# Experimental and Numerical Studies on Flowing Properties of Grouting Mortar Based on the Modified MPS Method

Authors: Ailifeila Aierken; Shilin Luo; Jianqing Jiang; Linlin Chong; Jin Chang; Rui Zhang; Xiangchao Zhang
Journal: Geofluids
 (2022)

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## Abstract

Grouting mortar has widespread application in engineering because of the advantages of good durability, low cost, and environmental friendliness. To study the flowing properties of grouting mortar, laboratory minislump tests taking different rheological parameters of grouting mortar into consideration were conducted to obtain the flow time and flow pattern of grouting mortar. A modified MPS method (moving-particle semi-implicit method) introduced two sections of Bingham rheological, and segregation was proposed. The effects of the plastic viscosity on the flow state of mortar were studied, and the relationship between the plastic viscosity and the flowing time was established. The numerical results show that the modified MPS method can be used to predict the flow time and flow pattern of mortar in the template, and plastic viscosity has a great influence on the flow time of concrete, and with the increase of plastic viscosity, the time to flow to the specified area also increases. Moreover, the analysis of the rheological parameters can provide the basis for the design of mix ratio in the construction practice.


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## Body

## 1. Introduction

In a variety of seismic reinforcement methods for concrete structures, the steel plate lining method is widely used to reinforce existing bridge piers or columns. In this method, the properties of grouting materials must be carefully controlled to fill the narrow gap between the steel plate and the structure. Otherwise, the reinforcement will not perform the expected strength in the design. Among the properties of grouting materials, fluidity and segregation-resistance are the most important properties to control the filling process.From a computational point of view, choosing the right strategy for the simulation is an important issue, and several approaches have been tried to simulate the flow of these approaches in previous studies [1–3]. The CFD software is usually employed to simulate the pipe flow to predict the velocity distribution and pressure loss. However, fresh concrete is considered a single-phase fluid in CFD methods, it is unable to provide information about the dynamic segregation of concrete in the pipe for predicting blockage [4–6]. Recently, the meshless particle method, such as Smoothed Particle Hydrodynamics (SPH) method [7], has been increasingly used in the flow simulation of fluid, which represents the fluid with particles. The meshless particle method has the potential to numerically discuss aggregate segregation [8]. The SPH method has been proved to be applied to the flow simulation of fresh concrete [9–12]. There is no doubt that accurate calculation of pressure is important for the grouting mortar simulation. However, since the magnitude of the time step is limited in the SPH, the accuracy and efficiency of calculation are greatly reduced. For analyzing the incompressible fluid problem, Koshizuka and Oka [13] developed a new meshless method named moving-particle semi-implicit method (MPS). The MPS has been successfully used in the simulation of multiphase flows under high pressure [14], and the flow simulations of fresh cementitious fluid [15, 16], and has special advantages in simulating flow and filling behavior. Li et al. [17] and Jing-jun et al. [18] used the MPS method to simulate the flow characteristics of SCC and RFC in the L-box test. The results show that this method can simulate the flow process of SCC in rock-filled bodies, predict the compactness of RFC, and ultimately provide a reference for the design and construction practice of RFC engineering [19–21]. The MPS method was also used to investigate the geohazard mechanism [22, 23] and flow performances of concrete [24, 25]. Thus, the numerical simulation method including MPS has been widely used in many geotechnical applications, such as mining engineering, tunnel engineering, and slope engineering [26–32].This paper was aimed at extending the MPS approach to simulate the flow of grouting mortar in a minislump test. This methodology provided a thorough understanding of whether a minislump test can satisfy the self-compatibility criterion of passing ability through narrow gaps beside the flowability criterion. The result is compared with the flow time and flow pattern of the minislump test in the laboratory. By using this method, the effects of the plastic viscosity on the flow state of mortar in the minislump test are studied. The relationship between the plastic viscosity, the flowing time, and the density of mortar which is calculated as the ratio of mass to volume of the test is established. On this basis, the material segregation constitutive model is proposed. The distribution of aggregates in the mixes will also be tracked during the simulation to check whether or not they are homogeneously distributed after the flow has stopped. Simultaneously, it can provide a reference basis for the design and construction practice of cement mortar in engineering.


## 2. Methodology

### 2.1. Experimental Method

#### 2.1.1. Material Preparation

In this study, grouting mortar was chosen as a grouting material and used minislump test to measure the flow properties. Three kinds of mix proportions of grouting mortar were presented according to the rational mix design method proposed in [19, 20]. All of the mix designation, cement, water, fine aggregate, and natural river sand among these three schemes were unchanged, which is 0.4, 969, 388, 621, and 156, respectively. The variations of rheological parameters of grouting mortar were changeable and controlled by different mix proportions of thickening agent and polycarboxylate superplasticizer. The detailed compositions of all mixes were given in Table 1. It should be noted that only W/C ratio of 0.4 was selected in this paper because we want to test the feasibility of the modified MPS method. Next step, we will try to simulate the flow pattern of mortar and concrete under different mix proportions.Table 1
Mix proportion of grouting mortar, kg/m3.
NoMix designationCementWaterFAaVMASbW×%HRWRcC×%FA∗FA∗∗10.49693886211560.00.920.452.030.10.7aFine aggregate <2.5 mm (note: a part of the fine aggregate is the coarser fraction of the limestone powder, FA∗125 μm–1.2 mm, whereas FA∗∗ refers to natural river sand ≥1.2 mm). bThickening agent. cPolycarboxylate superplasticizer.


