# Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

**V**

_{1}(Ω) is defined for a block to illustrate displacement of the block using the Virtual Element Method (VEM). Based on VEM theory, the total potential energy of the block system in DDA is formulated and minimized to obtain the global equilibrium equations. At the end of a time step, the vertex coordinates are updated by adding their incremental displacement to their previous coordinates. In the new method, no explicit expression for the displacement

**u**is required, and all numerical integrations can be easily computed. Four numerical examples originally designed by Shi are analyzed, verifying the effectiveness and precision of the proposed method.

## 1. Introduction

**d**in the traditional DDA were defined for a block by six independent variables, which consists of two incremental rigid translations, one incremental rigid rotation angle, and three incremental constant strain components. For a point

**x**in the block, the incremental displacement

**u**is calculated as the product of

**d**and the transformation matrix

**T**; then, the global equilibrium equation is derived from this displacement function. The displacement function uses the first-order approximation of sin r

_{0}= r

_{0}and cos r

_{0}= 1 for a small rotation angle r

_{0}, which induces accumulated errors over the time steps and sometimes causes issues. The most noticeable issue is the unreasonable volume expansion when a large rotation occurs. To restrain this issue, some scholars proposed various emendations [31,32,33], in which the post-adjustment strategy was most popular in the existing DDA codes [34,35]. This method, albeit simple and effective, resulted in a displacement different from the displacement evaluated in the equilibrium equation. Even with a tiny difference, the contact state might be entirely changed for a contact pair. Then, to simulate the continuous–discontinuous deformation, numerical methods for the continuum and non-continuum models were increasingly coupled. DDA was coupled with finite element method (FEM) by some scholars [9,22] for simulation of a rock failure process. In classical FEM, displacements at the element nodes are chosen as the degrees of freedom. This inconsistency in the degrees of freedom between DDA and FEM causes a barrier in developing a compact and efficient code for the coupling method.

**u**at the vertices of a block, an individual virtual element space

**V**

_{1}(Ω) is adopted to describe the displacements of points in the block, and the projector Π

_{P}

**u**from

**V**

_{1}(Ω) on the linear displacement space

**P**

_{1}(Ω) is deduced. Next, the total potential energy is investigated for the block system. In the potential energy, the bilinear forms of

**u**are expressed as the summation of the exact solution of Π

_{P}

**u**and an approximation of

**u**-Π

_{P}

**u**. The potential energy induced by the contact restraints are derived using the new degrees of freedom. Then, for the block system, the global equilibrium equation is derived based on the principle of minimum potential energy. Finally, the open–close iteration strategy is employed to resolve the global equilibrium equation as the original DDA. The proposed method avoids the issues attributable to first-order approximation for a small rotation angle r

_{0}in the original DDA and has a higher computational efficiency than vertex displacement-based DDA using displacement functions in PFEM. The validity and effectiveness of the proposed method are verified by several numerical examples.

## 2. Basic Principles of DDA

_{b}discrete blocks with their individual domains and boundaries. The domain Ω

^{I}of block I is bounded by ∂Ω

^{I}, which is usually composed of the Dirichlet boundary ${\Gamma}_{\mathrm{u}}^{\mathrm{I}}$ and the traction boundary ${\Gamma}_{\mathrm{t}}^{\mathrm{I}}$. The displacement on ${\Gamma}_{\mathrm{u}}^{\mathrm{I}}$ is prescribed as

**û**

^{I}, and the surface traction on ${\Gamma}_{\mathrm{t}}^{\mathrm{I}}$ is denoted by

**t**

^{I}. Here, the superscript I is the block index. The deformations and large displacements of a block in DDA are accumulated by the incremental displacements and deformations over time steps. The displacements of blocks are independent from each other, and contact constraints are imposed on the interactions between blocks.

