A 3D particle finite element model for the simulation of soft soil excavation using hypoplasticity

Abstract

A numerical model based on the particle finite element method (PFEM) combined with a hypoplastic constitutive model is proposed for the analysis of soft soil excavations by means of single and multiple excavation tools. The PFEM allows to efficiently account for large deformations of the excavated soil material and free surfaces characterizing the simulation of tool–soil interaction during excavation. The utilization of a hypoplastic model, formulated in rate form, allows for a straightforward coupling with the standard velocity-based PFEM. Furthermore, effects such as pressure and density dependency of the soil stiffness are consistently incorporated into the formulation. The solution of the governing equations is performed implicitly, while an adaptive sub-stepping scheme is employed for the explicit time integration of the constitutive equation. Thus, the accuracy of the solution is improved at both global and local (constitutive) levels. The performance of the method is evaluated by means of the numerical re-analysis of selected geotechnical benchmark examples and soft soil excavations in 2D and 3D setups.

Appendices

$\alpha$-Bossak time integration scheme

In the $\alpha$ -Bossak integration method [51], nodal accelerations $\dot{\bar{\varvec{v}}}$ are interpolated within a time step $[t_{n},t_{n+1}]$ and evaluated at time $t_{n+1-\alpha } = (1-\alpha )t_{n+1} + \alpha t_{n}$ via

$$\begin{aligned} \dot{\bar{\varvec{v}}}_{n + 1 - \alpha } = (1-\alpha ) \dot{\bar{\varvec{v}}}_{n+1} + \alpha \dot{\bar{\varvec{v}}}_{n}. \end{aligned}$$

(29)

The end-of-step nodal accelerations ${\dot{\bar{\varvec{v}}}}_{n+1}$ and displacements $\bar{\varvec{u}}_{n+1}$ are computed according to

$$\begin{aligned} \dot{\bar{\varvec{v}}}_{n+1}&= \frac{1}{\gamma \Delta t} (\bar{\varvec{v}}_{n+1} - \bar{\varvec{v}}_n) + \frac{1}{\gamma }\left( \gamma - 1\right) \dot{\bar{\varvec{v}}}_{n} , \end{aligned}$$

(30)

$$\begin{aligned} \bar{\varvec{u}}_{n+1}&= \bar{\varvec{u}}_{n} + \Delta t\bar{\varvec{v}}_{n} + \frac{\Delta t^{2}\left( 1 - 2\beta \right) }{2}\dot{\bar{\varvec{v}}}_{n} + \Delta t^{2}\beta \dot{\bar{\varvec{v}}}_{n+1}, \end{aligned}$$

(31)

where $\alpha$, $\beta$ and $\gamma$ are the integration parameters of the method.

Linearization of the balance of momentum equation

The rate of the element internal forces vector $~^e\!\dot{\varvec{f}}^{TL}_\mathrm{int}$ can be written in terms of the time derivative of the first Piola–Kirchhoff stress tensor $\dot{\varvec{\sigma }}_{PK1}$ as

$$\begin{aligned} ~^e\!\dot{\varvec{f}}^{TL}_\mathrm{int} = \int _{\varOmega _{0}^{e}}\frac{\partial \varvec{N}}{\partial \varvec{X}}^{T}\dot{\varvec{\sigma }}_{PK1}\mathrm{d}\varOmega _{0}^{e}, \end{aligned}$$

(32)

which is a non-symmetric stress measure, i.e., one base points to the initial configuration $\varOmega _{0}$ while the other base underlies in the current one $\varOmega _{n}$.

A full pull back transformation of $\dot{\varvec{\sigma }}_{PK1}$ yields the material rate of the symmetric second Piola–Kirchhoff stress tensor $\dot{\varvec{S}}$

$$\begin{aligned} \dot{\varvec{\sigma }}_{PK1} = \dot{\overline{\varvec{S}\varvec{F}^{T}}}, \end{aligned}$$