#### 2.1.2. Testing Apparatus

The minislump flow test is widely used in laboratories and construction sites for detecting grouting mortar flow characteristics. In this work, the slump cylinder was placed on the wet plate, and the mortar was injected into the cylinder, size, and shape of the cylinder as shown in Figure1. The linear scraper was used to smooth the upper surface of the mortar, and sector scraper was adapted to remove the residual mortar dripping on the glass plate. Dampen the cylinder mold and place it on a flat moist glass plate, fill the mold with mortar, and remove the mold immediately by raising it in a vertical direction after striking off the surface by screeding. Measure the time when the mortar reach 250 mm, the diameter of the two right angles was measured with a ruler, and the average value was obtained as the flow value. The volume of the mix in the mold is 1.96×105mm3 with a diameter and height of 50 mm and 100 mm, respectively.Figure 1
Size and shape of the minislump test.


### 2.2. Modified MPS Method

The accurate prediction of the mortar flowing behavior is not a simple task, particularly in the presence of a complex formwork shape. In this regard, the indispensable and inexpensive approach offering considerable potential is the numerical simulation of grouting mortar. MPS method applies a prediction–correction semi-implicit computation algorithm, which calculates external force and viscous term in the prediction step and calculates pressure term in the correction step. Since the MPS method has been brought forward for only two decades, it is considered one promising but immature method. Some drawbacks still exist with the MPS method, particularly in the investigation of the flowing properties of grouting mortar with different viscosity coefficients. Thus, a modified MPS method introduced two sections of Bingham rheological and segregation was developed. Actually, through the comparison between the numerical simulation and the experimental results by using modified MPS method, it is found that the simulation accuracy was improved. Simultaneously, it can provide a reference basis for the design and construction practice of cement mortar in engineering.


#### 2.2.1. Governing Equations

The governing equations for incompressible viscous flows are the mess conservation and Navier-Stokes equations:(1)DρDt=0,(2)DuDt=−1ρ∇P+υ∇2u+g,whereρ is the density, t is the time, u is the velocity vector, P is the pressure, υ is dynamic viscosity, and g is the acceleration due to gravity.In the MPS method, the density and pressure increase when particles come close to each other and vice versa. The interaction between particles uses the weight function:(3)wrijrerij−10<rij<re,0re≤rij,Otherwise,whererij is the distance between particle i and j, and re is the effective radius. re is chosen as 2.1rij [13]. All the forms of the weight function satisfy that the weight values are nonzero in the region of effective radius and the weight function increases as rij decreases and vice versa. It means that the rij will increases when the distance between particles is closer than re and becomes zero when the distance is farther than re.The particle number density, which is proportional to the mass density, at the position of the particlei is defined as
(4)ni=∑j≠iwrj−ri,whereri and rj are the position vector of the ith andjth particles, respectively, and ni is the particle number density of the ith particle.The discretization models of MPS method are derived from the Taylor expansion, and the gradient and Laplacian models in the MPS method are given as(5)∇∅i=dn0∑j≠i∅j−∅irj⟶−ri⟶2rj⟶−ri⟶wrj⟶−ri⟶,∇2∅i=2dλn0∑j≠i∅j−∅iwrj⟶−ri⟶,where∅j and ∅i are the quantities possessed by particles i and j, respectively, n0 is the constant particle number density, d is the number of spatial dimensions, and λ is the Laplacian model coefficient, which is defined as
(6)λ=∑j≠iwrj−rirj−ri2∑j≠iwrj−ri.This parameter adjusts the increase in the variance caused by the Laplacian model to that of the analytical solution. The gradient and Laplacian models are substituted into their corresponding operators in Equation (2). and the pressure, velocity, and position can be obtained. Substitute the velocity and pressure of each particle into the Lagrangian model. The source term of pressure can be represented by
(7)∇2Pi=2dλn0∑j≠iPj−Piwrj−ri,wherePj and Pi is the pressure of the ith and jth fluid particles, respectively. Substitute the pressure of particles into the gradient model. The pressure gradient model can be obtained
(8)∇Pi=2dn0∑j≠iPj−P^irj−rirj−riwrj−ri,whereP^i is the minimum pressure within the effective radius re. The purpose of replacing pressure Pi with P^i is to enforce repulsive force on all particles to stabilize the simulation.


