Hypoplastic model for sands with loading surface

Abstract

Although the hypoplastic models for sands have exhibited good predictive capability in monotonic loading, they are not able to reproduce memory effects and predict excessive plastic accumulation under cyclic loading. To overcome these issues, a loading surface has been incorporated into a hypoplastic model. This surface is capped and has two hardening variables. Notions from the bounding surface plasticity were borrowed in order to formulate the hardening functions. With this novel model, some salient features can be described: the model can account for the accumulation of plastic deformation, a memory effect is provided by the new surface, and stress-induced anisotropy effects observed in sands are successfully simulated. A short calibration guide of the parameters is given, and some simulations for Hostun RF loose sand and Toyoura sand are presented.

Appendices

Appendices

1.1 A Consistency condition of the loading surface

The loading surface function F _s depends on the stress ${{\varvec{\sigma}}}$ and the hardening variables denoted collectively by H. The evolution equations for the hardening variables are of interest. Similar to elastoplastic relations, we propose the following general relation for the evolution laws of the hardening variables:

$$ {\mathop{\bf H} \limits ^{\circ}}=\bar{\bf H}\dot{\lambda}, $$

(43)

where $\bar{\bf H}=\bar{\bf H}({{\varvec{\sigma}}},{\bf H}, e)$ is the set of hardening functions. For this model, the hardening variables are the backstress ratio tensor and preconsolidation pressure ${\bf H}=\{ \varvec{\alpha}, p_c\},$ and their objective time derivatives are represented with ${\mathop{\bf H} \limits ^{\circ}}.$ The consistency condition $\dot{F}_s=0$ is obtained by differentiating the yield condition F _s  = 0 and replacing ${\mathop{{\varvec{\sigma}}} \limits ^{\circ}}$ with Eq. 4:

$$ \begin{aligned} \dot{F}_s&=\frac{\partial{F_{s}}}{\partial{{\varvec{\sigma}}}}:{\mathop{{\varvec{\sigma}}} \limits ^{\circ}}+\frac{\partial{{F}_s}}{\partial{\bf H}}:{\mathop{\bf H}\limits ^{\circ}}=0\\ &\frac{\partial{{F}_s}}{\partial{{\varvec{\sigma}}}}:{\mathsf{ E}}:\left({\dot{\varvec{\epsilon}} }-Y {\bf m}^{\rm Hp}\parallel{\dot{\varvec{\epsilon}}}\parallel-{\dot{\lambda}}{\bf m}^{S}\right)+\frac{\partial{F}_s}{\partial{\bf H}}:\bar{\bf H}{\dot{\lambda}} =0 \end{aligned} $$

(44)

Solving for $\dot{\lambda}$ gives:

$$ {\dot{\lambda}}=\frac{{(\partial{F_s}}/{\partial{{\varvec{\sigma)}}}}:{\mathsf{ E}}:\left({\dot{\varvec{\epsilon}} }-Y {\bf m}^{\rm Hp}\parallel{\dot{\varvec{\epsilon}}}\parallel\right)}{H_s+({\partial{F_s}}/{\partial{{\varvec{\sigma)}}}}:{\mathsf{ E}}:{\bf m}^{S}}, $$

(45)

where $H_s=-({ \partial F_s}/{\partial {\bf H}}):\bar{\bf H}$ is the hardening modulus. One can rewrite Eq. 45 as:

$$ {\dot{\lambda}}=\frac{{(\partial{F_s}}/{\partial{{\varvec{\sigma)}}}}:{\mathop{{\varvec{\sigma}}} \limits ^{\circ}}^{\rm Hp}} {H_s+{(\partial{F_s}}/{\partial{{\varvec{\sigma)}}}}:{\mathsf{ E}}:{\bf m}^{S}}, $$

(46)

where ${\mathop{{\varvec{\sigma}}} \limits ^{\circ}}^{\rm Hp}={\mathsf{ E}}:\left({\dot{\varvec{\epsilon}} }-{Y}{\bf m}^{\rm Hp}\parallel{\dot{\varvec{\epsilon}}}\parallel\right)$ is the hypoplastic stress rate (see Eq. 2).

