An unresolved SPH-DEM model for simulation of ductile and brittle surface erosion by abrasive water-jet (AWJ) impact

Abstract

The abrasive water-jet (AWJ) erosion process involves the complex interaction between fluid medium, abrasive particles and solid material, which brings great challenges to the establishment of numerical model. Because traditional grid-based methods are not suitable for the problems of local deformation and material removal, the meshfree method smoothed particle hydrodynamics (SPH), based on the unresolved coupling and the discrete element method (DEM), is adopted to establish the model for AWJ study. The fluid medium is treated as a weakly compressible viscous liquid, the solid material is treated as an elastic-plastic material, and the abrasives are treated as rigid bodies. The fluid and solid phases are discretized with SPH particles, and the abrasives are described with DEM particles. The Johnson-Cook (J-C) and Johnson-Holmquist-II (JH-2) constitutive models are used to describe the stress-strain behavior of ductile and brittle materials, respectively. The effectiveness of the numerical model is further verified by AWJ impact experiments. The plastic deformation and cumulative failure characteristics of ductile materials, and the crack formation and propagation characteristics of brittle materials are systematically analyzed. The results provide insight for the AWJ research and lay a foundation for investigation of other complex fluid-particle flow in a numerical way.

Introduction

Abrasive water-jet (AWJ) cutting technology is a type of machining technology developed on the basis of pure water-jet cutting technology. With the same water pressure, the erosion ability of AWJ is much greater than that of pure water-jet¹. AWJ technology is initially used in the mining industry, and plays an important role in coalbed methane extraction and gas prevention operations. Since the 1980s, AWJ technology has been widely used in the machining field. Compared with traditional cutting tools and new thermal cutting technologies such as laser beam, electron beam and plasma, AWJ cutting technology has the advantages of no heat affected zone, low processing stress, high quality of slit surface, wide processing range and clean and environmental protection, which has been widely used in cutting operations of metals, rocks, ceramics and composite materials^2,3.

In early stage of AWJ technology application, erosion experiment is the main research method. However, the complex interaction between different jet parameters makes the experimental process time-consuming and expensive. In addition, due to the short impact duration and small erosion area, it is difficult to directly observe the plastic deformation and material removal during impact process, which makes the study of AWJ erosion mechanism quite difficult. Therefore, in the past two decades, various numerical methods have been widely developed and used to predict the erosion performance and reveal the erosion mechanism.

The traditional grid-based method of Finite Element Method (FEM) was initially used to describe the stress-strain behavior of the solid material, and there were various ways to describe the water-jet. In the Coupled Eulerian Lagrangian Method (CEL), the AWJ was modelled by Euler grid, and the solid phase was modelled by FEM based on Lagrangian framework, as shown in Fig. 1(a). CEL method can deal with the deformation and fragmentation of fluid phase during the impact process, but the introduction of additional fixed Euler calculation domain led to very low the calculation efficiency low^4,5,6. In the pure FEM method, the fluid medium, the abrasive particles, and the solid phase were all described by FEM grid, as shown in Fig. 1(b). Ma et al.⁷ used pure FEM method to study the erosion behavior of the high-speed water-jet. Due to the grid distortion during the impact process, the numerical accuracy and stability were affected, and the calculation was even suspended. Junkar et al.⁸ adopted pure FEM method to model the abrasive particles and solid materials and studied the material removal process was studied by AWJ impact, but the influence of fluid medium was ignored.

Figure 1

Numerical models for AWJ simulation. (a) CEL method (Applied in Ref.^4,5,6) (b) Pure FEM method (Applied in Ref^7,8). (c) SPH-FEM coupled method (Applied in Ref.^{9,10,11,12,13,14,15,16}) (d) SPH-DEM coupled method (Applied in present study).

At present, SPH-FEM coupled method is widely used in AWJ simulation, as shown in Fig. 1(c). The fluid medium is modelled by SPH method, while the abrasive particles and the solid materials are described by FEM method. As a meshfree Lagrange method, SPH has been used in particle impact study. During the simulation, the continuous medium (fluid and solid material) is discretized into a series of regularly arranged SPH particles, replacing the nodes and grid elements in FEM method. These particles have their own field variables and material properties, and move according to their own velocity and acceleration. The SPH method has more advantages than FEM when dealing with large displacement and deformation problems, such as single-particle impact¹⁷, multi-particle impact¹⁸ and particle embedding in the targeted material¹⁹. Wang et al.⁹ proposed a SPH-FEM model to predict the AWJ cutting performance on ductile materials within the working pressure range of 100 ~ 350 MPa. Liu et al.^10,11 investigated the influence of nozzle position on rock-breaking of the cutter with water-jet assistance. Results showed that the effect of the rear water-jet was better than the front water-jet, and the optimal incident angle was 10°. Jiang et al.^12,13 used SPH-FEM coupled method to study the crack initiation and expansion mechanism inside the rocks by water-jet impact, and concluded that the rock-breaking volume of pulsed water-jet was higher than that of continuous water-jet in the same condition of energy loss. Ren et al.^14,15,16 investigated the rock failure mechanism by particle water-jet impact based on SPH-FEM method. Results showed that the rock failure occurred instantaneously (within microseconds), presenting a cyclic process of “damage-cumulative damage-failure-cumulative damage-failure”. The optimal jet incident velocity and particle volume fraction were also obtained.

In above researches, due to the limitations of grid-based methods, the numerical accuracy and stability of FEM heavily depend on the quantity and quality of grid. When dealing with large deformation and crack propagation problems, only the methods of mesh deletion or mesh repartition can be used to overcome highly distorted elements. For the mesh deletion method, it is difficult to reproduce the chip separation phenomenon of solid material by particle impact. While the grid repartition method needs to remesh the solid material at each time step, the computational efficiency can be greatly reduced. Coupling with the Discrete Element Method (DEM) may be an effective approach. According to solving modes of fluid-abrasive interaction, the coupling of SPH and DEM methods can be mainly categorized into two groups: the fully resolved and unresolved methods. In the fully resolved methods, each abrasive particle is arranged by a group of DEM particles, which serve as the boundaries for the abrasive particle in the fluid phase. Peng et al.²⁰ studied the suspension flows containing solid particles, indicating that the fully resolved SPH-DEM coupling model can reproduce the interaction between non-Newtonian fluid flows and fixed/moving solid particles with complex irregular shapes. Joubert et al.²¹ proposed a fully resolved model with SPH gradient correction measures to investigate the drag characteristics of several particle shapes and topologies in the fluid phase. Results showed that the model can solve various fluid behaviors related to the geometry of solid particles even at low resolutions. Hashemi et al.²² investigated the movement of suspended solid bodies in Oldroyd-B fluid flows by using a modified boundary treatment technique, which could fully resolve the interaction between the fluid phase and the solid particles. Shi et al.²³ established a soil–water SPH coupling model to simulate the landslide process, and the generated wave of the landslide body could be resolved satisfactorily.

Since the fluid-solid interaction takes into account the real particle shape and spacing, the fully resolved method is particularly useful when the phenomena is concentrated on the behavior of a single solid particle or flow disturbances of a few particles. However, when simulating the fluid-particle flow contains a large number of solid particles, or the research scale is much larger than the particle diameter, the computational load to fully resolve the fluid-particle interaction may be extremely high. In these applications, the unresolved coupling method can be an effective alternative, in which the local averaged governing equations are solved based on the local porosity, and the interactions between phases are considered through semi-theoretical correlations or empirical force correlations. Robinson et al.²⁴ used an unresolved SPH-DEM coupling method to simulate the single/multiple particle sedimentation of fluid–particle flow, which was suitable for both dilute and dense particle flow problems. Jo et al.²⁵ conducted a gas–liquid–solid analysis on the packed bed flow, indicating that the local averaged techniques could accurately achieve the momentum exchange between three phases. Wu et al.²⁶ investigated the single/multiple particle sedimentation and dam-break processes based on the unresolved SPH-DEM coupling model, which can handle the solid structural fractures caused by fluid impact. Sun et al.²⁷ performed three-dimensional simulations of a quasi-steady flow in a rotating cylindrical tank, proving the unresolved model was suitable for simulating fluid-particle flow.

