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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fluid density due to applied deformation. The simulation results showed that the dependences of  c on the viscosity of fluid  , permeability k and sample radius R have similar nonlinear logistic-like profiles (at fixed value of strain rate   ). In particular, an increase in  or a decrease in k contribute to the slowing down of fluid redistribution (outflow from the sample) and thus play a role similar to an increase in the applied strain rate. An increase in the transverse dimension (radius R ) of the sample leads to a decrease in the value of radial gradient of pore pressure and, consequently, to a decrease in the rate of redistribution of pore fluid. Analysis of simulation results showed that uniaxial compressive strength of the sample is an unambiguous nonlinear and monotonically decreasing function of the ratio of strain rate to fluid flow rate. This ratio is expressed by the following parameter combination (Fig. 1b): Samples have the same compressive strength if they are characterized by the same value of R , (even if the specific values of the parameters   , k,  and R differ by orders of magnitude). The parameter R has the dimension of pressure. Hereinafter we will use its normalized value R n = R / P init , where P init is the initial value of pore pressure in the sample (atmospheric pressure in the considered case). Fig. 1b shows that a significant decrease in strength begins in the region R n >10 (the transition from region I to region II; the latter is the transient interval to the so-called undrained hydrological regime for the sample material and characterized by a strong decrease in fluid outflow with increasing R n ). The strength of the sample approaches the minimum value in the region R n >10 4 (region III). Note that the minimum value of  c corresponds to the undrained deformation mode and can be estimated analytically from failure criterion (2) and constitutive equations (1) for the solid-phase skeleton and interstitial fluid: k R R    2  . (3)

 0 1 0 5 1     uct .  





.

(4)

bK

c ,min

fl

K aK 

p

fl

a

The key result of the study is the revealed logistic nature of the decreasing dependence  c ( R n ). This is a direct result of the competition between the processes of deformation of the skeleton and fluid flow, which lead to a oppositely oriented changes in the pore pressure. The shown logistic dependence is approximated with good accuracy by a sigmoid in the following form (Fig. 1b):

 R R 0 1     n n uct

,

c ,min

(5)

c ,min    

 m

c

where R n 0 is the normalization constant, m is the exponent. Despite the fact that the effect of a substantial and nonlinear decrease in the strength of fluid-saturated brittle samples with an increase in the strain rate was mentioned in many experimental studies, the logistic dimensionless form of this dependence was obtained by the authors for the first time. The parametric analysis has shown that the parameters of the logistic curve depend not only on the combination of the parameters that determine the value of R n in (3), but also on the influence of the pore fluid on the magnitude of local stresses at which local fracture begins. This influence is characterized by the dimensionless parameter b in the strength criterion (2) for the fluid-saturated material. In the study, we varied the value of the parameter b within the interval between b =0.1 ( b is equal to the porosity; this estimate is the lower bound proposed by Terzaghi for materials with ensembles of shallow rounded pores) and b =1 (the upper estimate; this is the most common approximation applicable for complex porosity, including multiscale porosity). Analysis of the simulation results showed that increase in b (at fixed values of other material parameters) is accompanied by a decrease in the values both of the minimum strength  c,min (up to several times) and of the characteristic width of the region II (where the main change in the strength value occurs). In the expression (5) this corresponds to an increase of the exponent m . It was found that the set of numerically derived points corresponding to different b can be reduced with acceptable accuracy to a single (unified) curve using the normalized strength values (  c -  c,min )/(  uct -  c,min ) and the parameter R n b (Fig. 2a).