#### 2.2.2. Derivation of MPS

The semi-implicit scheme is adopted in MPS method. The Navier-Stokes equation is solved in two steps:(9)DuDtk+1=−1ρ∇Pk+1+υ∇2uk+gk.The superscriptk represents the kth time step, and the k+1 represents the k+1−th time step. The left side of Equation (9) can be divided into two parts
(10)DuDtk+1=uk+1−u∗Δt+u∗−ukΔt.The explicit process is conducted. The intermediate velocity is only calculated by viscosity and gravity term.(11)u∗=uk+Δtυ∇2uk+Δtgk.The intermediate position can be ascertained by(12)r∗=rk+u∗Δt.Then, the incompressible condition is enforced in the MPS method, namely, the particle number density must remain constant, as(13)ni=n0whereni is the particle number density of the ith fluid particle.The following Poisson equation is derived, and the pressure of particles is implicitly evaluated using this equation(14)∇Pik+1=−ρΔtni∗−n0n0,whereni∗ is the temporary particle number density calculated after the explicit step. After calculating the pressure of particles, the gradient of pressure is computed by Equation (8). The velocity of the next time step is corrected by
(15)uk+1=u∗−Δt−1ρ∇Pk+1.The position of particles at thek+1−th time step can be calculated
(16)rk+1=r∗+uk+1−u∗Δt.The Dirichlet boundary condition is enforced in the Poisson equation on the free surface. The pressure of particles is set to zero when the particle number density satisfies the equation(17)ni∗<βn0,whereβ is a parameter for surface detection and β=0.97 is adopted in this study.The solid wall boundary condition is represented by fixed boundary particles which are composed of one layer of boundary particles and several layers of dummy particles.The flow chart of the MPS method is shown in Figure2.Figure 2
Flow chart of the MPS method.


#### 2.2.3. Biviscosity Section

The behavior of the mortar in the flow test was simulated using MPS numerical method. The governing equation is the momentum equation of the Bingham fluid. The non-Newtonian viscosity of a biviscosity model is used in this study. The momentum equation for theith direction is as follows:
(18)DuiDt=Fi+1ρ−∇P+η+τy1−e−mIIII∇2ui+2ε̇ij∂τy1−e−mII/II∂xj,whereFi represents the body forces (N/m3), ρ is the density of mortar (kg/m3), P is the pressure (Pa), η is the plastic viscosity (Pa·s), τy is the yield value (Pa), m is the fitting index [6], ε̇ij is the strain rate, and II=2ε̇ijε̇ij.
(19)ε̇ij=12∇u+∇uT.Velocity gradient ofith particle at time level k is calculated using the following gradient model
(20)∇uik=dn0∑j≠iujk−uikrjk−rikrjk−rik2wrjk−rik,whered is the number of space dimensions.The viscosity coefficient is calculated between the particlesi and j and average of particle i and particle j.
(21)μij=η+τy1−e−mIIII,νij=μi+μj2ρ.When calculating viscous items after calculating the viscous coefficient, the formulas for viscous terms are discretized using the Laplacian model as(22)ui∗=uik+∆t2dλn0∑j≠iνijkuj∗−ui∗wrj∗−ri∗.


#### 2.2.4. Segregation Section

According to the model of the previous studies proposed [33–37], segregation of material occurred because the pressure force from the difference of fresh concrete caused the mortar flows from a high-pressure area to low pressure area is used to simulate in this paper as shown as Figure 3.
(23)Mit+Δt=Mit+α×∑j≠iPj−Piwrj−ri∑j≠iwrj−ri×Mit×∆tPj>Pi,Mit+Δt=Mit−α×∑j≠iPj−Piwrj−ri∑j≠iwrj−ri×Mit×∆tPj<Pi,Figure 3
Conceptual diagram of the segregation model.wheret is the time, Mit is the mix-proportion of point i, and Mit+Δt is the proportion of mortar. α is the material segregation resistance coefficient, which is based on the change of the rheological constant assumed by this study.The maximum and minimum of mix-proportion need to be set. The maximum of mix-proportion was set as 0.99, and minimum of mix-proportion was set as (1~solidcontentinaggregate/100) in this paper. The mortar no longer flows into central area when the mix-proportion of point i reaches the maximum value, and the mortar will no longer flows out when the mix-proportion of point i reaches the minimum.


## 2.1. Experimental Method

### 2.1.1. Material Preparation

In this study, grouting mortar was chosen as a grouting material and used minislump test to measure the flow properties. Three kinds of mix proportions of grouting mortar were presented according to the rational mix design method proposed in [19, 20]. All of the mix designation, cement, water, fine aggregate, and natural river sand among these three schemes were unchanged, which is 0.4, 969, 388, 621, and 156, respectively. The variations of rheological parameters of grouting mortar were changeable and controlled by different mix proportions of thickening agent and polycarboxylate superplasticizer. The detailed compositions of all mixes were given in Table 1. It should be noted that only W/C ratio of 0.4 was selected in this paper because we want to test the feasibility of the modified MPS method. Next step, we will try to simulate the flow pattern of mortar and concrete under different mix proportions.Table 1
Mix proportion of grouting mortar, kg/m3.
NoMix designationCementWaterFAaVMASbW×%HRWRcC×%FA∗FA∗∗10.49693886211560.00.920.452.030.10.7aFine aggregate <2.5 mm (note: a part of the fine aggregate is the coarser fraction of the limestone powder, FA∗125 μm–1.2 mm, whereas FA∗∗ refers to natural river sand ≥1.2 mm). bThickening agent. cPolycarboxylate superplasticizer.