The unresolved coupling method has been widely used in fluid-particle interaction study such as debris flow simulation, but rarely involves the AWJ impact process. In this study, an unresolved SPH-DEM model is proposed to investigate the AWJ erosion performance on ductile and brittle materials based on our previous research²⁸, as shown in Fig. 1(d). The fluid and solid phases are described by SPH method, and the abrasives are described by DEM method. The unresolved coupling between SPH and DEM particles is carried out by the local averaging technique in fluid and abrasive phases, and the calculations are conducted by introducing empirical formulas for hydrodynamic and drag force estimation. The model enables the detailed interactions between water-jet, abrasives and solid materials.

The paper is organized as follows: in Sect. 2, the basic theories of SPH and DEM methods are introduced respectively. In Sect. 3, the unresolved coupling between SPH and DEM particles is involved. In Sect. 4, the proposed model is verified by comparing with the impact experiment, and the erosion process of AWJ impact on ductile and brittle materials is compared and discussed. Finally, the conclusions are summarized in Sect. 5.

Description of DEM and SPH methods

In this study, the fluid and solid phases are discretized with SPH particles, while each abrasive particle is modelled by a single DEM particle. The interactions among these three phases are shown in Fig. 2. For the 2D model, the abrasive particles are treated as circular rigid bodies, which can be described by rigid motion equations. The motion of fluid and solid phases is described by N-S equations. In the unresolved SPH-DEM coupled method, the coupling force $\:{\overrightarrow{\text{F}}}_{\text{a}\text{}\text{f}}$ between abrasive (DEM particle) and fluid (SPH particle) phases is calculated by local averaging techniques. The interaction force $\:{\overrightarrow{\text{F}}}_{\text{f}\text{}\text{s}}$ between fluid and solid phases is solved by SPH integration, while the force $\:{\overrightarrow{\text{F}}}_{\text{a}\text{}\text{s}}$ between abrasive and solid phases is solved by contact algorithm. The interaction forces mentioned above will be discussed in following paragraphs.

Figure 2

Interactions among the phases during AWJ impact.

DEM motion equations

For rigid bodies, Newton’s second law can be used to describe the motion of abrasive DEM particles, as shown below:

$$\left\{ {\begin{array}{*{20}{c}} {{m_i}\frac{{d{{\vec {v}}_i}}}{{dt}}=\vec {F}_{{a - a}}^{{}}+{{\vec {F}}_{a - s}}+{{\vec {F}}_{a - f}}+{m_i}\vec {g}} \\ {{I_i}\frac{{d{{\vec {\omega }}_i}}}{{dt}}=\vec {T}_{{a - a}}^{{}}+{{\vec {T}}_{a - s}}} \end{array} }, \right.$$

(1)

where m_i, I_i, $\:{\overrightarrow{\text{v}}}_{\text{i}}$ and $\:{\overrightarrow{\omega}}_{\text{i}}$ are the mass, inertia, centroid velocity and angular velocity of the DEM particle i, respectively. $\:{\overrightarrow{\text{T}}}_{\text{a}\text{}\text{a}}$ and $\:{\overrightarrow{\text{T}}}_{\text{a}\text{}\text{s}}$ are the total contact torques for abrasive-abrasive and abrasive-solid interaction, respectively. For ease of reading, the subscripts i and j represent abrasive DEM particles, while the subscripts a and b are labeled as SPH particles.

DEM contact modelling between abrasive particles

DEM method was first proposed by Cundall and Strack to investigate the discontinuous mechanical effects inside the rocks²⁹. The continuous rock medium was discretized into a series of circular (2D) and spherical (3D) DEM particles. The method was widely used in the study of geotechnical engineering and geological hazards to track the movement of solid particles inside the fluid phase³⁰. In DEM calculation process, the size and shape of each particle remains unchanged, and the contact force is generated by squeezing and overlapping. Considering the energy conversion and loss during the interaction, the soft-sphere contact model can be used to handle the contact force between DEM particles³¹, as shown in Fig. 3. The contact force $\:{\overrightarrow{\text{F}}}_{\text{ij}}^{\text{c}}$ between DEM particle i and j is composed by a spring, a damper and a slider, which can be divided into normal contact force $\:{\overrightarrow{\text{F}}}_{\text{ij}}^{\text{n}}$ and tangential contact force $\:{\overrightarrow{\text{F}}}_{\text{ij}}^{\text{t}}$, expressed as:

$$\left\{ {\begin{array}{*{20}{c}} {\vec {F}_{{ij}}^{n}= - {k_n}{\delta _n}\vec {n} - {d_n}\vec {v}_{{ij}}^{n}} \\ {\vec {F}_{{ij}}^{t}= - {k_t}{\delta _t}\vec {t}{\text{ }} - {d_t}\vec {v}_{{ij}}^{t}} \end{array} } ,\right.$$

(2)

where the superscripts n and t represent the normal and tangential directions, respectively. $\:\overrightarrow{\text{n}}$ and $\:\overrightarrow{\text{t}}$ are the unit normal and unit tangent vectors, respectively. k and η are the stiffness and damping coefficients, respectively. δ and $\:{\overrightarrow{\text{v}}}_{\text{ij}}$ denote the overlap and relative velocity between DEM particle i and j, respectively.

Figure 3

Soft-sphere contact model for DEM particles.

In addition, the tangential contact force $\:{\overrightarrow{\text{F}}}_{\text{ij}}^{\text{t}}$ is limited by the sliding condition. When $\:{\overrightarrow{\text{F}}}_{\text{ij}}^{\text{t}}$ is greater than the sliding friction force, the two particles will slide relative to each other, and the expression of $\:{\overrightarrow{\text{F}}}_{\text{ij}}^{\text{t}}$ in Eq. (2) can be rewritten as:

$$\left\{ {\begin{array}{*{20}{c}} {\vec {F}_{{ij}}^{t}= - {k_t}{\delta _t}\vec {t} - {d_t}\vec {v}_{{ij}}^{t}{\text{ }}\left| {\vec {F}_{{ij}}^{t}} \right| \leqslant \mu \left| {\vec {F}_{{ij}}^{n}} \right|} \\ {\vec {F}_{{ij}}^{t}= - \mu \left| {\vec {F}_{{ij}}^{n}} \right|\vec {t}{\text{ }}\left| {\vec {F}_{{ij}}^{t}} \right|>\mu \left| {\vec {F}_{{ij}}^{n}} \right|} \end{array} }, \right.$$

(3)

where µ is the sliding friction coefficient.

In summary, for abrasive-abrasive interaction, the total contact force $\:{\overrightarrow{\text{F}}}_{\text{a}\text{}\text{a}}$ and total contact torque $\:{\overrightarrow{\text{T}}}_{\text{a}\text{}\text{a}}$ of DEM particle i can be calculated as:

$$\left\{ {\begin{array}{*{20}{c}} {\vec {F}_{{a - a}}^{{}}=\sum\limits_{j} {(\vec {F}_{{ij}}^{n}+\vec {F}_{{ij}}^{t})} } \\ {{{\vec {T}}_{a - a}}=\sum\limits_{j} {(\vec {F}_{{ij}}^{t} \times {R_i}\vec {n})} } \end{array}}, \right.$$

(4)

where j represents all DEM particles in contact with particle i, and R_i is the radius of the particle i.

Determination of overlap δ and relative velocity $\:{\overrightarrow{\text{v}}}_{\text{ij}}$

As shown in Fig. 3, the dotted circle indicates the position of DEM particle i during the previous time step. With the relative motion of i and j, the two particles overlap and generate the contact force. The normal overlap $\delta_n$is given by:

$${\delta _n}={R_i}+{R_j} - \left| {{{\vec {x}}_{ij}}} \right|,$$

(5)

where $\:\left|{\overrightarrow{\text{x}}}_{\text{ij}}\right|=\left|{\overrightarrow{\text{x}}}_{\text{i}}-{\overrightarrow{\text{x}}}_{\text{j}}\right|$ is the distance between particle i and j.

