Solution of stress intensity factor for mode I centre crack in undermatched butt joint with certain reinforcement

Abstract

In order to improve the fracture resistance of undermatched butt joint, the influence of joint shape parameters on the stress intensity factor (SIF) of undermatched butt joint with mode I centre crack was studied, and the solution of SIF was obtained by combining the analytical method with the finite element method as well. The results are useful to guide the shape design of undermatched butt joint with mode I centre crack under both static load and fatigue load conditions.

Keywords

Undermatched butt joint Mode I centre crack Stress intensity factor Analytical method Finite element method

List of symbols

crack dimension

a _c

critical crack dimension

a ₀

initial crack dimension

material property constant

Young's modulus

height of weld reinforcement

imaginary part of

J integral

K _I

stress intensity factor of mode I crack

K _IC

fracture toughness

material property constant

N _f

fatigue remaining life

weld toe smooth transition radius

Re[Z_I(z)]

real part of Z_I(z)

half of base metal thickness

half of weld width

shape factor

complex variable

Z_I(z)

Westergaard complex stress function

derived function of Z_I(z)

Δσ

cyclic stress

Poisson's ratio

external stress

σ _c

critical stress

σ _y

stress of y direction

Introduction

In order to save the weight of structure, high strength steels have been widely used in ship building, pressure vessel producing and other industrial fields. However, welding cold cracking has become one of the main failure modes of high strength steel welded structures.1 Preheat and post-weld heat treatment is one of the main solutions to avoid welding cold cracking.2 Unfortunately, the potential of embrittlement and softening of the heat affected zone increased remarkably, while the strength and toughness of heat affected zone decreased evidently after preheat and post-weld heat treatment. 3 3,4 To improve joint performance, it is significant to apply working conditions without preheat and post-weld heat treatment. In order to prevent cold cracking and decrease the temperature of preheat and post-weld heat treatment, more attention has been paid to undermatched joint design. 5 5,6 Traditionally, the undermatched joint has lower load carrying capacity compared with equal matched joint. To make the load carrying capacity of undermatched joint not less than that of the base metal or equal matched joint, an undermatched butt joint form with certain reinforcement, which is shown in Fig. 1, has been proposed by Fang et al.7 Its joint shape parameters include reinforcement h, weld width 2w, weld toe smooth transition radius r and base metal thickness 2t. The shaded area indicates the low strength filler metal. The design principle is adjusting the stress distribution by making height of reinforcement loaded and reducing the stress concentration factor of the weld toe zone by adding a joint shape parameter, which is a smooth transition radius of the weld toe.

Figure 1

Shape parameters of centre cracked undermatched butt joint with certain reinforcement

As a matter of fact, brittle failures often occur under low stress due to the sensitive and sudden break-up of defects in the welded joint, especially in high strength steel joint.8^– 10 Fracture mechanics can deal with problems of structure with cracks or other defects. The safety of welded joint with cracks can be assessed by comparing its stress intensity factor (SIF) result with fracture toughness, which is a constitutional property of certain material. Therefore, it is meaningful to obtain the solution of SIF for undermatched butt joint with certain reinforcement. Nowadays, the SIF can be obtained by analytical, numerical or experimental standardisation method. The numerical method is the most widely used method of these three methods due to its efficiency and accuracy. As one of the numerical methods, the finite element method has been widely applied in engineering fields.11^– 17

Basic concept and theory

It is well known that SIF for mode I crack can be described as (1) where σ is the external stress, a is the crack dimension and Y is the shape factor that depends on structural shape parameters, crack location and loading mode.18 Obviously, K_I only depends on structural shape parameters when crack dimension and location, loading mode and magnitude are known. Therefore, the shape parameters of the undermatched butt joint shown in Fig. 1 will have a great effect on its SIF.

So far, the SIF solution for mode I crack in a fixed width finite plate has been given in some textbooks, but the SIF solution for mode I crack in a width changed finite plate has not been proposed yet.

