PSI - Issue 77

Victor Rizov et al. / Procedia Structural Integrity 77 (2026) 397–404 Author name / Structural Integrity Procedia 00 (2026) 000–000

399 3

the beam structure is denoted by 1 t and 2 t , respectively. The temperature varies linearly along the beam thickness. The mechanical behaviour of the beam is non-linear elastic and is treated by using the constitutive law written in Eq. (1) (Tsankov (1996)). (1) where σ is the stress, ε is the strain, L and H are material properties. The beam is continuously inhomogenous along its length. As a result of this, the material properties, L and H , are distributed continuously in longitudinal direction. This distribution is described by the laws written in Eqs. (2) and (3), respectively. ε ε σ L H + = ,

1 β l L L rght −

lft

1 β

x

L L

= +

,

(2)

lft

2 l H H rght − β

lft

β

H H

x

= +

,

(3)

2

lft

where

x l ≤ ≤ 0 .

(4)

Here, lft L and rght L are the values of L in the left-hand and right-hand ends of the beam, rght H are the values of H in the left-hand and right-hand ends of the beam, 1 β and 2 β are parameters, x is the beam longitudinal axis (shown in Fig. 1), l is the beam length (apparently, 1 2 l l l = + ). The coefficient of thermal expansion, t α , is distributed along the length of the beam according to the law written in Eq. (5). (5) where tlft α and trght α are the values of t α in the left-hand and right-hand ends of the beam, 3 β is a parameter. The longitudinal fracture due increased temperature in the beam is analyzed by the integral J (Broek (1986)). In this relation, we study the beam behaviour under increased temperature via Eqs. (6) – (22). The angle of rotation, 1 D ϕ , of the left-hand end of the beam is zero. This fact is used for composing of Eq. (6). 0 1 = D ϕ . (6) 1 D ϕ is determined by Eq. (7). dx dx h u u l a l D D l a D D D g D f D ∫ ∫ − + + + − = 2 1 1 3 4 0 1 3 1 1 1 1 κ κ ϕ , (7) where D g u 1 and D f u 1 are the longitudinal displacements of the upper and lower surfaces of the beam in the left hand end, 1 h is the thickness of the upper arm of the crack, 1 3 D D κ and 3 4 D D κ are the curvatures in portions, 1 3 DD and 3 4 DD , of the beam, the lengths, 1 l and 2 l , are shown in Fig. 1. Equations (8) and (9) are used for deriving D g u 1 and D f u 1 , respectively. t dx u l l t D g 1 0 1 1 2 ∫ + = α , (8) t dx t dx u m l l l t l t D f ∫ ∫ + + = 1 2 1 1 2 0 1 α α , (9) lft H and 3 3 β β α α trght − α α x l tlft tlft t = + ,