Arching development above active trapdoor: insight from multi-scale analysis using FEM–SPH

Abstract

Underground excavation is usually accompanied by complex soil-structure interaction problems in practical engineering. This paper develops a novel multi-scale approach for investigating the soil arching effect through trapdoor tests. This approach adopts the finite element method (FEM) and smoothed particle hydrodynamics (SPH) method to handle the particle-rigid body interaction in the trapdoor tests, incorporating a micromechanical 3D-H model to derive the nonlinear material response required by the SPH method. The variation of the earth pressure on the trapdoor in simulations exhibits good agreement with those of the experiments. Extensive parametric analyzes are performed to assess the effects of soil height and inter-particle friction angle on the evolution of load transfer and soil deformation. Three deformation patterns are observed under different buried conditions, including the trapezoid, the triangle, and the equal settlement pattern. Results indicate that the planes of equal settlement develop progressively with the trapdoor movement and then enter the range of experimentally observed values. Additionally, three failure mechanisms are identified that correspond to the three deformation patterns. Due to the advantages of the micromechanical model, mesoscale behavior is captured. The anisotropy of stress distribution in the plastic region is found during the arching process.

1 Introduction

With the rapid growth of urbanization in major cities, the demand for urban underground space has sparked a variety of deep-level construction projects. Due to the complexity of soil properties, structural characteristics, and construction sequences, the responses of surrounding soil mass and structures in deep underground operations are highly nonlinear and unpredictable [25, 33]. These activities may induce an unloading region near the excavation face, accompanied by localized differential settlement [5] and stress concentration on the adjoining structures [9]. This phenomenon of stress redistribution is caused by the shear stress at the interface between the mobilized and stationary portions of soil mass [35], which is referred to as the “soil arching effect”.

This yielding trapdoor between two stationary parts can be a prototype for investigating soil arching, which can be seen in many natural scenarios and construction processes such as the formation of subterranean voids [18] or the unloading of vertical excavation. The conventional trapdoor test was first conducted by Terzaghi [51], which has proven its effectiveness in investigating the soil arching effect. The plane-strain trapdoor tests, with the advantages of easy observation and low cost, have made great contributions to the thorough and in-depth understanding of the arching mechanism [2, 26, 68].

The formation and evolution of soil arching, characterized by a variation of stress, strain, and deformation within the soil mass, is a major focus of the trapdoor experiments. The ground reaction curve (GRC) has been a widely used approach for reflecting the evolution of arching stress with the trapdoor movement [3, 5, 27]. King et al. [29] adopted four phases of arching stress development, including initial, maximum, load recovery, and creep strain, to describe the stress variation in the field studies. Han et al. [22] proposed a simplified GRC consisting of three lines, requiring only the minimum arching ratio, the final arching ratio, and the corresponding relative trapdoor displacement. In terms of the observed arch shapes, three evolution stages were proposed by Iglesia et al. [27]: (a) the maximum arching stage with curved arches; (b) the intermediate stage with triangular arches; and (c) the terminal stage with rectangular arches. Additional, more deformation patterns of soil arch including triangular, trapezoid, parabolic shape were accurately captured by PIV system [56, 74, 75]. More recently, George and Dasaka [19] discovered the cubic polynomial contour of a soil arch in the yielding zone above the trapdoor in terms of the trajectories of major principal stresses. Besides these obvious patterns, some scholars [30, 32, 47, 48] observed a latent deformation pattern characterized by equal settlement planes (ESP) in the trapdoor tests with relatively high soil height. Moreover, several relevant deformation features of soil, such as the height of ESP [48, 73], surface subsidence [47], settlement trough [49], and profile curves of settlement [70], have also been extensively studied.

Understanding the formation and development of failure surfaces can also contribute to the identification of the arching phases in GRC [18].Al Hattamleh et al. [1] supported that soil arching only maximize its effect after the strain localization. Similarly, by observing the failure region and arching ratio of the soil at different depth ratios, Rai et al. [44] discovered that as the soil depth increases, the failure region transitions from a global to a localized state, signifying the complete development of soil arching. In previous research, there have been several methods for determining the failure surface. Boundaries between the arching-unloaded region and the arching-loaded region can be used to identify the failure surfaces [15]. Rui et al. [46] and Tao et al. [49] regarded the shear strain bands as slip surfaces and compared them according to the inclination angle in tests with varying soil heights. George and Dasaka [19] employed the Mohr’s circle representations to determine the direction and magnitude of the principal stresses along the soil arch including the failure surface. Furthermore, Liu et al. [36] and Chen et al. [8] identified the principal failure surface, known as the shear band which is characterized by a high gradient in particle rotation and a sudden change in the direction of particle contact forces. Concerning the failure mechanisms of soil arching, Xu et al. [68] and Khatami et al. [28] visualized the shear strain field using the digital image correlation (DIC) and particle image velocimetry (PIV) techniques, respectively, and conducted an analysis on how the strain localization develops. Two kinds of failure surface appeared above the single trapdoor including internal and external shear bands [16, 18, 36, 54, 60]. Moreover, Wu et al. [56] identified typical progressive failure modes, such as local punching shear failure, vertical slip, and inclined slip mechanism. However, recent research to provide insight into the failure mechanisms pays too little attention to the micro mechanism within these shear bands.

Since computer technology has been undergoing rapid development, numerical modeling has become a promising and effective tool for engineering problems [12, 23, 31, 43, 67, 77]. The traditional grid-based Finite Element Method (FEM) has been widely used to analyze the soil arching effect due to its efficiency and reliability [13, 20, 78]. Considering the limitations of traditional continuum mechanics, several methods based on non-continuum mechanics have been proposed. The Discrete Element Method (DEM) can address discontinuous problems and capture the complex mechanical properties of granular materials [53, 61,62,63, 66, 76]. It has also been proven successful in revealing the soil arching mechanism [4, 48, 71], but requires demanding computational resources in handling large-scale boundary value problems. The micro-mechanical models, based on the statistical description of directional contacts between particles, have been proposed to avoid excessive computational cost [10, 11, 64, 65]. Moreover, the smoothed particle hydrodynamic (SPH) method has recently been adopted to deal with geotechnical engineering problems involving large deformation and geomaterial post-failure issues [6, 34, 57, 72], with the potential for application to the analysis of soil arching. The fact that both the FEM and SPH methods are in the Lagrangian frame makes it possible to couple the two approaches into the simulation of trapdoor tests, taking advantage of the complementarities. Although the SPH method provides a discrete solution to the large deformation problem, it still requires a considerable number of calculations to solve sophisticated constitutive formulations [58]. In view of this, a micromechanical model is introduced in the derivation of the material response to avoid considering phenomenological constitutive relations in SPH method.

This paper harnesses the advantages of the multi-scale approach to deepen our understanding of soil arching behavior. The SPH particles are used to simulate soil subjected to large deformations, while the remaining parts are meshed with the finite elements. The 3D-H model acting on the SPH particles is to derive the nonlinear material response of soil. The proposed framework is validated by comparing the variations of the pressure on the trapdoor with the experimental results from the literature. Then, an extensive parametric analysis is conducted to explore the influence of the material properties and geometry on the soil arching behaviors. Specifically, the evolution of soil arching in terms of deformation pattern, stress redistribution, and failure surfaces are investigated. Finally, a mesoscopic analysis is performed to reveal the microscopic behaviors hiding behind the macroscopic scale.

