A Hypoplastic Constitutive Model for Granular Materials with Particle Breakage

Acknowledgments

Notation

a_e, b, a_p

breakage parameters;

B_p

breakage potential;

B_r

relative breakage;

B_t

total breakage;

C_i

parameter of the basic hypoplastic model;

D

strain rate tensor;

D_c

relative density;

D_q

deviatoric strain rate;

D_v

volumetric strain rate;

d_M

maximum diameter;

d_m

minimum diameter;

E

tangent modulus;

$\dot{e}$

time derivation of the void ratio;

e

void ratio;

e_crt

void ratio at a critical state;

e_min

minimum void ratio;

e₀

initial void ratio;

I_e

density function;

I_se

function of stiffness;

M

stress ratio;

M₀

stress ratio without breakage;

p_a

atmospheric pressure;

p₀

initial confining pressure;

p

mean stress;

q

deviatoric stress;

r

stress ratio;

T

stress tensor;

T*

deviatoric stress tensor;

$\stackrel{\circ }{\mathbf{T}}$

Jaumann stress rate tensor;

$\stackrel{\bullet }{\mathbf{T}}$

Cauchy stress rate tensor;

tr

trace of a tensor;

W

spin tensor;

β

stiffness parameter;

${\epsilon }_1$

axial strain;

λ, ξ, λ_br, α

four critical-state parameters;

ϕ_crt

friction angle at the critical state;

ʋ

initial Poisson’s ratio; and

ψ

dilation angle.

References