(33)

where $\varvec{F} = \frac{\displaystyle \partial \varvec{x}}{\displaystyle \partial \varvec{X}}$ is the gradient of deformation tensor. In the following, minor symmetries of the fourth-order stiffness tensor $\varvec{C}$ is assumed in the pre and post-multiplication of operators by the deformation gradient tensor $\varvec{F}$. Inserting Eq. (33) into (32) and applying the chain rule of differentiation yields

$$\begin{aligned} ~^e\!\dot{\varvec{f}}^{TL}_\mathrm{int}= & {} \int _{\varOmega _{0}^{e}}\frac{ \partial \varvec{N}}{\partial \varvec{X}}^{T}\dot{\varvec{\sigma }}_{PK1}\mathrm{d}\varOmega _{0}^{e}\nonumber \\= & {} \underbrace{\int _{\varOmega _{0}^{e}}\frac{ \partial \varvec{N}}{\partial \varvec{X}}^{T}\dot{\varvec{S}}\varvec{F}^{T}\mathrm{d}\varOmega _{0}^{e}}_{^e\!\dot{\varvec{f}}^{TL}_{m}} + \underbrace{\int _{\varOmega _{0}^{e}}\frac{ \partial \varvec{N}}{\partial \varvec{X}}^{T}\varvec{S}\dot{\varvec{F}}^{T}\mathrm{d}\varOmega _{0}^{e}}_{^e\!\dot{\varvec{f}}^{TL}_{g}}. \end{aligned}$$

(34)

In the following, the subscript $()_\mathrm{int}$ will be dropped and introduced only when necessary. The term $^e\!\dot{\varvec{f}}^{TL}_{m}$ in the right-hand-side of Eq. (34) describes the material response of the body via a constitutive law, written in compact form as

$$\begin{aligned} \dot{\varvec{S}} = \varvec{C}_{0}\dot{\varvec{E}}, \end{aligned}$$

(35)

where $\varvec{C}_{0}$ is a fourth-order constitutive tensor with its basis in the undeformed configuration and $\dot{\varvec{E}}$ the rate of the Green Lagrangian strain tensor.

Substitution of Eq. (35) into Eq. (34) yields

$$\begin{aligned} ~^e\!\dot{\varvec{f}}_{m}^{TL} = \int _{\varOmega ^{e}_{0}}\frac{ \partial \varvec{N}}{\partial \varvec{X}}^{T}\varvec{F}\varvec{C}_{0}\dot{\varvec{E}}\mathrm{d}\varOmega ^{e}_{0}. \end{aligned}$$

(36)

In order to extract the nodal velocities from $~^e\!\dot{\varvec{f}}_{m}^{TL}$ in Eq. (36), it is now convenient to introduce the time rate of the Green Lagrangian strain $\dot{\varvec{E}}$ expressed by

$$\begin{aligned} \dot{\varvec{E}}&= \frac{1}{2}\left( \dot{\varvec{F}^{T}}\varvec{F} + \varvec{F}^{T}\dot{\varvec{F}} \right) =\frac{1}{2}\left( \frac{\partial \dot{\varvec{x}}}{\partial {\varvec{X}}}^{T}\varvec{F} + \varvec{F}^{T}\frac{\partial {\dot{\varvec{x}}}}{\partial {\varvec{X}}} \right) . \end{aligned}$$

(37)

The algebraic manipulation of the right-hand-side of Eq. (37) together with the definition of the time derivative of spatial coordinates $\dot{\varvec{x}} = \varvec{v} = \varvec{N}\bar{\varvec{v}}$, allows to re-write $\dot{\varvec{E}}$ as

$$\begin{aligned} \dot{\varvec{E}} = \varvec{F}^{T}\frac{\partial {\varvec{N}}}{\partial \varvec{X}}\bar{\varvec{v}} = \varvec{B}_{0}\bar{\varvec{v}}, \end{aligned}$$

(38)

where $\varvec{B}_{0}$ is the strain–displacement matrix computed in the initial configuration $\varOmega _{0}$. Inserting Eq. (38) into Eq. (36) yields

$$\begin{aligned} ~^e\!\dot{\varvec{f}}^{TL}_{m}&= \int _{\varOmega ^{e}_{0}}\frac{\partial \varvec{N}}{\partial \varvec{X}}^{T}\varvec{F}\varvec{C}_{0}\varvec{F}^{T}\frac{\partial \varvec{N}}{\partial \varvec{X}}\mathrm{d}\varOmega ^{e}_{0}\bar{\varvec{v}}, \end{aligned}$$