2 Methodology

2.1 Finite element method (FEM)

In this study, the load–displacement response of the trapdoor is evaluated using FEM. To realize the coupling of the FEM–SPH approach, the explicit time method is used. In the explicit dynamic procedure, the duration of the trapdoor test is divided into a large number of small time increments, making it relatively easy and efficient to calculate each small increment. One incremental calculation consists of nodal calculations and element calculations, by which the kinematic conditions are updated for the next increment. First, the dynamic equilibrium is written with the balance of internal and external forces at the start of the increment t [38, 59].

$$\begin{aligned} \textbf{M} {\ddot{\textbf{u}}}|_{(t)} = \textbf{P}|_{(t)} - \textbf{I}|_{(t)} \end{aligned}$$

(1)

where $\textbf{M}$ is the nodal mass matrix, ${\ddot{\textbf{u}}}$ is the nodal acceleration, $\textbf{P}$ is the external forces acting on nodes, $\textbf{I}$ is the internal element forces. According to a central difference rule, the equations of velocities and displacements can be explicitly integrated through time using Eq. (2):

$$\begin{aligned} \left\{ \begin{array}{l} \dot{\textbf{u}}|_{(t+\frac{\Delta t}{2})} = \dot{\textbf{u}}|_{(t-\frac{\Delta t}{2})}+\frac{(\Delta t|_{(t+\Delta t)}+\Delta t|_{(t)})}{2} {\ddot{\textbf{u}}}|_{(t)} \\ \textbf{u}|_{(t+\Delta t)} = \textbf{u}|_{(t)}+\Delta t|_{(t+\Delta t)} \dot{\textbf{u}}|_{(t+\frac{\Delta t}{2})} \end{array} \right. \end{aligned}$$

(2)

where $\dot{\textbf{u}}$ is the nodal velocity, $\textbf{u}$ is the displacement, $\Delta t$ is the time increment. In element calculation, the incremental displacement is used to obtain the element strain, which is then used to compute the stress and internal force through material constitutive relationships. Finally, the state of the current increment is updated for the next incremental calculation.

2.2 Smoothed particle hydrodynamics (SPH)

SPH is a meshless and Lagrangian approach for solving systems of partial differential equations (PDEs) that can be used to simulate extreme soil deformation. The problem domain is discretized into a limited number of particles, each of which carries field variables like mass, density, and velocity. The motion of each particle is initiated by external forces and interactions between particles. Through tracking the motion and location of particles and approximating them as moving interpolation points, the mechanical behavior of the entire system can be evaluated [21]. The derivation of the SPH formulation is based on interpolation theory. The derivation procedure is divided into two stages: kernel approximation and particle approximation. The first step is to design a continuous function f that uses the integral representation to approximate the physical variables of particles, as follows:

$$\begin{aligned} f(\varvec{x})\approx \int \limits _{\Omega }f(\varvec{x}^{'})W(\varvec{x}-\varvec{x}^{'},h)\rm{d}\varvec{x}^{'} \end{aligned}$$

(3)

where h is the smoothing length that defines the range of the kernel support, $\varvec{x}'$ denotes the location vector of the adjacent particle in the support area, and $\Omega$ is the volume of the integral. The kernel function W, which could be considered a weighting function because its value represents the extent of influence of the surrounding particle, plays a central role in this equation. The determination of the h is performed automatically during the simulation, by controlling the number of SPH particles in the support domain within a certain range, which is generally between 50 and 60 in order to achieve accurate results. The most commonly used cubic spline function is employed in this research [42] as follows:

$$\begin{aligned} W(\varvec{x}-\varvec{x}^{'},h)=\alpha _{d} \left\{ \begin{array}{l} \frac{2}{3}-R^{2}+\frac{1}{2}R^{3}, \qquad \, 0\le R<1 \\ \frac{1}{6}(2-R^{3}),\qquad \qquad 1\le R <2 \\ 0,\qquad \qquad \qquad \qquad 2\le R \end{array} \right. \end{aligned}$$

(4)

where $\alpha _{d}$ denotes the normalization factor. In three dimensional space, $\alpha _{d}=3/2 \pi h^{2}$. R is the relative distance between two points, $\varvec{x}$ and $\varvec{x}'$.

Fig. 1

Particle approximation based on kernel function W

Furthermore, in the stage of particle approximation presented in Fig. 1, the continuous function $f(\varvec{x})$ is transformed into a discrete form using a set of discrete particles. Only the particles within the support domain with a radius of kh are taken into account in the particle approximation [55]. The function $f(\varvec{x})$ and its derivative $\nabla f(\varvec{x})$ are approximated as follows:

$$\begin{aligned}{} & {} f(\varvec{x}_{i})\approx \overset{N}{\underset{j=1}{\sum }}\dfrac{m_{j}}{\rho _{j}}f(\varvec{x}_{j})\cdot W(\varvec{x}_{i}-\varvec{x}_{j},h) \end{aligned}$$

(5)

$$\begin{aligned}{} & {} \nabla f(\varvec{x}_{i})\approx \overset{N}{\underset{j=1}{\sum }}\dfrac{m_{j}}{\rho _{j}}f(\varvec{x}_{j})\cdot \nabla _{i} W(\varvec{x}_{i}-\varvec{x}_{j},h) \end{aligned}$$

(6)

where i is the particle of interest, j represents those neighboring particles within the smoothing length, $m_{j}$ and $\rho _{j}$ represent the mass and density of the particle j, respectively; N is the total number of neighboring particles. Then, the continuity and momentum equations in the continuous description can be discretized and converted into the SPH form using the above-mentioned particle approximation approaches:

$$\begin{aligned}{} & {} \dfrac{\rm{d}\rho _{i}}{\rm{d}t}= \overset{N}{\underset{j=1}{\sum }}m_{j} (\varvec{v}_{i}^{\alpha }-\varvec{v}_{j}^{\alpha })\dfrac{ \partial {W(\varvec{x}_{i}-\varvec{x}_{j},h)}}{\partial {\varvec{x}_{i}^{\alpha }}} \end{aligned}$$

(7)

$$\begin{aligned}{} & {} \dfrac{\rm{d}\varvec{v}_{i}^{\alpha }}{\rm{d}t}= \overset{N}{\underset{j=1}{\sum }}m_{j} \left( \dfrac{\varvec{\sigma }_{i}^{\alpha \beta }}{\rho _{i}^{2}}+\dfrac{\varvec{\sigma }_{j}^{\alpha \beta }}{\rho _{j}^{2}}+\varPi _{ij}\delta ^{\alpha \beta }\right) \nonumber \\{} & {} \dfrac{\partial {W(\varvec{x}_{i}-\varvec{x}_{j},h)}}{\partial {\varvec{x}_{i}^{\beta }}}+\textbf{F}_{i} \end{aligned}$$