The direction of the unit normal vector $\:\overrightarrow{\text{n}}$ points from i to j, expressed as:

$$\vec {n}=\frac{{{{\vec {x}}_j} - {{\vec {x}}_i}}}{{\left| {{{\vec {x}}_j} - {{\vec {x}}_i}} \right|}}.$$

(6)

The relative velocity $\:{\overrightarrow{\text{v}}}_{\text{ij}}$ depends on the position vector, the centroid velocity, and the angular velocity of the two particles, which is calculated follows:

$$\left\{ {\begin{array}{*{20}{c}} {{{\vec {v}}_{ij}}={{\vec {v}}_i} - {{\vec {v}}_j}+({R_i}{{\vec {\omega }}_i}+{R_j}{{\vec {\omega }}_j}){\text{ }} \times \vec {n}} \\ {\vec {v}_{{ij}}^{n}=({{\vec {v}}_{ij}} \cdot \vec {n}){\text{ }}\vec {n}} \\ {\vec {v}_{{ij}}^{t}={{\vec {v}}_{ij}} - \vec {v}_{{ij}}^{n}} \end{array} }. \right.$$

(7)

Therefore, the unit tangent vector $\:\overrightarrow{\text{t}}$ of DEM particle i can be determined as:

$$\vec {t}=\frac{{\vec {v}_{{ij}}^{t}}}{{\left| {\vec {v}_{{ij}}^{t}} \right|}}.$$

(8)

Different from the normal overlap $\:{\delta}_{\text{n}}$, the tangential overlap $\:{\delta}_{\text{t}}$ is an integral variable, obtained by the accumulation of the tangential overlap increment $\:\varDelta\:{\delta}_{\text{t}}$ in each time step, and can be expressed as:

$${\delta _t}={\delta _t}^{\prime }+\Delta {\delta _t}={\delta _t}^{\prime }+\vec {v}_{{ij}}^{t} \cdot \Delta t,$$

(9)

where $\:{\varDelta\:\delta}_{\text{t}}^{{\prime\:}}$ is the tangential overlap of the previous time step, and $\:{\delta}_{\text{t}}^{{\prime\:}}$ = 0 when the contact does not exist. So $\:{\delta}_{\text{t}}^{{\prime\:}}$ obtained in each time step needs to be saved as the basis for the calculation in the next integration period.

Determination of stiffness coefficient k and damping coefficient d

According to Hertz contact theory, the normal stiffness coefficient k_n and the tangential stiffness coefficient k_t are related to the size, elastic modulus and Poisson’s ratio of DEM particles. In the case of ignoring particle deformation, k_n and k_t can be treated as constants, shown as below³²:

$$\left\{ {\begin{array}{*{20}{c}} {{k_n}=\frac{{2{E_i}{R_i}{E_j}{R_j}}}{{{E_i}{R_i}+{E_j}{R_j}}}} \\ {{k_t}=\frac{{2{E_i}{R_i}\nu _{i}^{{}}{E_j}{R_j}\nu _{j}^{{}}}}{{{E_i}{R_i}\nu _{i}^{{}}+{E_j}{R_j}\nu _{j}^{{}}}}} \end{array} }, \right.$$

(10)

where E, ν, G and R are the elastic modulus, Poisson’s ratio, shear modulus and radius of the DEM particle, respectively.

In the soft-sphere contact model, the collision of DEM particles can be regarded as the movement of spring vibrator. For a spring oscillator with mass m, the normal damping coefficient d_n and tangential damping coefficient d_t depend on the stiffness coefficients, which can be expressed as³³:

$$\left\{ {\begin{array}{*{20}{c}} {{d_n}=2\xi \sqrt {m{k_n}} } \\ {{d_t}=2\xi \sqrt {m{k_t}} } \end{array} } ,\right.$$

(11)

where $\xi = - \ln e/\sqrt {{\pi ^2}+{{\ln }^2}e}$ is the damping ratio, and e is the restitution coefficient, which needs to be measured by experiment.

When e = 1.0, ξ = d_n = d_t = 0, the contact between DEM particles are completely elastic without any damping effect. Therefore, there is no energy loss during the conversion process of mechanical energy and elastic potential energy. When e ≈ 0, ξ ≈ 1, the particles are in the critical damping conditions, and the kinetic energy is damped at the maximum speed. The effect of restitution coefficient e on DEM particle collisions is shown in Fig. 4. In the simulation, a single DEM particle has a radius of 5 mm and an initial height of 50 mm, falling freely at t = 0, and makes contact with the bottom boundary at about t = 0.096s. As shown in Fig. 4, when e = 1.0, the particle can rebound to the initial height and cycle repeatedly, which means that the particle mechanical energy is conserved during the collisions. With the decrease of e, the height of the particle rebound decreases, and the damping coefficient increases gradually, leading to the attenuation effect on the kinetic energy. Finally the mechanical energy is exhausted, making the particle stop at the bottom boundary.

Figure 4

Effect of recovery coefficient e on DEM particle collisions.

DEM contact modelling between abrasive and solid phases

The contact illustration between abrasive and solid phases is shown in Fig. 5. The abrasive particle is circular (2D) and consists of a DEM particle with a well-defined boundary, which is treated as a rigid body during the collision. The contact type between solid SPH particles and rigid abrasive DEM particles belongs to point-line contact. The solid particles move relative to each other under the action of rigid boundary, resulting in deformation and surface damage. Therefore, the penalty function algorithm is adopted to calculate the contact force according to the overlap between the solid SPH particle and the rigid DEM boundary, which can be divided into the following 4 steps³⁴:

• (1) Contact detection: when the solid SPH particle a (x_a, y_a) is close enough to the boundary of the rigid DEM particle (inside the dotted circle in Fig. 5), the contact is detected.

• (2) Determination of contact point p: contact point p is located on the boundary of DEM particle i, and also on the line between DEM particle i and SPH particle a, as shown in Fig. 5.

• (3) Determination of unit normal vector $\:{\overrightarrow{\text{n}}}_{\text{p}}$ and unit tangent vector $\:{\overrightarrow{\text{t}}}_{\text{p}}$: the unit normal vector direction $\:{\overrightarrow{\text{n}}}_{\text{p}}$ points from i to a. The unit tangent vector $\:{\overrightarrow{\text{t}}}_{\text{p}}$ is determined by the tangential component of relative velocity $\:{\overrightarrow{\text{v}}}_{\text{pa}}$, as follows:

$$\left\{ {\begin{array}{*{20}{c}} {{{\vec {v}}_{pa}}={{\vec {v}}_p} - {{\vec {v}}_a}} \\ {{{\vec {v}}_p}={{\vec {v}}_i}+{{\vec {\omega }}_i} \times ({{\vec {x}}_p} - {{\vec {x}}_i})} \end{array} } ,\right.$$

(12)

where $\:{\overrightarrow{\text{v}}}_{\text{p}}$ is the velocity on point p. $\:{\overrightarrow{\text{x}}}_{\text{i}}$ is the position vector of DEM particle i.

• (4) Determination of overlap d_n and contact force $\:{\overrightarrow{\text{F}}}_{\text{ia}}$: the distance d_p between the solid SPH particle a and the contact point p is obtained at first. If d_p<d_ini (d_ini is the initial spacing of SPH particles), the overlap is determined by d_n = d_ini-d_p. The contact force $\:{\overrightarrow{\text{F}}}_{\text{ia\:}}$ can be divided into the normal contact force $\:{\overrightarrow{\text{F}}}_{\text{ia}}^{\text{n}}$ and the tangential contact force $\:{\overrightarrow{\text{F}}}_{\text{ia}}^{\text{t}}$, which can be written as:

$$\left\{ {\begin{array}{*{20}{c}} {\vec {F}_{{ia}}^{n}=(1 - \chi )\left[ {\frac{{2{m_a}}}{{{{\left( {\Delta t} \right)}^2}}}{d_n}} \right]\vec {n}} \\ {\vec {F}_{{ia}}^{t}=\hbox{min} \left( {\frac{{2{m_a}}}{{\Delta t}}\left( {{{\vec {v}}_{pa}} \cdot {{\vec {t}}_p}} \right){{\vec {t}}_p},\mu \left| {\vec {F}_{{ia}}^{n}} \right|{{\vec {t}}_p}} \right)} \end{array} }, \right.$$

(13)

where χ is the overlap coefficient (χ = 0 means no overlap is allowed). m_a is the mass of SPH particle a, and Δt is the length of a single time step. Tangential contact force $\:{\overrightarrow{\text{F}}}_{\text{ia}}^{\text{t}}$ is limited by the sliding friction force µ$\:\left|{\overrightarrow{\text{F}}}_{\text{ia}}^{\text{n}}\right|$.