### 2.1.2. Testing Apparatus

The minislump flow test is widely used in laboratories and construction sites for detecting grouting mortar flow characteristics. In this work, the slump cylinder was placed on the wet plate, and the mortar was injected into the cylinder, size, and shape of the cylinder as shown in Figure1. The linear scraper was used to smooth the upper surface of the mortar, and sector scraper was adapted to remove the residual mortar dripping on the glass plate. Dampen the cylinder mold and place it on a flat moist glass plate, fill the mold with mortar, and remove the mold immediately by raising it in a vertical direction after striking off the surface by screeding. Measure the time when the mortar reach 250 mm, the diameter of the two right angles was measured with a ruler, and the average value was obtained as the flow value. The volume of the mix in the mold is 1.96×105mm3 with a diameter and height of 50 mm and 100 mm, respectively.Figure 1
Size and shape of the minislump test.


## 2.1.1. Material Preparation

In this study, grouting mortar was chosen as a grouting material and used minislump test to measure the flow properties. Three kinds of mix proportions of grouting mortar were presented according to the rational mix design method proposed in [19, 20]. All of the mix designation, cement, water, fine aggregate, and natural river sand among these three schemes were unchanged, which is 0.4, 969, 388, 621, and 156, respectively. The variations of rheological parameters of grouting mortar were changeable and controlled by different mix proportions of thickening agent and polycarboxylate superplasticizer. The detailed compositions of all mixes were given in Table 1. It should be noted that only W/C ratio of 0.4 was selected in this paper because we want to test the feasibility of the modified MPS method. Next step, we will try to simulate the flow pattern of mortar and concrete under different mix proportions.Table 1
Mix proportion of grouting mortar, kg/m3.
NoMix designationCementWaterFAaVMASbW×%HRWRcC×%FA∗FA∗∗10.49693886211560.00.920.452.030.10.7aFine aggregate <2.5 mm (note: a part of the fine aggregate is the coarser fraction of the limestone powder, FA∗125 μm–1.2 mm, whereas FA∗∗ refers to natural river sand ≥1.2 mm). bThickening agent. cPolycarboxylate superplasticizer.


## 2.1.2. Testing Apparatus

The minislump flow test is widely used in laboratories and construction sites for detecting grouting mortar flow characteristics. In this work, the slump cylinder was placed on the wet plate, and the mortar was injected into the cylinder, size, and shape of the cylinder as shown in Figure1. The linear scraper was used to smooth the upper surface of the mortar, and sector scraper was adapted to remove the residual mortar dripping on the glass plate. Dampen the cylinder mold and place it on a flat moist glass plate, fill the mold with mortar, and remove the mold immediately by raising it in a vertical direction after striking off the surface by screeding. Measure the time when the mortar reach 250 mm, the diameter of the two right angles was measured with a ruler, and the average value was obtained as the flow value. The volume of the mix in the mold is 1.96×105mm3 with a diameter and height of 50 mm and 100 mm, respectively.Figure 1
Size and shape of the minislump test.


## 2.2. Modified MPS Method

The accurate prediction of the mortar flowing behavior is not a simple task, particularly in the presence of a complex formwork shape. In this regard, the indispensable and inexpensive approach offering considerable potential is the numerical simulation of grouting mortar. MPS method applies a prediction–correction semi-implicit computation algorithm, which calculates external force and viscous term in the prediction step and calculates pressure term in the correction step. Since the MPS method has been brought forward for only two decades, it is considered one promising but immature method. Some drawbacks still exist with the MPS method, particularly in the investigation of the flowing properties of grouting mortar with different viscosity coefficients. Thus, a modified MPS method introduced two sections of Bingham rheological and segregation was developed. Actually, through the comparison between the numerical simulation and the experimental results by using modified MPS method, it is found that the simulation accuracy was improved. Simultaneously, it can provide a reference basis for the design and construction practice of cement mortar in engineering.


### 2.2.1. Governing Equations

The governing equations for incompressible viscous flows are the mess conservation and Navier-Stokes equations:(1)DρDt=0,(2)DuDt=−1ρ∇P+υ∇2u+g,whereρ is the density, t is the time, u is the velocity vector, P is the pressure, υ is dynamic viscosity, and g is the acceleration due to gravity.In the MPS method, the density and pressure increase when particles come close to each other and vice versa. The interaction between particles uses the weight function:(3)wrijrerij−10<rij<re,0re≤rij,Otherwise,whererij is the distance between particle i and j, and re is the effective radius. re is chosen as 2.1rij [13]. All the forms of the weight function satisfy that the weight values are nonzero in the region of effective radius and the weight function increases as rij decreases and vice versa. It means that the rij will increases when the distance between particles is closer than re and becomes zero when the distance is farther than re.The particle number density, which is proportional to the mass density, at the position of the particlei is defined as
(4)ni=∑j≠iwrj−ri,whereri and rj are the position vector of the ith andjth particles, respectively, and ni is the particle number density of the ith particle.The discretization models of MPS method are derived from the Taylor expansion, and the gradient and Laplacian models in the MPS method are given as(5)∇∅i=dn0∑j≠i∅j−∅irj⟶−ri⟶2rj⟶−ri⟶wrj⟶−ri⟶,∇2∅i=2dλn0∑j≠i∅j−∅iwrj⟶−ri⟶,where∅j and ∅i are the quantities possessed by particles i and j, respectively, n0 is the constant particle number density, d is the number of spatial dimensions, and λ is the Laplacian model coefficient, which is defined as
(6)λ=∑j≠iwrj−rirj−ri2∑j≠iwrj−ri.This parameter adjusts the increase in the variance caused by the Laplacian model to that of the analytical solution. The gradient and Laplacian models are substituted into their corresponding operators in Equation (2). and the pressure, velocity, and position can be obtained. Substitute the velocity and pressure of each particle into the Lagrangian model. The source term of pressure can be represented by
(7)∇2Pi=2dλn0∑j≠iPj−Piwrj−ri,wherePj and Pi is the pressure of the ith and jth fluid particles, respectively. Substitute the pressure of particles into the gradient model. The pressure gradient model can be obtained
(8)∇Pi=2dn0∑j≠iPj−P^irj−rirj−riwrj−ri,whereP^i is the minimum pressure within the effective radius re. The purpose of replacing pressure Pi with P^i is to enforce repulsive force on all particles to stabilize the simulation.