(8)

where $\rho _{i}$ denotes the density of particle i with velocity $v_{i}$ and $\rho _{j}$ is the density of particle j with velocity $v_{j}$, $m_{j}$ is the mass of particle j, $\sigma _{i}$ and $\sigma _{j}$ are the stress tensors of particles i and j, $\delta$ is the Dirac delta function, $\varvec{F}_{i}$ is external force, the superscript $\alpha$ and $\beta$ denote the spatial directions, and $\varPi _{ij}$ is additional artificial viscosity used to prevent the penetration of particles among each other, as proposed by Monaghan [41]:

$$\begin{aligned} \varPi _{ij}= \left\{ \begin{array}{l} \dfrac{-ac_{ij} \phi _{ij}+\beta \phi _{ij}^{2}}{\rho _{ij}} \quad \varvec{v}_{ij}\cdot \varvec{x}_{ij}<0\\ \,\,\qquad \quad 0 \qquad \qquad \; \varvec{v}_{ij}\cdot \varvec{x}_{ij}\ge 0 \end{array} \right. ,\qquad \phi _{ij}=\dfrac{h_{ij}\varvec{v}_{ij}\cdot \varvec{x}_{ij}}{\left| \varvec{x}_{ij}\right| ^{2}+0.01h^{2}} \end{aligned}$$

(9)

where $c_{ij}=(c_{i}+c_{j})/2$, $\rho _{ij}=(\rho _{i}+\rho _{j})/2$, $h_{ij}=(h_{i}+h_{j})/2$, $\varvec{v}_{ij}=\varvec{v}_{i}-\varvec{v}_{j}$, and $\varvec{x}_{ij}=\varvec{x}_{i}-\varvec{x}_{j}$, $c_{i}$ and $c_{j}$ refer to the speed of sound in the soil; $\alpha$ and $\beta$ are artificial viscosity terms which are often set as 1.0.

2.3 FEM–SPH coupling algorithm

In this study, SPH particles are utilized to simulate the soil, while FEM elements are employed to represent rigid boundaries and the movable trapdoor. Consequently, the stress variations resulting from the formation of the soil arch can be obtained through the elements located above the trapdoor. The interaction between FE meshes and SPH particles can be realized by the node-surface contact algorithm, as seen in Fig. 2. The contact forces comprise the normal force ($\varvec{f}_{Ni}$) and tangential force ($\varvec{f}_{li}$), which can be calculated through the penalty method. The SPH particles specified as slave nodes will be examined for penetration into the FEM elements that are considered the master surface in each increment. If the penetration condition is fulfilled, the contact forces are applied and calculated according to the extent of penetration and contact stiffness. The $\varvec{f}_{Ni}$ and $\varvec{f}_{li}$ applied to the node i are defined by the equations below:

$$\begin{aligned} \varvec{f}_{Ni}= & {} -\varvec{f}_{s}k_{i}l_{i}\varvec{n}_{i} \end{aligned}$$

(10)

$$\begin{aligned} \varvec{f}_{li}= & {} \mu \left| \varvec{f}_{Ni} \right| \end{aligned}$$

(11)

where $\varvec{f}_{s}$ is the scale factor for contact stiffness; $l_{i}$ is the penetration depth; $\varvec{n}_{i}$ is the direction vector normal to the master surface; $\mu$ is the friction coefficient; $k_{i}$ is the stiffness factor for the master segment, which can be computed as follows:

$$\begin{aligned} k_{i}= k_{1}\dfrac{K_{i}s_{i}^{2}}{V_{i}} \end{aligned}$$

(12)

where $K_{i}$ is the bulk modulus, $s_{i}$ is the face area of the element, $V_{i}$ is the volume, and $k_{1}$ is a penalty scale factor.

Fig. 2

Interaction between SPH particles and Lagrangian FE meshes

2.4 A brief introduction to the 3D-H model

Referring to the previous study on the 3D-H model, which has been implemented in the SPH method [65], an intermediate granular assembly is selected to build a bridge between macro-scale and mesoscale. Based on this micromechanical model, it is possible to calculate the macroscopic stress tensor by decomposing the constitutive relationship of the micro-directional model. The calculation process has three main steps, as shown in Fig. 3.

Fig. 3

General homogenization scheme of 3D-H model

• (1) Kinematic localization: a global coordinate system ($\varvec{x_{1},x_{2},x_{3}}$) and a local frame (n,t,w), which are used to describe the directions of both stress and strain on the macro- and meso-scale, are given in Fig. 4. A rotation matrix $\varvec{P}$ is employed to complete the coordinate transformation from a global frame to a local frame. The vector is a key feature of the dimension of the mesostructure: L=[${l_{1},l_{2},l_{3}}$]$^{T}$, where $l_{1}, l_{2}, l_{3}$ denote the lengths along with directions n,t,w, respectively. Hence, the following equation, based on the kinematic localization assumption, assumes that the material is isotropic and undergoes affine deformation:

  Fig. 4

  

  Global and local coordinate systems

  $$\begin{aligned} \left\{ \begin{array}{l} \frac{\delta l_{1}}{l_{1}} =\varvec{n^t \cdot \delta \varepsilon \cdot n}=({\varvec{{P}}}\cdot \varvec{\delta \varepsilon } \cdot {\varvec{{P}}}^{-1})_{11} \\ \frac{\delta l_{2}}{l_{2}} =\varvec{t^t \cdot \delta \varepsilon \cdot t}=({{\varvec{P}}}\cdot \varvec{\delta \varepsilon } \cdot {\varvec{{P}}}^{-1})_{22} \\ \frac{\delta l_{3}}{l_{3}} =\varvec{w^t \cdot \delta \varepsilon \cdot w}=({\varvec{{P}}}\cdot \varvec{\delta \varepsilon } \cdot {\varvec{{P}}}^{-1})_{33} \\ \end{array} \right. \end{aligned}$$

  (13)

  where $\delta \varvec{\varepsilon }$ is the incremental macroscopic strain tensor. In this step, the macroscopic strain tensor is transformed into the hexagon deformation described by the vector $\delta \varvec{L}$ which can be used as the known variable to compute the relative displacement at each contact and contact forces in the subsequent step.

• (2) Meso-structure behavior: the 3D-H model (Fig. 5) can be divided into two 2D multiscale constitutive model for the calculation in this step. For Hexagon A, though combining the balance equation of force and moment, and the contact law, as well as performing a series of coordinate transformations, the incremental constitutive relation can be obtained (see detail in Appendix 1):

  Fig. 5

  

  The 3D mesostructure and its decomposition procedure in the 3D H-model

  $$\begin{aligned} \dfrac{1}{\left| D\right| ^{a}}\begin{bmatrix} K_{11}^{a} &{}\quad K_{12}^{a} \\ K_{21}^{a} &{}\quad K_{22}^{a} \end{bmatrix} \begin{bmatrix} \delta l_{1}\\ \delta l_{2} \end{bmatrix}=\begin{bmatrix} \delta F_{1}^{a}\\ \delta F_{2}^{a} \end{bmatrix} \end{aligned}$$

  (14)

  The deduction of the relational expression between $\delta F$ and $\delta l$ is the same for Hexagon A and B. Hence, the total incremental force along direction n can be obtained by superimposing Hexagon A and Hexagon B ($\delta \varvec{F}_{1}=\delta \varvec{F}_{1}^{a}+\delta \varvec{F}_{1}^{b}$). Consequently, the relationship between meso-strain and incremental force is identified.