According to Newton’s third law, the contact force $\:{\overrightarrow{\text{F}}}_{\text{ai}}$ of abrasive DEM particle i is applied in the opposite direction to $\:{\overrightarrow{\text{F}}}_{\text{ia}}$, and the force functional point is contact point p. Therefore, the total contact force $\:{\overrightarrow{\text{F}}}_{\text{a}\text{}\text{s}}$ and the total contact force $\:{\overrightarrow{\text{T}}}_{\text{a}\text{}\text{s\:}}$for abrasive-solid interaction can be obtained by:

$$\left\{ {\begin{array}{*{20}{l}} {{{\vec {F}}_{a - s}}=\sum\limits_{a}^{{}} {\vec {F}_{{ai}}^{{}}} } \\ {{{\vec {T}}_{a - s}}=\sum\limits_{a}^{{}} {\left[ {\vec {F}_{{ai}}^{{}} \times ({{\vec {x}}_p} - {{\vec {x}}_{\text{i}}})} \right]} } \end{array}} ,\right.$$

(14)

where a represents all solid SPH particles in contact with DEM particle i.

Figure 5

Contact force between abrasive and solid phases.

SPH modelling for fluid phase based on local average technique

Unlike the grid-based numerical methods, SPH method uses a finite number of particles to represent the continuous medium. The SPH particles have their own field variables and material properties, such as velocity, mass, density, stress, etc³⁵. The motion equations for SPH particles are based on the continuum mechanics conservation laws, and the particle properties change over time due to the interactions with other neighboring particles within the support domain³⁶. These characteristics make SPH method suitable for large impact and deformation problems. The advantages of SPH method lie in the conceptual simplicity, straightforward implementation, and easy introduction of new physical quantities³⁷.

Two key steps are needed for SPH formulation: integral representation (kernel approximation) and particle approximation. By the process of these two steps, the field function f$\:\text{(}\overrightarrow{\text{x}}\text{)}$ and the special derivative $\:{\triangledown}\text{f}\text{(}\overrightarrow{\text{x}}\text{)}$ of SPH particle a can be approximated as:

$$\left\langle {f({{\vec {x}}_a})} \right\rangle =\sum\limits_{{b=1}}^{N} {\frac{{{m_b}}}{{{\rho _b}}}} f({\vec {x}_b}) \cdot W({\vec {x}_a} - {\vec {x}_b},h),$$

(15)

$$\left\langle {\frac{{\partial f({{\vec {x}}_a})}}{{\partial \vec {x}}}} \right\rangle =\sum\limits_{{b=1}}^{N} {\frac{{{m_b}}}{{{\rho _b}}}} f({\vec {x}_b}) \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{{}}}},$$

(16)

where $\:{\overrightarrow{\text{x}}}_{\text{a}}$ is the position vector of SPH particle a. Subscript b means the neighboring particles in the support domain of particle a. m and ρ are the mass and material density of the SPH particle, respectively. N is the total number of the neighboring particles. $\frac{{\partial {W_{ab}}}}{{\partial {{\vec {x}}_a}}}$ is the kernel gradient, which can be written as$\frac{{\partial {W_{ab}}}}{{\partial {{\vec {x}}_a}}}=\frac{{{{\vec {x}}_a} - {{\vec {x}}_b}}}{{{r_{ab}}}}\frac{{\partial {W_{ab}}}}{{\partial {r_{ab}}}}=\frac{{{{\vec {x}}_{ab}}}}{{{r_{ab}}}}\frac{{\partial {W_{ab}}}}{{\partial {r_{ab}}}}$, and r_ab is the distance between particle a and b.

In Eq. (15), $\:\text{W}\text{(}{\overrightarrow{\text{x}}}_{\text{a}}-\text{}{\overrightarrow{\text{x}}}_{\text{b}}\text{,}\text{h}\text{)}$ is the smoothing function for SPH formulation, and h is the smoothing length. There are many possible choices of the smoothing function, and the cubic spline function is selected in this study, expressed as³⁸:

$$W({\vec {x}_a} - {\vec {x}_b},h)={W_{ab}}={a_d} \times \left\{ {\begin{array}{*{20}{l}} {\frac{2}{3} - {q^2}+\frac{1}{2}{q^3},{\text{ 0}} \leqslant q<1} \\ {\frac{1}{6}{{\left( {2 - q} \right)}^3},{\text{ 1}} \leqslant q<2} \\ {0,{\text{ }}q \geqslant {\text{2}}} \end{array}} ,\right.$$

(17)

where a_d is the normalization factor with a_d = 15/(7πh²) for the two-dimensional problem. q = r_ab/h is the normalization distance between particle a and b. When q ≥ 2, W_ab = 0, which indicates that the radius of the support domain is 2h. The selection of h value is very important for the calculation accuracy and efficiency. In this study, h = 1.25d_ini, which is the most widely used in SPH formulation.

Local average variables in SPH

When the fluid medium contains a large number of abrasive particles, it requires a lot of computation to solve the N-S equation and Newton’s equation directly by using the fully-resolved coupled method. In this case, the unresolved SPH-DEM coupled method may be a better choice for modeling the abrasive-fluid interaction. The local average technique is adopted to solve the coupling force between fluid phase (SPH) and abrasive particles (DEM). The concept was first proposed by Anderson and Jackson³⁹ to deal with the momentum transitions between different phases in suspension by defining local average variables on fluid and particle phases instead of some field variables (such as fluid density, fluid velocity, and particle velocity, etc.). These local average variables at any point x in the fluid domain can be obtained by convolution with the smoothing function, as shown below:

$$\left\langle{\bar{f}(x)}\right\rangle=\epsilon(x)f(x)=\int_{{{\text{fluid}}}}{\bar{f}({x^{\prime}})}W(x-{x^{\prime}})dV,$$

(18)

$$\epsilon(x)=1-\int_{{{\text{particle}}}}{W(x-{x^{\prime}})dV},$$

(19)

where $\:\stackrel{\text{-}}{\text{f}}\text{(}x\text{)}$ is the local average value of the field function f$\:\text{(}x\text{)}$. $\:\epsilon\text{(}\text{x}\text{)}$ is the local porosity at point x, which depends on the number and distance of neighboring DEM particles. If there are no DEM particles nearby, $\:\epsilon\text{(}\text{x}\text{)}$= 1.

By the particle approximation process in Eq. (19), the local porosity of SPH particle a can be expressed as⁴⁰:

$$\epsilon {\ominus _a}=1 - \sum\limits_{j} {{W_{aj}}({h_c}){\text{ }}} {V_j},$$

(20)

where V_j is the volume of DEM particle j near SPH particle a, and h_c is the coupling smoothing length.

Therefore, the locally averaged density $\:{\stackrel{\text{-}}{\rho}}_{\text{a}}$ of the fluid SPH particle a is written as:

$${\overline {\rho } _a}={\varepsilon _a}{\rho _a},$$

(21)

where ρ_a is the actual density of SPH particle a.

SPH motion equations for fluid phase

The motion equations for fluid phase are based on the N-S equations, and the continuity and momentum equations in SPH formulation based on the local average technique are given by:

$$\frac{{{\text{d}}{{\overline {\rho } }_a}}}{{{\text{d}}t}}={\overline {\rho } _a}\sum\limits_{{b=1}}^{N} {\frac{{{m_b}}}{{{{\overline {\rho } }_b}}}(} \vec {v}_{a}^{\alpha } - \vec {v}_{b}^{\alpha }) \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\alpha }}},$$

(22)

$$\frac{{{\text{d}}\vec {v}_{a}^{\alpha }}}{{{\text{d}}t}}=\sum\limits_{{b=1}}^{N} {{m_b}\left[ { - \left( {\frac{{{P_a}}}{{\overline {\rho } _{a}^{2}}}+\frac{{{P_b}}}{{\overline {\rho } _{b}^{2}}}} \right)+\Pi _{{ab}}^{{visc}} - {\pi}_{{ab}}^{{art}}} \right]} \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\alpha }}}+\vec {F}_{{f - a}}^{\alpha }+\vec {F}_{{f - s}}^{\alpha }+{\vec {g}^\alpha },$$

(23)

where $\:{\overrightarrow{\text{v}}}_{\text{a}}$ and P_a are the velocity and pressure of SPH particle a, respectively. $\:{\overrightarrow{\text{F}}}_{\text{f}-\text{a}}$ and $\:{\overrightarrow{\text{F}}}_{\text{f}-\text{s}}$ are the abrasive-fluid coupling force and the solid-fluid contact force, respectively.

