Issue 49

A. Akhmetov et alii, Frattura ed Integrità Strutturale, 49 (2019) 190-200; DOI: 10.3221/IGF-ESIS.49.20

Figure 5: Strength of the geomedium layers as a function of depth.

M ATHEMATICAL MODEL

T

he description of continuum deformation includes the set of following equations: fundamental conservation laws and constitutive relations. Fundamental conservation laws:



d

    , i i u

– Mass

(2)

0

dt

     , i ij j u

(3) (4)

– Impulse



g

i

 E









– Energy

ij ij

Here ρ is the material density, u i

is the stress tensor components, i g

is the i-component of the displacement vector,  ij

Fis the i-component of the gravity acceleration, E is the internal energy,  ij is the strain tensor component. The constitutive equations of the first group are written down in the rate form (5)–(6) in which the stress rate is proportional to elastic strain rate         e t p ij ij ij .

Dt s

D

   1

 

  

ij

   e ij

  e kk ij



2

(5)

3

    • e P K

(6)

kk

where

(7)

     ( ) ij ij ij P s

Ds

ij

 

s

      s s

(8)

ij

ik jk jk ik

Dt

1 (

   ij

 u u   , i j

(9)

)

, j i

2

1 (

   ij

 u u   , i j

(10)

)

, j i

2

195