On invariant integrals in the analysis of cracked plates

International Journal of Fracture 40:111-126, 1989. © 1989 Kluwer Academic Publishers. Printed in the Netherlands. 1 1 1

On invariant integrals in the analysis of cracked plates

H O R A C I O S O S A 1 and G E O R G E H E R R M A N N 2 ~Mechanical Engineering and Mechanics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA 2Division of Applied Mechanics, Stanford University, Stanford, California 94305, USA

Received 6 January 1987; accepted in revised form 26 July 1988

Abstract. Bending of cracked plates is considered within a formulation in which transverse shear deformation is taken into account. Path-domain-independent (or invariant) integrals are constructed by separating the strain energies of bending and shear. This separation permits the expression of the three stress intensity factors, which can occur simultaneously under certain loadings, in terms of the integrals mentioned. A numerical implementation is proposed and illustrated by means of a known example. Future applications are discussed in order to solve more complicated configurations.

1. Introduct ion

Despite its practical importance, the subject of plate structures containing crack-like defects and subjected to out-of-plane loading, seems far from being completely resolved. Due to inherent difficulties, current emphasis is placed on providing numerical procedures in order to solve a crack problem. These procedures are usually "direct", for example by extrapolating the displacements that have been found by a finite element solution, or in an "indirect" manner by the use of so-called path-independent integrals.

The problem of whether to include the effect of shear deformations has received particular attention because it is crucial in the determination of the asymptotic behaviour of stresses. Although the classical plate theory renders misleading field quantities in the neighborhood of the crack, many formulations are still based upon such an approach.

It seems that most of the available work addressing the subject of cracked plates including the effect of transverse shear deformations, has been constrained to mode I fracture. The apparent reasons for this are: (a) mode I is undoubtedly the most common mode to occur, and (b) numerical procedures are often not able to give a systematic and simple approach to mixed-mode bending crack problems. Recently, Sosa and Eischen [1] presented a method based on the use of a path-independent integral expressed in terms of plate resultant quantities, within the Reissner's plate theory. In the same reference a review of previous methods is given.

One of the motivations of this paper is due to an article by Bergez [2] recently found by the authors, even though it was published almost 10 years ago. In the aforementioned reference, path-independent integrals are given for shell structures. Relationships of the integrals with stress intensity factors were proposed based on a separation of the former into bending and shear contributions, in order to render the same number of integrals as fracture parameters. As a conclusion it was claimed that any mixed-mode fracture problem could be solved with such an approach. It is apparent to the authors that the line integrals as proposed in [2] are not path-independent. In fact, this has been checked through several numerical experiments.

112 H. Sosa and G. Hermann

The present paper proposes, along the same line of thinking used in [2], expressions which are given in terms of line and surface integrals, yet being independent of the path and enclosed domain when surrounding a crack tip, hence the name ofinvariant integrals. These integrals are also related to fracture parameters, and for that purpose the asymptotic behaviour of relevant field quantities when a crack is present, are determined by the method of eigenfunction expansions as suggested by Hartranft and Sih [4].

As a first step towards the validation of such an approach a numerical implementation for mode I problems has been carried out. The results, although they are not conclusive, show the feasibility of the method as a good alternative to existing ones and of apparent potential for the treatment of more complicated configurations. This being the subject of a forthcoming article.

2. The transverse shear plate theory

With reference to an (x~, x2, x3) rectangular coordinate system, a plate is considered as an elastic body composed of a cylindrical surface bounded by two parallel planes located at x3 = +_ hi2. In the present work the theory of plates due to Mindlin [3] is used. This means that the components of the displacement vector are assumed to be given by

Ul = X 3 ~ I ( X I ' X2)' ~/2 ~- X31~g2(XI' X2)' L/3 = W(XI, X2) , (1)

where w is a weighted average (taken over the thickness) of the transverse displacement, while ~1 and ~2 are average rotations (or equivalent change of slope) of the normal to the undeformed middle surface of an element about the x 2- and xraxes, respectively. Collectively, w, ~1 and ~2 are called the plate displacements. The rest of the governing plate equations are presented in the next section with a more appropriate coordinate system.