**u**

^{1},

**u**

^{2}, …,

**u**

^{nb}) to be the unique minimizer of the total potential energy of the block system:

**V**is the incremental displacement space of the block system:

**J**’(v) represents the energy due to the contact constraints. A key feature of DDA is that rigorous contact constraints are used to manage the interactions between blocks. For the contact pair marked in Figure 1,

**J**’(v) induced by the contact constraints is determined by

**u**

^{I}∈

**V**and

^{I}**u**

^{L}∈

**V**

^{L}.**V**and v∈

**V**by

**u**,

**v**) for

**u**∈

**V**and

**v**∈

**V**represents the energy due to elastic deformation and the inertial force:

**ε**is the incremental constant strain decided by

**u**:

**σ**is the incremental Cauchy stress related to

**ε**by the constitutive equation:

**V**the velocity of block I at the origin of a time step and by ∆, the time interval of the time step, the acceleration is computed as

_{0}**v**) is defined by

**b**denotes the constant body force and

**σ**

_{0}is the constant stress accumulated over the previous time steps.

**d**

^{I}is the degrees of freedom concerning block I.

**F**

^{I}is the generalized load vector with the same dimension of

**d**

^{I}. Denoting the dimension of

**d**

^{I}with dim(

**d**

^{I}),

**K**

_{II}is a dim(

**d**

^{I}) × dim(

**d**

^{I}) matrix and

**K**

_{I}

_{L(I≠}

_{L)}is a dim(

**d**

^{I}) × dim(

**d**

^{L}) matrix denoting the contact restraints on blocks I and L. If no contact pair is provided by blocks I and L in the current time step,

**K**

_{I}

_{L}is zero.

## 3. Demerits Caused by the Original Degrees of Freedom and Previous Attempts to Construct a Vertex Displacement-Based DDA

#### 3.1. Demerits Caused by the Original Degrees of Freedom in DDA

**d**within one time step:

_{0}and v

_{0}represent the increments of the rigid horizontal and vertical translations, respectively; r

_{0}denotes the increment of the rigid rotation angle around the central point (x

_{0}, y

_{0}) of block Ω; and (ε

_{x}, ε

_{y}, γ

_{xy}) denotes the increments of the constant strains. Under small deformation assumption, the increments of the displacement

**u**= (u, v)

^{T}at a point

**x**= (x, y) in block Ω are computed as

**T**is the translation matrix as follows:

_{0}= 1 and sin r

_{0}= r

_{0}are adopted for the small rotation angle r

_{0}in Equation (11). The approximation errors accumulated over time steps can lead to false volume expansion when a large rotation occurs. Various modifications were suggested to remedy this defect, in which the post-adjustment strategy is most popular in the existing DDA codes. After vector

**d**is obtained, the post-adjustment strategy employs Equation (13) to calculate the incremental displacement

**u**. Although simple and effective in most cases, the resulting displacement must be different from the displacement estimated in the equilibrium equation. Sometimes, for a contact pair, the state adopted in the equilibrium equation may be not coincident with the geometry relationship of the resulted configurations. An example is provided in Section 5.1 to demonstrate this issue.

#### 3.2. Previous Attempts to Construct Vertex Displacement-Based DDA

**u**

_{i}= (u

_{i}, v

_{i})

^{T}at the vertex

**x**

_{i}constitute the new degrees of freedom:

**u**of the displacement at a point

**x**in a block Ω can be computed in a similar method to FEM, that is

_{i}is the shape function in terms of the vertex

**x**

_{i}. Because the block in DDA might have more than four vertices, the shape functions developed in PFEM were introduced to construct N

_{i}[36]. In the recent study [36], Wachspress interpolation was selected as the shape function for developing vertex displacement-based DDA.

_{i}to develop an effective PFEM [38]. As shown in Figure 2, a block Ω has n vertices located at

**x**

_{i}and its boundary is composed of n straight edges. The red edges with vertex

**x**

_{i}are denoted by Γ

_{i}. N

_{i}should be satisfied.

