PSI - Issue 77

Victor Rizov et al. / Procedia Structural Integrity 77 (2026) 389–396 Author name / Structural Integrity Procedia 00 (2026) 000–000

390

2

community around the globe. Promising type of structural materials are multilayered materials as combinations of layers of different materials (Finot and Suresh (1996), Kaul (2014), Rzhanitsyn (1986), Tokova et al. (2016)). The concept of multilayered materials allows one to take advantage of varied properties of the layers and to obtain a multilayered system that is superior for a given engineering application (Lloyd and Molina-Aldareguia (2003), Sy Ngoc Nguyen et al. (2020)). Modernization of the multilayered structural materials represents a complex and multilateral task which requires research and development by the united efforts of engineers and specialists in different scientific areas (material science, structural mechanics, fracture mechanics, etc.). The current theoretical paper represents a contribution in the area of delamination fracture of multilayered load bearing structural systems. The interest towards delamination fracture behaviour is conditioned by the high vulnerability of multilayered systems to delamination (Dolgov (2005), Narisawa (1997), Rizov (2018), Rizov (2018), Rizov (2019)). The paper deals with analysis of delamination in a planar multilayered structural member of circular cross-section. The member who has a vertical and a horizontal portion is under angles of twist and bending that increase with time at constant velocities. The layers are inhomogeneous along the thickness. The basic purpose of the paper is to analyze the SERR with considering the velocity of loading and non-linear elastic behaviour of the material. 2. Analysis of the SERR The planar structural member in Fig. 1 has two portions, 1 3 LL and 3 4 L L . The vertical portion is rigidly fixed in its upper end. The cross-section of the member is a circle of radius, D R . The member is made by concentric layers that are inhomogeneous along their thickness. There is a delamination crack between the layers in portion, 1 2 LL , of the member (Fig. 1). The free end of the internal delamination arm is under angle of twist, T ϕ , that is a linear function of time, t , i.e. v t T T ϕ ϕ = , (1) where T v ϕ is the velocity. Besides, the section, 3 L , in which the horizontal and vertical portions of the structural member are connected is under angle of bending, M ϕ , varying linearly with time with velocity, M v ϕ , i.e. v t M M ϕ ϕ = . (2) Obviously, portion, 1 3 LL , of the member in Fig. 1 is loaded in torsion, while portion, 3 4 L L , is loaded in bending. Formula (3) presents the non-linear constitutive law for the mechanical behaviour of the layers under torsion (Lukash (1997)) .

   ,

  

γ



n

1

Ti G H

S

τ i

= − γ

γ

+

(3)

i

i

i

γ



0

where i τ is the shear stress, Ti G ,

i H , i S , i n and 0 γ  are material properties, γ is the shear strain, γ  is the

derivative with respect to time. The nonlinear constitutive law (4) is applied for the layers under bending.

   ,

  

ε



m

1

E Q

i P

σ i

ε ε

= −

+

(4)

i

i

i

ε



0

where i σ is the stress, ε is the strain, ε  is the strain first derivative with respect to time, i E , i Q , i P and 0 ε  are material properties. Formulas (5) – (10) present the variation of the material properties along the thickness, ∆ , of inhomogeneous layer. ( ) Gi Gi b i b Ti Ti Ti Ti R R G G G G − ∆ − = + α β ε , (5)