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

190

3

The conservation of mass and momentum:

x   σ

dt d ν

V = V ρ ρ 0 0 ,

(1)

=

j ij

ρ

i

Strain rate-velocity relations:

v

x v

 

 

   x = v 2 1  

   ,

   x = v 2 1  

  

+

j

i j

ε 

ω 



(2)

i

i

x

ij

ij

j

i

j

Here ρ 0 and ρ are the initial and current density, respectively; V 0 and V are the initial and current elementary volume, respectively, v i is the velocity vector components, σ ij is the stress tensor components, x i is the Cartesian coordinates, ij   is a component of the strain rate tensor, ij   is a component of the rotation rate tensor. The system of equations does not include the energy conservation law because we consider the barotropic medium. The relations of an isotropic el astic- plastic m edium with Drucker–Prager yield criterion were used as constitutive equations (Alejano and Bobet, 2012) . The stress tensor is presented as the sum of spherical (pressure P ) and deviatoric ( s ij ) parts ij ij ij = P + s δ σ  , where δ ij is the Kronecker symbol. The constitutive equations describing elastic deformation are given by Eqs . (3)

3 1

T P       ij ij

   

T P ), θ θ (  

T P ) θ δ - θ (  

T T P P   θ ,     

P= K 

s = G 

s

θ , ω ω kj ik +s   

ε ε 2

 

(3)

ij

ij

kj ik

ii

ii

In these equations K is the bulk modulus, T θ  is a total volumetric strain rate, P θ  is an inelastic volumetric strain rate, G is the shear modulus, kj ik kj ik s + s ω ω   is the term providing the Jaumann corotational derivative. The constitutive equations describing inelastic deformation include the yield surface f (σ ij ) and the plastic potential g (σ ij ) (4), together with the non- associated plastic flow rule (5 ):

f

α 2     J P Y

J P g ij    2 ( )

,

(4)

) σ ( ij

0

 g 

σ ) σ ( ij ij

p

( 5 )

if ( ) 0   f

ε 

 

ij

ij



Here J 2 is the second invariant of the deviatoric stress tensor, α is the coefficient of internal friction, Y is the cohesion (or shear strength),   is a plastic multiplie r, β is the coefficient of dilatancy. Two criteria were utilized for describing the fracture of the material. The first criterion is based on the accumulated inelastic strain. The second criterion relies on the tensile pressure. After meeting any fracture criterion in a calculation cell , all components of the stress tensor are equated to zero in the cell, and then the material ceases to resist tension but not compression. Thus, destroyed material arises in the calculation cells . The union of the destroyed adjacent cells forms a crack. The stemming of these cells simulate s crack propagation. Finite difference method was used to solve the set of differential equations of solid mechanics (Wilkins, 1999) . The simulation was carried in a two- dimensional formulation under conditions of plane strai n. Recently, a vast number of papers apply the approach to derive a representative volume element (RVE) of material by binarization or digitization of SEM or CT images. In our case, a RVE of the three- phase ceramic composite proceeds from a SEM image of the composite whose microstructure consists of an Al 2 O 3 matrix, inclusions of two phase s ZrB 2 and SiC, and por es . In Fig . 1, one can see the RVE of the three- phase porous composite. Here, the white color corresponds to aluminum oxide, the cyan color fits zirconium diboride, the grey color correspond s to silicon carbide and the dark blue color fit s the por es . The porosity of Al 2 O 3 –ZrB 2 –SiC