Constitutive Modeling for Two Sands under High Pressure

Acknowledgments

Notation

B_r

relative breakage index (unit: %);

d

dilatancy;

dp′ and dq

mean effective stress increment and deviatoric stress increment, respectively (unit: MPa);

d₀, α ${\vartheta }_d$, and k_d

dilatancy parameters;

dɛ_a

axial strain increment (unit: %);

dɛ_v and dɛ_s

volumetric strain and shear strain increments, respectively (unit: %);

$d{\epsilon }_v^e$ and $d{\epsilon }_s^e$

elastic volumetric and shear strain increments, respectively (unit: %);

$d{\epsilon }_v^p$ and $d{\epsilon }_s^p$

plastic volumetric and shear strain increments, respectively (unit: %);

e

current void ratio;

e₀

initial void ratio;

e_cs

critical state void ratio;

e_cs0

initial critical state void ratio;

G₀

elastic shear modulus parameter;

H_p

plastic modulus;

h₀

pressure-dependent plastic modulus parameter;

I_ep

void ratio–pressure state parameter;

$I_{ep}^0$

initial void ratio–pressure state parameter;

K_e and G_e

elastic bulk modulus and elastic shear modulus, respectively;

k_b

bounding stress-ratio parameter;

k_W

breakage-energy parameter;

M_b

bounding stress ratio;

M_cs

critical state stress ratio;

$n_v^f$ and $n_s^f$

compression-related and shearing-related components of the loading unit vector, respectively;

$n_v^g$ and $n_s^g$

compression-related and shearing-related components of the plastic flow unit vector, respectively;

p_a

atmospheric pressure (unit: MPa);

p′

mean effective stress (unit: MPa);

$p_{cs}^{\mathrm{\prime }}$

critical-state mean effective stress (unit: MPa);

p₀

initial confining pressure (unit: MPa);

q

deviatoric stress (unit: MPa);

S_b

coefficients of bounding stress ratio;

S_d

coefficient of dilatancy ratio;

W_p

plastic work (unit: MJ/m³);

η

stress ratio;

λ, e_B0, k_B and χ_B

breakage-dependent critical-state parameters;

${\sigma }_1^{\mathrm{\prime }}$ and ${\sigma }_3^{\mathrm{\prime }}$

maximum and minimum principle stress, respectively (unit: MPa);

ϕ_cs

critical state friction angle (unit: degree);

ϕ_m and ψ_m

mobilized friction angle and dilatation angle, respectively (unit: degree);

χ_h and k_h

plastic modulus parameters;

χ_W

breakage-energy parameter (unit: MJ/m³); and

υ

Poisson’s ratio.

References