^{∞}being within block Ω while C

^{0}is on the boundary, N

_{i}must be piece-wise linear along Γ

_{i}but vanishes at the other edges. This property ensures that the boundary of a straight line is still a straight line after deformation, which benefits contact detection and contact condition imposition.

_{i}is defined for a point

**x**within a polygon as follows.

**x**

_{1},

**x**

_{2}, …,

**x**

_{n}are the vertices of the polygon arranged counterclockwise and A(

**x**

_{i},

**x**

_{j},

**x**

_{k}) represents the area of a triangle connecting three points, such as in Figure 3. By observation, the basic principles required for N

_{i}are completely satisfied by Definition (19).

_{i}in PFEM are the rational functions, which make the derivation and integration process more cumbersome when computing the block matrices. Though some enhancements [38] were made to simplify the formulas of N

_{i}, the rational function is indispensable to definite N

_{i}, which creates difficulties in the integration over a polygon. Up until now, the most popular way to solve this issue was partitioning the polygon into subdomains with certain shapes. Then, the Hammer or Gauss integration scheme was employed to compute the integration on each sub-triangle or sub-rectangle. The summation of the integrations on subdomains was considered the integration over the original polygon. While the integration error was considerable, some remedies were proposed to address the integration error for PFEM [42]. In conclusion, the additional work in the derivation and programming is very expensive when using the interpolation function N

_{i}in PFEM. When Equation (15) is adopted to govern the block displacement, some advantages of the original DDA can compromised.

## 4. Vertex Displacement-Based DDA Using VEM

#### 4.1. Virtual Element Spaces

**V**

_{k}(Ω) can be defined for a block as follows:

_{k}(Ω) is the space of polynomials of degree less than or equal to k in Ω. Obviously, [P

_{k}(Ω)]

^{2}⊆

**V**

_{k}(Ω). For simplicity, [P

_{k}(Ω)]

^{2}is denoted as

**P**

_{k}(Ω).

**V**

_{1}(Ω) only involve the values of

**v**at the vertices of Ω, in accordance with the new degrees of freedom defined in Equation (14). (ii) The incremental displacements are expected to be piece-wise linear at the block boundary, which is satisfied by

**V**

_{1}(Ω). (iii) In the view that the original DDA adopted a complete first-order displacement function, the degree of accuracy is not threatened by using

**V**

_{1}(Ω). Therefore, an individual virtual element space

**V**

_{1}(Ω) is defined for a block, and the associated degrees of freedom are the new degrees of freedom defined in Equation (14). The dimension of the space

**V**

_{1}(Ω) is 2n, where n denotes the number of vertices for the block.

_{b}blocks in the block system, the total virtual element space can be given as

**V**in Equation (1) is replaced by

**V**

_{1}.

#### 4.2. The Projection Operator Π_{P}**u**: **V**_{1}(Ω)→**P**_{1}(Ω)

**u**,

**v**) from

**V**

_{1}(Ω) ×

**V**

_{1}(Ω) to R defined in Equation (4) is inevitable, which can be accomplished in the framework of VEM theory. Referring to [47], two projectors Π

^{∇}

**u**:

**V**

_{1}(Ω)→

**P**

_{1}(Ω) and Π

^{0}

**u**:

**V**

_{1}(Ω)→

**P**

_{1}(Ω) are respectively defined using the orthogonality conditions:

^{0}

**u**cannot be obtained from (23) because the zero- and first-order moments of

**u**in Ω are unknown. To solve this issue, an additional property stating that the moments of order 0 and 1 of

**u**and Π

^{∇}

**u**coincide was added by B. Ahmad [44] to the space

**V**

_{1}(Ω), resulting in the projection operator of Π

^{0}

**u**being the same as the projection operator of Π

^{∇}

**u**. For simplicity, Π

_{P}

**u**is adopted to express Π

^{∇}

**u**and Π

^{0}

**u**uniformly. In this study, Π

_{P}

**u**is deduced using an approach different from the standard procedure [46].

**w**, the average of the values it assumes at the vertices of Ω is denoted by

**w**

^{∗}:

