Undrained Solution for Cylindrical Cavity Expansion in Structured Clays Using a Hypoplastic Model

Data Availability Statement

Notation

$\mathcal{A}$

fourth-order tensor in hypoplastic model formulation;

a

cavity radius after expansion;

A₀

initial cavity radius;

A_m

variable in hypoplastic model formulation;

a_f

variable in hypoplastic model formulation;

a_y

hypoplastic variable equal to 0.3; the parameter controls the shape of the asymptotic state boundary surface;

a₁, a₂, a₃, a₄, a₅

parameters of transversely isotropic elasticity model;

cos 3θ

lode angle function;

D

Euler stretching tensor;

D

cone diameter;

d

normalized second-order tensor specifying asymptotic strain rate direction;

d^A

second-order tensor specifying asymptotic strain rate direction;

e

porosity;

F_m

factor of Matsuoka–Nakai yield condition;

f_d

pyknotropy factor of hypoplastic equation;

$f_d^A$

pyknotropy factor of hypoplastic equation, asymptotic state value;

f_s

barotropy factor of hypoplastic equation;

I₁, I₂, I₃

stress invariants;

k, A, s_f

structural parameters;

$\mathcal{L}$

hypoplasticity fourth-order tensor;

$\mathbf{N}$

hypoplasticity second-order tensor;

N

hypoplastic model parameter (position of normal compression line);

n

unit vector normal to the plane of symmetry;

OCR

overconsolidation ratio;

O_c

hypoplastic variable is equal to 2; wet of critical;

p′

mean effective stress;

p_e

value satisfies the relationship: 0 = q² + M²p′² − M²p′p_e;

$p_e^{*}$

Hvorslev equivalent pressure;

p_ij

second-order tensor, p_ij = n_in_j;

p_r

reference stress equal to 1 kPa;

q

deviatoric stress;

r

radial distance from the cavity position;

r_c

cone radius;

subscript “0”

initial state;

subscript “a”

at cavity wall;

subscript “l”

limit value;

subscript “s”

at the structure degradation boundary;

subscript “sd”

structural degradation;

subscript “su”

structural undegradation;

s

sensitivity;

$\stackrel{\mathrm{o}}{\mathrm{T}}$

stress rate tensor;

u

pore pressure;

α_E

anisotropy ratio of Young’s moduli;

α_f

hypoplastic variable controlling rate of stiffness decrease; may be considered as a model parameter;

α_G

anisotropy ratio of shear moduli;

${\alpha }_{\nu }$

anisotropy ratio of Poisson’s ratios;

Δu

excess pore pressure;

Δu_a

excess pore pressure at cavity wall;

ɛ_r, ɛ_θ

radial strain and circumferential strain;

ɛ_v, ɛ_s

volumetric strain and deviatoric strain;

${\kappa }^{*}$

hypoplastic model parameter controlling volumetric unloading response;

${\lambda }^{*}$

hypoplastic model parameter (slope of normal compression line);

ν_pp

Poisson’s ratio;

ω

variable controlling asymptotic state boundary surface shape;

σ_a

internal cavity pressure;

${\sigma }_r^{\mathrm{\prime }},{\sigma }_{\theta }^{\mathrm{\prime }},{\sigma }_z^{\mathrm{\prime }}$

effective radial, circumferential, and vertical stress components;

${\sigma }_{ra}^{\mathrm{\prime }}$

radial effective stress at cavity wall;

φ_c

critical state friction angle;

ξ

variable controlling asymptotic strain rate direction; and

$\mathbf{1}$

second-order identity tensor.

References