Plastic deformation ahead of a plane stress tensile crack growing in an elastic-perfectly-plastic solid

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Vol. 28, No. 2, pp. 139-146.1987

0013-7944187 @I 1987 Pergamon

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PLASTIC DEFORMATION AHEAD OF A PLANE STRESS TENSILE CRACK GROWING IN AN ELASTIC-PERFECTLY-PLASTIC SOLID Department

GUO QUANXIN and LI KERONG of Mechanics, Huazhong University of Science and Technology, The People’s Republic of China

Wuhan, Hubei,

A~~ct~uasistatically propagating plane stress tensile and anti-plane strain cracks in an elastic-~~ectIy-plastic solid have been studied. In the plastic loading zone, based on the basic equations of the Prandtl-Reuss flow rule and the Huber-Mises yield criterion, the stresses and particle velocities have been expanded in a power series of the distance y to the crack-line. By matching the stresses and particle velocities with the dominant terms of appropriate elastic fields at the elastic-plastic boundary near the crack-line, the arbitrary functions that enter in the expansions have been determined and a complete solution for the strain on the crack line has been obtained. For an anti-plane strain crack, the present solution coincides with that given by Achenbach and Dunayevsky, but not for a plane stress tensile crack. Numerical calcuiation has been carried out for the far-field stress level that would be required for a steadily propagating crack.

1. INTRODUCTION

materials, crack initiation is usually followed by stable crack growth. Hence, more attention should be given to fracture instability after some stable crack growth than to the initiation of growth in a safety analysis of a degraded structure component. The asymptotic structure of quasistatic near-tip fields of stress and deformation for a growing crack in an elastic-plastic material has been discussed intensively. Generally speaking, certain arbitrary functions are remained in the asymptotic solutions. Hence, in order to obtain the complete elastic-plastic solution for quasistatically growing cracks, numerical calculations must be introduced to determine the arbitrary functions. In a recent paper, Achenbach and Dunayevsky[l] investigated the near crack-line fields of stress and defo~ation for a growing crack in an elastic-perfectly-plastic material. They obtained the analytical expression for the strain on the crack line by use of an expansion and matching technique. In a more recent paper, Achenbach and Li[2] reconsidered the results of ref. [l]. However, in both of the two papers the assumption II = (cos 9, sin +!J)was made, at least for a small I,&,for the unit normal vector of the elastic-plastic boundary, where the polar coordinates R, t,Qwas centered at the center of the elastic field. This implies that the center of curvature of the elastic-plastic boundary at the point J, = 0 coincides with that of the elastic field. Guo and Li[3] have shown, however, that this assumption may violate the Huber-Mises yield criterion. In the present paper, the general expression has been obtained for the unit normal vector II of the elastic-plastic boundary by employing expansion of the elastic-plastic boundary RP( r, $) in power series of $. The expansion technique of refs [l] and [2] is applied to obtain the power series expansions of the stresses and particle velocities in the plastic loading zone. Matching the relevant stress and deformation components in the plastic zone with the dominant terms of appropriate elastic fieids at the elastic-plastic boundary shows that, for an anti-plane strain crack the unit normal vector of ref. [ 1J coincides with that given in the present paper, but not for a plane stress tensile crack. The plastic strain on the crack line has been used in conjunction with the crack growth criterion of critical plastic strain, which was proposed by McClintock and Irwin[4], to determine the stress intensity factor of the far-field that would be required for a steadily propagating crack. Numerical calculation shows that the present results are less than those of refs [l] and [2]. Hence, the safety analysis based on the results of refs [l] and [2] is not always safe. FOR DUCTILE

139

GUO QUANXIN

140

and LI KERONG

2. BASIC EQUATIONS As shown in Fig. 1, x1, x2, x3 is a stationary coordinate system with x3-axis parallel to the crack front. A moving coordinate system x, y, z is centered at the crack tip, with its axes parallel to the x1, x2 and x3 axes. The position of the crack tip is defined by x1 = a(t). In such two coordinate system, one has

a a -=ax, ax' In the moving coordinate

a a -=ax2 ay'

a a -=ax3 az'

system material time derivative

(2.la,b,c)

is

(')=a()/at-aa()/ax,

(2.2)

where a = daldt is the crack growing speed. In the stationary coordinate system, the equilibrium equations are Uij,j = 0

The Huber-Mises

(i, i = 1, 2, 3).