**w**over Ω is denoted by ⟨

**w**⟩:

**P**

_{1}(Ω) is the space of linear displacements over Ω, which can be split into two displacement spaces due to rigid motions and constant strain. We denote the space regarding rigid motions by

**R**(Ω) and the space concerning constant strain by

**C**(Ω):

**P**

_{1}(Ω) is the summation of

**R**(Ω) and

**C**(Ω). Both

**R**(Ω) and

**C**(Ω) are subspaces of

**V**

_{1}(Ω). Two projection maps, Π

_{R}:

**V**

_{1}(Ω)→

**R**(Ω) and Π

_{C}:

**V**

_{1}(Ω)→

**C**(Ω), are defined to extract the rigid motions and constant strain of

**u**

**∊V**

_{1}(Ω). Considering that elements of

**C**(Ω) should contain no rigid motion and that elements of

**R**(Ω) should contain no constant strain, the following orthogonality conditions are imposed on the two maps:

_{R}and Π

_{C}satisfying the above properties can be given by

**p**in Equation (22) contains only constant elements

**.**Obviously, the projection map Π

_{P}

**u**given by Equation (32) satisfies Equation (22).

**u**= (u, v) is linear on each edge of Ω. Denoting the length of edge i connecting vertex i−1 and vertex i as l

_{i}, as Figure 4 shows, and the outward unit normal vector on edge i as

**n**

_{i}= (n

_{ix}, n

_{iy})

^{T}, 〈∇

**u**〉 can be computed as

_{P}

**u**is given as

**ε**(Π

_{P}

**u**) can be computed as follows:

**ε**= (ε

_{x}, ε

_{y}, γ

_{xy}) can be given as

**P**

_{c}is actually composed by the latter three rows in

**H**.

**u**∈

**V**

_{1}(Ω) can be decomposed into Π

_{P}

**u**and the residual item

**u**−Π

_{P}

**u**. Then, a bilinear form from

**V**

_{1}(Ω) ×

**V**

_{1}(Ω) to R can be analyzed without the explicit expression of

**u**, which will be demonstrated in the next section.

#### 4.3. Computation of α(**u**, **v**) and f(**v**)

**u**,

**v**) in Equation (4) and f(

**v**) in Equation (8) are rewritten as

**V**

_{0}is considered an element in

**V**

_{1}(Ω) to estimate

**V**

_{0}within the block because

**V**

_{0}at the block vertices are directly related to

**u**. Consequently, α

^{V}is a bilinear form.

**V**

_{0}can also be evaluated using the rigid body motion and the constant strain associated with Π

_{P}

**u**. That will induce

**V**

_{0}to become an element in

**P**

**(Ω) and α**

_{1}^{V}to become a linear form.

**u**,

**v**) regarding two elements

**u**,

**v**∈

**V**

_{1}(Ω) can be rewritten as

**u**−Π

_{P}

**u**to the bilinear form. The stabilization term cannot produce an exact solution, so the second term in VEM is usually denoted as s(

**u**−Π

_{P}

**u**,

**v**−Π

_{P}

**v**) to indicate that it is an approximation. The standard scheme of s(

**u**−Π

_{P}

**u**,

**v**−Π

_{P}

**v**) was provided in classical VEM literature [46], but it is quite free in the construction of s(

**u**−Π

_{P}

**u**,

**v**−Π

_{P}

**v**) in practice [48,49]. The stabilization term proposed by Veiga [46] is adopted in this study:

^{E}is rewritten as follows:

**ε**(Π

_{P}

**u**) into Equation (47), the consistency term is

**D**is the elastic matrix

**I**is a 2n × 2n unit matrix and

**P**

_{u}is a 2n × 2n constant matrix:

^{E}is the approximation of α

^{E}, the positive parameters η

^{E}can be decided upon by requiring that s

^{E}and α

^{E}are comparable. The consistency term α

^{E}(Π

_{P}

**u**, Π

_{P}

**u**) can be estimated using s

^{E}as follows:

