Clay hypoplasticity with explicitly defined asymptotic states

Abstract

A new rate-independent hypoplastic model for clays is developed. The model is based on the recently proposed approach enabling explicit incorporation of the predefined asymptotic state boundary surface and corresponding asymptotic strain rate direction into hypoplasticity. Several shortcomings of the existing hypoplastic model for clays are identified and corrected using the proposed approach. Thanks to the independent formulation of the individual model components, the new model is more suitable to form a basis for further developments and enhancements than the original one.

Appendix

Complete formulation of the proposed hypoplastic model. The general rate formulation reads

$$ \mathop{{\bf T}}\limits^{\circ}=f_s{\varvec{\mathcal L}}:{\bf D}-\frac{f_d}{f_d^A}\varvec{\mathcal A}:\varvec{d}\|{\bf D}\| $$

(43)

with

$$ \varvec{\mathcal L}=\varvec{\mathcal I}+\frac{\nu}{1-2\nu}{\bf 1}\otimes{\bf 1} $$

(44)

$$ \varvec{\mathcal A}=f_s{\varvec{\mathcal L}}+\frac{{\bf T}}{\lambda^*}\otimes{\bf 1} $$

(45)

$$ f_s=\frac{3p}{2}\left(\frac{1}{\lambda^*}+ \frac{1}{\kappa^*}\right)\frac{1-2\nu}{1+\nu} $$

(46)

where ν, λ^* and κ^* the are model parameters, p =  − tr T/3, and $\varvec{1}$ and ${\varvec{\mathcal I}}$ are the second- and fourth order unity tensors, respectively. The factor f _d reads

$$ f_d=\left(\frac{2p}{p_e}\right)^\alpha $$

(47)

with α = 2 and the equivalent pressure

$$ p_e=p_r\exp\left[\frac{N-\ln(1+e)}{\lambda^*}\right] $$

(48)

where N is a parameter and p _r is a reference stress equal to 1 kPa. The factor $f_d^A$ reads

$$ f_d^A=2^\alpha(1-F_m)^{\alpha/\omega} $$

(49)

where F _m is the Matsuoka--Nakai factor calculated from

$$ F_m=\frac{9I_3+I_1I_2}{I_3+I_1I_2} $$

(50)

and the exponent ω reads

$$ \omega=-\frac{\ln\left(\cos^2\varphi_c\right)}{\ln 2}+a\left(F_m-\sin^2\varphi_c\right) $$

(51)

where $\varphi_c$ is a parameter and a = 0.3. The stress invariants I ₁, I ₂ and I ₃ are given by

$$ I_1={\rm tr}{\bf T} $$

(52)

$$ I_2=\frac{1}{2}\left[{\bf T}:{\bf T}-\left(I_1\right)^2\right] $$

(53)

$$ I_3={\rm det}{\bf T} $$

(54)

Finally, the asymptotic strain rate direction $\varvec{d}$ is calculated as

$$ \varvec{d}=\frac{\varvec{d}^A}{\|\varvec{d}^A\|} $$

(55)

where

$$ \begin{aligned}{\varvec d}^A&=-\hat{{\bf T}}^*+\\ &\varvec{1}\left[\frac{2}{3}-\frac{\cos 3\theta+1}{4}F_m^{1/4}\right] \frac{F_m^{\xi/2}-\sin^\xi\varphi_c}{1-\sin^\xi\varphi_c} \end{aligned} $$

(56)

with the Lode angle θ

$$ \cos3\theta=-\sqrt{6}\frac{{\rm tr}\left(\hat{{\bf T}}^\ast\cdot \hat{{\bf T}}^\ast\cdot\hat{{\bf T}}^\ast\right)}{\left[\hat{{\bf T}}^\ast: \hat{{\bf T}}^\ast\right]^{3/2}} $$

(57)

exponent ξ

$$ \xi=1.7+3.9\sin^2\varphi_c $$

(58)

and the stress measure $\hat{{\bf T}}^\ast={\bf T}/\hbox{tr}{\bf T}-\varvec{{\bf 1}}/3$. The model requires five parameters $\varphi_c, \lambda^*, \kappa^*, N$ and ν, and state variables T and void ratio e.