considerations, eq. (8) becomes -<,=

+

X

<.,,

=

3

0,, +

(13)

and eq. (6) simplifies to (14)

X = ]-(1-~).~ The first of eqs. (9) gives

(15) All strains e,~ and stresses o,/ are referred to the rectangular Cartesian coordinates defined in Fig. 2. Referring to a system of stationary coordinates xj ( j = 1, 2), the strain rates 6f are related to the displacement rate /t; for small deformation by the expressions k,i=~-

~ + ~

.

(16)

The equation of motion is given by 0oij 0/t i 0x i - P 0t

(17)

with p being the mass density of the material. If the crack moves in a steady state manner with velocity v, a system of moving coordinates (x, y ) may be defined such that x = xl -

vt,

(18)

Y = x2.

[ x2

Crack ~.

°'~,__ v t

Ay

1 _l -[ o

-x

Fig. 2. Movingand stationary coordinates.

~--- X!

Y.C. Gao / Strain singularitv behavior 0,[ moving cracks in elasnc-eiscop[astl{ materials

236

Introduced in Fig. 2 are the polar coordinates (r, 0) for describing the asymptotic behavior, For an incompressible material, the displacements may be expressed in terms of a single function .9 a~2 r ~0"

1

u~-

u°-

Or"

19}

Consider the form (2 = r2-a g( O ).

2O )

Equation (20) may be substituted into eqs. (19) and differentiated twice with respect to ume. This leads to fir = -t,2r

] a[sinOf"(O)+ScosOf'(6)+(l-8)

iio=-Sv2r

sin0~'(0)],

] 8[sin0~"(0)-(l-8)cos0~'(0)].

(21)

Remember that the dot operation d 0 dt - - v~-x

(221

holds for the steady-state moving crack problem. The function ~(0) is related to .g(0) as ~'(0) = sin 0 g ' ( O ) - ( 2 - 8) cos 0 g(O}.

(23}

Application of eq. (16) in conjunction with eqs. {19) and (2) further gives 8r =

--EO

=

t, Sr ' af'(O),

e*,o"

-- -- ~ r

1 ~[£"(0)+(l--82)~(0)].

I~.4,''

The prima notation denote differentiation with respect to 0. 3.2. Material with artificial viscosi O, Let vl denote the artificial viscosity of the material considered in [9]. It has the property that ( B{} r,

for r < R{}

~,rh},

for r >/R {}

= ~~

(25)

in which ~0 is a constant and R{} determines the size of zone within which the viscosity is nonhomogeneous. Let the singular portion of the stress field be expressed as or=o,~

a[p(o)+s(o)],

oo=o,~r a [ P ( O ) - - S ( O ) I ,

ore=o,/

ST(O).

(26)

Equations (13) when expressed in terms of (r, 0) can then be employed together with eqs. (24) and (26) to satisfy the stress dynamic equilibrium condition in eq. (17). There results the differential equations for all nonzero values of r: S' - 2T+

c~ + 8 cos 0 8 sin 0 S - si~./"

-- (}

(T'+2S)(1-m2sin20)-8(P+S)-m2sin0(a+6cos0)T -

+ TSM~'c°s 0 1'

(27)

--62(1 - 8 ) M 2 sin 0 f = 0 sin0

(f"+f)(1-M --8(8

S-ST-

sin---O

1

sin 20 f ' ~-

(1-8)

2 sin20)+28 sin0(P+S)+2(a+Scos0)T-SM

-- M 2 s i n 2 0 ) f = 0.

sin0cos 0/

=li

2 sin0cos0f'

Y.C. G a o / Strain singularity behavior of mot, ing cracks in elastic-t,iseoplastic materials

237

r

o.E

\

\ 0.4

Region:

r -~

\

\

\

E n

\

0.2

0

0.2

0.4 M a c h no.

Fig. 3. D o m a i n

0.6 M

divided with different stress singularities.

Keep in mind that the four quantities P, S, T and f are functions of 0 where f is related to ~" in eq. (23) as

f(O) =

2~'(0).

