PSI - Issue 41

Efstathios E. Theotokoglou et al. / Procedia Structural Integrity 41 (2022) 361–371 Efstathios E.Theotokoglou/ Structural Integrity Procedia 00 (2022) 000–000

364

4

The relation of strains and the nodal displacements of an element is (Cook, et al., (2001)).       d   

(9)

The stiffness matrix is given by (Cook, et al., (2001)), (Burlayenko, et al., (2017)), (Li, et al., (2000)).

1 1

  k

     T B B t det J D d d    

1 1     

(10)

In the case of graded elements the constitutive matrix is a function of the coordinates (Burlayenko, et al., (2017)), (Santare, et al., (2003)), (Martínez-Pañeda, (2019)), (Paulino & Kim, (2007)), (Asemi, et al., (2014)), (Nojumi & Wang, (2020)), (Li, et al., (2000)) and the stiffness matrix can be written as (Cook, et al., (2001)), (Burlayenko, et al., (2017)), (Paulino & Kim, (2007)), (Li, et al., (2000)).           1 1 T 1 1 k B , , B , t det J D d d                          (11) 4. The problem considered In this work we consider two configurations, a sandwich beam with metal core and FGM face sheets and a sandwich beam with ceramic core and FGM face sheets (Figure 1) (Belarbi, et al., (2022)). We also examined the discontinuous cases of the beam with ceramic core and metal face sheets and of the beam with metal core and ceramic face sheets in order to compare them with the other sandwich beams. The load was a distributed load on the upper face sheet of the beam. In this study an 8-node two-dimensional quadrilateral isoparametric element was used. The dimensions used are b = 50 mm, h = 200 mm, L = 2000 mm, the load was q z = 5000 N / mm (Belarbi, et al., (2022)). The properties of the materials are given in the Table 1 (Belarbi, et al., (2022)). For the study, 500 elements are used in each layer. The elements in the FGM layers are graded elements, while for the core were used homogeneous elements. The beam is supported by fixed supports in both left and right ends. The problem is solved under plane stress conditions.

Table 1. Material properties (Belarbi, et al., (2022)). Properties

Metal: Ti-6Al-4V

Ceramic: ZrO 2

E (Young's modulus)

70 (GPa)

380 (GPa)

v (Poisson ratio)

0.3

0.3

Figure 1Model of our problem, dimensions, load, constraint and cross sections (Belarbi, et al., (2022))