^{E}(Π

_{P}

**u**, Π

_{P}

**u**) is given in Equation (48). Equating the traces of the two matrices, η

^{E}is determined:

^{E}, the bilinear forms α

^{M}and α

^{V}are formulated as

**P**

_{m}is a 2n × 2n constant matrix as

^{σ}, f

^{b}, and f

^{t}in Equation (44) are investigated. Obviously, f

^{t}can be computed directly. Because

**u**is linear on each edge of Ω, a definite shape function matrix

**N**(

**x**) can be easily determined for a point

**x**on ∂Ω to compute

**u**(

**x**) using Equation (6). Thus, we have

_{P}

**u**can be valued in the first-order VEM, so

**σ**

_{0}is a constant stress in this study. Using the integration by parts, the linear form f

^{σ}can be processed as

**b**(x, y) as an element in

**P**

**(Ω), we have**

_{1}**L**

^{b}is a 2 × 2 n matrix as follows:

#### 4.4. Computation of **J**’(v) Due to the Contact Constraints

_{B}and an entrance line jk of Ω

_{A}is plotted in Figure 5a. Point o is the closest point from line jk to point i. At the start of a time step, the coordinates of points i, j, k, and o are (x

_{i}, y

_{i}), (x

_{j}, y

_{j}), (x

_{k}, y

_{k}), and (x

_{o}, y

_{o}), respectively. Within this time step, their displacements are (u

_{i}, v

_{i}), (u

_{j}, v

_{j}), (u

_{k}, v

_{k}), and (u

_{o}, v

_{o}), respectively. Then, the normal penetration d

**and tangential relative displacement d**

_{n}**in Figure 5b can be measured using their new coordinates when the time step terminates.**

_{τ}**has a negative value, the contact pair must be “open” and no contact constraint applies, which results in no spring being added into the block system. Otherwise, a normal stiffness spring is added to oppose the penetration, which results in a deformation energy J**

_{n}**. Suppose that the two blocks contact each other along a coincident edge. The Coulomb friction law is adopted to determine whether the tangential relative motion is permitted. If the tangential relative motion is inadmissible, the contact pair is “lock” and a stiffness spring along the tangential direction is introduced to resist the tangential relative motion, leading to a deformation energy J**

^{n}**. Otherwise, the contact pair is “slide” and a pair of friction forces is adopted along the tangential direction, inducing a potential energy J**

^{τ}**. The allowable upper bound of the Coulomb friction law is considered the value of the friction forces. For the block system,**

^{f}**J**’(v) in Equation (1) can be obtained by collecting J

**, J**

^{n}**and J**

^{τ},**of all contact pairs.**

^{f}**has an approximation (61) by neglecting the second-order infinitesimal value:**

_{n}**, the normal spring resisting the normal penetration induces the deformation energy as follows:**

_{n}**causes three second-order vectors to be added to the global load vector and nine second-order square matrices to be added into the global stiffness matrix.**

^{n}**, the tangential relative displacement d**

_{n}**can be formulated as**

_{τ}**, the tangential spring results in the deformation energy:**

_{τ}**also leads to three second-order vectors being added to the global load vector and nine second-order square matrices being added into the global stiffness matrix.**

^{τ}_{up}. For d

**in Figure 5b, the friction force f**

_{τ}_{up}

**τ**acts on Ω

_{B}at point i and the friction force—f

_{up}

**τ**acts on Ω

_{A}at point o. The potential energy caused by the pair of friction force can be formulated as

**causes three second-order vectors to be added to the global load vector. It is noticed that the Coulomb model of friction assumes that the sliding frictional force is proportional to the normal contact force. Provided that the joints between blocks have the mechanical properties of the friction angle φ and the inner cohesion c, f**

^{f}_{up}is equivalent to |ρ

**d**

_{n}**|tanφ + c. If Equation (64) is adopted to estimate f**

_{n}_{up}, the frictional energy term will contribute not only to the global force vector but also to the stiffness matrix, leading to non-symmetry of the stiffness matrix. To avoid this issue, f

_{up}is estimated by using d

**obtained in the previous “open–close” iteration. In the view that the variation in d**

_{n}**is small when the “open–close” iteration is about to converge, such a simplification is acceptable.**

_{n}**J**’(v) to the global equations are obtained and Equation (9) is fulfilled.