(39)

$$\begin{aligned}&=\int _{\varOmega ^{e}_{0}}\varvec{B}_{0}^{T}\varvec{C}_{0}\varvec{B}_{0}\mathrm{d}\varOmega ^{e}_{0}\bar{\varvec{v}}. \end{aligned}$$

(40)

Next, the geometric part of the rate of element internal forces $~^e\!\dot{\varvec{f}}_{g}^{TL}$ (see Eq. (34) is considered. Inserting the definition of the rate of the deformation gradient $\dot{\varvec{F}} = \frac{\displaystyle \partial \varvec{N}}{\displaystyle \partial \varvec{X}}\bar{\varvec{v}}$ into $~^e\!\dot{\varvec{f}}_{g}^{TL}$ yields

$$\begin{aligned} ~^e\!\dot{\varvec{f}_{g}^{TL}} = \int _{\varOmega _{0}^{e}}\frac{\partial \varvec{N}}{\partial \varvec{X}}^{T}\varvec{S}\frac{\partial \varvec{N}}{\partial \varvec{X}}\mathrm{d}\varOmega _{0}^{e}\bar{\varvec{v}}. \end{aligned}$$

(41)

The consistent linearization of the material and geometric parts of the rates of internal forces Eqs. (40) and (41), respectively, is performed by means of the Gateaux derivative. The end-of-step increment $~^e\!\Delta \dot{\varvec{f}}^\mathrm{int}_{n+1}$ expressed in TL framework (the index $(\cdot )^{TL}$ is herein omitted) is computed as follows

$$\begin{aligned} \Delta ~^e\!\dot{\varvec{f}}^\mathrm{int}_{n+1}&= \Delta ~^e\!\dot{\varvec{f}}^{m}_{n+1} + \Delta ~^e\!\dot{\varvec{f}}^{g}_{n+1}\nonumber ,\\&= \frac{\partial ~^e\!\dot{\varvec{f}}^{m}_{n+1} }{\partial \bar{\varvec{v}}_{n+1}}\Delta \bar{\varvec{v}}_{n+1} + \frac{\partial ~^e\!\dot{\varvec{f}}^{g}_{n+1} }{\partial \bar{\varvec{v}}_{n+1}}\Delta \bar{\varvec{v}}_{n+1}\nonumber ,\\&= \int _{\varOmega ^{e}_{0}}\left[ \varvec{B}_{0}^{T}\varvec{C}_{0}\varvec{B}_{0}\right] _{n+1}\mathrm{d}\varOmega ^{e}_{0}\Delta \bar{\varvec{v}}_{n+1}\nonumber \\&\quad + \int _{\varOmega _{0}^{e}}\left[ \frac{\partial \varvec{N}}{\partial \varvec{X}}^{T}\varvec{S}\frac{\partial \varvec{N}}{\partial \varvec{X}}\right] _{n+1}\mathrm{d}\varOmega _{0}^{e}\Delta \bar{\varvec{v}}_{n+1}. \end{aligned}$$

(42)

Integrating Eq. (42) over a time increment $\Delta t$ yields

$$\begin{aligned} \Delta {~^e\!\varvec{f}}^\mathrm{int}_{n+1}&= \int _{\varOmega ^{e}_{0}}\left[ \varvec{B}_{0}^{T}\Delta t\varvec{C}_{0}\varvec{B}_{0}\right] _{n+1}\mathrm{d}\varOmega ^{e}_{0}\Delta \bar{\varvec{v}}_{n+1}\nonumber \\&\quad + \int _{\varOmega _{0}^{e}}\left[ \frac{\partial \varvec{N}}{\partial \varvec{X}}^{T}\Delta t\varvec{S}\frac{\partial \varvec{N}}{\partial \varvec{X}}\right] _{n+1}\mathrm{d}\varOmega _{0}^{e}\Delta \bar{\varvec{v}}_{n+1}. \end{aligned}$$

(43)