(2.3)

yield criterion may be stated as 1 Sijsij= k2, 2

stress tensor, k is the yield stress in pure shear.

where sij are the components of the deviatoric The strain rate is defined by ijj =

(2.4)

k

(lii,j

+

tij,i),

(2.5)

where Ui is the components of displacement vector. Using the Prandtl-Reuss flow rule, we have iii = E; + EC,

(2.6)

where 1 iTj = - bij -; 2cL ii

=

&&jij,

#i&j,

(2.7)

(2.8)

in which CL, E and Y are the elastic shear modulus, Young’s modulus and Poisson’s ratio, respectively, while i is a non-negative factor.

Fig. 1. Geometry for crack tip, center of elastic field E, and the elastic-plastic boundary.

Plane stress tensile cracks

3. A MODE-III QUASItVATICALLY

141

PROPAGATING

In the moving coordinate system, the only non-vanishing Thus, eqs (2.3) and (2.4) become

CRACK

displacement

is u3 = w(x, y, t).

(3.1)

and

(3.2) The Prandtl-Reuss

flow rule, eqs (2.5)-(2.8),

reduce to (3.3a)

iaa

1

28~

2~

--=_

ayz+ ia,,.

(3.30)

3.1 The expansions of stress and deformation in plastic zone In the plastic loading region y/x 6 1, 0 < x < x,, where x = x,, defines the elastic-plastic boundary on the crack line, the stresses and deformation can be expressed by (3.4a)

o;z = TIC%OY + O(Y3), uyz =

so(x, t) + s2(x, dY2+ 0oJ4),

ti = t&(x, t)y + i =

O(y3),

(3.4b) (3.5)

&(x, t) + l&(x, t)y2 + O(y4).

(3.6)

Here we have taken into account that a,, and ti are antisymmetric with respect to y = 0, while a,,* and i are symmetric. Substituting of eqs (3.4)-(3.6) into eqs (3.1)-(3.3) and collecting terms of the same order in y yield a system of nonlinear ordinary differential equations for the coefficients of the expansions. Solving this equation system gives so= k,

lk s2=-2-p

k 71=--

(3.7a,b,c)

X

and (3.8) Hence, in the plastic loading zone we have (3.9a)

uxz =-Sy+O(y3),

uyz =

(3.9b)

k-;;y2+O(y4),

ti=+[-tln(t)+$]

y+O(y3),

(3.10)

142

GUO QUANXIN and LI KERONG

where A(t) is an integral constant and can be determined elastic-plastic boundary.

from the continuity conditions at the

Near the crack line, the stresses and deformation in the plastic loading zone should be matched with the dominant terms of elastic fields. In polar coordinates R, 1,4centered at the center of the elastic fields, the elastic singular fields near the crack line are (3.11)

u,, = --

1 1 112 &II+ + OM3), 2 C-J 27rR

(3.12a)

(3.12b) The elastic-plastic bound~y is defined by R = RP(t, It). Since Rrff, +) is s~metric respect to + = 0, for a small $ we have

R&, JI)= RoW+R&M2+O((C14)It follows from (3.13) that the unit normal vector n = (n,, n,) of the elastic-plastic n, =

1-kB:qJ*+ O(Jp),

ny =

6

JI+ wJ3h

with

(3.13) boundary is (3.14a) (3.14b)

where B

_1_2R,

1--

RO

(3.15)

In refs [l] and [2], the assumption n = (cos J/, sin $) has been made. However, eqs (3.14) and (3.15) show that, in the meaning of O( 4k3)the assumption of n = (cos I,%, sin JI> is correct if and only if R2 = 0. The center of curvature of the elastic-plastic boundary at point $I = 0 will not coincide with the center of the elastic fields if R2 # 0. Substituting (3.12) and (3.13) into (3.2) we can obtain (3.16)

Rz= 0.

(3.17)

From (3.17), (3.14) and (3.15) one can find that, in the meaning of O(tft’), the assumption n = (cos JI,sin JI) of ref. [l] is correct. Hence, similar to ref. [l], from the condition that a,, and rG are continuous across the elastic-plastic boundary we can obtain xi,= 2R0,

(3.18)

A(t) = h(t) + XJt).

(3.19)

For more details see refs [l], [5) or [6].