• (3) Stress averaging: In this step, a specimen of volume V is selected for analysis and its macro-stress tensor $\varvec{\sigma }$ can be calculated by averaging the mesostress which is generated by all the mesostructures in this specimen:

  $$\begin{aligned} \varvec{\sigma } =\dfrac{1}{V}\iiint \omega (\theta ,\varphi ,\psi ) \varvec{P}^{-1}\widetilde{V} {\widetilde{\varvec{\sigma }}} (\varvec{n},\varvec{t},\varvec{w})\varvec{P}\sin \varphi {\text{d}}\theta {\text{d}}\varphi {\text{d}}\psi \end{aligned}$$

  (15)

  where the distribution function $\omega (\theta ,\varphi ,\psi )$ is constant because the specimen is isotropic and $\theta \in [0,2\pi )$, $\varphi \in [0,\pi ]$, $\psi \in [0,2\pi )$ ($\theta$, $\varphi$, $\psi$ are the Euler angles shown in Fig. 4). ${\widetilde{V}}$ is the volume of meso-structure along the local frame ($\varvec{n}$, $\varvec{t}$, $\varvec{w}$). The mesostress with respect to the local frame can be computed from the local variables (Fig. 22 and Fig. 23) through the Love–Weber formula [14, 17, 37, 40]:

  $$\begin{aligned} \left\{ \begin{array}{l} {\widetilde{V}}\tilde{\sigma }_{11}=4N_{1}d_{1}\cos ^{2}\alpha _{1}+4T_{1}d_{1}\cos \alpha _{1}\sin \alpha _{1}+2N_{2}d_{2}\\ \quad \qquad +4N_{3}d_{3}\cos ^{2}\alpha _{2}+4T_{3}d_{3}\cos \alpha _{2}\sin \alpha _{2}+2N_{4}d_{4}\\ {\widetilde{V}}\tilde{\sigma }_{22}=4N_{1}d_{1}\sin ^{2}\alpha _{1}-4T_{1}d_{1}\cos \alpha _{1}\sin \alpha _{1}\\ {\widetilde{V}}\tilde{\sigma }_{33}=4N_{3}d_{3}\sin ^{2}\alpha _{2}-4T_{3}d_{3}\cos \alpha _{2}\sin \alpha _{2}\\ \end{array} \right. \end{aligned}$$

  (16)

  Notably, the value of off-diagonal components is always zero because of the anti-symmetry of the mesostructure. The initial opening angle is represented as $\alpha _{0}$ in a virgin specimen and $\alpha _{2}=\alpha _{1}=\alpha _{0}$. The initial void ratio $e_{0}(\varvec{n},\varvec{t},\varvec{w})$ of each mesostructure in the local frame $(\varvec{n},\varvec{t},\varvec{w})$ can be expressed as a function of $\alpha _{0}$ as blow [58]:

  $$\begin{aligned} e_{0} =-\dfrac{4}{\pi }\cos ^{3}\alpha _{0}-\dfrac{6}{\pi }\cos ^{2}\alpha _{0}+\dfrac{4}{\pi }\cos \alpha _{0}+\dfrac{6}{\pi }-1 \end{aligned}$$

  (17)

  It is worth noting that in the 3D-H model, the local void ratio $e_{0}$ is not correlated with the macroscopic void ratios of the specimen considered, but rather associated with the initial opening angle $\alpha _{0}$ in the mesostructures (as shown in Fig. 6). The $\alpha _{0}$ is limited within the range of 45°–90° due to the initial configuration of the meso-structure in 3D conditions, which ensures there is no overlapping between grains. As $\alpha _{0}$ increases from 45° to 74.9°, the $e_{0}$ grows from 0.4 to 1.09, reaching its maximum value. Beyond this point, further increase in $\alpha _{0}$ would result in a reduction in $e_{0}$. These key parameter $\alpha _{0}$ and $e_{0}$ can be obtained through experimental testing, determining the geometric fabric of the meso-structure.

  Fig. 6

  

  Evolution of initial void ratio as a function of initial opening angle

A detailed diagram of the hybrid FEM–SPH approach combined with the 3D-H model is flowcharted in Fig. 7, with three modules presenting the detailed computational procedures. In the iterative process, the penetration between SPH particles and the faces of finite elements will be checked in each iteration. The portion of FEM exchanges information with the portion of SPH only when contact occurs. Then, contact forces are generated and fed back into the governing equation to renew the parameters of SPH particles and FEM nodes.

Fig. 7

Computation flowchart for the multi-scale framework

3 Multi-scale modeling of trapdoor test

3.1 Establishment of the numerical model

The numerical example used to demonstrate the proposed multi-scale technique is a typical trapdoor problem in which soil particles settle with the movement of a trapdoor under gravity. The initial setup of the 3D numerical model for this research is presented in Fig. 8. The model consists of a soil region, a soil receptacle, and a base loading system including a trapdoor and two rigid bases. The soil receptacle has interior dimensions of 700 mm in height, 700 mm in length, and 300 mm in width. At the base of the receptacle, a moveable trapdoor with a width of B = 150 mm was embedded between two stationary portions. The soil, with a height H, simulated by SPH particles, rests on the rigid bases and trapdoor. The linear-elastic material model was assigned to the trapdoor. The specific characteristic parameters of the soil and the trapdoor are outlined in the following section. Except for the soil and trapdoors, the receptacle and base are simulated with a rigid wall, which does not need to be given material parameters. The trapdoor, meshing eight-node hexagonal solid elements, was discretized with 1218 elements (average size of 10 mm). At the beginning of the test, the SPH particles in the box were evenly dispersed throughout the entire soil region. The friction coefficient between the soil particles and the container is set to 0.05 to avoid problems with the boundary effect.

Fig. 8

Model setup: a Plan view; b side view (units: millimeters)

Since it takes some time for SPH particles to stabilize their motion and homogenize their distribution in space, the process of numerical simulation is divided into two stages: (I) equilibrium stage; (II) loading stage. Initially, in the equilibrium stage, the test begins with the generation of the SPH particles. After all the particles have completed stabilization under gravity, the average stress above the trapdoor is expected to equal the theoretical stress value. In the loading stage, the trapdoor is moved downward with a speed of 15 mm/s until it reaches the prescribed displacement of 20% of the trapdoor width. The numerical model is performed under normal gravity. The fill container and the stationary part of the base are assumed to be fixed and rigid in both stages. The total computation time for the coupled simulation of over 100,000 SPH particles using a CPU device (Intel(R) Xeon(R) Platinum 8268 CPU @ 2.90 GHz) is around 40 h. The coupling of FEM and SPH is implemented within ABAQUS and the 3D-H model model acts on SPH particles by coding a VUMAT subroutine to solve constitutive relations.

