PSI - Issue 77

Victor Rizov et al. / Procedia Structural Integrity 77 (2026) 382–388 Author name / Structural Integrity Procedia 00 (2026) 000–000

385

4

z h − ≤ ≤ .

2 h

2 (12) In formulas (8) – (11), the subscripts, VS and NS , refer to the upper and lower surface of the beam construction, respectively (for instance, VS E and NS E are the values of E at the upper and the lower surface, et cetera), z is the vertical centric axis of the beam (Fig. 1), 1 g , 2 g , 3 g and 4 g are material parameters regulating the distribution of E , η , 0 β E and 0 γ η , respectively. From the analysis of the balance of the energy, we get the following expression of the strain energy release rate, G , for the crack in the beam structure:

b a b M Q − ∂ ∂ 1 3 ϕ

a U

∂

G

=

,

(13)

∂

3 Q ϕ is the angle of rotation of section, 3 Q , of the beam, U is the strain energy in the

where b is the beam width,

beam construction. We determine U by addition of the strain energies,

2 3 Q Q U and

1 2 Q Q U ,

3 4 Q Q U , cumulated in the beam

sectors, i.e.

2 3 Q Q U U U U + = + 1 2

.

(14)

3 4 Q Q Q Q

We use formula (15) for obtaining of the strain energy in the upper crack arm u dV U Q Q V Q Q Q Q 0 1 2 ) ( 1 2 1 2 ∫∫∫ = ,

(15)

where 0 1 2 Q Q u is the unit strain energy,

1 2 Q Q V is the volume of the upper crack arm. 0 1 2 Q Q u is derived as

ε

u Q Q ∫ = 0 0 1 2

σ ε d .

(16)

The curvature and the coordinate of the neutral axis which are needed for describing of the distribution of stresses and strains in the beam sectors are determined with the help of the following equations: 0 1 = Q ϕ , (17) 0 1 = Q u , (18) dA dA Q Q A Q Q A Q Q 2 3 ( ) 1 2 ( 1 2) σ σ ∫∫ ∫∫ = , (19) zdA z dA Q Q A Q Q A Q Q 2 3 ( ) 1 2 1 ( 1 2) σ σ ∫∫ ∫∫ = , (20) dA dA Q Q A Q Q A 3 4 ( ) 2 3 ( ) σ σ ∫∫ ∫∫ = , (21) zdA zdA Q Q A Q Q A Q Q 3 4 ( ) 2 3 ( 2 3) σ σ ∫∫ ∫∫ = , (22) where 1 Q ϕ and 1 Q u are the angle of rotation and longitudinal displacement of section, 1 Q , of the beam. Equations (19) – (22) represent equilibrium conditions of sections, 2 Q and 3 Q , of the beam construction. We derive the strain energy release rate also by applying formula (23) to verify the results found-out by the energy balance.