In Eq. (23), the first term is the pressure term, which is determined by the equation of state (EOS). For the fluid phase, the Mie-Gruneisen EOS equation is adopted to describe the pressure wave effect of the fluid medium during the impact, which can be expressed as⁴¹:

$$P=\frac{{{\rho _0}c_{0}^{2}\eta \left[ {1+\left( {1 - \frac{{{\Gamma _0}}}{2}} \right)\eta - \frac{a}{2}{\eta ^2}} \right]}}{{{{\left[ {1 - ({S_1} - 1)\eta - {S_2}\frac{{{\eta ^2}}}{{\eta +1}} - {S_3}\frac{{{\eta ^2}}}{{{{(\eta +1)}^2}}}} \right]}^2}}}+({\Gamma _0}+a\eta )e,$$

(24)

where η = $\:\stackrel{\text{-}}{\rho}$/ρ₀ − 1, and ρ₀ is the reference density. e is the internal energy per unit of mass. The EOS parameters of the fluid phase are listed in Table 1.

Table 1 EOS parameters for fluid phase⁴².

The second term $\textstyle\prod_{ab}^{visc}$  in Eq. (23) is the physical viscous force term, which is used to calculate the viscous force in Newtonian fluids and can be expressed as⁴³:

$$\Pi _{{ab}}^{{visc}}=\sum\limits_{{b=1}}^{N} {{m_b}\left[ {\frac{{4{\mu _f}{{\vec {r}}_{ab}}}}{{({{\bar {\rho }}_a}+{{\bar {\rho }}_b}){{\left| {{{\vec {r}}_{ab}}} \right|}^2}}}} \right]} \cdot {\vec {v}_{ab}}$$

(25)

where µ_f is the fluid kinematic viscosity.

The third term $\:{\pi}_{\text{ab}}^{\text{art}}$ is the artificial viscosity term, which is proposed by Monaghan⁴⁴ to suppress the unphysical pressure oscillations in the simulation, as shown below:

$${{\varvec{\uppi}}}_{{ab}}^{{art}}=\left\{ {\begin{array}{*{20}{l}} {\frac{{ - {\alpha _{{\varvec{\uppi}}}}{{\bar {c}}_{ab}}{\phi _{ab}}+{\beta _{{\varvec{\uppi}}}}{\phi _{ab}}^{2}}}{{{{\bar {\rho }}_{ab}}}},{{\vec {v}}_{ij}} \cdot {{\vec {x}}_{ij}}<0} \\ {0,{{\vec {v}}_{ij}} \cdot {{\vec {x}}_{ij}} \geqslant 0} \end{array}} ,\right.$$

(26)

where c_a is the sound speed of SPH particle a, $\:{\stackrel{\text{-}}{\text{c}}}_{\text{ab}}\text{=}\text{(}{\text{c}}_{\text{a}}\text{+}{\text{c}}_{\text{b}}\text{)}\text{/}\text{}\text{2}$. $\:{\stackrel{\text{-}}{\rho}}_{\text{ab}}\text{=}\text{(}{\stackrel{\text{-}}{\rho}}_{\text{a}}+{\stackrel{{-}}{\rho}}_{\text{b}}\text{)}\text{/}\text{2}$. The coefficient ${\phi _{ab}}$ is expressed as:

$${\phi _{ab}}=\frac{{{{\bar {h}}_{ab}}\left( {{{\vec {v}}_{ab}} \cdot {{\vec {x}}_{ab}}} \right)}}{{{{\left| {{{\vec {x}}_{ab}}} \right|}^2}+{{\left( {0.1{{\bar {h}}_{ab}}} \right)}^2}}},$$

(27)

where $\:{\stackrel{\text{-}}{\text{h}}}_{\text{ab}}\text{=}\text{(}{\text{h}}_{\text{a}}\text{+}{\text{h}}_{\text{b}}\text{)}\text{/}\text{2},\:{\overrightarrow{\text{v}}}_{\text{ab}}\text{=}{\overrightarrow{\text{v}}}_{\text{a}}-{\overrightarrow{\text{v}}}_{\text{b}}\:\text{a}\text{n}\text{d}\:{\overrightarrow{\text{x}}}_{\text{ab}}={\overrightarrow{\text{x}}}_{\text{a}}-{\overrightarrow{\text{x}}}_{\text{b}}.$

α_π and β_π in Eq. (26) are the energy dissipation coefficients, which are determined by medium characteristics. α_π is related to the volumetric viscosity, while β_π is applied to prevent the penetration between SPH particles during high-speed impacts. Monaghan chose α_π = β_π = 0.01 when dealing with the free surface flow problems, and suggested that values of α_π and β_π can be around 1.0 in most cases⁴⁵. Randles et al. chose α_π = β_π = 2.5 when studying the high-speed impact (~ 1 km/s) of metal materials such as copper and steel⁴⁶. The total system energy is not conserved due to the dissipation of the particles’ kinetic energy with artificial viscosity term. Therefore, on the premise of eliminating non-physical oscillations, the energy dissipation coefficients should be selected as small as possible to ensure the numerical stability. In this study, α_π = β_π = 0.1 is used for the fluid phase and α_π = β_π = 1.0 for the solid phase.

SPH modelling for solid phase

Similar to the fluid phase, the motion equations for the solid SPH particles are shown as follows:

$$\frac{{{\text{d}}{\rho _a}}}{{{\text{d}}t}}={\rho _a}\sum\limits_{{b=1}}^{N} {\frac{{{m_b}}}{{{\rho _b}}}\vec {v}_{{ab}}^{\alpha }} \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\alpha }}},$$

(28)

$$\frac{{{\text{d}}\vec {v}_{a}^{\alpha }}}{{{\text{d}}t}}=\sum\limits_{{b=1}}^{N} {{m_b}\left[ { - \left( {\frac{{{P_a}}}{{\rho _{a}^{2}}}+\frac{{{P_b}}}{{\rho _{b}^{2}}}} \right) - {\pi}_{{ab}}^{{art}}} \right]} \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\alpha }}}+\sum\limits_{{b=1}}^{N} {{m_b}\left( {\frac{{\vec {\tau }_{a}^{{\alpha \beta }}}}{{\rho _{a}^{2}}}+\frac{{\vec {\tau }_{b}^{{\alpha \beta }}}}{{\rho _{b}^{2}}}} \right)} \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\beta }}}+\vec {F}_{{a - s}}^{\alpha }+\vec {F}_{{f - s}}^{\alpha }+{\vec {g}^\alpha },$$

(29)

where the second term in Eq. (29) is the shear stress term, and ${\overrightarrow\tau}_a$ is the deviatoric stress tensor.