_{P}

**u**over the foregoing time steps is considered the initial stress for the following time step.

## 5. Numerical Examples

**was prescribed as 100E and ρ**

_{n}**was prescribed as 40E for all methods. Here, E is Young’s modulus of the block.**

_{τ}#### 5.1. Rotating Triangular Block Problem

^{3}, E = 20 GPa, and Poisson’s ratio ν = 0.25. The gravity force of this model was ignored, and a concentrated load P = 10 kN acted at point 5 along the horizontal direction. All vertices of the foundation were fixed as the boundary conditions.

_{0}is very tiny in each time step, the difference is sometimes considerable in the two normal penetration values. Moreover, the normal penetration value is negative in the resulting configurations at steps 5, 6, and 7, which violates the actual contact conditions completely.

#### 5.2. Sliding Problem

^{3}kg/m

^{3}, E = 35 MPa, and ν = 0.3. The external load only came from the gravitational force with gravity acceleration g = 9.8 m/s

^{2}. No inner cohesion was involved in this model. All vertices of the ramp were fixed as the boundary conditions.

_{P}

**u**can be determined within the block. To investigate the difference in stress results of DDA with the post-adjustment strategy and the proposed method, the sliding problem was reanalyzed. Because the stress is unstable in a dynamic analysis, five friction angles, 50°, 55°, 60°, 65°, and 70° were adopted and static analysis with 50 time steps was executed to obtain a stable stress result. Firstly, little difference occurs in the resulting stress values of the two methods. When φ = 50°, the maximal principal stress σ

_{1}= 0.50 kPa and the minor principal stress σ

_{2}= −6.91 kPa in the original DDA and σ

_{1}= 0.63 kPa and σ

_{2}= −6.66 kPa in the proposed method. When φ = 70°, σ

_{1}= 2.22 kPa and σ

_{2}= −5.97 kPa in the original DDA and σ

_{1}= 2.22 kPa and σ

_{2}= −5.91 kPa in the proposed method. Then, the minimum principal stress direction appears to increase gently with the friction angle increasing, as shown in Figure 8a, and it stabilizes when the friction angle increases to 65°. The variations in the minimum principal stress direction with the friction angles are plotted for the two methods in Figure 8b. The proposed method has a gentler direction for the minimum principal stress. The stable direction for the minimum principal stress is −76.93° in the original DDA and −76.60° in the proposed method.

#### 5.3. Surrounding Rock Problem

^{3}, ν = 0.20, and E = 200 MPa. Only the gravitational force with g = 10 m/s

^{2}acts on the model. The hoop joints in the surrounding rock divided the model into three layers. The outermost boundaries of this model were fixed as the displacement boundary conditions.

_{1}= −0.4 kPa and σ

_{2}= −109.7 kPa, which induces a minimum principal stress direction of 6.7°. The bottom two rocks in the inner layer suffer σ

_{1}= −15.6 kPa and σ

_{2}= −138.6 kPa, and the direction of the minimum principal stress is 11.4°.

_{1}= −6.2 kPa and σ

_{2}= −117.6 kPa, and the minimum principal stress direction is 9.2°. The bottom two rocks in the inner layer suffer σ

_{1}= −24.8 kPa and σ

_{2}= −227.5 kPa, which has a minimum principal stress direction of 8.4°. The reason for this is that the stability of the arch only relies on the compression between blocks when the joints friction effect is absent.