To this end, the linearization of the relevant terms has been performed in TL description. To construct the UL formulation, push-forward transformations are applied over Eq. (43) (see reference [25]). The end-of-step increment $\Delta {~^e\!\varvec{f}}^\mathrm{int}_{n+1}$ expressed in UL description reads

$$\begin{aligned} \Delta {~^e\!\varvec{f}}^\mathrm{int}_{n+1}&= \int _{\varOmega _{n}^{e}}\left[ \varvec{B}^{T}\Delta t\varvec{C}\varvec{B}\right] _{n+1}\mathrm{d}\varOmega _{n}^{e}\Delta \bar{\varvec{v}}_{n+1}\nonumber \\&\quad + \int _{\varOmega _{n}^{e}}\left[ \frac{\partial \varvec{N}}{\partial \varvec{x}}^{T}\Delta t\varvec{\sigma }\frac{\partial \varvec{N}}{\partial \varvec{x}}\right] _{n+1}\mathrm{d}\varOmega _{n}^{e} \Delta \bar{\varvec{v}}_{n+1},\nonumber \\&= \left[ ~^e\!\varvec{K}^{m}_{n+1} + ~^e\!\varvec{K}^{g}_{n+1} \right] \Delta \varvec{v}_{n+1}, \end{aligned}$$

(44)

where $~^e\!\varvec{K}^{m}_{n+1}$ and $~^e\!\varvec{K}^{g}_{n+1}$ are defined as in Sect. 2.1. The remaining terms to be linearized from Eq. (5) correspond to the element vector of dynamic forces $~^e\!\varvec{f}^{UL}_\mathrm{dyn}$, written in lumped format as

$$\begin{aligned} ~^e\!\varvec{f}^{UL}_\mathrm{dyn} = \frac{1}{3}\varvec{I}\int _{\varOmega _{n}^{e}}\rho \mathrm{d}\varOmega _{n}^{e}\dot{\varvec{v}} \end{aligned}$$

(45)

The increment vector of the dynamic forces is evaluated at time $t_{n+1-\alpha }$ by means of the directional derivative of $~^e\!\varvec{f}^{UL}_\mathrm{dyn}$ with respect to the nodal accelerations $\Delta \dot{\bar{\varvec{v}}}_{n+1-\alpha }$

$$\begin{aligned} \Delta ~^e\!\varvec{f}^\mathrm{dyn}_{n+1-\alpha } = \frac{\partial ~^e\!\dot{\varvec{f}}^\mathrm{dyn}_{n+1-\alpha } }{\partial \dot{\bar{\varvec{v}}}_{n+1-\alpha }}\Delta \dot{\bar{\varvec{v}}}_{n+1-\alpha }. \end{aligned}$$

(46)

We attempt to reformulate the increment of nodal accelerations $\Delta \dot{\bar{\varvec{v}}}_{n+1-\alpha }$ in Eq. (46) in terms of the end-of-step velocities increment $\Delta {\bar{\varvec{v}}}_{n+1}$, by means of the $\alpha$-Bossak approximations (29), (30) and the chain rule of differentiation

$$\begin{aligned} \Delta \dot{\bar{\varvec{v}}}_{n+1-\alpha }&= \frac{\partial \dot{\bar{\varvec{v}}}_{n+1-\alpha }}{\partial \dot{\bar{\varvec{v}}}_{n+1}}\Delta \dot{\bar{\varvec{v}}}_{n+1},\nonumber \\&=\frac{\partial \dot{\bar{\varvec{v}}}_{n+1-\alpha }}{\partial \dot{\bar{\varvec{v}}}_{n+1}} \frac{\partial \dot{\bar{\varvec{v}}}_{n+1}}{\partial {\bar{\varvec{v}}}_{n+1}}\Delta \bar{\varvec{v}}_{n+1}\nonumber ,\\&= (1-\alpha )\frac{1}{\gamma \Delta t}\Delta \bar{\varvec{v}}_{n+1}. \end{aligned}$$

(47)