143

Plane stress tensile cracks

4. A MODE-I PLANE STRESS QUASISTATICALLY

GROWING

For a state of plane stress, a,, oXxzand uYZ vanish identically. coordinates the equilibrium equations are

au, aa,,_

dx+-ay The Huber-Mises

0,

CRACK

Hence,

in the moving

aa;,+aa,=(). ax

(4.la,b)

ay

yield criterion (2.4) becomes &+a;-uxuY+3u;,=3k2.

The Prandtl-Reuss

(4.2)

equations (2.5)-(2.8) reduce to (4.3a)

ad 1 ay=E(“-~~‘)+~~(2uy-u~),

(4.3b)

(4.3c) Similar to Section 3.1, based on the Prandtl-Reuss equations, the Huber-Mises yield criterion and equilibrium equations a system of ordinary differential equations can be obtained for the coefficients of the power series expansions of stress components and particle velocities t.i, lj. Under the assumption that the cleavage stress is uniform on the crack line, i.e.

uy(x,0, 4 = 4(t),

(4.4)

solving the equation system gives[2]

u~(x,y,f)=k(l-~~)+O(y4).

(4Sa)

uy(x, y, t) = 2k + O(y4),

(4.5b)

uxyb,

y,

t) = -k

$ +O(y5),

ri(x, y, r)=k(D(A+($[I-ln(t)]+kF-

(4Sc)

C(t)x] yz]+O(y4),

(4.6a)

(4.6b) where the functions B(t), C(t), D(t) can be obtained elastic-plastic boundary.

from the continuity

conditions

at the

4.1 The elastic-plastic boundary near the crack line In the polar coordinates R, +,, the elastic-plastic boundary R = R,(t, +) has been expanded as the power series (3.13) of t,G.Its unit normal vector n has been given by (3.14). Near the crack line, the stress and deformation components in the plastic zone should be matched with the dominant terms of elastic fields at the elastic-plastic boundary. As has been done by Achenbach and Dunayevsky[ l] and Achenbach and Li[2], for the elastic fields we take the elastic dominant

144

GUO QUANXIN and LI KERONG

fields for a notch with p as radius of curvature at its tip. The elastic dominant fields for a Mode-I blunt crack have been given by Creager and Paris[7]. For a small t/t we have (4.7a)

+2--)1’2K’[(1++-)+(~-$JQ’]+W% ax,

=

(4.7b)

(L-)“’ zq;-gJ $+0(V)-

(4.7c)

From the condition that the elastic fields should just reach the yield criterion elastic-plastic boundary, by using of (4.7), (4.2) and (3.13) we can obtain

at the

(4.8)

(4.9) It follows from (4.9) that R2 # 0. One can find from (3.14) and (3.15) that the assumption n = (cos +, sin I+$)in refs [l] and [2] is incorrect if RZ # 0. At the elastic-plastic boundary, by using of (3.13) we have

q2+ W4), Y =

(4.10a)

Rollr+ W3).

From the condition that U” and a,, are continuous (4.10), (4.7), (4.5) and (3.14) we can obtain

(4.10b) across the elastic-plastic

boundary,

using

(4.11)

+fk.~+~B:-(~-~)+(l-Q~)B,]

(4.12)

(4.13) From (4.8) and (4.11) it follows that the yield criterion and the continuity of o;, are incompatible if p = 0. That is why we take the elastic field for a notch rather than for a crack. Solving the equation system (4.8), (4.9) and (4.11)-(4.13) gives 2

P2 Ro

Ro=&: 3'

, ()

$z;,

xp= AR,.

(4.14a,b,c,d)

0

4.2 Plastic deformation near the crack line For a Mode-I blunt crack, the elastic fields for stresses have been given by Creager and Paris[7]. The corresponding displacements u, u can be obtained from the stress fields, near the crack line they are

145

Plane stress tensile cracks

(4.15a)

(4.15b) where ic = (3 - u)/(l + v). At the elastic-plastic boundary, from the continuity of displacement-rates obtain by using (4.15), (4.14) (4.10), (4.6) and (3.13) B(t)=

blci(+i-b,i,(t),

C(t) = Ec1ci(t) +

ti and ti we can

(4.16a) (4.16b)

~2~p(~)3t&)3,

where bl = (1 + &!) - 2/3,

(4.17a)

b2=1+1/Jz,

(4.17b)

cl = (l/A

- 1) -t 2/3,

(4.17c) (4.17d)

a=&-1.