3. The crack problem

Consider that an infinite plate of constant thickness h occupies the entire Xl - x 2 plane, except for the semi-infinite segment x 2 = 0, - c o < xl ~< 0, where a through-thickness crack is placed. Consequently, the x z-axis coincides with the crack front. In order to study the redistribution of stresses due to the presence of such a discontinuity, it is convenient to introduce polar coordinates (r, 0) located at x3 = 0, with r having its origin at xl = x2 = 0 and 0 measured so that the crack faces are located at 0 = _ re.

In such a coordinate system, the constitutive relations between the plate stresses (i.e., the bending and twisting moments and the shear forces) and displacements are given by

Mrr = D L a r +-r

Moo = D @ + - + v r-gg ar J

Analysis o f cracked plates 113

Mro - D . + 0o r 00 r

(ow ) Qr = Cs ~r q- Or

) Qo = Cs \ r 00 + 0o , (2)

where D and C s are the plate flexural and shear rigidity, respectively. That is

Eh 3 D - 12(1 - v2) ' C, = ~6Gh (3)

with E, v and G being Young's modulus, Poisson's ratio and shear modulus, respectively. The equilibrium equations (in the absence of external pressure) are expressed by

c~ Mr,. 1 O Mro Mr,. - Moo - - + - - - - + - - Qr Or r 00 r

c3 Mro 1 (?Moo 2 0~-~- + -r - - 0 0 + -r M'° = Qo (4)

_ _ 1 OQo 1 ~?Q" + - - - + - Q , . = O. Or r 00 r

Hartranft and Sih [4] have shown that the three-dimensional crack problem can be solved by representing the displacement fields as a double infinite series in the form

m~O n=O

where the functions of Un, V,,, and W~ depend on 0, x 3 and 2,=, and are equal to zero when n < 0 in order to avoid unbounded displacements in the vicinity of the crack. The eigen- values 2m (m = 0, 1 . . . . ) as powers of r are assumed to be constants. For the present problem an expression similar to (5) is required. However, some simplifications apply. A similar analysis to that in [4] can be carried out to show that appropriate values for 2m are m/2 (m = 0, 1, . . . ). Therefore the displacements can be represented by a power series involving one-half and integer powers of r. The point of departure here is that (1) specifies the dependence of the displacements through the thickness of the plate, and since the solutions will be sought in terms of~b~, 02 and w, rather than ul, u2 and u3, (5) can be replaced by

(1 -- v){D~r, D~o, Cshw ] = ~ r"/2[.f~(O), g=(O), h=(O)], (6) n=O


where the functions.f~, g~, and h. vanish for n < 0. When (6) is substituted into (2) one obtains

Mrr 1 - - V n = 0

1 r n/a-' 1 + ~ v f,,(O) + g'.(O) Moo - 1 - V.=o

Mr° = 2 r"/2- 5 - 1 g.(0) + f,,,'(0) (7) n = 0

Qr - 1 - v.=0 ~ h.(O) + h2 f._2(0)

] 1 r "/~-~ h'.(O) + h~ g._~(O) , Qo - 1 - v.=0

where the primes indicate derivatives with respect to 0. Substitution of the above expressions in (4) leads to a system of three ordinary differential equations, namely

= - 1 ~ h,,_2 + 5(1 - v)

h 2 L-4

gj + (3 - v) + ~ (1 + v) L ' + ---5---

1 5 (1 - v) !

h hn_2 + h2 - - g n 4

n 2 5(1 - v ) [ n 1 h~' + ~-h~ - h ~ L - 2 + g~-2 .

Equation (8) provides recurrence relations for the functions fn, g. and hn. It can be verified that for n ~> 2 the three equations are coupled, and as a consequence, some of the coefficients appearing in the solutions for n = 1 are also present in the solutions corresponding to n = 3. It can be shown [5] that after lengthy algebra, the solutions for the functions f . , g. and hn for various values of n lead to

( l - - Y)Dt/I r = B~ II cos 0 + B0 (2~ sin 0

Analysis o f cracked plates 1 1 5

- - cos + sin ÷ r l/2 B} ° cos 2 1 + v 2 3(1 + v) sin

l - v + rB~l) 1 + v - - + cos 201 + r3/2 I 4A] 2) 0 15(1 -- -v2)h sin

- - c o s + sin + 5(1 + v~ sin + O(r 2) + B~ l) c o s ~ - + 1 + v

(1 -- v)DOo [ ( 3o = - B o (1) s in0 + Bo (2) c o s 0 + ?/2 BIo - s i n ~ - + 1 + v