Finally, replacing Eq. (47) into Eq. (45) yields the linearized form of the vector of dynamic forces

$$\begin{aligned} \Delta ~^e\!\varvec{f}^\mathrm{dyn}_{n+1-\alpha }&= \frac{1}{3}\varvec{I}\int _{\varOmega ^{e}}\rho \mathrm{d}\varOmega _{n}^{e}(1-\alpha )\frac{1}{\gamma \Delta t}\Delta \bar{\varvec{v}}_{n+1},\nonumber \\&= (1-\alpha )\frac{1}{\gamma \Delta t}~^e\!\varvec{M}_{n+1}\Delta \bar{\varvec{v}}_{n+1}. \end{aligned}$$

(48)

Von Wolffersdorff Hypoplastic model

In this work, the formulation proposed in [22] is employed. The stress evolution is described by the expression

$$\begin{aligned} \mathring{\varvec{\sigma }} = f_{b}(p_{s})f_{e}(e)\left[ \varvec{L}:\varvec{d} + f_{d}(e)\varvec{N}||\varvec{d}||\right] , \end{aligned}$$

(49)

where $\varvec{L}$ and $\varvec{N}$ are written as

$$\begin{aligned} \varvec{L}&= \dfrac{1}{\dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })}: \dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })}}a^{2} \left[ \left( \dfrac{F}{a}\right) ^{2}\varvec{II} + \dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })}\otimes \dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })} \right] , \end{aligned}$$

(50)

$$\begin{aligned} \varvec{N}&= \dfrac{1}{\dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })}:\dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })}}a^{2} \left( \dfrac{F}{a}\right) \left[ \dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })} + \varvec{\sigma }^{*}\right] , \end{aligned}$$

(51)

with $\frac{\displaystyle \varvec{\sigma }}{\displaystyle \text {tr}(\varvec{\sigma })}$ as the normalized stress ratio, $\varvec{II}$ is the fourth-order symmetric unit tensor, $\varvec{\sigma }^{*} = \dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })} -\dfrac{1}{3}\varvec{I}$ is the deviatoric normalized stress ratio, $\varvec{I}$ the second-order unit tensor and the term $\dfrac{1}{\dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })}: \dfrac{\varvec{\sigma }}{\text {tr}(\varvec{\sigma })}}$ is a factor employed to reflect the reduction of incremental stiffness for stress paths deviating from the isotropic path [58].

The rate of the Cauchy stress $\dot{\varvec{\sigma }}$ is related to the Jaumann stress rate $\mathring{\varvec{\sigma }}$ via

$$\begin{aligned} \dot{\varvec{\sigma }} = \varvec{w}\varvec{\sigma } + \varvec{\sigma }\varvec{w}^{T} + \mathring{\varvec{\sigma }}, \end{aligned}$$

(52)

where $\varvec{w}= \frac{1}{2}\left( \nabla \varvec{v} - \nabla \varvec{v}^{T}\right)$ is the spin tensor, i.e., the skew-symmetric part of the velocity gradient.

The concept of incremental nonlinearity, relevant for the description of granular soils, is connected to the continuous dependence of the stiffness tensor $\varvec{C}$ on the direction of deformation $\dfrac{\varvec{d}}{||\varvec{d}||}$ [54]

$$\begin{aligned} \mathring{\varvec{\sigma }}&= \varvec{C}:\varvec{d}, \end{aligned}$$

(53)

$$\begin{aligned} \varvec{C}&= \dfrac{\partial \mathring{\varvec{\sigma }}}{\partial \varvec{d}} = \varvec{L} + \varvec{N}\otimes \frac{\varvec{d}}{||\varvec{d}||}. \end{aligned}$$

(54)

A hypoplastic critical state surface $f_{H}(\varvec{\sigma })$ is explicitly defined by means of the functions $a(\varphi _{c})$ and $F(\varvec{\sigma }^{*})$

$$\begin{aligned} a&= \frac{\sqrt{3}\left( 3 - \text {sin}\varphi _{c} \right) }{2\sqrt{2}\text {sin}\varphi _{c}}, \end{aligned}$$

(55)

$$\begin{aligned} F&= \sqrt{ \frac{1}{8}\text {tan}^{2}\psi + \frac{2 - \text {tan}^{2}\psi }{2 + \sqrt{2}\text {tan}\psi \text {cos}3\theta }} - \frac{1}{2\sqrt{2}}\text {tan}{\psi },\end{aligned}$$

