Strength of Materials, Vol. 26, No. 1I, November. 1994

M E T H O D O F S I N G U L A R F I N I T E E L E M E N T S I N T W O - D I M E N S I O N A L

P R O B L E M S W I T H S T R E S S R A I S E R S A T T H E I N T E R F A C E S O F

D I S S E V I I L A R M A T E R I A L S

S. M. Bilichuk UDC 539.3

A description is presented of a singular finite element for two-dimensional problems with stress raisers at the

interfaces of dissimilar materials. The stress raisers are in the form of V-notches with unloaded edges. The

stress-intensity factors o f finite plates with an inclined edge crack and a V-notch are determined for different

geometric and physical parameters.

The methods of fracture mechanics are widely used to solve problems concerning the strength of bodies with cracks.

Evaluating the strength of structures with cracks requires knowledge of the stress-intensity factors and their limiting values.

Here, we evaluate the stress-intensity factors around defects in the form of cracks and V-notches at the interface of materials

with different elastic characteristics.

Examples of use of the method of singular finite elements to numerically analyze problems involving cracks at

interfaces were presented in [1, 2], while similar problems were examined in [3] by the method of boundary elements. We

will construct a different singular finite element which employs expressions for the displacements that allow for an oscillating

sin~larity.

We will assume that the body consists of two wedge-shaped plates connected to one another along one edge. The

plates are made of materials with different elastic characteristics. Zero stresses are assigned on the other edges (Fig. 1), and

an external load acts on the remaining part of the lateral surface. Point O is a singular point. In accordance with [4], the

expressions for the displacements in the neighborhood of this point can be represented in the form

u x, = A ~ , +BUdl;

uy, = AW2 + B02.

(i)

Here, A and B are arbitrary constants; the functions q5 i and ~i are defined as follows:

tDi = ai,coM,Och220 + aizsin21Osh2zO § ai3cos(2 - Az)Och2zO + + ai~siu(2 - 2,)0sMfl;

ql i = bi,sin,l~OchRzO + bi2cosR~Osl'd20 + bi3sin(2 - ,,l~)0cl.lJ.z0 + + bucos(2 - 20Osh2z0,

(2)

where aij and bij (j = 1, 4) are constants dependent on the elastic characteristics of the materials (the elastic moduli and

Poisson's ratios) and the angles e~ and/3; X = X l + iX 2 is the singularity index, determined from transcendental equation (19) in [5].

The displacement field of a triangular singular finite element with its apex i at a singular point (Fig. 2) cart be written in the form:

Kishinev Polytechnic Institute. Translated from Problemy Proclmosti, No. 11, pp. 75-78, November, 1994. Original article submitted April 20, 1992.

846 0039-2316/94/2611-0846512.50 �9 Plenum Publishing Corporation

r'\ . ..-1"

0

Fig. 1 Fig. 2

Fig. 1. Wedges composed of two plates with different elastic characteristics.

Fig. 2. Singular finite element with the apex i at a singular point.

E I ,qf

,?;2 ~= r I

I

[ t

p

P I_ ~r -I I- i

Fig. 3. Plates of two dissimilar materials with a V-notch (a) and an inclined

edge crack (b).

U x R x + /Zx,

s Uy = U ; "F try,

(3)

Here, Ux* and Uy* are continuous functions; UxS and Uy s are singular complements,

,S u x = c (* , 2)" (Ux, COS~O - uy, sin~,);

Uy s = c ( r , 2). (ux,sin~o + Uv, COS~O), (4)

where c(r, X) is a function having no effect on the asymptote and being equal to zero at the points i, j, k.

In accordance with [6], the stiffness matrix of the s in~lar finite element is determined as

[K] = f [DIr[EI [Olds, (5) $

where [D] is the strain matrix; [E] is the matrix of the elastic constants of the material; s is the area of the triangle. The stiffness matrix has the form

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TABLE 1. Values of the Dimensionless Quantity F = (Kt 2 + K22)'/'-p(Tra) '/2 with 3'