### 2.2.2. Derivation of MPS

The semi-implicit scheme is adopted in MPS method. The Navier-Stokes equation is solved in two steps:(9)DuDtk+1=−1ρ∇Pk+1+υ∇2uk+gk.The superscriptk represents the kth time step, and the k+1 represents the k+1−th time step. The left side of Equation (9) can be divided into two parts
(10)DuDtk+1=uk+1−u∗Δt+u∗−ukΔt.The explicit process is conducted. The intermediate velocity is only calculated by viscosity and gravity term.(11)u∗=uk+Δtυ∇2uk+Δtgk.The intermediate position can be ascertained by(12)r∗=rk+u∗Δt.Then, the incompressible condition is enforced in the MPS method, namely, the particle number density must remain constant, as(13)ni=n0whereni is the particle number density of the ith fluid particle.The following Poisson equation is derived, and the pressure of particles is implicitly evaluated using this equation(14)∇Pik+1=−ρΔtni∗−n0n0,whereni∗ is the temporary particle number density calculated after the explicit step. After calculating the pressure of particles, the gradient of pressure is computed by Equation (8). The velocity of the next time step is corrected by
(15)uk+1=u∗−Δt−1ρ∇Pk+1.The position of particles at thek+1−th time step can be calculated
(16)rk+1=r∗+uk+1−u∗Δt.The Dirichlet boundary condition is enforced in the Poisson equation on the free surface. The pressure of particles is set to zero when the particle number density satisfies the equation(17)ni∗<βn0,whereβ is a parameter for surface detection and β=0.97 is adopted in this study.The solid wall boundary condition is represented by fixed boundary particles which are composed of one layer of boundary particles and several layers of dummy particles.The flow chart of the MPS method is shown in Figure2.Figure 2
Flow chart of the MPS method.


### 2.2.3. Biviscosity Section

The behavior of the mortar in the flow test was simulated using MPS numerical method. The governing equation is the momentum equation of the Bingham fluid. The non-Newtonian viscosity of a biviscosity model is used in this study. The momentum equation for theith direction is as follows:
(18)DuiDt=Fi+1ρ−∇P+η+τy1−e−mIIII∇2ui+2ε̇ij∂τy1−e−mII/II∂xj,whereFi represents the body forces (N/m3), ρ is the density of mortar (kg/m3), P is the pressure (Pa), η is the plastic viscosity (Pa·s), τy is the yield value (Pa), m is the fitting index [6], ε̇ij is the strain rate, and II=2ε̇ijε̇ij.
(19)ε̇ij=12∇u+∇uT.Velocity gradient ofith particle at time level k is calculated using the following gradient model
(20)∇uik=dn0∑j≠iujk−uikrjk−rikrjk−rik2wrjk−rik,whered is the number of space dimensions.The viscosity coefficient is calculated between the particlesi and j and average of particle i and particle j.
(21)μij=η+τy1−e−mIIII,νij=μi+μj2ρ.When calculating viscous items after calculating the viscous coefficient, the formulas for viscous terms are discretized using the Laplacian model as(22)ui∗=uik+∆t2dλn0∑j≠iνijkuj∗−ui∗wrj∗−ri∗.


### 2.2.4. Segregation Section

According to the model of the previous studies proposed [33–37], segregation of material occurred because the pressure force from the difference of fresh concrete caused the mortar flows from a high-pressure area to low pressure area is used to simulate in this paper as shown as Figure 3.
(23)Mit+Δt=Mit+α×∑j≠iPj−Piwrj−ri∑j≠iwrj−ri×Mit×∆tPj>Pi,Mit+Δt=Mit−α×∑j≠iPj−Piwrj−ri∑j≠iwrj−ri×Mit×∆tPj<Pi,Figure 3
Conceptual diagram of the segregation model.wheret is the time, Mit is the mix-proportion of point i, and Mit+Δt is the proportion of mortar. α is the material segregation resistance coefficient, which is based on the change of the rheological constant assumed by this study.The maximum and minimum of mix-proportion need to be set. The maximum of mix-proportion was set as 0.99, and minimum of mix-proportion was set as (1~solidcontentinaggregate/100) in this paper. The mortar no longer flows into central area when the mix-proportion of point i reaches the maximum value, and the mortar will no longer flows out when the mix-proportion of point i reaches the minimum.


