PSI - Issue 83

Victor Rizov et al. / Procedia Structural Integrity 83 (2026) 95–104

99

The acceleration, 5 D a in point, 3 L , are oriented as shown in Fig. 1. The bars of the frame-like component are functionally graded across their thickness. Therefore, the mass per unit area, m , changes continuously across the thickness. The change of m in the bar, 1 2 DD , is defined by Eq. (16).  3 L a , and the projections of

h 2

z



1



m me 

h

,

(16)

1

z h    .

2 h

(17)

1

2

1 m - mass per unit area in the left surface of the bar;  - parameter;

The notations used in the above equations are:

h - thickness of the bar; 1 z - centric axis of the bar cross-section. The intensity of the inertia load at any point on the frame-like component can be derived by multiplying the mass per unit area by the acceleration, taken with a minus sign. The frame-like structural component deforms as a non-linear elastic body. The constitutive law of non-linear elastic behaviour applied in this paper is (Tsankov (1996))







.

(18)

R S 



In Eq. (18) the following notations are used:  - stress;  - strain; R - parameter; S - parameter. The laws for change of R and S across the thickness of the bar, 1 3 DD , are defined by Eqs. (19) and (20).

h 2

z



1



R Re 

h

,

(19)

1

h 2

z



1



S S e 

h

,

(20)

1

where

z h    .

2 h

(21)

1

2

The notations used in the above equations are as follows: 1 R - value of R on the left surface of bar; 1 S - value of S on the left surface of bar;  - parameter;  - parameter. A lengthwise crack splits portion, 2 3 D D , of the bar, 2 4 D D . The length of the crack is denoted by a . The thicknesses of the upper and lower arms of the crack are 1 h and 2 h . The SERR, G , for this crack is extracted by differentiating the complementary strain energy, * U , in the frame-like component under the inertia loading by making use of Eq. (22).

bda dU *

G



,

(22)