= 0 and Different Values of E1/E 2

EIIE2 F F [31 1.0 2.051 2.1 I0 2.0 2.100 2.117 4.0 2.125 2.130 10.0 2.138 2.146

TABLE 2. Values of the Dimensionless Quantities KI* = K1/P'/Tra and K2* =

K2/P'/'xa for a V-notched Plate Made of Different Materials

~,. E~IE2. equal to

deg 1.0 171 1.0 2.0 3.0 4.0. 5.0 0 2.113 2.051 2.094 2.104 2.106 2.106

0,000 0,000 0.155 0,235 0,285. 0,318 30 2,131 2,111 2,303 2,359 2,342 2,312

0.000 0.000 0.163 0.246 0.301 0.339 60 2.248 2.182 2.263 2.376 2.507 2.679

0,000 0,000 0,150 0,239 0.300 0,340 90 2,576 2,301 2,356 2.432 2,502 2,563

0,000 0,000 0,142 0,223 0,276 0,313 120 3,360 3.356 3,384 3,429 3,469 3,505

0.000 0.000 0.061 0.105 0.136 0.160

10.0 2.100 0.394 0.755 0.321 2.405 0.451 2.778 0.367 3.618

�9 0.225

100.0 2.076 0.418 2.576 0.557 3.005 0.612 3.467 0.371 3.813 0.325

Note: Here and in Table 3, the numerator shows values of K 1 , the denominator

values of K' 2

TABLE 3. Values of the Dimensionless Quantities Kl* = K1/pv/Tra and K2* =

K2/P'C'~ra for a Plate Made of Different Materials and Having an Inclined Edge

Crack

e,. EtlEz. equal to deg 1:0 210 3.0- 4.0 5.0

0 2.069 2.111 '2.120 2.121 2.120 0.01~ 0.155 0.235 0.284 0.317

30 1.988 2.168 2.268 2.333 2.378 0.389 0.672 . 0.856 0.984 1.075

60 1.444 1.463 1.460 1.453 1.447 0.436 0.581 0.657 0.704 0.735

r

90 0,993 [ 0,984 0,960. 0,939 I 0,922 i

0,510 [ 0,670 0,755 0,807 I 0,842 120 0.507 0.476 0.432 0.393 0.360

0.416 0.567 0.648 0.697 0.731

10.0 2.111 0.392 2.483 1.297 1.428 0.806 0.873 0.921 0.261 0.807

100.0 2.081 0.416 2.612 1.171 1.382 0.778 0.785 0.885 0.084 0.789

[K.I [K,z] ] 1~ = LIK,2]" [K221 �9 (6)

In (6), [Kll ] is the stiffness matrix of the triangular finite element for two-dimensional problems of the theory of elasticity;

[K12] and [K22] are the contributions of the singular part to the stiffness matrix.

Since integration over the area of an isosceles triangle has been replaced by integration over the area of a sector with

a radius (Fig. 2) and since UxS and Uy s are represented in the form (4), we can perform the integration in Eq. (5) in closed

form and avoid having to use formulas of integration that would make the computation more difficult. The elements of the

matrices [KI: ], [K22] are awkward in form and are not presented here. We will examine the following problems as examples: a plate composed of different materials with a V-notch (Fig.

3a); a plate with an inclined edge crack at the interface (Fig. 3b). The notch and the crack are subjected to uniformly

distributed uniaxial tensile stresses P. Tables 1-3 show the results of calculations performed for a plate with a plane stress

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state and a/W = 0.4, v 1 = u 2 = 0.3. The mesh of the finite-element grid was 0.5a in both cases. To compare our results

with the data of other authors, Table 1 shows values of the dimensionless quantity F = (K12 + K22)t/2/pv/ra with 3' = 0

and different values of E J E 2. Column 1 in Table 2 shows the results calculated in [7] for identical materials Et/E 2 = 1 and different values of the divergence angle 3' of the V-notch.

As can be seen, the sing-ular finite element constructed here yields satisfactory results even when a coarse grid of elements is used.

REFERENCES

1.

2.

3.

4.

5.

6,

E. P. Chen, "Finite element analysis of a bimaterial interface crack," Teor. Appl. Fract. Mech., 3, No. 3, 257-262

(1985).

K. Y. Lin and J. W. Mar, "Finite element analysis of stress-intensity factors for cracks at a bimaterial interface,"

Int. J. Fract., 12, No. 4, 521-531 (1976).

R. Yuuki and S. B. Cho, "Efficient boundary element analysis of stress factors for interface cracks in dissimilar

materials," Eng. Fract. Mech., 34, No. 1, 179-188 (1989).

V. Z. Parton and P. I. Perlin, Methods of the Mathematical Theory of Elasticity [in Russian], Nauka, Moscow

(1981).

D. B. Body, "Two edge-bonded elastic wedges of different materials and wedge angle tractions," J. Appl. Mech.,

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R. Gallagher, Finite-Elements Method [Russian translation], Mir, Moscow (1984).

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