(30 E2 + B} 2) cos 2 3(1 + v~C°S + r ~ _

+ r3/2 I 4(2 + 3v) A~2) 0 _ B ~ ) ( s i n 5 0 9 + Vsin0 ) 15(1 - v2)h c o s ~ 2 1 + v

+ B~ 2) cos .2 5(1 + v ~ c ° s + O(r 2)

0[ (1 -- v)Cshw = A(01) ÷ rl/ZA{ 2) s in~ + r A~ 1) cos 0 5(1 -- V) B~2) sin 0~

h d

(1 + v)h ~ (7 + v ) c o s ~ - - (1 - v)cos

lO(1-V)2B~2)s in 0 A~2) 300~ O(r2 ) + 3(1 + v)h 2 + sin z _ ] + " (9)

These expressions were obtained by imposing the following traction-free boundary conditions at the crack surfaces:

Moo = Mro = Qo = 0 f o r 0 = _+~. (10)

The different coefficients AI .j), BI .j) (i = 0, 1, 2, 3 a n d j = 1,2) are unknowns which remain to be determined for each particular boundary value crack problem. In particular if the following definitions, for those coefficients which lead to singular resultants, are introduced:

BI~) _ K, BI2) _ 3K 2 AI2) _ 2(1 - v)hK 3 (11) 2x~ ' 2x/2' ~f2


by virtue of (7) one obtains

_ 1 K - - - 5 c o s ~ l 3K2 s in~ l } Mrr 4 , ~ { - 1 [ cOs302 + [ sin302 35 + B~l)[1 + cos 20]

+ ~x/r B~ l~ cos~- + 3cos + s in~- + ~sin

2 0 + ~ x /~ K3 sin~ + . . .

Moo ,~2r~{[30-~ ~ E - 4 '-- KI cos + 3cos - 3/£2 sin 302 + sinai} ~ { E ~0 ~j

q- B~l)[1 - - COS 20] - ~- x/r B~ 1) c o s T - - 5 c o s

E '° ~1} + B~ 2~ s i n - 2 - sin + . . .

~{E 30 ~ Mro - 4x/~ gl sin-}- + sin

E 3O + 3/£2 cos ~-

1 ~} + .~ cos

- B~ ° s i n 2 0 + ~x/r - sin - 2 - sin

+ B ? cos -2 - ~ cos + ~ ~ K3 cos ~ + . . .

1 0 [ A~ 1) 5 Bo(O Qr - x / ~ K 3 s i n ~ + (1 ~ v ) h +~5 cos0

+ 2(1 + v)h 2 KI 3 c o s - 2 + cos + ~ K 2 (1 + v) s i n - 2 - - ~

3,/-; 30 + 2(1 - v)h A~2~ s in-2 + . . .

1 o [ A ~ 5 1 Qo - , , ~ K3 c°s ~ - (1 - v)h + ~ B~°l~ sin 0

~ { [ ~o ~ 1 3 i + ~2(1 + v)h: -K~ 3 s in -2 + 3 sin + ~ K 2 (1

34; 3o 2(1 - v)h A~2~ cos -~- + . . . ,

30 ol} + v) c o s - 2 - 2 cos

(12)

Analysis of cracked plates 117

where the parameters K~ and K 2 are referred to as the moment intensity factors, while/£3 is the shear force intensity, which are defined as

K 1 = lim ~ 7 M22 (r, O) r~O

/£2 = lim x/27 M,2(r, O) (13) r~O

K3 = lim x/27 Q2(r, 0), r ~ 0

where M22 , M~2 and Q2 are the Cartesian components of the expression given by (12). The relationship of the above parameters with the local stress intensity factors can be established as follows:

kl(x3) - h3 K1, k2(x3) = h3 K2, k3(x3) = 1 - /£3. (14)

These relations are consistent with the variation of the stresses through the thickness of the plate, provided a linear and parabolic distribution is assumed for the bending and shear stress, respectively. In fact, the first identity in (14) has been confirmed experimentally [6].

