PSI - Issue 13

Evgeny V. Shilko et al. / Procedia Structural Integrity 13 (2018) 1508–1513 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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Fig. 2. The dependences of the normalized value of the dynamic strength of the samples of fluid-saturated brittle material on the product of the dimensionless parameter R n on the quadratic root of the coefficient b for the two “ limiting ” values of b : 3D cylindrical samples (a) and both 3D cylindrical and 2D rectangular samples (b). Points denote numerically derived values of strength at different strain rates, permeability values, sample radii and fluid viscosities. Dashed lines are common approximations of the sets of 2D and 3D points by sigmoid function (5) with product ( R n  b 1/2 ) as an argument instead of R n . The obtained result is very important because it allows: 1) quantitative estimation of the dynamic compressive strength for brittle permeable materials filled with a fluid at initial pore pressure P init ; 2) estimation of the value of the fluid effect parameter b for real brittle permeable materials on the basis of matching the available set of experimental points and the approximating curve (5). Note that approximating relationship (5) for brittle solids can be further generalized by means of taking into account the magnitudes of the mechanical characteristics of the solid phase skeleton, its porosity and the poroelasticity coefficient a . The argument of such a generalized dependence should be the parameter C solid R n b , where C solid is a material coefficient, which determines the effect of deformation of the enclosing skeleton on pore volume change. To reveal the influence of dimensionality of the problem on the parameters of sigmoid (5) we also simulated uniaxial compression of two-dimensional rectangular samples of permeable fluid-saturated brittle materials under the condition of plane stress. Outflow of the fluid from 2D samples was possible through the lateral faces only in the considered plane of the sample. Analysis of simulation results revealed a significant difference in the parameters of the approximating sigmoid  c ( R n b ) for three-dimensional (cylindrical) and two-dimensional (rectangular) samples. Fig. 2b shows the corresponding dependencies. In the case of a “plane” fluid outflow (along the axis transverse to the loading axis), the width of the transition region from fully drained ( R n  0) to undrained ( R n  ) conditions for the sample is much narrower (by 1-1.5 orders of magnitude). Analogous regularities also hold for the dependence of the effective Young's modulus E eff of fluid-saturated cylindrical and planar samples on the value of the parameter R n (Fig. 3). The obtained sets of points E eff ( R n b ) for the fluid-saturated brittle samples are approximated with good accuracy by sigmoid of the following form:

K

   

      1

E E 

K G K G max   3

fl

K E E K max  0

,

E E

max

, where

(6)

 

K K max



 p

eff

max

3

K

0 

R R

1 

0

max

n n

Here E max and K max are Young’s modulus and bulk modulus of fluid-saturated material under undrained hydrological conditions ( R n  ), p  1 for both approximating curves in Fig. 3. Note that the given relations for E max and K max are derived analytically on the basis of the constitutive equations of the Biot linear poroelasticity model. The simulation results indicate that two-dimensional estimates of the dynamic mechanical characteristics of samples of contrast (fluid-saturated) materials provide the correct form of the functional dependencies, however, obtained quantitative values are accurate only for some characteristics in “ limiting ” cases ( R n  0 or R n  ).