(56)

$$\begin{aligned} \text {tan}\psi&= \sqrt{3}|\varvec{\sigma }^{*}|,\end{aligned}$$

(57)

$$\begin{aligned} \text {cos}3\theta&= -\sqrt{6}\frac{\text {tr}\left( {\varvec{\sigma }^{*}\cdot \varvec{\sigma }^{*}\cdot \varvec{\sigma }^{*}}\right) }{\left( \varvec{\sigma }^{*}:\varvec{\sigma }^{*}\right) ^{\frac{3}{2}}}, \end{aligned}$$

(58)

where $\theta$ denotes the Lode angle. It is noted that the function $F(\varvec{\sigma }^{*})$ is derived in compliance with the Matsuoka–Nakai limit criterion $f_{MN}$[58].

To complete the model description, the evolution of the void ratio $\dot{e}$ is formulated as

$$\begin{aligned} \dot{e} =\left( 1 + e\right) \text {tr}(\varvec{d}). \end{aligned}$$

(59)

The admissible range of the void ratio e is bounded by three pressure-dependent compression curves $e_{i}(p_{s})$, $e_{c}(p_{s})$, $e_{d}(p_{s})$ [white area in Fig. 17)], defined by the relation

$$\begin{aligned} \frac{e_{i}}{e_{i0}} = \frac{e_{{c}}}{e_{{c0}}} = \frac{e_{{d}}}{e_{{d0}}} = \text {exp}\left[ -\left( \frac{p_{s}}{h_{s}}\right) ^{n}\right] . \end{aligned}$$

(60)

$e_{i}(p_{s})$ and $e_{d}(p_{s})$ define the upper and lower bounds for e at a given mean skeleton pressure $p_{s}$, respectively. $e_{c}(p_{s})$ denotes the pressure-dependent critical void ratio, $e_{i0}$, $e_{c0}$, $e_{d0}$ are the corresponding values for $e_{i}(p_{s})$, $e_{c}(p_{s})$, $e_{d}(p_{s})$ at zero pressure, $h_{s}$ is the skeleton granular hardness, and the coefficient n reflects the sensitivity of the granular skeleton to the variations of $p_{s}$.

Fig. 17

Hypoplastic constitutive model: characteristic pressure-dependent void ratio functions: The gray zone lies outside the admissible range for e

Characteristic features observed in the behavior of granular materials, such as pressure- and density-dependent stiffness and volumetric dilatancy, are properly accounted for through the stiffness scaling factors $f_{b}(p_{s})$ (the so-called barotropy factor) and $f_{e}(e)$, $f_{d}(e)$, denoted as density factors [56]:

$$\begin{aligned} f_{b}&= \frac{h_{s}}{n}\left( \frac{1+e_{i}}{e_{i}}\right) \left( \frac{e_{i0}}{e_{c0}}\right) ^{\beta }\left( \frac{3p_{s}}{h_{s}}\right) ^{1-n} \left[ 3 + a^{2}\right. \nonumber \\&\quad \left. - \sqrt{3}a\left( \frac{e_{i0} - e_{d0}}{e_{c0} - e_{d0}}\right) \right] , \end{aligned}$$

(61)

$$\begin{aligned} f_{e}&= \left( \frac{e - e_{d}}{e - e_{c}}\right) ^{\alpha },\end{aligned}$$

(62)

$$\begin{aligned} f_{d}&= \left( \frac{e_{c}}{e}\right) ^{\beta }. \end{aligned}$$

(63)

$f_{b}(p_{s})$ and $f_{e}(e)$ control the incremental stiffness of the soil in nonlinear dependence of the pressure $p_{s}$ and void ratio e, respectively.

The scalar function $f_{d} (e)$$(0\le f_{d}(e)\le 1)$, which can be interpreted as a weighting function of the nonlinear component $\varvec{N}||\varvec{d}||$ in Eq. (49), governs the transition of the soil to the critical state. In the case of $f_{d}(e) =0$ (implying $e=e_{d}$), the soil is in the densest possible state and its behavior can be approximated as hypoelastic [56]. When $f_{d}(e) =1$ (implying $e=e_{c}$), the soil reached the critical state.