If a distributed load acting on a face of the plate were present, then the third equilibrium equation would have a non-vanishing right-hand side. If this load were constant, then the solutions for the functions hn would be altered by accommodating a particular solution in terms of the load. A more general solution can also be analysed if the transverse pressure is given by p = p(x~, x 2). In this situation an appropriate form for p must be considered. For example, an expression of the form p = End_0/"pn(O) would be suitable to express the functions fn, g, and hn, in terms of the prescribed p,(O).

To conclude this section, it is important to emphasize that (9), containing terms beyond those with power ?/2, is of practical importance, in order to improve accuracy, when a direct numerical procedure such as regression analysis [7] is used to find stress intensity factors.

4. Proposed invariant integrals

In the present section plate-invariant integrals are derived, following a procedure that has already been described in [1] and [8]. However, in contrast to these references, the point of departure here is to consider the strain energy density of the plate as the sum of two independent contributions, namely bending and shear. To this end one introduces the following matrix definitions1 :

[m] = M22 ] ,

M12 /

.. Fll (iv 0 i bl = t 2J, i cb l = 1 0

\F,2 / 0 (1 - v)/2

(15)

Where superscripts b and s denote bending and shear contributions.


[q] = Q~ [7 *] = , [c ' ] = c , 0 Q2 ' I~23// , (16)

where

( ~ 1 r22 61//2 r l 2 (~01 a~/2 ]711 -- (~Xl, -- (~X2, -- a S 2 - ~ - ? X 1

c3w 0 w F,3 = O , - I - __G'~X--' r23 = 02 -~ 2 U'~X-"

(17)

Denoting by m~, C~/~, ql, etc. (c~,// = 1, 2, 3 and i, j = 1, 2) the elements of the matrices defined in (15) and (16), the bending and shear constitutive relations are given by

b b s s rn~ = Q~7~, qi = C~yj. (18)

It is appropriate to write the strain energy as

W = W b + W s = lm~b~ + ½q/7~ = ½7bCb~7~ + ½7~C~j7~. (19)

As a consequence of (18), it follows that

W b 0 W s m ~ - ~7b, q i - (77~ (20)

The first step is to obtain so-called balance laws, which are deduced through simple calculus operations on the strain energy of the plate. For this purpose consider the strain energy given by

W = W e(Tb~,xi) + W'(7~,xi). (21)

Equation (21) expresses the possibility of material properties that change with position. For the present needs, the only relevant operation is to take the gradient of (21). If so, the k-component is then given by 2

Wk = W~ + W~, (22)

which by means of the chain-rule leads to

aX--~ + aX k -- 637b= (~X k -~- k (~xk //expl -}- (~7~- (~Xk -~- \ (~Xk //expl " (23)

2 Where a comma indicates differentiation with respect to the material coordinate.

Analysis o f cracked plates 119

The subscript "expl" refers to the explicit derivative of the strain energy with respect to coordinate, while the strains are kept constants. By making use of the constitutive relations, the following identity is obtained

[W~ - m~7~,k] + [W~ - qi7~',k] = (W~)expl + (W~)expl. (24)

It can be shown that expanding, the second terms within brackets can be written as

m~Tb, k = (MHO,,k + M, zO2,k),l + (M~2O~,~ + M=Oz,k),2 -- (QltP,,k + Q2Oz,~) (25)

qiT~,k = (QlW,~),~ + (Qzw,k),2 + (pw),k - p , k w + (QlOi,k + Qz~z,k), (26)

where simple rules of differentiation and the equilibrium equations have been used. Combin- ing the above relations with (24) one finally obtains

{[Wbb~k -- (M~10~,k + m121~2,k)b,i- (M~2O~,k + M2202,~)62i1,i + (Q~01,k + Q2tPz,k)}

q-" {[ m s . PW)~Sik - - (Q1W,k){)li - (Qzw,k)62,],g- (Q,O,,k + Q202,k) + P,k w}

= (W, kb)expl @" (W,k)exp 1. (27)

Equation (27) is the local (or differential) form of a balance law. If the above expression is integrated over a domain ~ which is free from any defects or nonhomogeneities, then the right-hand side will vanish. Furthermore, the use of the divergence theorem where possible leads to

