PSI - Issue 32

Andrey Yu. Fedorov et al. / Procedia Structural Integrity 32 (2021) 194–201 A.Yu. Fedorov et al. / Structural Integrity Procedia 00 (2021) 000–000

197

4

P y

y

x

l y

t 1

l x

P x

t

a

Fig. 2. Computational scheme of a plate with a substrate.

provided that the behavior of the materials is linear, the mechanical parameters are Poisson’s coe ffi cients ( ν , ν 1 ) and the ratio of elastic moduli ( E / E 1 ) of the structure and substrate materials. The assumptions used in this study are as follows: 1. All materials are linearly elastic and isotropic. 2. All materials are perfectly connected and there is no relative slipping. These assumptions are part of the basic assumptions used in the studies by Wu et al. (2014); Shen et al. (2018); Wang et al. (2012). The following ratios of geometric parameter were used in the calculations: a / l x = 10, l y / l x = 0 . 15, t 1 / l x = 0 . 0075. In terms of geometry, we can change the substrate dimensions whenever it is necessary , but with fixed substrate dimensions, the strains near the substrate are determined by the plate-to-substrate thickness ratio t / t 1 . The stress-strain state of the plate with a substrate is calculated by the finite element procedure implemented with the use of the ANSYS package. The examined plate is a composite structure, which is made of elastic isotropic materi als. The modeling domain was discretized into prismatic 20-node elements SOLID186 with a quadratic approximation of the unknowns. A variant of the finite-element mesh near the substrate is shown in Fig. 3. The essence of the numerical study is to compare the strain values obtained at the center of the outer surface of the substrate (point with coordinates [0 , 0 , 0]) and at the center of the plate without a substrate (point with coordinates [0 , 0 , − t 1 ]) under the same conditions of tensile loading. Two loading conditions were considered: extension along the x -axis ( P x = 1, P y = 0) and extension along the y -axis ( P x = 0, P y = 1). Before the numerical study, the convergence of the solution was estimated depending on the number of elements of the finite-element mesh. Since the computational scheme contains singular points and lines, at which di ff erent materials come into contact, the convergence of the solution was estimated by considering the extreme ratios of elastic moduli of the plate and substrate materials ( E / E 1 = 0 . 001 and E / E 1 = 1000). Fig. 4 shows the dependences of stresses σ x / P x obtained at the center of the substrate surface on the number of elements of the finite-element mesh at t / t 1 = 10.

Fig. 3. Variant of the finite-element mesh near the substrate (a quarter of the mesh is shown).