The shear strain in rock fracture is derived using the Coulomb-Navier's theory, which is expressed by the following equations:
|τ|=S
0-μσ
T
0=2S
0/√μ
2+1+μ
C
0=2S
0/√μ
2+1-μ
where
π: shear stress
σ: normal stress
μ: coefficient of internal fraction
S
0 : shear strength
T
0 : tensile strength
C
0 : compressive strength
Between the shear strain γ and the normal strain ε the next relation holds for isotropic materials,
|γ|/2=S
0′-με
where
S
0′=1/2G(S
0-μλλ)
λ=ε
1+ε
2+ε
3 and
λ, G : Lame's parameters The strain quantities T
0′ and C
0′ corresponding to the tensile and compressive strengths, are expressed as follows :
T
0′=2(γ+G)/G(3γ+2G) (√;μ
2+1-μ)S
0 C
0=-2(γ+G)/G(3γ+2G) (√;μ
2+1+μ)S
0 The quantities of S
0′, T
0′ and C
0′ are in order of 10
-4 under suitable assumptions. It is in harmony with strain quantities obtained by the geodetic surveys before and after earthquakes.
These equations are represented by the Mohr's circle diagram in Fig. 2. The Mohr's circle moves depending on (ε
1+ε
3)/2 and the position of S0' depending on Δ=ε
1+ε
2+ε
3. Accordingly, the state in which the dilatation is larger, is nearer to the rock fracture, even if the quantity of (ε
1-ε
3) takes the same value.
There is a strong correlation between the anomalous dilatation of the rhombus base lines at Mitaka, Tokyo and the occurrence of earthquakes of intensity more than V (JMA) at Tokyo. It leads him to suggest the relation between dilatation and fracture of the earth's crust.
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