3.2 Calibration and verification

A series of trapdoor tests is conducted using the multi-scale approach to calibrate and verify the three material parameters of soil and one microgeometrical parameter. The former includes the normal stiffness ($k_{n}$), tangential stiffness ($k_{t}$), and inter-particle friction angle ($\varphi _{g}$), while the latter is the opening angle ($\alpha _{0}$) calculated from the initial void ratio. The active arching effect is characterized by the reduced vertical stress on the trapdoor, which is discussed later by normalizing the pressure on the trapdoor $P_{m}$ with the theoretical overburden stress $P_{0} = \gamma H$, where $\gamma$ is the unit weight of soil. The calibration is achieved by comparing the variation of the pressure on the trapdoor in the numerical model with the corresponding trapdoor experiment results [75]. The arching ratio $\rho = P_{m}/P_{0}$ versus normalized displacement $\Delta = \delta _{z}/B$ curves which is consistent with the experimental results are obtained by adjusting the above four parameters as well as particle spacing s Fig. 9a. The calibration tests are listed in Table 1 and the corresponding parameters are used as input in the verification, as shown in Table 2. Several researches [50, 69] indicated that the particle spacing s has an impact on the material response. This is because the s is an integral parameter to determine the smoothing length. As mentioned previously, by controlling the smoothing length so that the number of neighboring particles is between 50 and 60, the simulation results can achieve a level of accuracy. A reduction in s can improve the model’s accuracy and provide more data points for subsequent analysis, but a too-small s will result in a considerable amount of computational time. To investigate the influence of particle spacing on the average vertical stress over the trapdoor, a series of tests with a varied number of SPH particles are conducted, including PN11550, PN18850, PN27202, and PN46240. As shown in Fig. 9a, the gravity-induced pressure on the trapdoor in the tests of PN11550 and PN18850 deviates significantly from the theoretical overburden stress ($\gamma H$) in the equilibrium stage. As the number of particles increases to 27202, the pressure is almost identical to the theoretical value. When 0%$<\Delta<$10%, the load–displacement curves of all the tests exhibit a basically similar trend. Once the trapdoor displacement exceeds 10%, a significant difference in the variation of the pressure emerges among these tests, where the pressure in the PN46240 appears to still be growing but gradually levels off, reaching a plateau in other tests.

Table 1 List of calibration tests

Table 2 Parameters of soil particles selected in calibration

To verify that the parameters obtained by the calibration would also apply in other situations, tests with different normalized depths (H/B) are established. As depicted in Fig. 9b–d, the load–displacement relation in the tests with H/B = 2, 3, and 4 match closely with the experiment results, indicating that the ability of the multi-scale approach to simulate the load transfer capacity of soil arching is validated.

Fig. 9

The calibration and verification of the numerical model with different H/B

4 Numerical results and analysis

Once the coupled FEM–SPH approach is validated with the above experimental results and provides confidence in capturing the soil arching effect, extensions can be made for further investigation of the arching development in terms of load transfer, deformation behavior, and failure mechanism. Various parameters for the numerical model tests are listed in Table 3. Note that only one parameter was changed while keeping all the others the same in these tests.

Table 3 List of tests

4.1 Deformation pattern

Figure 10 presents the development of the vertical displacement field within a width of 400 mm above the trapdoor for different normalized depths (H/B = 1, 2, 3, and 4) with $\varphi _{g}$ = 20°. The magnitude of the vertical displacement ($U_{\rm{z}}$) is symbolized by the different colors of particles. The soil mass above the trapdoor can be divided into three zones depending on their vertical displacement. The active zone is the red-highlighted area that contains particles with the same displacement as the trapdoor. The area unaffected by the trapdoor movement is defined as the undisturbed zone (blue zone), and the rest of the soil is the so-called disturbed zone. Under shallow conditions (H/B = 1), the active zone extends to the ground surface with a trapezoid shape when the trapdoor movement is relatively small. As the trapdoor moves further down, the range of the disturbed zone enlarges, indicating that more particles are involved in the mobilization. Under deep conditions (H/B = 2, 3, and 4), the active zone is limited to a triangle area without extension to the ground surface. With increasing movement of the trapdoor, the active zone expends upward slightly. Particularly in the tests with H/B = 4 (Fig. 11), the undisturbed zone exists only at the bottom of the soil and the upper soil block experiences relatively minor overall settlement, which implies the disappearance of differential settlement.

Due to the differential settlement between trapdoor and rigid bases, the soil above these two parts also settles differently. Two vertical profiles I–I and II–II between which the differential settlement of soil is expected to be the maximum are selected to investigate the deformation behavior of soil. According to the above results of soil displacement, the differential settlement between the two profiles is observed to exist in the tests with H/B = 1, 2, and 3, while for H/B = 4, it may disappear. As shown in Fig. 12, the normalized settlement ($U_{z}/\delta _{z}$) curve along profile I–I appears to converge with that along with profile II–II at a certain height for tests with $\varphi _{g}$ = 5°, 10°, and 25°. For $\varphi _{g}$ = 30°, although the curve tends to converge, a small but noticeable differential settlement can still be observed, indicating that $\varphi _{g}$ may facilitates differential settlement of soils.

Fig. 10

Development of vertical displacement field for H/B = 1–3

Fig. 11

Development of vertical displacement field for H/B = 4

Fig. 12

Soil settlement along the profiles I–I and II–II with different $\varphi _{g}$ at H/B = 4

The soil arching effect is conducive to reducing the differential settlement induced by the vertical unloading [39], especially in deep-buried conditions. Based on the above results, it is found that when the soil height is sufficiently high, a plane of equal settlement (ESP) will develop at a critical height ($H_{c}$) above which differential settlement is no longer measurable [30]. Through plane strain trapdoor experiments in normal gravity, Terzaghi [52] found that the ESP is located at a height of 2–2.5 times the trapdoor width B. A minimum embankment height was determined as 1.0 times the clear spacing of piles by Carlsson [7] to avoid the differential settlement. Horgan and Sarsby [24] reported that the plane of equal settlement exists at about 1.545–1.92B. More recently, Al-Naddaf et al. [3] observed the ESP developing at the heights of 1.40B and 1.85B when the maximum trapdoor movement is set at 8 mm and 15 mm, respectively. Unlike previous studies, George and Dasaka [20] found that the differential settlement does not completely disappear, instead of existing at a negligible value. Note that the difference in the determination of $H_{c}$ may be related to material properties of soil, trapdoor movement, and even the accuracy of the measuring technique [30]. How to define the ‘measurable’ in the definition of ESP remains obscure, making it difficult to determine the $H_{c}$ with reasonable certainty. If the definition is too rigorous, $H_{c}$ will be too high or even disappear. Oppositely, an overly inclusive definition of the ESP will make itself meaningless. Hence, an angular distortion of 1/1000 is used to approximately define the non-measurable differential settlement, given by Eq. (18):

$$\begin{aligned} \dfrac{\Delta U}{S_{tb}} < \dfrac{1}{1000} \end{aligned}$$

(18)

where $S_{tb}$ and $\Delta U$ are the distance and the differential settlement between profiles I–I and II–II, respectively. Figure 13 presents the critical height in tests with H/B = 4 compared with experimented ranges suggested by [52] and [24]. The plane of equal settlement rises gradually with the trapdoor movement and is also consistent with the range of experimental observations. However, the $H_{c}$ at the end of the tests does not reach a stable plateau, implying that it may continue to increase with further trapdoor movement.