According to the solid mechanics, $\:{\overrightarrow{\tau}}_{\text{a}}$ is determined by the deviator stress rate $\dot {\vec {\tau }}_{a}^{{}}$. $\dot {\vec {\tau }}_{a}^{{}}$ is proportional to the strain rate $\dot {\vec {\varepsilon }}$, and the proportional coefficient is the shear modulus G. For explicit dynamic simulation, the deviator stress rate is obtained by Hook’s law with the Jaumann rate correction, expressed as³⁵:

$$\dot {\vec {\tau }}_{a}^{{}}=\frac{{{\text{d}}\vec {\tau }_{a}^{{\alpha \beta }}}}{{{\text{d}}t}}=2G(\dot {\vec {\varepsilon }}_{a}^{{\alpha \beta }} - \frac{1}{3}{\delta ^{\alpha \beta }}\dot {\vec {\varepsilon }}_{a}^{{\gamma \gamma }})+\vec {\tau }_{a}^{{\alpha \gamma }} \cdot \dot {\vec {R}}_{a}^{{\beta \gamma }}+\vec {\tau }_{a}^{{\gamma \beta }} \cdot \dot {\vec {R}}_{a}^{{\alpha \gamma }},$$

(30)

where $\dot {\vec {\varepsilon }}_{a}^{{}}$ and $\dot {\vec {R}}_{a}^{{\beta \gamma }}$ are the strain rate tensor and rotation rate tensor, respectively, as shown in follows⁴⁷:

$$\left\{ {\begin{array}{*{20}{c}} {\dot {\vec {\varepsilon }}_{a}^{{\alpha \beta }}=\frac{1}{2}\sum\limits_{{b=1}}^{N} {\left( {\frac{{{m_b}}}{{{\rho _b}}}\vec {v}_{{ba}}^{\alpha } \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\beta }}}+\frac{{{m_b}}}{{{\rho _b}}}\vec {v}_{{ba}}^{\beta } \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\alpha }}}} \right)} } \\ {\dot {\vec {R}}_{a}^{{\alpha \beta }}=\frac{1}{2}\sum\limits_{{b=1}}^{N} {\left( {\frac{{{m_b}}}{{{\rho _b}}}\vec {v}_{{ba}}^{\alpha } \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\beta }}} - \frac{{{m_b}}}{{{\rho _b}}}\vec {v}_{{ba}}^{\beta } \cdot \frac{{\partial {W_{ab}}}}{{\partial \vec {x}_{a}^{\alpha }}}} \right)} } \end{array}} .\right.$$

(31)

The deviatoric stress tensor $\:{\overrightarrow{\tau}}_{\text{a}}$ can be obtained by integrating Eq. (30) with time, and the integration is carried out when the material is in the elastic deformation stage. If the stress exceeds the yield limit, the plastic deformation occurs. The Von-Mises yield criterion is adopted in this study. For solid SPH particle a, when the Von-Mises equivalent stress ${\bar {\sigma }_{{\text{VM}}}}=\sqrt {\frac{3}{2}\vec {\tau }_{a}^{{\alpha \beta }}\vec {\tau }_{a}^{{\alpha \beta }}}$ exceeds the yield stress $\:{\sigma}_{\text{y}}$, the deviatoric stress tensor ${\overrightarrow\tau}_a$ should be brought back to the yield surface:

$${\vec {\tau }_a}=\frac{{{\sigma _y}}}{{{{\bar {\sigma }}_{{\text{VM}}}}}}{\vec {\tau }_a}.$$

(32)

The yield stress $\:{\sigma}_{\text{y}}$ can be solved by the constitutive model, which is composed of strength equation, failure equation and EOS equation. It is necessary to select a suitable constitutive model for the solid phase to study the mechanical behavior of the targeted material by AWJ impact. For ductile materials, the surface erosion process is mainly plastic deformation caused by abrasive particle impact. For brittle materials, the erosion process is mainly the surface fragmentation caused by crack diffusion.

Ductile material

The J-C constitutive model is selected to model the ductile materials, which is widely used to describe the strength changes of ductile materials (such steel, copper and aluminum) in high strain, high strain rate or high temperature conditions. The strength equation of J-C model is written as⁴⁸:

$${\sigma _y}=\left[ {A+B{{\left( {{\varepsilon _p}} \right)}^N}} \right]\left[ {1+C\ln \left( {\frac{{\dot {\varepsilon }_{p}^{{}}}}{{\dot {\varepsilon }_{0}^{{}}}}} \right)} \right]\left[ {1 - {{({T^*})}^M}} \right],$$

(33)

where ${\varepsilon _p}=\int {\dot {\varepsilon }_{p}^{{}}} dt$ is the equivalent plastic strain, and $\dot {\varepsilon }_{p}^{{}}=\sqrt {\frac{2}{3}\dot {\vec {\varepsilon }}_{a}^{{\alpha \beta }}\dot {\vec {\varepsilon }}_{a}^{{\alpha \beta }}}$ is the equivalent plastic strain rate. $\:{\dot{\epsilon\:}}_{\text{0}}=\text{1.0}$s^-1 is the reference strain rate. A, B, C, N and M are material dependent constants. T^* is the normalized temperature given by:

$${T^*}=\frac{{T - {T_0}}}{{{T_m} - {T_0}}},$$

(34)

where T is the actual temperature. T₀ and T_m are the reference temperature and melting temperature of the ductile material, respectively.

When the plastic deformation occurs, the failure equation is used to describe the failure process of solid phase. The failure equation of J-C model is written as⁴⁸:

$${\varepsilon _f}=\left[ {{D_1}+{D_2}\exp \left( {{D_3}\left( {\frac{{{\sigma _m}}}{{{{\bar {\sigma }}_{{\text{VM}}}}}}} \right)} \right)} \right]\left[ {1+{D_4}\ln \left( {\frac{{\dot {\varepsilon }_{p}^{{}}}}{{\dot {\varepsilon }_{0}^{{}}}}} \right)} \right]\left[ {1+{D_5}{T^*}} \right],$$

(35)

where $\:{\epsilon}_{\text{f}}$ is the failure strain, and D₁ ~ D₂ are the material dependent constants. $\:{\sigma}_{\text{m}}$ is the mean principal stress, which is equal to the isotropic pressure P.

The cumulative damage failure criterion is adopted to model the material removal, and the damage factor D is introduced to describe the damage accumulation. For solid SPH particle a, when D ≥ 1.0, the material is considered to be completely fractured. The deviator stress $\:{\overrightarrow{\tau}}_{\text{a}}$ is set to zero, and the particle is deleted to represent the material removal, as shown in follows:

$$D=\sum {\frac{{\Delta {\varepsilon _p}}}{{{\varepsilon _f}}}},$$

(36)

where $\:\Delta{\epsilon}_{\text{p}}$ is the equivalent plastic strain increment of the current time step.

Similar to the fluid phase, the Mie-Grüneisen EOS equation is also applied for ductile materials written as⁴⁶:

$$P=\frac{{{\rho _0}c_{0}^{2}\eta \left[ {1+\left( {1 - \frac{{{\Gamma _0}}}{2}} \right)\eta } \right]}}{{\left[ {1 - \left( {{S_a} - 1} \right)\eta } \right]}}+{\rho _0}{\Gamma _0}e,$$

(37)

where S_a is the linear Hugoniot slope coefficient.

The J-C model parameters of two common ductile materials are listed in Table 2.

Table 2 J-C model parameters of common ductile materials for solid phase.

Brittle material

The JH-2 constitutive model is selected to model the brittle materials, which is developed to study the stress-strain behavior of brittle materials (such as ceramics, glass and rock) in large impact and large deformation conditions. Based on the JH-1 model, the JH-2 strength equation introduces the damage factor D to describe the softening effect of the brittle materials during the impact process, as shown below⁵⁰:

$$\left\{ {\begin{array}{*{20}{c}} {\sigma _{i}^{*}=A{{\left( {{P^*}+{T^*}} \right)}^N}\left( {1+C\ln {{\dot {\varepsilon }}^*}} \right){\text{ }}D=0{\text{ }}({\text{Intact}}){\text{ }}} \\ {{\sigma ^*}=\sigma _{i}^{*} - D\left( {\sigma _{i}^{*} - \sigma _{f}^{*}} \right){\text{ 0}}<D<1.0{\text{ }}({\text{Damaged}}){\text{ }}} \\ {\sigma _{f}^{*}=B{{\left( {{P^*}} \right)}^M}\left( {1+C\ln {{\dot {\varepsilon }}^*}} \right) \leqslant \sigma _{{f\hbox{max} }}^{*}{\text{ }}D=1.0{\text{ }}({\text{Fracture}})} \end{array} } ,\right.$$

(38)

where the superscript * indicates normalization processing. The normalized strengths ($\:{\sigma}^{\ast}$, $\:{\sigma}_{\text{i}}^{\ast}$, $\:{\sigma}_{\text{f}}^{\ast}$ and $\:{\sigma}_{\text{f}\text{\:}\text{max}}^{\ast}$) have the general form $\:{\sigma}^{\ast}={\sigma}_{\text{y}}/{\sigma}_{\text{HEL}}$, where $\:{\sigma}_{\text{HEL}}$ is the equivalent stress at Hugoniot Elastic Limit (HEL). A, N, C, B and M are material dependent constants. P^* = P/P_HEL and T^* = T/P_HEL are the hydrostatic pressure and the maximum tensile strength normalized by the pressure P_HEL at HEL, respectively. $\:{\dot{\epsilon}}^{\ast}\text{=}{\dot{\epsilon}}^{\text{p}}$ is the dimensionless strain rate, and $\:{\dot{\epsilon\:}}_{\text{0}}=\text{1.0}$s^-1 is the reference strain rate.

