PSI - Issue 41

Victor Rizov et al. / Procedia Structural Integrity 41 (2022) 115–124 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

116

2

(1999), Malzbender (2003), Markov and Dinev (2005), Mušálek et al. (2010)). By combining of layers properly, economical multilayered systems of a high load-bearing capacity can be developed. Multilayered systems are very suitable for reducing the weight of engineering structures. Therefore, it is not surprising that research and application of multilayered and inhomogeneous materials and structures in many areas of engineering is a topic of high interest (Chikh (2019), Marae Djouda et al. (2019), Nagaral et al. (2019), Saidi and Sahla (2019), Yu et al. (2003)). Although the multilayered systems are modern and highly efficient, they have some substantial drawbacks. For instance, the structural integrity of these systems depends largely on their delamination fracture behaviour. Delamination (separation of layers) reduces the strength and load-bearing capacity and affects the reliability and durability of the multilayered structures. Analysis of delamination in multilayered systems can have various aspects, considering the material properties, the geometry of the structural member, the external loading, the time dependence of the mechanical behaviour etc. (Rizov (2017), Ruzov (2018), Rizov (2019), Rizov and Altenbach (2020)). The area of interest of this paper is delamination in non-linear viscoelastic multilayered statically undetermined beam structures. The interest to non-linear viscoelastic behaviour is due to the fact that previous analyses of delaminaton are concerned with linear viscoelastic beams (Rizov (2020), Rizov (2021)). In reality, however, statically undetermined multilayered beams with delamination crack frequently exhibit non-linear viscoelastic behaviour that has to be taken into consideration in the delamination analysis. The aim of this paper is to derive the strain energy release rate for a delamination crack in a multilayered statically undetermined beam structure that has non-linear viscoelastic mechanical behaviour. The method of the J -integral is used for verification. 2. Delamination analysis of statically undetermined non-linear viscoelastic beam structure

1 2 H H , depicted in Fig. 1 is made of n non-linear viscoelastic layers.

The multilayered beam structure,

Fig. 1. Statically undetermined multilayered beam with a delamination crack.

The width, thickness and length of the beam are b , h and l , respectively. There is a delamination crack of length, a , in the beam as depicted in Fig. 1. The lower and upper delamination crack arms have thickness, 1 h and 2 h , respectively. The right-hand end of the beam is clamped. The end, 1 H , of the upper crack arm is constrained with an axial double rod. An axial force, F , and a bending moment, M , are applied at the free end of the lower crack arm (Fig. 1). The variations of the axial force and the bending moment with time, t , are expressed as F v t F  , (1) M v t M  , (2)