Fig. 13

Comparisons between observed critical height with experimental range

4.2 Load transfer behavior

Figure 14 shows that the evolution of arching ratio curves in the tests with different H/B and $\varphi _{g}$. As the trapdoor undergoes a slight downward movement, the arching ratio rapidly decreases down to a minimum arching ratio $\rho _\text {min}$, the minimum being smaller for a larger $\varphi _{g}$. Then, the $\rho$ gradually increases with further trapdoor movement, which is so-call the stick–slip behavior [45], and reaches a final value $\rho _\text {ult}$ at the end of trapdoor movement $\Delta _\text {ult}$. The ground reaction curve (GRC) can be divided into several states given the observed trend of the arching ratio as shown in Fig. 15a modified from [22] and [75]. The $\rho _\text {min}$ of GRC means that the load transfer ability of soil arching is exerted to the limit. With the increasing $\varphi _{g}$, there is a gradual drop in the $\rho _\text {min}$, indicating the enhancement of the strength of the soil arching. However, the effect of H/B on the strength of soil arching does not always seem to be positive, which may attribute to the transformation of deformation pattern. To describe the load recovery behavior (arching degradation) after the maximum arching, the load recovery index $\lambda _\text {rec}$ is introduced in this study, which is expressed as $\lambda _\text {rec} = (\rho _{f}-\rho _{m})/(\Delta _{f}-\Delta _{m})$ where the final state is considered to occur at the end of the test in this study. As shown in Fig. 15b, the $\lambda _\text {rec}$ is impacted by $\varphi _{g}$ to a lesser extent under deep conditions (H/B = 2, 3, and 4) but is more sensitive to the $\varphi _{g}$ under shallow condition ($H/B=1$). This may be related to the different deformation patterns of soils with different H/B.

Fig. 14

Variations of the arching ratio with different $\varphi _{g}$ at H/B = 1–4

Fig. 15

a The generalized ground reaction curve (modified from [22] and [75] and b variation of the load recovery index with different H/B and $\varphi _{g}$

The pressure distribution of soil particles on the bottom of the container and trapdoor is exhibited in Fig. 16 for three typical states: the maximum arching state, load recovery state, and final state. The high-stress regions appear in red color, while blue signifies low-stress regions. It is evident that the soil arch formed over the trapdoor transfers the pressure induced by gravity onto the bases on both sides, resulting in a significant low-stress region on the trapdoor. The striped stress bands displayed on the trapdoor are attributed to the voids between discrete SPH particles. When H/B = 1, stress concentrations are observed in the region of the bases closest to the trapdoor. As the H/B ratio increases, the concentration regions gradually shift toward both sides of the bases. When the ratio exceeds 3, the stress concentration region relocates to both sides of the foundation. Moreover, as the trapdoor descends, the stress concentration region gradually becomes less concentrated. The above evolution pattern of stress concentration areas is associated with the displacement of the center of gravity within the characteristic regions of soil particles. According to the conceptual representation proposed by [15], the soil can be categorized into the following regions: arching-unloading region, collapsing subregion, meta-stable subregion, and arching-loaded region based on the load distribution. More importantly, the delineation of these regions may vary with different soil heights. For instance, at H/B = 1, the collapsing region corresponds to the entire soil arch unloading area. With the slope of the collapsing region approaching vertical (Fig. 10), the arching ratio is consequently low, resulting in a lesser transfer of load. Most of the load transfer relies on shear forces in adjacent areas, occurring precisely at the junction between the trapdoor and the bases. As H/B increases to 2, the collapsing region becomes confined to a triangular shape. This indicates that particles above the collapsing region form a soil arch, transferring stress to both sides and not relying solely on the friction within the shear band.

Figure 17 illustrates the vertical stress evolution of SPH particles within the soil at various heights. Prior to the initiation of trapdoor motion, both particle arrangement and stress distribution are uniform. However, as the $\Delta$ reaches 10%, it becomes evident that the stress above the trapdoor is relieved, leading to a significant prominence in vertical stress on both sides. The region circumscribed by the red dashed line delineates the area where a majority of the stress exceeds the initial pressure at the soil bottom, namely the theoretical overburden stress at the corresponding soil height. When the H/B is relatively small, the area of this region decreases after the degradation of the soil arch. However, when the H/B is greater, there is no significant change in this area.

Fig. 16

Evolution of the contact pressure on the trapdoor for H1F20, H2F20, H3F20, and H4F20

Fig. 17

Development of vertical stress field for H/B = 1-4

Figure 18 presents the distribution of vertical stresses ($\sigma _{\rm{z}}$) along the two vertical profiles I–I (middle) and II–II (far right) with different $\varphi _{g}$ for H/B = 4. The theoretical values ($\gamma H$) are denoted by the black dotted lines. Due to the presence of the soil arching, the $\sigma _{\rm{z}}$ along profiles I–I gradually deviates from the theoretical overburden stresses with increasing depth and reaches a local minimum at a certain distance (about 100 mm) above the trapdoor. This trend is more pronounced in the tests with larger $\varphi _{g}$ in which the local minimum $\sigma _{\rm{z}}$ can even reach 0. Below the local minimum point, the $\sigma _{\rm{z}}$ is observed to increase at a faster rate than theoretical stress. Since the soil particles along profiles II–II are far from the yielding trapdoor, the distribution of $\sigma _{\rm{z}}$ is relatively less influenced by the soil arching, where the $\sigma _{\rm{z}}$ gradually deviates from the theoretical curve to the right with increasing depth. However, the distribution of $\sigma _{\rm{z}}$ shows no obvious changes with the movement of trapdoor due to the lower degradation of soil arching.

Fig. 18

Distribution of vertical stresses with different $\varphi _{g}$ for H/B = 4

4.3 Development of the failure surfaces

Development of the equivalent plastic strain ($\overline{\varepsilon }^{pl}$) for different H/B is presented in Figs. 19 and 20. Six particles are selected to represent the region with different $\overline{\varepsilon }^{pl}$, including three high-plastic regions (B, D, and E) and three low-plastic regions (A, C, and F). The appearance of plastic zones indicates the failure of the soil material and the formation of failure surfaces. The failure surfaces proceed from both edges of the trapdoor at maximum arching state, then extending upward into the soil material with the movement of the trapdoor. It is noteworthy that the failure pattern under different buried conditions can be categorized into three types, corresponding to the three different deformation patterns. When there is not enough soil height ($H/B = 1$) to complete the formation of the soil arching, only one type of failure surface appears to extend vertically, separating a trapezoidal zone occupied by particles in the elasticity stage. Under higher conditions (H/B = 2 and 3), two types of failure surfaces can be observed: internal failure surfaces that converge over the trapdoor and external failure surfaces that incline outward, which are located at the boundaries of the active and disturbed zones. Particularly in the tests with $H/B = 4$, the failure zones do not extend gradually upward due to the high stress level, instead being localized as found earlier by [44]. This results in an entire non-plastic zone in the upper part of the soil, giving rise to the equal settlement plane.