Sub-stepping integration algorithm

• 1. Initialize the scheme by setting $\alpha ^{1}= \alpha ^{0}$, $NS= 1$, ${e}^{i+1,1}_{n+1}= e^{i}_{n+1}$ and $\varvec{\sigma }^{i+1,1}_{n+1}= \varvec{\sigma }^{i}_{n+1}$. In this work, the initial sub-step reduction factor is set to $\alpha _{0} = 1/8$.

• 2. In sub-step k of increment $i+1$ two forward-Euler (FWE) evaluations of Eqs. (49) and (59) are performed: First, the rates $\dot{e}^{i+1,k}_{n+1}= \dot{e}^{i+1,k}_{n+1}({e}^{i+1,k}_{n+1}, \varvec{d}^{i+1}_{n+1})$ and $\mathring{\varvec{\sigma }}^{i+1,k}_{n+1} = \mathring{\varvec{\sigma }}^{i+1,k}_{n+1}\left( e^{i+1,k}_{n+1},\varvec{\sigma }^{i+1,k}_{n+1}\right)$ are computed. These estimations are utilized to determine the intermediate void ratio and the stresses at sub-step $k + \frac{1}{2}$

  $$\begin{aligned} \varvec{\sigma }^{i+1,k + \frac{1}{2}}_{n+1}&= \varvec{\sigma }^{i+1,k}_{n+1} + \Delta t^{i+1,k}_{n+1}\dot{\varvec{\sigma }}^{i+1,k}_{n+1},\end{aligned}$$

  (64)
  $$\begin{aligned} e^{i+1,k + \frac{1}{2}}_{n+1}&= e^{i+1,k}_{n+1} + \Delta t^{i+1,k}_{n+1}\dot{e}^{i+1,k}_{n+1}, \end{aligned}$$

  (65)

  with

  $$\begin{aligned} \dot{\varvec{\sigma }}^{i+1,k}_{n+1}= \mathring{\varvec{\sigma }}^{i+1,k}_{n+1} + \varvec{w}_{n}\varvec{\sigma }^{i+1,k}_{n+1} + \varvec{\sigma }^{i+1,k}_{n+1}\varvec{w}^{T}_{n}. \end{aligned}$$

  (66)

  In Eq. (66), the spin tensor $\varvec{w}_{n}$ is evaluated at the previous time step [25] and the rate of deformation tensor $\varvec{d}^{i+1}_{n+1}(\varvec{v}^{i+1}_{n+1})$ is considered to remain constant throughout the iteration step $i+1$ as in [49].

  The estimates from Eqs. (64) and (65) are inserted again into Eqs. (49) and (59) and further utilized to obtain the rates

  $$\begin{aligned} \mathring{\varvec{\sigma }}^{i+1,k+\frac{1}{2}}_{n+1}&= \mathring{\varvec{\sigma }}^{i+1,k+\frac{1}{2}}_{n+1}\left( e^{i+1,k + \frac{1}{2}}_{n+1},\varvec{\sigma }^{i+1,k + \frac{1}{2}}_{n+1}\right) , \end{aligned}$$

  (67)
  $$\begin{aligned} \dot{e}^{i+1,k + \frac{1}{2}}_{n+1}&= \dot{e}^{i+ 1,k + \frac{1}{2}}_{n+1}\left( e^{i+1,k + \frac{1}{2}}_{n+1}, \varvec{d}^{i+1}_{n+1}\right) . \end{aligned}$$

  (68)

  The material time derivative of the Cauchy stress computed as

  $$\begin{aligned} \dot{\varvec{\sigma }}^{i+1,k+\frac{1}{2}}_{n+1}&= \mathring{\varvec{\sigma }}^{i+1,k+\frac{1}{2}}_{n+1} + \Delta t^{i+1,k}_{n+1}\left( \varvec{w}_{n}\varvec{\sigma }^{i+1,k+\frac{1}{2}}_{n+1}\right. \nonumber \\&\quad \left. + \varvec{\sigma }^{i+1,k+\frac{1}{2}}_{n+1}\varvec{w}^{T}_{n}\right) . \end{aligned}$$

  (69)

