PSI - Issue 35

V.A. Zimina et al. / Procedia Structural Integrity 35 (2022) 188–195 V.A. Zimina, I.Yu. Smolin / Structural Integrity Procedia 00 (2021) 000–000

192

5

0 l l l 

 V

dV

  σ xx

V σ = 1

,

(9)

 

0

xx

Here angular brackets mean the volume-averaged values, l and l 0 are the initial and current lengths of the specimen along the loading axis, respectively. Numerical determination of the effective elastic moduli of three-phase composite is based on approximations of various stress-strain curves. The detailed algorithm used for determining effective elastic properties of heterogeneous materials is described in the paper by Mikushina and Smolin, (2019) and outlined blow. The stress strain curves considered are the dependences of the averaged values of the relevant parameters of the stress and strain states (stress intensity – strain intensity, pressure – volumetric strain, axial stress – axial strain). As an example, an effective value of shear modulus is a coefficient of the linear approximation of the dependency of stress intensity σ i on strain intensity ε i and the effective value of the bulk modulus is a coefficient of the linear approximation of the dependency of pressure and volumetric strain according to the formulas:

> 3 i G     

> i

,

(10)

>      P K

>

Young's modulus and Poisson's ratio are defined for the case of plane strain according to the formulas:

> > > xx

yy       

< >= E 

< > xx 

,

(11)

 =

xx

2

< >

(1

)

 

yy

3. Results and discussions 3.1. Simulation of the mechanical behavior of Al 2 O 3 –ZrB 2 –SiC composite

It is known that the ceramic composite materials are referred to the class of brittle materials. This type of material exhibits different strength behavior in tension and compression. In the papers by Balokhonov and Romanova (2009) and Balokhonov et al. (2020) it was shown, that the composite materials primarily fracture in the local regions of tension due to their heterogeneous structure. In this connection, the uniaxial tension of the RVE of three-phase composite was simulated in this paper. In Figs. 2–3, one can see the results of numerical simulation of mechanical behavior of Al 2 O 3 –ZrB 2 –SiC composite. The macroscopic stress–strain curve is shown in Fig. 2. The points on the stress-strain curve correspond to various stages of deformation: 1 – elastic deformation; 2, 3 – nucleation and propagation of cracks in the structure of composite; 4 – macroscopic fracture of the composite. When the deformation is less than 0.079 %, the stress increases linearly with the strain. Fig. 3 demonstrates the fracture patterns of the investigated structure of the composite. The colors in the RVE correspond to the legend in Fig. 1. The numbering of the fracture patterns in Fig. 3 corresponds to the stages of deformation which are presented as the numbers 1, 2, 3, 4 on the macroscopic stress-strain curve in Fig. 2. The first drop of the stress-strain curve is observed when the deformation reaches 0.079 %. As this takes place, a crack nucleates in the RVE (Fig. 3b). The crack in the ceramic composite is generated near the pore surface and propagates to the upper boundary of the structure of composite transversely to the applied loading direction. The place of crack nucleation is due to the stress concentration caused by the pore shape. When the crack reaches the upper boundary of the RVE, no new fracture occurs, and stress increases on the macroscopic stress-strain curve resulting in an apparent hardening of the composite material. Point 3 on the stress-strain curve in Fig. 2 corresponds to the second drop of the stress caused by new crack nucleation in the structure of composite (Fig. 3c). The crack also originated near the pore, i.e. at the place of stress concentration. The further crack propagation occurs transversely to the direction of tension. The macroscopic fracture of the ceramic composite takes place when the crack reaches the opposite side of the RVE (Fig. 3d). This instant corresponds to point 4 on the stress-strain curve and to the rapid drop of stress in Fig. 2.