Fig. 19

Development of equivalent plastic strain for H/B = 1–3

Fig. 20

Development of equivalent plastic strain for H/B = 4. Selected particles for meso-scale analysis is marked as white cross in tests with H/B = 4 and $\varphi _{g}$ = 20°

5 Multi-scale analysis

A meso-scale analysis can be conducted using the 3D-H model to explore the underlying critical microscopic behavior hidden behind the macroscopic scale, which is crucial to understand the evolution mechanism of the soil arching from a micro perspective. As shown in Fig. 20, six Gauss points of SPH particles are selected to represent the regions where different equivalent plastic strains occurred in the test with H/B = 4. The original mesostresses are integrated over the two Euler angles $\theta$ and $\psi$ as follows:

$$\begin{aligned} \begin{aligned} \tilde{\sigma }^{I}_{n}(\varphi ) = \iint \omega \tilde{\sigma }_{n} \rm{d}\theta \rm{d}\psi \\ \tilde{\sigma }^{I}_{t}(\varphi ) = \iint \omega \tilde{\sigma }_{t} \rm{d}\theta \rm{d}\psi \end{aligned} \end{aligned}$$

(19)

where $\tilde{\sigma }_{n}$ and $\tilde{\sigma }_{t}$ are normal and tangential mecrostress, respectively. On the basis of these formulas, the 3D mesostresses can be visualized on 3D plots with the help of $\tilde{\sigma }^{I}_{n}(\varphi )$ and $\tilde{\sigma }^{I}_{t}(\varphi )$ as a function of the last Euler angle $\varphi$. Figure 21 present the distribution of the above two microscopic variables at four critical states during the soil arching process, including initial state, maximum soil arch state, $\Delta$ = 10%, and final state. Prior to the trapdoor movement, all mesoscopic variables of each SPH particle were isotropic, with magnitudes determined by particle location. At maximum arching state, the distribution of the $\tilde{\sigma }^{I}_{n}(\varphi )$ and $\tilde{\sigma }^{I}_{t}(\varphi )$ at each point begins to exhibit varying degrees of anisotropy. It can be seen that the trapdoor movement aggravates the anisotropic distribution of $\tilde{\sigma }^{I}_{n}(\varphi )$ of all the SPH particles when the $\Delta$ reaches 10%. Obviously, the anisotropy of $\tilde{\sigma }^{I}_{t}(\varphi )$ distribution of the particles B, D, and E in the plastic region are exceptionally more severe than that of particles A, C, and F in the elastic region. At the final state, significant increase in $\tilde{\sigma }^{I}_{n}(\varphi )$ and $\tilde{\sigma }^{I}_{t}(\varphi )$ of particles B, D, and E can be seen, but there is little change in these two variables of particle A and C. Note that the stress evolution in the 3D-H model is not controlled by any parameter and is a natural process. This multi-scale approach is employed in the trapdoor test in an effort to simulate the macroscopic stress–strain response through the evolution of microscopic and mesoscopic data, by which we are informed that the failure mechanism of soil arching is accompanied by an anisotropic distribution of stress.

Fig. 21

Meso-scale variable distributions of 3D-H model inside SPH particles A, B, C, D, E, and F at four states

6 Conclusions

This paper developed a novel multi-scale approach incorporating FEM, SPH, and a multi-scale constitutive model to simulate the soil arching behavior during the unloading process of excavation. The proposed approach is proven to capture the load transfer behavior by comparing with experimental observations. Further, extensive parametric analysis is presented to reveal the deformation behavior and failure mechanism. Finally, several representative Gauss points of SPH particles are selected to provide a better understanding of the micro-mechanism embedding behind the macroscopic responses. The main conclusions can be summarized:

• 1. The variation of the arching ratio predicted by the proposed multi-scale approach is in good agreement with the experiment observations. The mesh dependency of this approach issue is found to be negligible.

• 2. Three possible deformation patterns with different H/B are revealed: the trapezoid deformation pattern, the triangle deformation pattern, and the equal settlement pattern. The soil height needs to be large enough to facilitate the formation of soil arching. Under deep condition (H/B = 4), a plane of equal settlement appears, representing the negligible differential settlement. The evolution of the $H_{c}$ is characterized by an increase with the downward movement of trapdoor.

• 3. The enhancement of the arch’s ability to transfer stresses from the regions above the trapdoor to both sides results from the increases in $\varphi _{g}$ and H/B. The load recovery resulting from arching degradation is more pronounced at lower soil heights and higher $\varphi _{g}$. The stress concentration regions on the foundation move away from the trapdoor with increasing H/B. The vertical stress along the centerline of the soil deviates from theoretical overburden stresses and the greater the $\varphi _{g}$, the more obvious the deviation.

• 4. The observed failure mechanisms can also be classified into three different modes, which correspond to the three deformation behaviors. With H/B = 1, 2, and 3 the failure surfaces gradually penetrate the ground surface with trapdoor movement, while under deep conditions (H/B = 4) the surfaces remain confined to the lower part of the soil. A complete elastic deformation zone develops in the upper region of the soil due to the limited extension of failure surfaces, facilitating the formation of equal settlement planes.

• 5. Six Gauss points of SPH particles were selected for mesoscopic analysis, which represent the regions with different plastic strains. The evolution of the mesoscopic variables under the trapdoor problem is characterized by the anisotropy of stress distribution during the arching process. The anisotropy of stress distribution in the region with high plastic strain is more severe than that in the elastic region.

Appendices

Appendix 1: The derivation of incremental constitutive relation of the Hexagon A

Through simplifying the 3D mesostructure, two independent and similar 2D hexagonal patterns, Hexagon A (Fig. 22) and Hexagon B (Fig. 23) are analyzed in this step. Due to the symmetry of the geometrical configuration and external forces applied to the mesostructure, two grains and two contacts, including contact 1 between grains 1 and 2 and contact 2 between grains 2 and 3, need to be analyzed in Hexagon A. The kinematic relations yield:

Fig. 22

Mechanical description of Hexagon A

Fig. 23

Mechanical description of Hexagon B

$$\begin{aligned} \left\{ \begin{array}{l} \delta u_{n}^{1}=\delta d_{1} \\ \delta u_{t}^{1}=d_{1}\delta \alpha _{1} \\ \delta u_{n}^{2}=\delta d_{2} \end{array} \right. \end{aligned}$$

(20)

where $u_{n}^{1}$ and $u_{t}^{1}$ are the normal and tangential components of the relative incremental displacement of contact 1, respectively. $u_{n}^{2}$ is the normal component of the relative incremental displacement of contact 2. The geometrical description for Hexagon A can be written:

$$\begin{aligned} \left\{ \begin{array}{l} l_{1}=d_{2}+2d_{1}\cos \alpha _{1}\\ l_{2}=2d_{1}\sin \alpha _{1} \end{array} \right. \end{aligned}$$

(21)

The force balance of grain 1 along direction n and of grain 2 along directions $\varvec{w}$ and $\varvec{n}$, together with the moment balance of grain 2, reads:

$$\begin{aligned} \left\{ \begin{array}{l} F_{1}^{a}=2(N_{1}\cos \alpha _{1}+T_{1}\sin \alpha _{1}) \qquad \qquad \qquad \qquad \qquad (22a)\\ F_{2}=N_{1}\sin \alpha _{1}-T_{1}\cos \alpha _{1} \qquad \qquad \qquad \qquad \qquad (22b)\\ N_{2}=N_{1}\cos \alpha _{1}+T_{1}\sin \alpha _{1}+G_{2} \qquad \qquad \qquad \qquad \qquad (22c) \\ G_{2}=T_{2} \qquad \qquad \qquad \qquad \qquad (22d) \end{array}\right. \end{aligned}$$

where $N_{i}$ and $T_{i}$ denote the normal and tangential contact forces of contact i, respectively. In terms of the elastic-perfect plastic interparticle contact law (B), we get two simplified formulas:

$$\begin{aligned} \left\{ \begin{array}{l} \delta N_{1}=-k_{n}\delta d_{1} \\ \delta N_{2}=-k_{n}\delta d_{2} \\ \delta T_{1}=B_{1}\delta \alpha _{2}-A_{1}\delta d_{2}+C_1 \\ \end{array} \right. \end{aligned}$$

(23)

where $C_{1}$ is equal to zero except during a transition from an elastic to a plastic regime, which can be ignored for very small strain increments as considered throughout this paper.