## 2.2.1. Governing Equations

The governing equations for incompressible viscous flows are the mess conservation and Navier-Stokes equations:(1)DρDt=0,(2)DuDt=−1ρ∇P+υ∇2u+g,whereρ is the density, t is the time, u is the velocity vector, P is the pressure, υ is dynamic viscosity, and g is the acceleration due to gravity.In the MPS method, the density and pressure increase when particles come close to each other and vice versa. The interaction between particles uses the weight function:(3)wrijrerij−10<rij<re,0re≤rij,Otherwise,whererij is the distance between particle i and j, and re is the effective radius. re is chosen as 2.1rij [13]. All the forms of the weight function satisfy that the weight values are nonzero in the region of effective radius and the weight function increases as rij decreases and vice versa. It means that the rij will increases when the distance between particles is closer than re and becomes zero when the distance is farther than re.The particle number density, which is proportional to the mass density, at the position of the particlei is defined as
(4)ni=∑j≠iwrj−ri,whereri and rj are the position vector of the ith andjth particles, respectively, and ni is the particle number density of the ith particle.The discretization models of MPS method are derived from the Taylor expansion, and the gradient and Laplacian models in the MPS method are given as(5)∇∅i=dn0∑j≠i∅j−∅irj⟶−ri⟶2rj⟶−ri⟶wrj⟶−ri⟶,∇2∅i=2dλn0∑j≠i∅j−∅iwrj⟶−ri⟶,where∅j and ∅i are the quantities possessed by particles i and j, respectively, n0 is the constant particle number density, d is the number of spatial dimensions, and λ is the Laplacian model coefficient, which is defined as
(6)λ=∑j≠iwrj−rirj−ri2∑j≠iwrj−ri.This parameter adjusts the increase in the variance caused by the Laplacian model to that of the analytical solution. The gradient and Laplacian models are substituted into their corresponding operators in Equation (2). and the pressure, velocity, and position can be obtained. Substitute the velocity and pressure of each particle into the Lagrangian model. The source term of pressure can be represented by
(7)∇2Pi=2dλn0∑j≠iPj−Piwrj−ri,wherePj and Pi is the pressure of the ith and jth fluid particles, respectively. Substitute the pressure of particles into the gradient model. The pressure gradient model can be obtained
(8)∇Pi=2dn0∑j≠iPj−P^irj−rirj−riwrj−ri,whereP^i is the minimum pressure within the effective radius re. The purpose of replacing pressure Pi with P^i is to enforce repulsive force on all particles to stabilize the simulation.


## 2.2.2. Derivation of MPS

The semi-implicit scheme is adopted in MPS method. The Navier-Stokes equation is solved in two steps:(9)DuDtk+1=−1ρ∇Pk+1+υ∇2uk+gk.The superscriptk represents the kth time step, and the k+1 represents the k+1−th time step. The left side of Equation (9) can be divided into two parts
(10)DuDtk+1=uk+1−u∗Δt+u∗−ukΔt.The explicit process is conducted. The intermediate velocity is only calculated by viscosity and gravity term.(11)u∗=uk+Δtυ∇2uk+Δtgk.The intermediate position can be ascertained by(12)r∗=rk+u∗Δt.Then, the incompressible condition is enforced in the MPS method, namely, the particle number density must remain constant, as(13)ni=n0whereni is the particle number density of the ith fluid particle.The following Poisson equation is derived, and the pressure of particles is implicitly evaluated using this equation(14)∇Pik+1=−ρΔtni∗−n0n0,whereni∗ is the temporary particle number density calculated after the explicit step. After calculating the pressure of particles, the gradient of pressure is computed by Equation (8). The velocity of the next time step is corrected by
(15)uk+1=u∗−Δt−1ρ∇Pk+1.The position of particles at thek+1−th time step can be calculated
(16)rk+1=r∗+uk+1−u∗Δt.The Dirichlet boundary condition is enforced in the Poisson equation on the free surface. The pressure of particles is set to zero when the particle number density satisfies the equation(17)ni∗<βn0,whereβ is a parameter for surface detection and β=0.97 is adopted in this study.The solid wall boundary condition is represented by fixed boundary particles which are composed of one layer of boundary particles and several layers of dummy particles.The flow chart of the MPS method is shown in Figure2.Figure 2
Flow chart of the MPS method.


## 2.2.3. Biviscosity Section

The behavior of the mortar in the flow test was simulated using MPS numerical method. The governing equation is the momentum equation of the Bingham fluid. The non-Newtonian viscosity of a biviscosity model is used in this study. The momentum equation for theith direction is as follows:
(18)DuiDt=Fi+1ρ−∇P+η+τy1−e−mIIII∇2ui+2ε̇ij∂τy1−e−mII/II∂xj,whereFi represents the body forces (N/m3), ρ is the density of mortar (kg/m3), P is the pressure (Pa), η is the plastic viscosity (Pa·s), τy is the yield value (Pa), m is the fitting index [6], ε̇ij is the strain rate, and II=2ε̇ijε̇ij.
(19)ε̇ij=12∇u+∇uT.Velocity gradient ofith particle at time level k is calculated using the following gradient model
(20)∇uik=dn0∑j≠iujk−uikrjk−rikrjk−rik2wrjk−rik,whered is the number of space dimensions.The viscosity coefficient is calculated between the particlesi and j and average of particle i and particle j.
(21)μij=η+τy1−e−mIIII,νij=μi+μj2ρ.When calculating viscous items after calculating the viscous coefficient, the formulas for viscous terms are discretized using the Laplacian model as(22)ui∗=uik+∆t2dλn0∑j≠iνijkuj∗−ui∗wrj∗−ri∗.


