Bounding-Surface Plasticity Model for Drained Cyclic Behaviors of Cohesionless Soil

Acknowledgments

Notation

C(ζ)

hardening measure;

c

ratio of M_e to M_c;

C_u

uniformity coefficient;

D

dilatancy;

D₅₀

median grain size;

${\mathbf{D}}^e$

hypoelastic stiffness;

${\mathbf{D}}^{ep}$

elastoplastic stiffness;

d₀

positive model constant;

e

current void ratio;

e₀

initial void ratio;

e_c

critical void ratio;

e_max

maximum void ratio;

e_min

minimum void ratio;

$\mathbf{e}$

deviatoric strain tensor;

${\mathbf{e}}^p$

plastic deviatoric strain tensor;

$e_{\mathrm{\Gamma }}$

parameter corresponding to CSL in the e − p plane;

F

bounding surface function;

G

hypoelastic shear modulus;

G₀

model parameter related to the shear modulus;

$g(\overline{\theta })$

interpolation function of Lode angle;

$\mathbf{I}$

fourth-order identity tensor;

J₂

second isotropic invariant of $\mathbf{s}$;

J₃

third isotropic invariant of $\mathbf{s}$;

K

hypoelastic bulk modulus;

K_p

plastic modulus;

${\overline{K}}_p$

plastic modulus of image point;

L

loading index;

$\mathbf{L}$

gradient of the bounding surface;

M

critical stress ratio;

M_c

value of M for compression;

M_e

value of M for extension;

M^d

stress ratio related to the phase transformation state;

M^p

virtual peak stress ratio;

m

material parameter corresponding to M^d;

$\mathbf{m}$

normalized direction vector of plastic flow;

m_p

component of $\mathbf{m}$ along p-axis;

m_q

component of $\mathbf{m}$ along q-axis;

N

number of loading cycles;

n

positive model parameter corresponding to M^p;

$\mathbf{n}$

unit tensor of the direction of plastic deviatoric strain;

p

effective mean stress;

p₀

initial confining pressure;

$\overline{p}$

image of p;

${\overline{p}}_c$

size of F;

q

deviatoric stress;

$\overline{q}$

image of q;

R

ellipse aspect ratio of the bounding surface;

$\mathbf{r}$

deviatoric stress ratio tensor;

${\mathbf{r}}^{in}$

value of $\mathbf{r}$ at the beginning of each loading stage;

$\mathbf{s}$

deviatoric stress tensor;

$\overline{\mathbf{s}}$

image of $\mathbf{s}$;

t

+1 for compression $(\overline{q}>0)$ or −1 for extension $(\overline{q}<0)$;

α

material constant corresponding to hardening measure;

β

material constant corresponding to hardening measure;

$\mathbf{\delta }$

second-order identity tensor;

$\epsilon$

strain tensor;

${\epsilon }^p$

plastic strain tensor;

ɛ_q

deviatoric strain;

ɛ_v

volumetric strain;

${\epsilon }_q^e$

elastic part of ɛ_q;

${\epsilon }_v^e$

elastic part of ɛ_v;

${\epsilon }_q^p$

plastic part of ɛ_q;

${\epsilon }_v^p$

plastic part of ɛ_v;

ζ_q

accumulated plastic deviatoric strain;

ζ_v

accumulated negative plastic volumetric strain;

η

stress ratio;

η^ave

average value of η during cyclic loading;

ηⁱⁿ

value of η at the beginning of each loading stage;

$\overline{\theta }$

lode angle;

λ

parameter corresponding to CSL in the e − p plane;

ν

Poisson’s ratio;

ξ

parameter corresponding to CSL in the e − p plane;

ρ

distance from the mapping origin to the stress point;

$\overline{\rho }$

distance from the mapping origin to the image point;

$\mathbf{\sigma }$

stress tensor;

φ_cs

critical friction angle;

χ

material constant corresponding to hardening measure;

ψ

state parameter; and

ω

material constant corresponding to hardening measure.

References