(28)

OrS

Denoted by/* is the shear modules of elasticity. In eqs. (27), the parameter a and Mach number M are given by 2/xRo -

where c 2 = ~

T]OU

u '

M

-

(29)

C2

is the shear wave velocity.

180 °

¢J

90°

go ° Angle e

80 °

Fig. 4. A n g u l a r v a r i a t i o n s o f a r c t a n g e n t o f ] 2 a r 0 / ( o r - o0) I f o r p o w e r l a w s i n g u l a r i t y w i t h a = 3, 6 = 8.5 x 10

2 a n d M 2 = 10

1

E C. Gao / Strain singularity behavior o/ moving cracks m elastic-viscoplastlc materta#,

238

180~

~=20 = 0.016 M2= 0.001

90 ~

//~

_ .............

90 ° Angle e

180

°

F i g . 5. A n g u l a r v a r i a t i o n s o f a r c t a n g e n t o f 1 2 o ~ 0 / ( o ,. - o0) j f o r p o w e r t a w s i n g u l a r i t y w i t h c~ = 20, 8 :~:,I.(-, >: I~i

: a n d .~I: ~ I0

The free surface crack conditions are %=o<0=0

for0=

+~r

(30)

while symmetry requires the vanishing of o~o along the x-axis. This requires (30) can thus be enforced by requiring f ( 0 ) = 0,

f ' ( 0 ) = (1 + f f / 8 ) S ( 0 )

p(0)

T(0)

=1,

=

-

= 0.

f(O)

to be odd in 0. Equations

(31)

The exponent 8 on r is chosen to satisfy eqs. (31). Equations (27) subjected to the conditions in (31) are solved numerically. Results will be discussed for different values of the parameters c~ and M in eqs. (29). Their values could lie to the left side, right side or on the boundary / ' in Fig. 3. The limit a ---, 0 corresponds to 8 - , 0.5 whereas 8 .... 0 corresponds to the approachment of F from the left. Numerical values of the function + ( 0 ) defined by

q,(o) =

-tan--1

2or0

_----go

(32}

are shown graphically in Fig. 4 for c~=3, 3 = 8 . 5 × 1 0 2 and M 2 = 0 . 1 and in Fig. 5 for ( , = 2 0 , 3 = 1.6 × 10 -2 and M 2 = 10 -3. These variations of ~p(0) with 0 correspond to a power law strain singularity of r -~.

4. Other limiting and singular solutions When the boundary F is approached from the left side with 6 ~ 0. it can be seen from the second of eqs. (31) that f ' ( 0 ) ~ 0. To this end, it is expedient to consider small values of 8 m an expansion for 1-

f= ~f+

3"f~ n=O

(33)

Y.C. Gao / Strain singularitybehavior of moving cracks in elastic-viscoplastic materials"

239

and the functions P, S and T:

(P, S, T ) =

~'~ 3"(P,,, Sn, Tn).

(34)

n~O

Equations (27) and (31) can thus be applied to give f = - 2 B sin 0

(35)

and the relations S(~- 2To + si~-~S0 + 2B cot 0 = 0 (To'+ 25;o)(1 - M 2 sin 2 0) - a M 2 sin 0 To - M2B cos 20 = 0 O~

Po + ~ S o

+2Bc°t

(36)

0=0

(f0" + f 0 ) ( 1 - M2 sin20) + 2aTo + 2BM2 sin 0 cos 20 = 0 in which B is a constant. Equations (36) are the same as those given in [9] for ~ --+ m. On the boundary F, the expansion f in eq. (33) is not valid. In fact, the stresses would contain other forms of singularity as given by ~X3

(o,,oO, OrO)=oysr 8 L ~ _ , L ~ [ ( P ° + S ° ) , ( P ° - S ° ) , T , , ° ]

(37)

n=0

in which L stands for L = log( -R° 7-)

(38)

Here, R 0 is a normalization length parameter for the asymptotically small distance r near the crack tip. The rate of the displacement function is = v r 1-8L/3 L q ) + q o + Y~ L - n %

.