## 2.2.4. Segregation Section

According to the model of the previous studies proposed [33–37], segregation of material occurred because the pressure force from the difference of fresh concrete caused the mortar flows from a high-pressure area to low pressure area is used to simulate in this paper as shown as Figure 3.
(23)Mit+Δt=Mit+α×∑j≠iPj−Piwrj−ri∑j≠iwrj−ri×Mit×∆tPj>Pi,Mit+Δt=Mit−α×∑j≠iPj−Piwrj−ri∑j≠iwrj−ri×Mit×∆tPj<Pi,Figure 3
Conceptual diagram of the segregation model.wheret is the time, Mit is the mix-proportion of point i, and Mit+Δt is the proportion of mortar. α is the material segregation resistance coefficient, which is based on the change of the rheological constant assumed by this study.The maximum and minimum of mix-proportion need to be set. The maximum of mix-proportion was set as 0.99, and minimum of mix-proportion was set as (1~solidcontentinaggregate/100) in this paper. The mortar no longer flows into central area when the mix-proportion of point i reaches the maximum value, and the mortar will no longer flows out when the mix-proportion of point i reaches the minimum.


## 3. Experimental Results

The plastic viscosity coefficient reflects the flow properties of concrete. Generally, the higher the plastic viscosity, the slower the mortar flow. As listed in Table2, the plastic viscosity for three kinds of mix proportions of grouting mortar was 2.14, 5.00, and 2.82 Pa·s, the mortar flow value within 5 seconds recorded by the camera was 245, 195, and 226 mm, respectively. The segregation resistance index y and segregation resistance coefficient α was described as Equations (24) and (25). The relationship between flow value and time with different plastic viscosity of minislump tests was shown in Figure 4. At the same time before the flow stops, with the increase of plastic viscosity, the flow distance of grouting mortar becomes shorter. The flow time with different plastic viscosity at the same distance and the flow velocity of mortar also increases with the decrease of plastic viscosity, which is a negative correlation trend. It also directly shows that plastic viscosity is an important factor affecting the flow velocity of a cement-based material.Table 2
Measured data of minislump test.
Noρ(kg/m3)ηp(Pa·s)τ0(Pa)Flow∗(mm)yα121332.1422.62451.236.1×10−4221565.0035.181951.577.9×10−4321402.8226.752261.326.6×10−4∗Flow value of minislump test in 5 s (flow pattern of mortar was recorded by the camera during the test).Figure 4
Relationship between flow value and time with different plastic viscosity.


## 4. Numerical Results and Discussions

### 4.1. The Effects of Viscosity Coefficient on Flow Value

The purpose of slump flow simulation is to verify the implementation of the Bingham model for non-Newtonian rheology. The numerical model has a total number of particles of 121,107. The density of the fluid is 2133 kg/m3, 2156 kg/m3, and 2140 kg/m3. The rheological parameters (plastic viscosity and yield stress) of the sample were measured by viscometer as listed in Table 2. The computational time step is 0.00002 seconds, and the total computational time is 5.0 seconds.Figure5 shows the numerical simulation of the minislump flow process by using the modified MPS method when the plastic viscosity is 2.14 Pa·s. As illustrated in this plot, the flow value presents an increasing tendency with the rise of time. The flow value at time of 0.5 s, 1.0 s, 2.0 s, and 5.0 s was 0.085, 0.120, 0.181, and 0.241 m, respectively. Figure 6 manifests the comparison of simulation and test results. It can be seen from Figure 6 that the simulated value is in good agreement with the experimental value, but in the early stage of flow, the experimental value is smaller than the simulated value. This is caused by the delay in the lifting time of the slump cone under the test state, and the numerical simulation can achieve the ideal state of the instant lifting of the slump cone.Figure 5
Snap shots of 3D minislump test at time of 0.5 s (a), 1.0 s (b), 2.0 s (c), and 5.0 s (d).Figure 6
Comparison between simulation results and experiment results.All of the above indicate that the flow time and pattern are affected by the rheological constant.


### 4.2. Assessment of Segregation Section

The numerical simulation of flow is a powerful tool for understanding the rheological behavior of flow patterns. In order to verify the availability of the segregation model depending on the rheological properties and pressure, the simulation results of the flow test are validated with the measured data. The parameters used in the calculation are shown in Table2.The material segregation resistance coefficient is used in this paper, which is based on the regression analysis of the material segregation resistance index and rheological constant as shown in Table2.
(24)y=0.072η+0.011τ+0.827,wherey is the segregation resistance index, η is the plastic viscosity, and τ is the yield value. According to this formula, it can be known that y increases with the increase of η and τ. In addition, the segregation resistance index y represents the index of the tendency of material segregation resistance in the experiment. In addition, the material factor alpha-factor used in the analysis indicates the proportion of concrete moving in unit time and the value of segregation resistance is consistent with the tendency of segregation resistance index. Therefore, the segregation resistance index y obtained by the heavy regression equation is used as the coefficient of w (w=5.0×10−4), which is used as the material segregation resistance coefficient α (Table 2).
(25)αηmp,τy=wy.There is no unified test method for quantitative measurement of segregation of material. Therefore, from the perspective of numerical simulation, this paper verifies the feasibility of segregation section of mortar.Numerical simulation is carried out for different test conditions, and the test flow value and simulated flow value are compared and analyzed. The errors of the two are shown in Table3, where Te is the test result, Ts is the simulation result, and E is the error rate. The results show that the maximum error of the simulation results is 5.1%, which shows that the segregation section established in this paper has a certain accuracy and can provide a reference basis for the flow state of mortar and concrete described by the Bingham rheological model.Table 3
Accuracy of simulation.
No(Te/s)(Ts∗/s)(E1/%)Ts∗∗/s(E2/%)12452335.02402.021951798.21855.132262116.62154.9Ts∗/s: Simulation result by modified MPS method without segregation section. Ts∗∗/s: Simulation result by modified MPS method with segregation section.