1.2 B Tangent (Jacobian) modulus

The Tangent or Jacobian modulus is defined as:

$$ {\sf J}=\frac{\partial{\mathop{{\varvec{\sigma}}} \limits ^{\circ}}}{\partial{\dot{\varvec{\epsilon}} } } $$

(47)

From Eqs. 4 and 5, it follows that ${\sf J}$ yields to

$$ {\sf J}={\sf J}^{\rm Hp}-\frac{{\mathsf{ E}}:{\bf m}^{S}\otimes{{(\partial{F_s}}/{\partial{{\varvec{\sigma)}}}}:{\sf J}^{\rm Hp}}}{{(\partial{F_s}}/{\partial{{\varvec{\sigma)}}}}:{\mathsf{ E}}:{\bf m}^{S}+H_s} , $$

(48)

where ${\sf J}^{\rm Hp}$ is the Jacobian modulus for the hypoplastic equation 2. In order to deduce an analytical expression for ${\sf J}^{\rm Hp},$ the constitutive equation 2 is rewritten as:

$$ {\mathop{{\varvec{\sigma}}} \limits ^{\circ}}={\mathsf{ E}}:{\dot{\varvec{\epsilon}} }+{\bf N}\parallel{\dot{\varvec{\epsilon}}}\parallel, $$

(49)

where ${\bf N}=-Y {\mathsf{ E}}:{\bf m}^{\rm Hp}$ is the non-linear stiffness tensor [22]. Using the derivative ${\partial \parallel{\dot{\varvec{\epsilon}} } \parallel}/{\partial {\dot{\varvec{\epsilon}} }}=\overrightarrow{\dot{\varvec{\epsilon}} }$ and Eq. 47, one can show that ${\sf J}^{\rm Hp}$ gives:

$$ {\sf J}^{\rm Hp}={\mathsf{ E}}+{\bf N}\otimes\overrightarrow{\dot{\varvec{\epsilon}}} $$

(50)

1.3 C Critical state slope M

The critical state slope M satisfies the Matsuoka–Nakai relation [19], and we use the explicit solution presented by Niemunis et al. [24]. Function M depends on the critical state friction angle $\varphi_c$ and the Lode’s angle θ _A of the tensor of interest A. In our model, A may be the stress tensor ${{\varvec{\sigma}}}$ or the effective stress ratio n (Eq. 10). In general, the computation of the Lode’s angle θ _A is given by

$$ \cos 3\theta_{\bf A}=-3\sqrt{6}\det{\overrightarrow{\bf A}^*}, $$

(51)

where $\overrightarrow{\bf A}^*={\bf A}^*/\parallel{{\bf A}^*}\parallel.$ According to [24], M can be computed with

$$ M={\frac{3\phi-9}{\phi\cos(3\theta)}}\left[{\frac{1}{2}}-\cos\left[{\frac{\alpha}{3}}+{\frac{\pi}{3}}H(\cos(3\theta_{\bf A}))\right]\right], $$

(52)

where H() is the Heaviside function (H(x) = 1 for x > 0) and ϕ is defined as

$$ \phi={\frac{9-\sin^2{\varphi_c}}{1-\sin^2\varphi_c}} $$

(53)

The angle α is computed with

$$ \alpha=\hbox{arccos}\left\{\hbox{sign}\left[\cos(3\theta_{\bf A}){\frac{b}{(-3+\phi)^3}}\right]\right\}, $$

(54)

where sign () extracts the sign of the argument. Finally, factor b reads

$$ b=-27+27\phi-9\phi^2+18\cos^2(3\theta_{\bf A})\phi^2+\phi^3-2\cos^2(3\theta_{\bf A})\phi^3 $$

(55)

At isotropic states η = 0, the solution is undetermined and M takes the value of the critical state slope for triaxial compression M = M _c .

Let us introduce function g defined as

$$ g={\frac{M(\theta_{\bf A})}{M_c}}, $$

(56)

where M _c is the critical state slope for triaxial compression. Notice that function g is non-smooth at η = 0 and might be not suitable for some functions in the constitutive model. Therefore, we introduce the smooth function F defined as

$$ F=1+{\frac{\eta}{M_c g(\theta_{{\varvec{\sigma}}})}}(g(\theta_{{\varvec{\sigma}}})-1), $$

(57)

where $\theta_{{\varvec{\sigma}}}$ is the Lode’s angle for the stress tensor ${{\varvec{\sigma}}}.$ A similar function was previously proposed by Wolffersdorff [38] denoted with the same symbol F but using a different Lode’s angle $\theta_W=\theta_{{\varvec{\sigma}}}-30^{\circ}.$