{fr [Wbnk -- + -- + dF (/1101,k MI2 I/./2,k )nl (Mi2 ~tl,k M2202,k)H2]

+ f~ (Q~,I,~ + Q~4,~,~)dn}

+ {~ [W ~ - pw)n~ - (Q, nl + Qin~)w s] dF - Io (Q101,, + Q~4'~,~) da

+ I~ p,,w dn} = 0, (28)

where F is the boundary of fl with outward normal of components n~ and n 2 . Assuming that the pressure load p is constant, the above results can be summarized as

I? + I~ = 0, (29)


where

I~ = ~r [Wbnk - ( M l l 0 1 , k -}- MI202'ktF/1 -- (M12~l'k -~- M22@2'k)n21 d r

+ fn (Q,O,,k + Qz~bz,k)dO (30)

IS = ~r [W' - pw)nk - (Q~n~ + Q2n2)wk] dF - In (Q,4q.k + Q24,2,k) d~. (31)

The consequence of (29) is that

/~ = g = 0 (32t

if and only if

I~ >~ 0 and I s >i 0. (33)

The conditions given by (33) are true within the type of problems one has in mind, provided k = 1. For this particular case the following definitions are made

J ~ . ' = f f and j s , = f f . (34)

It would be tempting, to start with W b and W ~ independently, and carry on the same steps as applied to (21) rather than using (32) and (33). But although writing W as the sum of W b and W s is physically correct, it is not possible to encounter a situation in which W is only given by W s, unless applied shear tractions acting on the faces of the plate are allowed.

5. lnvarianee of J~ and js

Provided J1 e and J1 s can be evaluated independently, the condition that they are identically zero, when evaluated on a path that excludes any defect, is used to show that they render the same result no matter which open path and enclosed domain is used when surrounding the tip of a traction-free crack. Suppose the crack placed along the xl-axis as shown in Fig. 1.

According to the figure F = F 0 + F~? + F~: + F~ and f~ = f~0 - f~.. Since F (with its enclosed domain f~ indicated by the shaded region) is a closed path free of defects, then

J ( (F) = J~(F) = 0 (35)

or for example J (

~v0 + It+ + ~r~ + fr2 + fa0 - fa. = 0, (36)

where the line and domain integrands are those of (30) when k = 1.


~ ~: 1

r = ro + + + + r 2

Fig. 1. Path F and associated domain ~ surrounding a crack tip.

It is to be noted the reverse sense of integration on F~. Along the crack faces the following conditions hold true

n I = 0, M22 = M12 = Q2 = 0, (37)

therefore the integrals on F,. vanish leading to

The same result holds for j s changing the signs of the domain integrals. Due to (38) J1B and j s can be evaluated along any open path F and the associated domain ~, which starts on the lower crack face and terminates on the upper one. The virtue of (38) is that, to avoid numerical uncertainties, F can be chosen sufficiently far from the crack.

It is important to realize that neither J~ nor j s represent the energy release rate due to crack extension when computed around a crack tip. Despite this loss of physical interpretation, the integrals are important from a practical point of view due to their relations to fracture parameters as shown in the sequel.

6. Relations of J1 and j s with/(1, /(2 and K 3

In order to find these relations, the asymptotic behaviour of the field quantities, in terms of r, 0 and Ki (i = 1, 2, 3) provided by (9) and (12), are introduced into (30) and (31) with F being a circular path, centered at the crack tip. Due to the invariance property, the algebra is greatly simplified by choosing the circle vanishingly small, i.e. of radius e ~ 0. Polar coordinates are used, such that

dF = r dO, dg2 = r dr dO, - re ~< 0 ~< re. (39)


First it will be shown that the integration on the domain f~. will vanish as e ---, 0. One needs to compute

lim f ~ I0 (QIOI,1 ÷ Q2~2,1) F dO dr, (40)

where the field variables in terms of r have at worst the following order: Qi = O(r ~/2), Oe -- 0(//2) and therefore ~i,1 = O(r 1/2), (i = 1, 2). Hence the above expression becomes

lim f~_ f~ O(1/r)rf(O) d0 dr = l ime f~ f(O) d0 = 0, (41) ~ . 0 ~ a ~ 0

where f (0) represents any of the angular distributions of the field quantities. Now, it remains to evaluate