#### 5.4. Block Wall Failure Problem

^{3}kg/m

^{3}, E = 500 MPa, and ν = 0.25. The external load only included the gravitational force with g = 10 m/s

^{2}. No friction occurred between the blocks. The outer vertices of two base blocks were fixed as the boundary conditions.

_{1}= 560 kPa and σ

_{2}= −880 kPa and the direction of the minimum principal stress is −48°.

_{1}= 2115 kPa and σ

_{2}= −2205 kPa, and the minimum principal stress has a direction of −44°. Rotation and inclination of the blocks in the left and right regions are smaller compared to the results of the proposed method. The difference in the displacement mode of the two methods has great effects on the difference between the results. The linear displacement function was adopted directly in DDA with the post-adjustment strategy. However, in the case when the vertex number was n > 3, the contribution of the higher-order displacement was added to the proposed method. Therefore, the proposed approach provides a block of n > 3 vertices with larger deformability than DDA with the post-adjustment strategy. Once the deformation of this block is not restrained by the adjacent blocks, a larger deformation occurs in the proposed method.

## 6. Conclusions

_{0}and sin r

_{0}in the displacement function of the original DDA is avoided in the proposed method. The degrees of freedom are directly added to the former coordinates in the renewal of the vertex coordinates, which induces a simpler expression regarding the contact restraint impositions. On the other hand, the contact detection and the contact theory can still be adopted as in the original DDA. Meanwhile, most of the numerical integrations are executed along the block boundary when calculating the global stiffness matrix and load vector. Only numerical integrations within the block can be conducted using the simplex integration as in the classical DDA.

**V**

_{1}(Ω) is taken as the virtual element space for a block in this study. If the stress variability within the block is expected, a higher-order virtual element space instead of

**V**

_{1}(Ω), e.g.,

**V**

_{2}(Ω) and

**V**

_{3}(Ω), can be adopted for a block. When using a higher-order virtual element space, the degrees of freedom should be augmented; refer to [46]. Although this study is limited to 2D analysis, the method to improve DDA by reformulating the block displacement using VEM can be mimicked in a 3D case.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A typical block system and an individual block in Discontinuous Deformation Analysis (DDA).

**Figure 6.**(

**a**) A triangular block on the foundation and (

**b**) its movement under a horizontal point load.

**Figure 7.**(

**a**) The configuration of the sliding problem and (

**b**) the relative displacement errors of the proposed method and DDA with the post-adjustment strategy.

**Figure 8.**(

**a**) The variations in minimum principal stress direction with the friction angles and (

**b**) the difference in minimum principal stress direction of the two methods.

**Figure 10.**Configurations and main stresses after 100 time steps under two different friction angles: (

**a**) φ = 30° and (

**b**) φ = 0°.

**Figure 13.**The configurations and main stress vectors after 7 s by DDA with the post-adjustment strategy.

**Table 1.**The penetrations of the contact pair point 7–edge 14 by DDA with the post-adjustment strategy and the new method.

Time Step | The Original DDA with the Post-Adjustment Strategy | The New Method | The Rotational Angle r_{0} | ||
---|---|---|---|---|---|

Penetrations in the Equilibrium Equations (m) | Penetrations in the Updated Configurations (m) | Penetrations in the Equilibrium Equations (m) | Penetrations in the Updated Configurations (m) | ||

1 | 8.3289 × 10^{−}^{9} | 8.3288 × 10^{−}^{9} | 8.32890 × 10^{−}^{9} | 8.32890 × 10^{−}^{9} | −0.000001 |

2 | 2.5026 × 10^{−}^{8} | 2.5026 × 10^{−}^{8} | 2.50260 × 10^{−}^{8} | 2.50260 × 10^{−}^{8} | −0.000000 |

3 | 7.4918 × 10^{−}^{8} | 7.4918 × 10^{−}^{8} | 7.49180 × 10^{−}^{8} | 7.49180 × 10^{−}^{8} | −0.000000 |