Similar to the ductile materials, the JH-2 failure equation for the brittle materials is given by⁵⁰:

$$D=\sum {\frac{{\Delta {\varepsilon _p}}}{{{\varepsilon _f}}}} =\sum {\frac{{\Delta {\varepsilon _p}}}{{{D_1}{{({P^*}+{T^*})}^{{D_2}}}}}},$$

(39)

where D₁ and D₂ are material constants. When the brittle material is completely fractured (D ≥ 1), the deviator stress $\:{\overrightarrow{\tau}}_{\text{a}}$ is constant at zero, and the equivalent plastic strain $\:{\epsilon}_{\text{p}}$ no longer accumulates.

The polynomial EOS equation is employed to solve the hydrostatic pressure P for brittle materials, which is actually another general form of the Mie-Grüneisen equation expressed as⁵¹:

$$P=\left\{ {\begin{array}{*{20}{l}} {{\text{ }}{K_1}\mu +{K_2}{\mu ^2}+{K_3}{\mu ^3}+\Delta p{\text{ }}\mu \geqslant 0{\text{ }}({\text{Compression}})} \\ {{\text{ }}{K_1}\mu {\text{ }}\mu <0{\text{ }}({\text{Tension}}){\text{ }}} \end{array}} ,\right.$$

(40)

where µ = ρ/ρ₀-1 is the relative volume factor. K₁ ~ K₃ are material constants. ΔP is the pressure increment, representing the transformation of elastic internal energy to hydrostatic potential energy, and reflecting the expansion effect of brittle materials during the damage process. It varies from ΔP = 0 at D = 0 to ΔP = ΔP_max at D = 1.0 given by:

$$\Delta {p_{n{\text{+1}}}}= - {K_1}\mu +\sqrt {{{({K_1}\mu +\Delta {p_n})}^2}+2\beta {K_1}\Delta U},$$

(41)

where β is the energy conversion coefficient (0 ≤ β ≤ 1.0), and ΔU = U_n+1-U_n is the internal energy loss during a single time step. The JH-2 model parameters of two common brittle materials are listed in Table 3.

Table 3 JH-2 model parameters of common brittle materials for solid phase.

The radial return algorithm is adopted to solve the deviator stress term ${\overrightarrow\tau}_a$for the solid phase. $\:{{\text{(}\overrightarrow{\tau}}_{\text{a}}\text{)}}_{\text{n}\bullet\Delta\text{t}}$ is the deviator stress of solid SPH particle a at the beginning of the nth time step. Each time step $\:\Delta\text{t}$ contains two small cycles, and each cycle is $\:\Delta\text{t}\text{/2}$. The implementation procedures of $\:{\overrightarrow{\tau}}_{\text{a}}$ for the solid phase is shown in Fig. 6. For ductile and brittle materials, different constitutive models are used to solve the isotropic pressure P_a, the yield stress $\:{\sigma}_{\text{y}}$, and the damage factor D, while the solving process for other parameters is the same.

Figure 6

Implementation procedures of deviatoric stress term ${\overrightarrow\tau}_a$for solid phase (In a single time step)

Unresolved coupling between SPH and DEM methods

Coupling force for abrasive-fluid interaction based on local averaging technique

The essence of SPH-DEM coupling is to realize the interaction between the two kinds of particles. In unresolved coupled method, the coupling force $\:{\overrightarrow{\text{F}}}_{\text{a-f}}$ for abrasive-fluid interaction in Eq. (1) is shown in Fig. 7. According to the smoothing function, the coupling radius between the two phases is R_c = 2h_c. For DEM particle i, $\:{\overrightarrow{\text{F}}}_{\text{a-f}}$ is expressed as⁴⁰:

$${\vec {F}_{a - f}}={V_i}{( - \nabla P+\nabla \cdot \vec {\tau })_i}+{\vec {F}_d},$$

(42)

where the first term V_i(−∇P+∇·$\:{\overrightarrow{\tau}}_{\text{i}}$) is the average hydrodynamic force, which represents the buoyancy and shear force of the fluid phase acting on DEM particles within R_c, and V_i is the particle volume. The second term $\:{\overrightarrow{\text{F}}}_{\text{d}}$ is the drag force.

The Shepard corrected SPH interpolation is employed to solve the average hydrodynamic force term, as show in follows⁵⁴:

$$\left\{ {\begin{array}{*{20}{l}} {{V_i}{{( - \nabla P+\nabla \cdot \vec {\tau })}_i}=\frac{1}{{\sum\limits_{a} {\frac{{{m_a}}}{{{\rho _a}}}{W_{ia}}({h_c})} }}\sum\limits_{a} {{m_a}{\theta _a}} {W_{ia}}({h_c})} \\ {{\theta _a}=\sum\limits_{b} {{m_b}\left[ { - \left( {\frac{{{P_a}}}{{\overline {\rho } _{a}^{2}}}+\frac{{{P_b}}}{{\overline {\rho } _{b}^{2}}}} \right)+\Pi _{{ab}}^{{visc}}} \right]} \cdot {\nabla _a}{W_{ab}}(h)} \end{array}} ,\right.$$

(43)

where the subscript a represents the SPH particle within the coupling radius of DEM particle i, and the subscript b represents the SPH particle within the support domain of particle a.

Figure 7

SPH-DEM unresolved coupling force for abrasive-fluid interaction.

The drag force term $\:{\overrightarrow{\text{F}}}_{\text{d}}$ acting on DEM particle i is due to the repulsion of surrounding fluid SPH particles, depending on average porosity $\:{\epsilon}_{\text{i}}$ and relative velocity $\:{\overrightarrow{\text{v}}}_{\text{af}}$ at point i, expressed as:

$${\vec {F}_d}=\frac{\beta }{{1 - \epsilon {\ominus _i}}}{V_i} \cdot {\vec {v}_{af}},$$

(44)

where $\:{\epsilon}_{\text{i}}$ and $\:{\overrightarrow{\text{v}}}_{\text{af}}$ are given by:

$$\left\{ {\begin{array}{*{20}{l}} {\epsilon {\ominus _i}=\frac{{\sum\limits_{a} {{\epsilon _a}{V_a}} {W_{ia}}({h_c})}}{{\sum\limits_{a} {{V_a}} {W_{ia}}({h_c})}}} \\ {{{\vec {v}}_{af}}={{\bar {v}}_f} - {{\vec {v}}_i}} \end{array}} ,\right.$$

(45)

where $\:{\overrightarrow{\text{v}}}_{\text{i}}$ is the center velocity of DEM particle i. The average fluid velocity $\:{\stackrel{\text{-}}{\text{v}}}_{\text{f}}$ at point i can be approximated by the Shepard correction method expressed as:

$${\bar {v}_f}=\frac{{\sum\limits_{a} {{{\vec {v}}_a}{V_a}} {W_{ia}}({h_c})}}{{\sum\limits_{a} {{V_a}} {W_{ia}}({h_c})}},$$

(46)

In Eq. (44), β is the momentum exchange coefficient between two phases, which is determined by the average porosity $\:{\epsilon}_{\text{i}}$⁵⁵:

$$\beta =\left\{ {\begin{array}{*{20}{l}} {150\frac{{{{(1 - {\epsilon _i})}^2}}}{{{\epsilon _i}}}\frac{{{\mu _f}}}{{d_{i}^{2}}}+1.75(1 - {\epsilon _i})\frac{{{\rho _f}}}{{{d_i}}}\left| {{{\vec {v}}_{af}}} \right|{\epsilon _i}0.8} \\ {0.75{C_d}\frac{{{\epsilon _i}(1 - {\epsilon _i})}}{{{d_i}}}{\rho _f}\left| {{{\vec {v}}_{af}}} \right|\epsilon _{i}^{{ - 2.65}}{\epsilon _i}0.8} \end{array}} ,\right.$$