Solution process

Analytical method

For the crack under unidirectional tension in an infinite plate, the Westergaard complex stress function is19 (2) where σ is the external stress, a is half of the crack length and z is a complex variable.

Considering the centre crack in an undermatched butt joint with certain reinforcement under unidirectional tension (as depicted in Fig. 1), its Westergaard complex stress function can be described as (3) where c is an undetermined coefficient.

For the joint shown in Fig. 1, the stress of the y direction can be deduced as (4) The coefficient c can be determined by the equilibrium of forces. Considering the symmetry of the joint, the half part is taken. Equation (5) is obtained according to the balance of internal and external forces on the y direction (5) Solving from equation (5) produces (6) Substituting equation (6) into equation (3), the Westergaard complex stress function of centre cracked undermatched butt joint with certain reinforcement can be obtained as follows (7) Thus, its SIF is available as (8) where ξ = z−a.

Numerical simulation

In equation (8), only the influence of h on SIF is considered. Actually, all the joint shape parameters have an effect on SIF. If all the shape parameters are taken into account, the solution of SIF for the joint shown in Fig. 1 should be (9) where f(w,h,r,t) is an undetermined coefficient function, it is related to w, r, h and t respectively. f(w,h,r,t) cannot be obtained through analytical method but by finite element method and regression analysis.

For the plane strain condition, the relationship of J integral and SIF K_I is (10) where E is Young's modulus, J is the J integral and ν is Poisson's ratio.

Combining equations (9) and (10), coefficients f of joints with different shape parameters can be written as (11) Here, the J integral can be calculated with the finite element method. Then, the coefficient function f(w,h,r,t) can be obtained by regression analysis of all the coefficients f acquired above.

Because of the symmetry, one-fourth of the joint is selected as the analysis model, as shown in Fig. 2a, in which plane strain elements are used. Meshes near the crack tip are refined, as shown in Fig. 2b. The crack length in relation to the base metal thickness is 0·286, and the minimum element size in relation to crack length is 0·005.

Figure 2

Finite element model of one-fourth centre cracked butt joint with certain reinforcement

Figure 3 shows the influence of shape parameters on coefficient f. In Fig. 3a, w in relation to t is 2, the ratio of h and t ranges from 0 to 0·524 and r ranges from 2 to 11 mm. In Fig. 3b, w in relation to t is 4·519, the ratio of h and t ranges from 0 to 1·096 and r ranges from 4 to 30 mm. All the coefficients f are obtained by equation (11). It can be seen that f increases with the increase in h and r, while f decreases with the increase in w.

Figure 3

Influence of joint shape parameters on coefficient f

The coefficient function f(w,h,r,t) is finally determined by regression analysis of all the coefficients f achieved above and the corresponding shape parameters (12) Substituting equation (12) into equation (9), the solution of SIF for mode I centre crack in undermatched butt joint with certain reinforcement takes the form (13)

Discussion

The SIF results of centre cracked butt joints with two different w/t values, i.e. 2 and 5, are shown in Fig. 4. Here, σ is 100 MPa, a is 2 mm and t is 8 mm. r ranges from 4 to 14 mm, and h/t ranges from 0 to 1. It can be seen that the SIF results vary with different joint shape parameters. The influences of each joint shape parameter on SIF are different. K_I decreases significantly with the increase in h and w. K_I increases slightly with the increase in r, especially when w is large. K_I will be stable as h is fixed, and w is large enough. When h or w is increased, the average stress of the crack section decreases due to the increase of the crack section capacity area, and then K_I decreases. As far as the centre crack is concerned, the weld toe is far away from the crack. r has little influence on changing the stress state of the crack tip. Therefore, the influence of r on K_I can be ignored especially when w is large enough. In Fig. 4, ‘h/t = 0’ represents the traditional result of K_I, where the reinforcement is not considered. The results in Fig. 4 show that K_I differs greatly considering the reinforcement, and the value of K_I calculated by considering the joint shape parameters is smaller than the one calculated by the traditional way. Thus, it is necessary to give a SIF solution for centre cracked butt joint considering the influence of joint shape parameters.