• 3. Compute the relative stress error $E_{\sigma }$ in terms of the Cauchy stress rates at sub-steps k and $k + \frac{1}{2}$:

  $$\begin{aligned} E_{\sigma } = \frac{\left| \left| \dot{\varvec{\sigma }}^{i+1,k+\frac{1}{2}}_{n+1} - \dot{\varvec{\sigma }}^{i+1,k}_{n+1}\right| \right| \Delta t^{i+1,k}_{n+1}}{\left| \left| \varvec{\sigma }^{i+1,k}_{n+1} + \dfrac{\Delta t}{2}\left( \dot{\varvec{\sigma }}^{i+1,k+\frac{1}{2}}_{n+1} + \dot{\varvec{\sigma }}^{i+1,k}_{n+1} \right) \right| \right| }, \end{aligned}$$

  (70)

  and check the stress error $E_{\sigma }\le {\hbox {tol}}_{\sigma }$ against the tolerance (usually $10^{-5}\le {\hbox {tol}}_{\sigma }\le 10^{-2})$.

• 4. If the relative stress error $E_{\sigma } > tol_{\sigma }$, the current sub-step is not accepted. A recalculation of the sub-stepping procedure, starting from step 2, is performed using a reduced trial value $\alpha ^{k}$, estimated according to

  $$\begin{aligned} \alpha ^{k} = \max \,\left\{ 0.1,0.8 * \left( \frac{{\hbox {tol}}_{\sigma }}{E_{\sigma }}\right) ^{\frac{1}{2}}\right\} \alpha ^{k}. \end{aligned}$$

  (71)

  If $E_{\sigma }\le {\hbox {tol}}_{\sigma }$ the sub-step is accepted. Compute the updated values of stresses and void ratio, at the end of sub-step $k+1$, employing the rates at sub-step k and $k + \frac{1}{2}$ given by:

  $$\begin{aligned} \varvec{\sigma }^{i+1,k+1}_{n+1}&{=} \varvec{\sigma }^{i+1,k}_{n+1} {+} \frac{1}{2}\Delta t^{k}\left( \dot{\varvec{\sigma }}^{i+1,k {+} \frac{1}{2}}_{n+1} {+} \dot{\varvec{\sigma }}^{i+1,k}_{n+1}\right) , \end{aligned}$$

  (72)
  $$\begin{aligned} {e}^{i+1,k+1}_{n+1}&= {e}^{i+1,k}_{n+1} + \frac{1}{2}\Delta t^{k}\left( \dot{e}^{i+1,k+\frac{1}{2}}_{n+1} + \dot{e}^{i+1,k}_{n+1}\right) , \end{aligned}$$

  (73)

  If $\sum ^{NS}_{k = 1} \alpha ^{k} < 1$ and $NS < NS_{\max }$, where $NS_{\max }$ is the user-defined maximum allowable number of sub-steps per iteration, increase the sub-step number NS by one $NS = NS + 1$ and proceed to the next sub-step $k+1$. An initial trial value for $\alpha ^{k+1}$ is made in accordance with

  $$\begin{aligned} \alpha ^{k+1} = \text {min}\left\{ \alpha ^{k}, 1 - \sum ^{k}_{p = 1} \alpha ^{p}\right\} , \end{aligned}$$

  (74)

• 5. When the sub-stepping process is complete (i.e., $\sum ^{NS}_{k = 1} \alpha ^{k} = 1$), the increments at the end of the iteration step $i+1$ (i.e., $\Delta {\varvec{\sigma }}^{i+1}_{n+1}$ and $\Delta {e}^{i+1}_{n+1}$) correspond then to the increments $\Delta {\varvec{\sigma }}^{i+1,k+1}_{n+1}$ and $\Delta {e}^{i+1,k+1}_{n+1}$ estimated from the last successful sub-step.

• 6. After the time integration of the relevant variable is complete for iteration step $i+1$, a projection to the nodes of the increments $\Delta {\varvec{\sigma }}^{i+1}_{n+1}$ and $\Delta {e}^{i+1}_{n+1}$ is performed (see Sect. 2.3). A loop over all elements is carried out, and the following operations are executed:

• Projection of the increments to the nodes and update of total values ($\varvec{\sigma }^{i+1}_{n+1}$, $e^{i+1}_{n+1}$) by means of Eq. (14).