Peak Stress Obliquity in Drained and Undrained Sands. Simulations with Neohypoplasticity

Abstract

The difference between undrained and drained peak friction angle is considerable (up tp $10^\circ$), despite identical densities and pressures (at peak). This cannot be explained using the elastoplastic formalism. An attempt is made to describe this effect with neohypoplasticity. For this purpose two types of nonlinearity are used, the well-known term $Y m_{ij} \Vert \dot{\epsilon }\Vert$ and the novel skew-symmetric correction tensor which is added to the elastic stiffness.

Appendix 1: Invertibility Condition for Neohypoplasticity

If $\mathbf{z}: {\varvec{\dot{\epsilon }}} > 0$, then the invertibility condition is $Y < 1$, identically as in the older versions of hypoplasticity, cf. [16]. In the following we examine the case $\mathbf{z}:{\varvec{\dot{\epsilon }}} <0$ only. First, we find two scalar variables $x = \Vert {\varvec{\dot{\epsilon }}}\Vert$ and $y= \mathbf{z}':{\varvec{\dot{\epsilon }}}$, where $\mathbf{z}' = \omega \mathbf{z}$. Let us denote ${\varvec{\dot{\epsilon }}}^\mathrm{el}= \mathsf{E}^{-1} : {\varvec{\dot{\sigma }}}$ which is a known tensor. The first scalar equation is obtained multiplying the constitutive equation

$$\begin{aligned} {\varvec{\dot{\sigma }}}= & {} \bar{\mathsf{E}}\left( {\varvec{\dot{\epsilon }}} - \mathbf m Y \Vert {\varvec{\dot{\epsilon }}} \Vert + {\vec {~{\mathbf{{1}}}}} \mathbf{z}': {\varvec{\dot{\epsilon }}} \right) \end{aligned}$$

(52)

by $\mathbf{z}':( )$

$$\begin{aligned} \mathbf{z}:{\varvec{\dot{\epsilon }}}^\mathrm{el}= & {} y - \mathbf{z}':\mathbf m Y x \quad \text {because}\quad \mathbf{z}' \perp ~{\vec {\mathbf{{1}}}}\end{aligned}$$

(53)

$$\begin{aligned} y= & {} \mathbf{z}':{\varvec{\dot{\epsilon }}}^\mathrm{el}+ \mathbf{z}':\mathbf m Y x \end{aligned}$$

(54)

The second scalar equation is obtained resolving the constitutive equation with substituted x, y for ${\varvec{\dot{\epsilon }}}$ and finding $x^2 ={\varvec{\dot{\epsilon }}} :{\varvec{\dot{\epsilon }}}$

$$\begin{aligned} {\varvec{\dot{\epsilon }}}= & {} \mathsf{C}: {\varvec{\dot{\sigma }}} + \mathbf m Y x - {\vec {~{\mathbf{{1}}}}} y \end{aligned}$$

(55)

$$\begin{aligned} x^2= & {} ({\varvec{\dot{\epsilon }}}^\mathrm{el}+ \mathbf m Y x - {\vec {~{\mathbf{{1}}}}} y) : ({\varvec{\dot{\epsilon }}}^\mathrm{el}+ \mathbf m Y x - {\vec {~{\mathbf{{1}}}}} y) \end{aligned}$$

(56)

$$\begin{aligned} 0= & {} {\varvec{\dot{\epsilon }}}^\mathrm{el}:{\varvec{\dot{\epsilon }}}^\mathrm{el}- 2 {\varvec{\dot{\epsilon }}}^\mathrm{el}: ~{\vec {\mathbf{{1}}}}y + y^2 + 2 {\varvec{\dot{\epsilon }}}^\mathrm{el}:\mathbf m Y x - 2 ~{\vec {\mathbf{{1}}}}: \mathbf m Y x y + (Y^2-1) x^2 ,\qquad \qquad \end{aligned}$$

(57)

where $\mathbf m :\mathbf m = ~{\vec {\mathbf{{1}}}}:~{\vec {\mathbf{{1}}}}= 1$ has been used. We take y from (54) and substitute it into (57). The result is a quadratic equation $A x^2 + B x + C = 0$ to be solved for x. For uniqueness of the inverse solution this quadratic equation should have two roots, one positive and one negative, i.e. $x_1 x_2 < 0$ and we take the positive one because $x = \Vert {\varvec{\dot{\epsilon }}}\Vert$. The condition $x_1 x_2 < 0$ corresponds to $C/A < 0$. Using abbreviated forms of (54) and (57), viz.

$$\begin{aligned} {\left\{ \begin{array}{ll}y = \alpha + \beta x \\ 0 = a x^2 + b xy + c x + d y^2 +e y + f, \end{array}\right. }\end{aligned}$$

(58)

we obtain the quadratic equation $A x^2 + B x + C = 0$ with

$$\begin{aligned} {\left\{ \begin{array}{ll}A = a + b \beta + d \beta ^2 \\ C = f + e\alpha + d \alpha ^2 \end{array}\right. }\end{aligned}$$

(59)

or

$$\begin{aligned} {\left\{ \begin{array}{ll}A = (Y^2-1) - 2( ~{\vec {\mathbf{{1}}}}: \mathbf m ) ( \mathbf{z}':\mathbf m ) Y^2 + ( \mathbf{z}':\mathbf m )^2 Y^2 \\ C = {\varvec{\dot{\epsilon }}}^\mathrm{el}:{\varvec{\dot{\epsilon }}}^\mathrm{el}- 2 ({\varvec{\dot{\epsilon }}}^\mathrm{el}: ~{\vec {\mathbf{{1}}}}) (\mathbf{z}:{\varvec{\dot{\epsilon }}}^\mathrm{el}) + (\mathbf{z}:{\varvec{\dot{\epsilon }}}^\mathrm{el})^2 \end{array}\right. }\end{aligned}$$

(60)

The value C must be positive by the following argument: let us replace ${\varvec{\dot{\epsilon }}}^\mathrm{el}:{\varvec{\dot{\epsilon }}}^\mathrm{el}$ by $({\varvec{\dot{\epsilon }}}^\mathrm{el}: ~{\vec {\mathbf{{1}}}}) ({\varvec{\dot{\epsilon }}}^\mathrm{el}: ~{\vec {\mathbf{{1}}}})$, which cannot increase C and hence

$$\begin{aligned} C \ge ( {\varvec{\dot{\epsilon }}}^\mathrm{el}:~{\vec {\mathbf{{1}}}}- \mathbf{z}':{\varvec{\dot{\epsilon }}}^\mathrm{el}):( {\varvec{\dot{\epsilon }}}^\mathrm{el}:~{\vec {\mathbf{{1}}}}- \mathbf{z}':{\varvec{\dot{\epsilon }}}^\mathrm{el}) >0 \end{aligned}$$

(61)

for any ${\varvec{\dot{\epsilon }}}^\mathrm{el}\ne 0$. The condition of invertibility can be therefore reduced to $A < 0$ or

$$\begin{aligned} \left[ 1 - (2 ~{\vec {\mathbf{{1}}}}: \mathbf m ) ( \mathbf{z}':\mathbf m ) + ( \mathbf{z}':\mathbf m )^2 \right] Y^2 < 1 \end{aligned}$$

(62)