PSI - Issue 52

Mengke Zhuang et al. / Procedia Structural Integrity 52 (2024) 690–698 Author name / Structural Integrity Procedia 00 (2023) 000–000

692

3

m . In this work, the design parameter m includes geometric parameters and curvature. Consequently, the system of equations can be rearranged as: AX , m = F , m − A , m X (1) where A , m represents the derivatives of the coe ffi cient matrix with respect to the design parameters and X , m contains the derivatives of the displacement and tractions.

2.2. Derivatives of the Stress Intensity Factors

The CSDE technique is used in this work for evaluating the crack tip stress intensity factors. The meshes near the crack tip by the discontinuous quadratic elements are represented in Fig.1. The crack tip SIF can be linearly extrapolated by substituting the boundary displacement and rotation results of nodes

Fig. 1. Example of crack tip nodal element. Dirgantara (2001)

near the crack tip such that:

r AA ′ r AA ′ − r BB ′ {

K } AA ′

r BB ′ r AA ′ {

K } BB ′ −

{ K } tip

(2)

=

=

=

where { K } AA ′ B −{ w } B ′ . w consists of rotation in x and y direction ϕ 1 , ϕ 2 , the out-of-plane displacement w 3 and the in-plane displacements u 1 , u 2 in x and y direction respectively. The stress intensity factor at the nodes on the crack surfaces near the tip for three bending failure modes and two membrane failure modes can be found by: 6 5 l [ C ] { w } A −{ w } A ′ and { K } BB ′ 2 l [ C ] { w }

Eh 3 48 √ r Eh √ 2 π 8 √ r

Eh 3 48 √ r

5 Eh 24(1 + ν ) √ r

π 2

π 2

π 2

∆ w 3

K 2 b =

K 3 b =

K 1 b =

∆ ϕ 2

∆ ϕ 1

Eh √ 2 π 8 √ r

(3)

K 1 m =

K 2 m =

∆ u 2

∆ u 1

where the subscript m , b represent membrane mode and bending mode respectively. by applying the extrapolation in equation 2, the sensitivities of the stress intensity factors are: { K } tip , m = RRB , m { K } BB ′ + RRB { K } BB ′ , m − RRA , m { K } AA ′ − RRA { K } AA ′ , m (4) where RRA = r BB ′ r AA ′ − r BB ′ RRB = r AA ′ r AA ′ − r BB ′ RRA , m = r BB ′ , m r AA ′ − r BB ′ − r BB ′ ( r AA ′ , m − r BB ′ , m ) ( r AA ′ − r BB ′ ) 2 RRB , m = r AA ′ , m r AA ′ − r BB ′ − r AA ′ ( r AA ′ , m − r BB ′ , m ) ( r AA ′ − r BB ′ ) 2 (5)