In order to obtain the incremental evolution of the external forces, a function is required to be established in order to express the mesostrain in term of $\delta d_{1}$, $\delta d_{2}$, and $\delta a_{1}$. Combining the Eqs. (21), (22c), and (23), the following algebraic system expressing the incremental changes in $\delta d_{1}$, $\delta d_{2}$, and $\delta a_{1}$ can be reached:

$$\begin{aligned} \begin{bmatrix}2\cos \alpha _{1}&{}1&{}-2d_{1}\sin \alpha _{1}\\ 2\sin \alpha _{1} &{} 0 &{} -2d_{1}\cos \alpha _{1}\\ \cos \alpha _{1}+\frac{A_{1}}{k_{n}}(\sin \alpha _{1}+1) &{} -1 &{} \frac{F_{2}-B_{1}(\sin \alpha _{1}+1)}{k_{n}} \end{bmatrix} \begin{bmatrix} \delta d_{1}\\ \delta d_{2}\\ \delta \alpha _{1} \end{bmatrix}=\begin{bmatrix} \delta l_{1}\\ \delta l_{2}\\ 0 \end{bmatrix} \end{aligned}$$

(24)

where $A_{1}$ and $B_{1}$ are given in B.

Differentiating (Equation 22a), (Equation 22b) and combining with (Eq. 23) gives:

$$\begin{aligned} \left\{ \begin{array}{l} \delta F_{1}^{a}=-k_{n}\cos \alpha _{1}\delta u_{n}^{1}+k_{t}\sin \alpha _{1}\delta u_{t}^{1}-F_{2}\delta \alpha _{1}\\ \delta F_{2}=-k_{t}\cos \alpha _{1}\delta u_{t}^{1}-k_{n}\sin \alpha _{1}\delta u_{n}^{1}+F_{1}^{a}\delta \alpha _{1} \end{array} \right. \end{aligned}$$

(25)

Substituting Eqs. (20) and (24) in (Eq. 25), the incremental constitutive relation for Hexagon A can be expressed as follows:

$$\begin{aligned} \dfrac{1}{\left| D\right| ^{a}}\begin{bmatrix} K_{11}^{a} &{} K_{12}^{a} \\ K_{21}^{a} &{} K_{22}^{a} \end{bmatrix} \begin{bmatrix} \delta l_{1}\\ \delta l_{2} \end{bmatrix}=\begin{bmatrix} \delta F_{1}^{a}\\ \delta F_{2}^{a} \end{bmatrix} \end{aligned}$$

(26)

where:

$$\begin{aligned} \left\{ \begin{array}{l} K_{11}^{a}=2(F_{2}\sin \alpha _{1}-k_{n} d_{1} \cos ^{2}\alpha _{1}-k_{t} d_{1} \sin ^{2}\alpha _{1}) \\ K_{12}^{a}=(k_{t} d_{1} \sin \alpha _{1}-F_{2})(\dfrac{A_{1}}{k_{n}} \sin \alpha _{1}+\dfrac{A_{1}}{k_{n}}+3\cos \alpha _{1}) \\ \qquad \quad -\cos \alpha _{1}(B_{1}\sin \alpha _{1}+B_{1}-F_{2}-2k_{n} d_{1} \sin \alpha _{1})\\ K_{21}^{a}=2(k_{t}-k_{n})d_{1} \sin \alpha _{1}\cos \alpha _{1}-2F_{1}^{a}\sin \alpha _{1})\\ K_{22}^{a}=(F_{1}-k_{t} d_{1} \cos \alpha _{1})(\dfrac{A_{1}}{k_{n}} \sin \alpha _{1}+\dfrac{A_{1}}{k_{n}}+3\cos \alpha _{1}) \\ \qquad \quad -\sin \alpha _{1}(B_{1}\sin \alpha _{1}+B_{1}-F_{2}-2k_{n} d_{1} \sin \alpha _{1})\\ \left| D\right| ^{a}=\dfrac{A_{2}}{k_{n}}(B_{1}\sin \alpha _{1}+A_{1}d_{1}\cos \alpha _{1})(\sin \alpha _{1}+1)\\ \qquad \quad \quad -\dfrac{A_{2}}{k_{n}}(F_{2}\sin \alpha _{1}+k_{n} d_{1} \cos ^{2}\alpha _{1}+2k_{n} d_{1} ) \end{array} \right. \end{aligned}$$

(27)

Appendix 2: Contact law

This elastic-perfect plastic model includes a Mohr-Coulomb criterion and can be expressed under the following incremental formalism by employing the notations depicted in Figs. 22 and 23:

$$\begin{aligned} \left\{ \begin{array}{l} \delta {N_i} = {k_n}\delta u_n^i\\ \delta {T_i} = \min \left\{ {\left\| {{T_i} + {k_t}\delta u_t^i} \right\| ,\tan {\varphi _g}\left( {{N_i} + {k_n}\delta u_n^i} \right) } \right\} \times \frac{{{T_i} + {k_t}\delta u_t^i}}{{\left\| {{T_i} + {k_t}\delta u_t^i} \right\| }} - {T_i} \end{array} \right. \end{aligned}$$

(28)

where: $i=1,2,3,4$ denotes the identifier of contact number.

According to Eqs. (20), (28) can be rewritten as follows:

$$\begin{aligned} \left\{ \begin{array}{lr} \delta {N_i} = - {k_n}\delta {d_i}&{}\\ \delta {T_i} = {k_t}{d_i}\delta {\alpha _j} &{}\text {elastic regime}\\ \delta {T_i}= \tan \varphi _g \left( {N_i} - {k_n}\delta {d_i}\right) \xi _i- {T_i} &{} \text {plastic regime} \end{array}\right. \end{aligned}$$

(29)

where: $\xi _i$ is the sign of ${T_i} + {k_t}{d_i}\delta {\alpha _j}$; $j=1$ when $i=1,2$; $j=2$ when $i=3,4$; plastic regime is reached when $\parallel {k_t}{d_i}\delta {\alpha _j}+T_i\parallel \geqslant \tan \varphi _g \left( {N_i} - {k_n}\delta {d_i}\right)$, otherwise it is in elastic regime.

To facilitate the derivation, $I_i^p$ and $I_i^e$ are introduced as indicator functions of the contact state, expressed as below:

$$\begin{aligned} I_i^p = \left\{ {\begin{array}{*{20}{c}} {1 \qquad \text {in plastic regime}}\\ {0 \qquad \text {in elastic regime}} \end{array}} \right. ; \quad I_i^e = 1 - I_i^p \end{aligned}$$

(30)

Thus, the constitutive relations can be expressed as follows:

$$\begin{aligned} \left\{ \begin{array}{l} \delta {N_i} = - {k_n}\delta {d_i}\\ \delta {T_i} = {B_i}\delta {\alpha _j} - {A_i}\delta {d_i} + C_i \end{array}\right. \end{aligned}$$

(31)

where:$\left\{ \begin{array}{l} {A_i} = I_i^p {k_n}{\xi _i}\tan {\varphi _g}\\ {B_i} = I_i^e {k_t}{d_i}\\ C_i=I_i^p(\xi _i \tan \varphi _g N_i - T_i) \end{array}\right.$