4 | 3.3772 × 10^{−}^{7} | 1.4587 × 10^{−}^{7} | 3.37720 × 10^{−}^{7} | 3.37720 × 10^{−}^{7} | −0.001073 |

5 | 1.0094 × 10^{−}^{6} | −7.2045 × 10^{−}^{7} | 1.00970 × 10^{−}^{6} | 1.00970 × 10^{−}^{6} | −0.003218 |

6 | 3.0067 × 10^{−}^{6} | −1.8379 × 10^{−}^{6} | 3.01240 × 10^{−}^{6} | 3.01240 × 10^{−}^{6} | −0.005375 |

7 | 8.9005 × 10^{−}^{6} | −7.2391 × 10^{−}^{7} | 8.92310 × 10^{−}^{6} | 8.92310 × 10^{−}^{6} | −0.007554 |

8 | 2.6114 × 10^{−}^{5} | 9.8896 × 10^{−}^{6} | 2.62290 × 10^{−}^{5} | 2.62290 × 10^{−}^{5} | −0.009768 |

9 | 7.5032 × 10^{−}^{5} | 5.0108 × 10^{−}^{5} | 7.54400 × 10^{−}^{5} | 7.54400 × 10^{−}^{5} | −0.012047 |

10 | 2.0631 × 10^{−}^{4} | 1.7005 × 10^{−}^{4} | 2.07880 × 10^{−}^{4} | 2.07880 × 10^{−}^{4} | −0.014444 |

11 | 4.9419 × 10^{−}^{4} | 4.4322 × 10^{−}^{4} | 4.97090 × 10^{−}^{4} | 4.97090 × 10^{−}^{4} | −0.017006 |

12 | 7.6165 × 10^{−}^{4} | 6.9448 × 10^{−}^{4} | 7.63320 × 10^{−}^{4} | 7.63320 × 10^{−}^{4} | −0.019372 |

13 | 6.5260 × 10^{−}^{4} | 5.7022 × 10^{−}^{4} | 6.50170 × 10^{−}^{4} | 6.50170 × 10^{−}^{4} | −0.021275 |

14 | 3.2463 × 10^{−}^{4} | 2.2251 × 10^{−}^{4} | 3.22710 × 10^{−}^{4} | 3.22710 × 10^{−}^{4} | −0.023478 |

15 | 2.6022 × 10^{−}^{4} | 1.2772 × 10^{−}^{4} | 2.68970 × 10^{−}^{4} | 2.68970 × 10^{−}^{4} | −0.026494 |

16 | 5.7111 × 10^{−}^{4} | 3.9996 × 10^{−}^{4} | 5.96770 × 10^{−}^{4} | 5.96770 × 10^{−}^{4} | −0.029810 |

17 | 9.6448 × 10^{−}^{4} | 7.5227 × 10^{−}^{4} | 9.91470 × 10^{−}^{4} | 9.91470 × 10^{−}^{4} | −0.032847 |

18 | 9.2970 × 10^{−}^{4} | 6.7950 × 10^{−}^{4} | 9.13440 × 10^{−}^{4} | 9.13440 × 10^{−}^{4} | −0.035286 |

19 | 3.3880 × 10^{−}^{4} | 4.7618 × 10^{−}^{5} | 2.94330 × 10^{−}^{4} | 2.94330 × 10^{−}^{4} | −0.037664 |

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**MDPI and ACS Style**

Luo, H.; Sun, G.; Liu, L.; Jiang, W.
Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method. *Materials* **2021**, *14*, 1252.
https://doi.org/10.3390/ma14051252

**AMA Style**

Luo H, Sun G, Liu L, Jiang W.
Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method. *Materials*. 2021; 14(5):1252.
https://doi.org/10.3390/ma14051252

**Chicago/Turabian Style**

Luo, Hongming, Guanhua Sun, Lipeng Liu, and Wei Jiang.
2021. "Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method" *Materials* 14, no. 5: 1252.
https://doi.org/10.3390/ma14051252