Figure 4

Influence of joint shape parameters on SIF K_I

In fact, welded joints always endure static loads or fatigue loads. When joints stand static load, the critical crack length and critical stress are ought to be known. Based on equation (13), the critical crack length a_c and the critical stress σ_c of centre cracked butt joint with certain reinforcement can be obtained as follows (14) (15) When the joint shown in Fig. 1 stands fatigue load, its fatigue remaining life can be gained as (16) where a₀ is half of the initial crack length, a_c is half of the critical crack length, which is calculated by equation (14), Δσ is cyclic stress and A and m are material property constants.

With knowledge of the above, if any one of equations (13)–(16) can be testified to be accurate, the other equations given in this paper will be accurate. As we know, the critical stress σ_c can be tested by tensile test easily. Two different undermatched butt joints shown in Fig. 5 are chosen to testify the accuracy of critical stress σ_c in equation (15). The yield strengths of the filler and base metals are 403 and 629 MPa respectively. The chemical composition of the undermatched joint metal is shown in Table 1. All undermatched butt joints apply X groove and manual arc welding. The base metal thickness is 10 mm, and the centre crack length is precracked to 5·6 mm after wire electrode cutting. The experiment temperature is 20°C. The tensile rate is 2 mm min⁻¹. The fracture toughness K_IC of the filler metal E5015 is 1295 MPa mm^1/2. The joint shape parameters and comparison results are illustrated in Table 2. It can be seen that the results of critical stress calculated by equation (15) and obtained by experiments show little difference. Hence, the solution of SIF for mode I centre crack in butt joint with certain reinforcement can be used to efficiently guide the shape design of the butt joint.

Figure 5

Two different undermatched butt joints with centre crack

Table 1

Chemical compositions of undermatched joint metal

Joint metal	C/wt-%	Si/wt-%	Mn/wt-%	P/wt-%	S/wt-%	Fe/wt-%
Base metal Q620-CF	⩽0·09	⩽0·50	⩽1·8	⩽0·02	⩽0·01	Balance
Filler metal E5015	⩽0·12	⩽0·75	⩽1·6	⩽0·04	⩽0·035	Balance

Table 2

Comparison of critical stress obtained by experiments and equation (15) given in this paper

Specimen	h/mm	w/mm	r/mm	σ_c by experiments/MPa	σ_c by equation (15)/MPa	σ_c difference/%
Joint 1	2·5	18	8	509·6	527·8	3·45
Joint 2	3·8	28	8	602·2	622·6	3·28

Conclusions

Height of reinforcement plays the most important role in the SIF of centre cracked butt joint with certain reinforcement. The SIF decreases significantly with the increase in height in reinforcement especially when weld width is large enough. Weld width has relatively great effect on the SIF of centre cracked butt joint with certain reinforcement. The SIF decreases with the increase in weld width. The SIF will be stable when the height in reinforcement is fixed and the weld width is large enough. The weld toe smooth transition radius has little effect on SIF of centre cracked butt joint with certain reinforcement. The influence of weld toe smooth transition radius can be ignored when weld width and height in reinforcement are both large enough. It would be useful for improving the load carrying capacity of undermatched butt joint with centre crack to choose large enough height in reinforcement and weld width and also an appropriate weld toe smooth transition radius. The solutions of SIF, critical crack length, critical stress and fatigue remaining life for mode I centre crack in undermatched butt joint with certain reinforcement are obtained to guide its shape design under static and fatigue load conditions accurately.

Footnotes

Acknowledgements

Project no. 2007DFR70070 supported by Sino-Russian Inter-governmental Cooperation Projects is acknowledged.

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