(39)

n=]

Making use of eqs. (16), (19), (22) and (39), there results

ii, = - v 2 r ' a L ' { L sin 0 [ q ) " + (1 - 8)q)] + 3L cos 0 q~'+ sin 0[ep" + (1 - 8)qo] + 8 cos 0 ep' + (1 + / ? ) ( c o s O r b ' - sin O q~) + . . . }

iio = v2r 1-aL ¢{ - 3 L [sin O eb' - ( 1 - 3 ) cos O q ~ ] - 3 [ s i n O e p ' - ( 1 - 3 ) c o s O q p ]

(40)

- ( 1 + / ? ) [ s i n 0 q ~ ' - (1 - 26) cos O q~] + ... } and

kr = -~o = v r - l - ~ L ' [ 3 L ~ ' + 3q0'+ (1 + / ? ) q ) ' + . . . ] ~ro=lvr ' e L " { - L [ q ) " + ( 1 - 3 2 ) q ) ] - q o " - ( 1 - 8 2 ) q p +

23(1+/?)q)+...}.

(41)

Two limiting cases of the above expressions may be discussed; they correspond to 8L >> 1 and 8L << 1. Since 8 is a constant and L is a function, it would be more convenient to consider the asymptonic behavior of the solution in certain domain such as ro < r < R 0 with r0 being microscopic in size. It has been referred to in [10] as the radius of the crack tip core region within which variations of the material microstructure must be accounted for. An estimate of its size has been made [11] from a knowledge of the macroscopic damage free zone. This was accomplished by considering the interaction of the rate change of volume with surface for each material elements [12]. Such an effect is particularly significant in the region

240

Y, C. Gao / Strain singularity behatqor oJ moving cracks in elastic-t~iscoplasm" materials

18~T~

.

.

.

.

.

.

/

¢( = 1 0 0 A = 0,12071575 M2= 0 . 1

/ /

/

/

/ /_ .....

/

/

1 /

b

c°s2~ - M sin2e= O - - i ' - . /

'

i 90 °

0

Angle

Fig. 6. Angular variations of arctangent of 12o~0/(or

~__

__ 180 ° e

oCj} I for logarithm singularity with (~ = lO0, ,t :-- L12(}7 and M:'

l(}

r0 < r ~< R 0 next to the crack tip. As mentioned with reference to eq. (25), R~, is the region outside of which the artificial viscosity is no longer homogeneous. Hence, if 3 L >> 1, eqs. (37), (40) and (41) may be applied in conjunction with (13) so that eq. (17) is satisfied. By collecting terms with like L" for n = 0 and n = 1, the condition ~ - 0 is obtained and the equations governing P~j), S(O~, T~ ~ and q, are the same a,~ those given by eqs. (27) and (31). The same procedure applies to the case when 3 L--~- 1 and the terms containing 3L can be neglected. This gives eqs. (36) and

=-2csin0,

42)

3L<
180

"--A

oL = 1 0 0 0 = 2.52 . . . . . . . . . . . . . . . . M2= 0.001

//!I

/; /

] i

.... i+

i1 •

90 °

E

cos2@ - M2sin28 = 0

I 90 ° Angle

180 °

~--

e

Fig. 7. Angular variations of arctangent of 12ore/( or - oell for logarithm singularity with c~ = 10 ~, ,4 ~ 2,52 and M : = 1(}

K C. G a o / Strain singulari(v behatrior of moeing cracks in elastic-I?iscoplastic materials

241

In passing, it should be p o i n t e d out that when F is a p p r o a c h e d from the right side in which the logarithm strain singularity e 0

e - A log-)-

(43)

prevails, the coefficient A ---, m . If A >> [ l o g ( R o / r ) ] ~, then eqs. (37) a n d (39) should be used. N u m e r i c a l d a t a of ~p(0) in (32) c o r r e s p o n d i n g to l o g a r i t h m singularity are d i s p l a y e d in Figs. 6 a n d 7 for two different sets of values of c~, A a n d M. In c o m p a r i s o n with the results in Figs. 4 a n d 5 for the p o w e r law singularity, the inflection p o r t i o n of the curves in Figs. 6 and Fig. 7 are extended. This p o r t i o n coincides with the d o t t e d curve and satisfy the c o n d i t i o n cos2~ - M

