Anisotropic visco-hypoplasticity

Abstract

Apart from time-driven creep or relaxation, most viscoplastic models (without plastic and viscous strain separation) generate no or a very limited accumulation of strain or stress due to cyclic loading. Such pseudo-relaxation (or pseudo-creep) is either absent or dwindles too fast with increasing OCR. For example, the accumulation of the pore water pressure and eventual liquefaction due to cyclic loading cannot be adequately reproduced. The proposed combination of a viscous model and a hypoplastic model can circumvent this problem. The novel visco-hypoplasticity model presented in the paper is based on an anisotropic preconsolidation surface. It can distinguish between the undrained strength upon triaxial vertical loading and horizontal loading. The strain-induced anisotropy is described using a second-order structure tensor. The implicit time integration with the consistent Jacobian matrix is presented. For the tensorial manipulation including numerous Fréchet derivatives, a special package has been developed within the algebra program MATHEMATICA (registered trade mark of Wolfram Research Inc.). The results can be conveniently coded using a special FORTRAN 90 module for tensorial operations. Simulations of element tests from biaxial apparatus and FE calculations are also shown.

List of symbols

1.1 The following symbols are used in this paper

A :

Auxiliary variable for the intensity of viscous and hypoplastic strain rates

A, B, C :

Auxiliary subexpresssions for the definition of Jacobian $\mathsf{H}$

a :

Hypoplastic constant

b :

Auxiliary term in M(θ) function

C ₁ :

Material constant for hypoplastic strain

C ₂ :

Material constant for evolution of anisotropy

C ₃ :

Material constant limiting anisotropy

${\bf c}_T$ :

Stress correction tensor (RMI)

c _u :

Undrained shear strength

c _uComp :

Undrained shear strength for triaxial compression

${\bf c}_{\Omega}$ :

Correction on anisotropy tensor (RMI)

D :

1-d strain rate (extension > 0)

D ^vis :

1-d viscous strain rate

D _v :

Roscoe’s volumetric strain rate

D _q :

Roscoe’s deviatoric strain rate

D _r :

Reference creep rate

D :

Strain rate tensor

${\bf D}^*$ :

Deviatoric strain rate tensor

${\bf D}^{\rm e}$ :

Elastic strain rate tensor

${\bf D}^{\rm Hp}$ :

Hypoplastic strain rate tensor

${\bf D}^{\rm vis}$ :

Viscous strain rate tensor

$\mathsf{E}$ :

Hypoelastic stiffness tensor

$\hat{\mathsf{E}}$ :

Dimensionless hypoelastic stiffness tensor

e :

Void ratio

e ₀ :

Reference void ratio

e ₁₀₀ :

Reference void ratio preconsolidation p _B  = 100 kPa

$\dot{e}$ :

Void ratio rate

e _B0 :

Void ratio on the reference isotach (1-d model)

F :

Preconsolidation surface

F ₊ :

Surface affine to F passing through current stress

F _crit :

Critical state surface

F _M :

Function of the Lode angle used in hypoplasticity

G :

Alternative definition of the preconsolidation surface

H():

Heaviside function

$\mathsf{H}$ :

Jacobian tensor

$\mathsf{H}$ _w :

Stiffness tensor of water

h :

Current height of the sample

h ₀ :

Initial height of the sample

h ₁, h ₂, h ₃ :

Auxiliary variables for $p_{B+}^0$

$\mathsf{I}$ :

Fourth order symmetric identity tensor

I _v :

Viscosity index

$\mathsf{J}$ :

Fourth order identity tensor

K _w :

Bulk modulus of water

K ₀ :

Coefficient of earth pressure at-rest

K _2/1, K _3/1 :

Stress ratios

$\mathsf{K}, \mathsf{N}, \mathsf{R}, \mathsf{S}, \mathsf{W}, \mathsf{Z}$ :

Auxiliary fourth order tensors for the definition of the Jacobian $\mathsf{H}$

$\hat{\mathsf{L}}$ :

Dimensionless hypoelastic stiffness used in hypoplasticity

M :

Slope of the critical state line (a function of θ)

M _C :

Slope of the critical state line for triaxial compression

M _E :

Slope of the critical state line for triaxial extension

M _Ω :

Critical state slope for the current $\varvec{\Omega}$

m :

Flow rule (a function of T and $\varvec{\Omega}$)

$\hat{\bf N}$ :

Dimensionless hypoplastic relaxation tensor

OCR:

Overconsolidation ratio

p :

Roscoe’s mean effective stress $p=-\hbox{tr}\;{\bf T}/3$

$\bar{p},\bar{q}$ :

p, q Scaled so that MCC ellipse becomes a circle with a unit diameter

p _B :

Equivalent (preconsolidation) pressure

p _B+ :

pseudo equivalent (preconsolidation) pressure

p _w :

Pore water pressure

Q :

Surcharge uniform pressure for calculation of the strip foundation

q :

Roscoe’s deviatoric stress $q=\sqrt{\frac{3}{2}}||{\bf T}^*||$

${\bf r}_T$ :

Stress error tensor

${\bf r}_\Omega$ :

Anisotropy error tensor

s :

Settlement of a footing

T :

1-d stress (compression negative)

$\dot{T}$ :

1-d stress rate

t :

Time

T ₀ :

1-d reference value of stress

T _B :

1-d equivalent stress(T _B  > 0)

$\dot{T}_B$ :

1-d equivalent stress rate

T _B0 :

Equivalent 1-d stress on the reference isotach

T _max :

Maximum principal stress

T _min :

Minimum principal stress

t ₀ :

Reference time

T :

Cauchy effective stress tensor

${\bf T}^*$ :

Deviatoric stress tensor

${\bf T}^B$ :

Equivalent stress tensor

$\dot{\bf T}$ :

Stress rate tensor

${\mathring{\bf T}}$ :

Zaremba-Jaumann stress rate

$\hat{\bf T}$ :

Dimensionless stress tensor $\hat{\bf T}={\bf T}/\hbox{tr}\;{\bf T}$

${\bf T}_{K0}$ :

Uniaxial compression stress

x ₁, x ₂ :

Horizontal and vertical coordinates

w _L :

Liquid limit

w _P :

Plastic limit

α:

strain invariant from [62] $\alpha= \sqrt{6} \overset{\rightarrow}{{\bf D}^*} \cdot \overset{\rightarrow}{{\bf D}^*} : \overset{\rightarrow}{{\bf D}^*}$

β:

Auxiliary term in M(θ) function

γ′:

Buoyant unit weight

δ _ij :

Kroenecker symbol

$\Updelta \sqcup$ :

Increment of $\sqcup$

$\epsilon$ :

1-d logarithmic strain

$\epsilon_0$ :

1-d reference value of logarithmic strain

θ:

Lode’s angle

λ:

Compression index

κ:

Swelling index

μ _i :

Coefficients of the general equation of an ellipse

ϕ:

Matsuoka-Nakai’s constant term $\phi=\frac{9-\sin^2{\varphi_c}}{1-\sin^2{\varphi_c}}$

ϕ(p):

Matsuoka-Nakai analogous term for the alternative preconsolidation surface

ϕ_max :

Constant of the alternative pre- consolidation surface

φ _c :

Critical friction angle

χ:

Positive scalar homogeneity

Ψ:

Angle between the current stress state and the isotropic axis

ψ:

Coefficient of secondary compression

$\varvec{\Omega}$ :

Anisotropy tensor

ω:

Anisotropy of the ellipse in $\bar{p}-\bar{q}$ space

1 :

Second order identity tensor