~r~ (") d r = f~_~ ( . ) r dO. (42)

The following intermediate results are obtained after integration:

3're f~_~ Wbn, r dO - (1 - v)(K~ - K~) (43.a) Eh 3

f ~ [(M,~61,t + M12~t2,1)nl ÷ (M12~tl,i ÷ M221~2,1)n2]r dO

-3 re Eh 3

[(3 ÷ v)/~l + (5 - v)K 21 (43.b)

f~ (W ~ - pw)n~r dO = 0 (43.c)

y~ (Q1nl + Qzn2)wl r dO - 2G

which leads to

(43.d)

j ( _ 12re [K~ + K~2 ] (44) Eh 3

j s _ 6(1 + v)rc 5Eh K2' (45)

Equations (44) and (45) provide two measure numbers for three unknowns. One more equation is obtained by computing J2 which is defined as

J2 '= I~ + I s (46)


or

J2 = fF { ( W - p w ) n 2 - - [[email protected] -]- M12022 -t- Q1w,2]]71

- - [M121//2,i + m221P2,2 - [ - Qzw,z]n2} d r - f~ [ W - pw~ dxl , (47)

where F is an open path enclosing one crack tip. The second integral is a number obtained by integrating the quantity ( W - pw) on the upper and lower crack faces, and the notation ~. ~ denotes the difference or jump of such quantity. As given by (47), J2 is path-independent, and its measure number is then related to the fracture parameters by the expression

24~ J2 - Eh 3 K, K2, (48)

which has been obtained following the same procedure as described above.

7. Proposed numerical implementation. Examples

In order to verify the validity of (44) and (45), several examples with known solutions were analyzed. The numerical implementation required to compute (30) and (31) is virtually the same as the one described in [1]. However for completeness a brief description is provided. The main difference resides in the fact that now an area integral must be computed. From a numerical standpoint, this additional evaluation does not represent any significant dis- advantage because the integrand contains only first derivatives of the average rotations.

All the field quantities involved in J~ and j s were evaluated numerically by a finite element procedure. The advantage of using invariant integrals resides in the simplicity of the mesh, which does not require any special elements and can be relatively coarse in order to solve a crack problem. For the bending of plates, the choice of the element is a crucial step. In the calculations performed, an eight-node serendipity quadilateral with four nodes at the corners and the other four at the midsides of the element was used. Each node possesses three degrees of freedom. This element already proved to be quite efficient in [1], in particular in analyzing moderately thick plates.

Uniform reduced integration was used to compute the bending and shear stiffness matrices. All the field quantities were evaluated at the 2 x 2 Gaussian integration points on the interior of the element. The line integrals, which are obtained by summing contributions from edges of all elements on F, were computed numerically using a two-point Gaussian quadrature. It is important to remark that in order to obtain the required data on the edges of the element, the smoothing procedure suggested by Hinton and Campbell [9] has been employed. The area integral, as the sum of all the elements within f2, was computed by a 2 x 2 Gaussian quadrature rule.

The two problems that are presented correspond to the case of a square plate with a central crack, in one case loaded by edge moments and in the other case being simply-supported along the edges parallel to the crack and subjected to uniform pressure. Normalized stress intensity factors for these cases have been provided in [1].


mmmmam

f

!

\ / \ / N / \ / \ /1

I- Xl

Fig. 2. Finite element mesh and integration paths to compute J~ and j s .

Due to the symmetry of the load and geometry, only one quarter of the plate of dimensions 2b x 2b needed to be analyzed. The mesh and possible integration paths are shown in Fig. 2, where the crack (of total length 2a) is located along the x~-axis.