2 sin20 = 0

(44)

5. Conclusions F o r the e l a s t i c - v i s c o p l a s t i c m a t e r i a l p r o p o s e d [9], the singular b e h a v i o r of a c o n s t a n t velocity crack is f o u n d to d e p e n d on the viscosity a n d M a c h n u m b e r . A l o g a r i t h m singularity prevails for low viscosity while a p o w e r law singularity is o b t a i n e d for high viscosity. A c o m m o n limit exists for b o t h solutions at a critical viscosity. The quasi-static results is recovered when d y n a m i c effects are neglected. W h a t has been established is that i n h o m o g e n e i t y of the viscosity local to the crack can affect the singular b e h a v i o r of the stresses. Influence of material i n h o m o g e n e i t y on crack tip stresses has been discussed [13] in c o n n e c t i o n with the results o b t a i n e d f r o m the classical t h e o r y of plasticity which imposes the same functional form of the constitutive relation everywhere. A d d i t i o n a l investigations are n e e d e d to fully u n d e r s t a n d how the crack tip b e h a v i o r is affected b y the c o n s t i t u t i v e coefficients, p a r t i c u l a r l y for materials that behave n o n l i n e a r l y a n d are dissipative in character.

Acknowledgement The a u t h o r wishes to thank Professor G.C. Sih for his help in i m p r o v i n g the p r e s e n t a t i o n of the paper.

References [1] A.D. Chitaley and F.A. McClintock, Elastic-plastic mechanics of steady crack growth under anti-plane shear, J. Mech. Phys. Solids 19 (3) (1971) 147-163. [2] Y.C. Gao, Influence of compressibility on the elastic-plastic field of a growing crack, in: Elastic Plastic Fracture Second Symposium, A S T M S T P 803, Vol. 1, 1981, pp. 176 190. [3] Y.C. Gao, Elastic plastic field at the tip of a crack growing steadily in perfect plastic medium, Acta Mech. Sinica 12 (1) (1980) 48-56 (in Chinese). [4] Y.C. Gao, B. Hart and K.C. Hwang, The contradictions in the quasi-static asymptotic solution to a growing crack, Acta Mech. Sinica 18 (1) (1986) 88-92 (in Chinese). [5] Y.C. Gao and S. Nemat-Nasser, Dynamic fields near a crack-tip growing in an elastic-perfectly-plastic solid, Mech. Mater 2 (1983) 47-60. [6] Y.C. Gao, Asymptotic dynamic solution of Mode I propagating crack-tip field, Int. J. Fract. 29 (4) (1985) 171-180. [7] Y.C. Gao, Plane stress dynamic plastic field near a propagating crack tip, Int. J. Fract. 34 (2) (1987) 111-129. [8] J.T. Leighton, C.R. Champion and L.B. Freund, Asymptonic analysis of steady dynamic crack growth in an elastic-plastic material, J. Mech. Phys. Solids 35 (5) (1987) 541-563. [9] Y.C. Gao, Uniparameter plastic field near a dynamic crack tip, Mech. Res. Commun. 15 (5) (1988) 307-313. [10] G.C. Sih, The state of affairs near the crack tip, in: Modelling Problems in Crack Tip Mechanics, ed. J.T. Pindera (Martinus Nijhoff: The Netherlands, 1983) pp. 65-90.

242

Y.C. G a o / Strain singularity behavior of moving cracks in elastic-viscoplastic rnaterta[s ~

[11] G.C. Sih and D.Y. Tzou, Heating preceded by cooling ahead of crack: macrodamage free zone, 7heor. AppL Fract. Mech. O ~2) (1986) 103-111. [12] G.C. Sih, Mechanics and physics of energy density and rate of change of volume with surface, Theor~ Appl. Fract. Mech. 4 (3) (1985) 157-173. [13] O.C. Sih and D.Y. Tzou, Plastic deformation and crack growth behavior, in: Plasticity and Failure Beha~tor of Solids. eds. G.C. Sih, A.J. Ishlinsky and S.T. Mileiko (Kluwer Academic Publishers: The Netherlands, 1990) pp. 91-114