## 4.1. The Effects of Viscosity Coefficient on Flow Value

The purpose of slump flow simulation is to verify the implementation of the Bingham model for non-Newtonian rheology. The numerical model has a total number of particles of 121,107. The density of the fluid is 2133 kg/m3, 2156 kg/m3, and 2140 kg/m3. The rheological parameters (plastic viscosity and yield stress) of the sample were measured by viscometer as listed in Table 2. The computational time step is 0.00002 seconds, and the total computational time is 5.0 seconds.Figure5 shows the numerical simulation of the minislump flow process by using the modified MPS method when the plastic viscosity is 2.14 Pa·s. As illustrated in this plot, the flow value presents an increasing tendency with the rise of time. The flow value at time of 0.5 s, 1.0 s, 2.0 s, and 5.0 s was 0.085, 0.120, 0.181, and 0.241 m, respectively. Figure 6 manifests the comparison of simulation and test results. It can be seen from Figure 6 that the simulated value is in good agreement with the experimental value, but in the early stage of flow, the experimental value is smaller than the simulated value. This is caused by the delay in the lifting time of the slump cone under the test state, and the numerical simulation can achieve the ideal state of the instant lifting of the slump cone.Figure 5
Snap shots of 3D minislump test at time of 0.5 s (a), 1.0 s (b), 2.0 s (c), and 5.0 s (d).Figure 6
Comparison between simulation results and experiment results.All of the above indicate that the flow time and pattern are affected by the rheological constant.


## 4.2. Assessment of Segregation Section

The numerical simulation of flow is a powerful tool for understanding the rheological behavior of flow patterns. In order to verify the availability of the segregation model depending on the rheological properties and pressure, the simulation results of the flow test are validated with the measured data. The parameters used in the calculation are shown in Table2.The material segregation resistance coefficient is used in this paper, which is based on the regression analysis of the material segregation resistance index and rheological constant as shown in Table2.
(24)y=0.072η+0.011τ+0.827,wherey is the segregation resistance index, η is the plastic viscosity, and τ is the yield value. According to this formula, it can be known that y increases with the increase of η and τ. In addition, the segregation resistance index y represents the index of the tendency of material segregation resistance in the experiment. In addition, the material factor alpha-factor used in the analysis indicates the proportion of concrete moving in unit time and the value of segregation resistance is consistent with the tendency of segregation resistance index. Therefore, the segregation resistance index y obtained by the heavy regression equation is used as the coefficient of w (w=5.0×10−4), which is used as the material segregation resistance coefficient α (Table 2).
(25)αηmp,τy=wy.There is no unified test method for quantitative measurement of segregation of material. Therefore, from the perspective of numerical simulation, this paper verifies the feasibility of segregation section of mortar.Numerical simulation is carried out for different test conditions, and the test flow value and simulated flow value are compared and analyzed. The errors of the two are shown in Table3, where Te is the test result, Ts is the simulation result, and E is the error rate. The results show that the maximum error of the simulation results is 5.1%, which shows that the segregation section established in this paper has a certain accuracy and can provide a reference basis for the flow state of mortar and concrete described by the Bingham rheological model.Table 3
Accuracy of simulation.
No(Te/s)(Ts∗/s)(E1/%)Ts∗∗/s(E2/%)12452335.02402.021951798.21855.132262116.62154.9Ts∗/s: Simulation result by modified MPS method without segregation section. Ts∗∗/s: Simulation result by modified MPS method with segregation section.


## 5. Conclusions

Based on the results obtained in the experiment and the numerical simulation, it is clear that plastic viscosity and yield stress have the great influence on the flow time of mortar, and with the increase of plastic viscosity, the time to flow to the specified area also increases.A good agreement between numerical and experimental data was found in the comparison of the final shape of the sample. It can be known that the modified MPS method in this paper is feasible for the flow simulation of grouting mortar and has high accuracy. The investigated results indicate that the modified MPS method by setting reasonable particle spacing can simulate and predict the flow process of non-Newtonian fluid with a free surface. The effectiveness of the segregation section depending on the rheological properties and pressure proposed in this paper was verified. Through the comparison between the numerical simulation and the experimental results, it is found that the simulation accuracy was improved from 91.8% to 94.9%, which showed that the segregation section was suitable for the simulation of the grouting mortar.



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*Source: 4042418-2022-06-22.xml*