Here the main concern was to show that in such problems, where mode I is the only relevant mode, j s is negligible in comparison with J•. Tables 1 through 4 display the relevant data, where three domains have been used. The invariance of Ji B (js is virtually zero) is to be noted. Furthermore, the integrals suggested in [2] are those denoted by J~ (line) and js (line), i.e., contributions coming only from F. In all cases the following data has

Table 1. Square plate of dimension 2b with a central crack of length 2a subjected to bending moments M22 = 1 at x2 = _+ b

Domain J~ (line) Jl B (area) J~

1 0.0685 0.0043 0.0728 2 0.0670 0.0054 0.0724 3 0.0683 0.0042 0.0725

Table 2. Square plate of dimension 2b with a central crack of length 2a subjected to bending moments M22 = 1 at x 2 = _+ b

Domain j s (line) j s (area) j s

1 0.0043 0.0043 0.0000 2 0.0056 0.0054 0.0002 3 0.0042 0.0042 0.0000


Table 3. Simply-supported square plate of dimension 2b with a central crack of length 2a subjected to uniform pressurep = 1

Domain JIB (line) j e (area) J~

1 0.0172 0.0007 0.0179 2 0.0166 0.0013 0.0179 3 0.0158 0.0020 0.0178

Table 4. Simply-supported square plate of dimension 2b with a central crack of length 2a subjected to uniform pressurep = 1

Domain Jl s (line) js (area) js

1 0.0007 0.0007 0.0000 2 0.0013 0.0013 0.0000 3 0.0021 0.0020 0.0001

been used: E = 1000, v = 0.3, a/b = 0.55 and b/h = 2. The values of K~ can be readily obtained by (44) setting K 2 = 0. Similar behavior was observed for thinner plates, al though it should be emphasized that the present plate element is more accurate with increased thickness.

Considering that the results in Tables 1-4 are qualitatively satisfactory, it seems clear that with the combined use of JIB, j s and J2, any mixed-crack problem could be solved properly in a simple manner. In a forthcoming paper the use of J2 will be considered. This quantity needs special care because of the presence of an integral to be evaluated along the crack faces. Another example that could be analyzed with the information presented in this paper is the case of a finite plate subjected to all around twisting moments. As shown by Wang [10] for an infinite plate, in such a case modes II and III are present. Therefore, (44) and (45) (with 321 = 0) will be sufficient to solve a problem which apparently has not been treated before.

8. Conclusions

The separation of the total strain energy into an energy of bending and shear allowed relating new plate path-domain-independent integrals to the three stress intensity factors. A proposed numerical implementation of the expressions newly derived have permitted ascertaining the good accuracy which is attainable in calculating the desired fracture parameters.

On the basis of these results, it appears that any mixed-mode crack problem could be solved in an expeditious manner. Further numerical examples and their discussion are relegated to a forthcoming paper.

Acknowledgement

Support of this work by the Department of Energy under DOE Contract 12040 to Stanford University is greately appreciated.


References

1. H.A. Sosa and J.W. Eischen, Engineering Fracture Meehanics 25 (1986) 451-462. 2. D. Bergez, International Journal of Fraeture 12 (1976) 587-593. 3. R.D. Mindlin, ASME Journal of Applied Mechanics 73 (1951) 31-38. 4. R.J. Hartranft and G.C. Sih, Journal of Mathematics and Mechanics 19 (1969) 123-138. 5. H.A. Sosa, "On the Analysis of Bars, Beams, and Plates with Defects," Ph.D thesis, Stanford University

(1986). 6. B.R. Mullinix and C.W. Smith, International Journal of Fracture 10 (1974) 337-351. 7. J.W. Eischen, "Fracture of Nonhomogeneous Materials," Ph.D thesis, Stanford University (1986). 8. H. Sosa, P. Rafalski and G. Herrmann, "Conservation Laws in Plate Theories," to appear in Ingenieur-Archiv

(Archive of Applied Mechanics). 9. E. Hinton and J. Campbell, International Journal for Numerical Methods in Engineering 8 (1974) 461-480.

10. N.M. Wang, International Journal of Fracture 6 (1970) 367-378.

lt~sum& On consid+re la flexion de t61es flssur6es selon une formulation off la d6formation par clsaillement transversal est prise en consid6ration. On construit des int6grales ind6pendantes du domaine ou du parcours (ou int6grales invariantes) en s6parant les 6nergies de d6formation en flexion et le cisaillement. Cette s6paration permet d'identifier trois facteurs d'intensit6 de contraintes qui peuvent se pr6senter simultan6ment sous certaines charges et qui s'expriment selon les int6grales mentionn6es. On propose une pr6sentation num6rique et on l'illustre au moyen d'un exemple connu. On discute les applications futures en vue de r6soudre des configurations plus complexes.

On invariant integrals in the analysis of cracked plates

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