In the stationary coordinate crack line is (see ref. [5])

system, from (4.6) and (4.16) it follows that the strain on the

E,,(x~, o, t) = k(2 - V)/E + E;YP’(xI,

Ol+

fE;~;P(xI, JI* 0, $ ds,

(4.18)

where (4.19a) E;P[~(XI,l)l=$

$zpP(-%

fExl- 49, ~p(s)l=

I

b&-l)-&*

(

$-1

)

,

4 = ; $if[x1- a(s), x,(s)],

Xl -

’4s) {2 In Lh

s) + bl - b25(xl, s) f

(4.19b)

(4.19c)

t4-19d) [{(XT: $3” - [icxf: s)]*I*

t* = max(i& f,,), t is the time that crack propagation starts, tp is the time that the elastic-plastic boundary reaches the position x1 and the plastic strain starts to accumulate. As was done by Achenbach and Li[2], we extend the crack growth criterion, which was originally proposed by McClintock and Irwin[4] for anti-plane strain cracks, to plane stress tensile cracks. The crack growth criterion states that the crack will grow if the plastic strain at a fixed distance xf ahead of the crack tip reaches its critical value Epf Y. For plastic strain below lPf at xi = a(t) -i- or the crack cannot grow. Applying this criterion to crack initiation, one can obtain from (4.18) (4.20) If all fields are assumed to be time-invariant to an observer traveling with the crack tip, i.e. for a steady-state case, we have It(t) = constant, x,(t) = xpS= constant. Applying the crack growth criterion to steady crack growth, we can obtain from (4.18)

GUO QUANXIN

146

and LI KERONG

Table 1. Values of stress intensity factor Kp/K,= Ef

results of ref. [2] results of this paper

for quasistatic and steady crack growth

3.4

10.2

17

30

50

70

90

100

1.046 1.039

1.292 1.280

1.618 1.602

2.417 2.394

4.165 4.128

6.730 6.672

10.390 10.304

12.738 12.634

(4.2 1) It follows from (4.14b) and (4.14d) that

WlKI,

= bpsIx,(ti)l”2

(4.22)

where Kr, and Ki.” are stress intensity factors for crack initiation and steady crack growth, respectively. From (4.20)-(4.22) one can find that the Poisson’s ratio v does not effect the relationship between K3’”and et = $‘Elk. Some numerical results are listed in Table 1 for the stress intensity factor that would be required for a crack growing quasistatically and steadily for a given value of ef. From Table 1 one can find that the present results are slightly lower than those given by Achenbach and Li[2]. As been pointed out before, the differences arise due to the inappropriate treatment of the unit normal vector of the elastic-plastic boundary in refs [1] and [2]. Hence, a safety analysis based on the results of refs [l] and [2] is not always safe. Ack~wledge~n~Helpf~ discussions with Dr 2. L. Li of Mechanics Department, Huazhong University of Science and Technology, Wuhan, China, are gratefuily acknowledged.

REFERENCES [1] J. D. Achenbach and V. Dunayevsky, Crack growth under plane stress conditions in an elastic-perfectly-plastic material. J. Me&. PZrys. Sotids 32, 89-100 (1984). [2] J. D. Achenbach and Z. L. Li, Plane stress crack-line fields for crack growth in an eiastic-perfectly-plastic material. Engng Fracture Mech. 20, 534-544 (1984). [3] Q. X. Guo and K. R. Li, Near crack-line fields for plane stress tensile cracks growing in elastic-perfectly-plastic solids. Proc. Znt. Conf. Fracture and Fracture Mech. April 21-24, 1987, Shanghai, China, pp. 349-353 (1987). ]4] F. A. McClintock and G. R. Irwin, Plastic aspects of fracture mechanics. Fracture Toughness Testing and Its ApF~icutio~s, ASTM STP 381 (1965).

[SJ Z. L. Li, Plastic deformation on the crack-line in an elastic-perfectly-plastic material. Ph.D Thesis, Northwestern University (1985). [6] Q. X. Guo, Near crack-line field analysis for crack growth in an elastic perfectly-plastic solid. Sci.M. Thesis, Huazhong University of Science and Technology (1985). [7] M. Creager and P. C. Paris, Elastic field equations for blunt cracks with reference to stress-corrosion cracking. Znt. J. Fracture Mech. 3, 247-252 (1967). (Receioed

15 October 1986)