(47)

where C_d is the drag coefficient given by:

$${C_d}=\left\{{\begin{array}{l}{\ast{20}{l}}{\frac{{24}}{{R{e_i}}}(1+0.15Re_{i}^{{0.687}}),R{e_i}1000}\\{0.44,R{e_i}1000} \end{array}},\right.$$

(48)

where Re_i is the Reynolds number around DEM particle i expressed as:

$$R{e_i}=\frac{{\left| {{{\vec {v}}_{af}}} \right|{\epsilon _i}{\rho _f}{d_i}}}{{{\mu _f}}}.$$

(49)

In order to satisfy Newton’s third law, the coupling force $\:{\overrightarrow{\text{F}}}_{\text{f-a}}$ of SPH particle a subjected to DEM particles in Eq. (23) is the weighted average of $\:{\overrightarrow{\text{F}}}_{\text{a-f}}$ acting on DEM particles nearby, which can be given by:

$$\left\{ {\begin{array}{*{20}{l}} {{{\vec {F}}_{f - a}}= - \frac{{{m_a}}}{{{{\bar {\rho }}_a}}}\sum\limits_{i} {\frac{1}{{{S_i}}}} {{\vec {F}}_{a - f}}{W_{ai}}({h_c})} \\ {{\text{ }}{S_i}{\text{ }}=\sum\limits_{b} {\frac{{{m_b}}}{{{{\bar {\rho }}_b}}}} {W_{bi}}({h_c})} \end{array}} ,\right.$$

(50)

where the subscript b represents the SPH particle within the coupling radius R_c of DEM particle i.

Time integration scheme

The time step length Δt is crucial for the explicit dynamic simulation. If Δt is too large, it will result in too few time steps for SPH&DEM particles to come into contact, which affects the numerical stability, and even causes the calculation to abort. If Δt is too small, it needs more time steps to simulate the same physical time period, which increases time cost and reduces computational efficiency. For SPH particles, the Courant-Friedrichs-Levy (CFL) condition is adopted, which requires Δt to be proportional to the smallest spatial resolution. For DEM particles, Δt is related to the normal stiffness coefficient k_n and can be expressed as⁵⁶:

$$\left\{ {\begin{array}{*{20}{c}} {\Delta {t_{}}=\hbox{min} \left( {\Delta {t_{sph}},\Delta {t_{dem}}} \right)} \\ {\Delta {t_{sph}}={{\hbox{min} }_{a{\text{th}} - {\text{particle}}}}\left( {\frac{{{C_F}{h_a}}}{{{c_a}+\left| {{{\vec {v}}_a}} \right|}}} \right)} \\ {\Delta {t_{dem}}=\frac{{2\pi }}{{{\alpha _{tn}}}}\sqrt {\frac{{{m_i}}}{{{k_n}}}} } \end{array} } ,\right.$$

(51)

where C_F = 0.2 is the CFL coefficient, h_a, c_a, and $\left| {{{\vec {v}}_a}} \right|$ are the smoothing length, sound velocity, and maximum velocity of the SPH particle, respectively. α_tn = 300 is the model coefficient, and m_i is the mass of the DEM particle. If the difference between Δt_sph and Δt_dem is less than one order of magnitude, the smaller one will be selected as the time step of SPH-DEM coupled model to ensure the numerical stability.

As shown in Eq. (51), the time step Δt_sph for SPH particles is determined by the sound velocity c_a and the maximum velocity $\left| {{{\vec {v}}_a}} \right|$. In this paper, C_F = 0.2, h_a = 1.25d_ini = 6.25 × 10^-5m, c_a = 3940 m/s (sound velocity of OFHC copper), $\left| {{{\vec {v}}_a}} \right|$ = 600 m/s, and Δt_sph is calculated as 2.75 × 10^-9s. For DEM particles, Δt_dem is a constant value, which is related to the diameter and material properties. In this paper, d_dem = 0.2 mm, ρ_dem = 4120 kg/m³, E = 248GPa, υ = 0.3, and Δt_dem is calculated as 6.34 × 10^-9s. Therefore, considering computational efficiency and numerical stability, the time step is set to Δt = 2.0 × 10^-9s.

The flow chat of unresolved SPH-DEM coupled model is shown in Fig. 8, which is implemented by a Fortran code. The leap-frog method is applied for the time integration scheme, and the link-list method is used for neighboring particle-pair searching (NPPS). The internal forces of SPH particles include pressure term P_a, viscosity term $\Pi _{{ab}}^{{visc}}$& $\:{\pi}_{\text{ab}}^{\text{art}}$, and deviator stress term ${\overrightarrow\tau}_a$, which have been shown in Eqs. (23) and (29).

Figure 8

Flowchart of unresolved SPH-DEM coupled model for AWJ impact.

Results and discussion

In this section, several simulation cases of AWJ impact are conducted based on the unresolved SPH-DEM model above, including the single AWJ impact on rigid wall (Sect. 4.1), continuous AWJ impact on ductile materials (Sect. 4.2) and brittle materials (Sect. 4.3).

The 2D AWJ model is shown in Fig. 9, including geometrical parameters (model thickness & width) and AWJ incident parameters (incident angle α, incident velocity v_jet, AWJ diameter d_jet and distance between adjacent abrasives in Y direction y_dis). The geometrical parameters depend on AWJ incident parameters. Fluid SPH particles and abrasive DEM particles are continuously generated at the inlet and exit the computation domain through outlets on both sides to simulate the continuous impact process over a long time scale. In the actual AWJ cutting process, the solid specimen is fixed by surrounding fixtures. Therefore, the left and right sides of the solid model are set as fixed constraints during the simulation process, while the upper and lower boundaries are not constrained.

Figure 9

Illustration of unresolved SPH-DEM model for continuous AWJ impact.

Single abrasive water-jet impact on rigid wall

For the unresolved SPH-DEM model, the most critical numerical parameter is the coupling radius R_c between SPH and DEM particles. Two type of particles within R_c interact through the local porosity. If R_c is too small, the coupling force between abrasive and fluid phases is not enough to reflect the abrasive-fluid interaction. If R_c is too large, the local characteristics of the flow field and the abrasives cannot be captured, and a large amount of calculation is required. In general, R_c should be larger than the diameter of the DEM particle at least, but also small enough to capture the local characteristics of the flow field.

In order to determine the optimal value of R_c, a model of single abrasive water-jet impact is established, as shown in Fig. 10(a). The water-jet contains only one abrasive particle to study the motion under the action of fluid SPH particles around. The circular abrasive particle has a diameter of d_ab = 0.20 mm, and consists of a group of DEM particles (Fully resolved) or a single DEM particle (Unresolved). The fully resolved SPH-DEM model for AWJ impact has been discussed in our previous research⁵⁷. The solid phase is set as the rigid body, ignoring the deformation during the impact process. Other numerical parameters are listed in Table 4.

Table 4 Numerical parameters for continuous AWJ impact simulation.

When the water-jet impacts the solid surface, the shape of water-jet undergoes a drastic change and ultimately leaves from the outlets on both sides. The velocity and pressure distribution of fluid phase, once the water-jet shape has stabilized, is shown in Fig. 10(b) and 10(c). At the center of the water-jet flow, the fluid SPH particles have the highest pressure and the lowest velocity. It indicates that as the abrasive particles gradually approach the solid surface from inlet, the pressure and drag effects from the fluid phase located at the center are greatly enhanced, causing the abrasive particles to decelerate before they contact with the solid phase.

Figure 10

Single abrasive water-jet numerical simulation.

Time history of the single abrasive particle’s velocity is shown in Fig. 11, for the time period from t = 0 to contact with the rigid solid wall. In the fully resolved model, the velocity of abrasive particle in contact with the solid is about 162 m/s. For the unsolved model, the abrasive contact velocity is affected by the coupling radius R_c. When R_c = 1.0d_dem, the abrasive velocity fluctuates in a stepwise manner, with greater amplitude. Due to the small size of R_c, the number of fluid SPH particles interacting with the single abrasive DEM particle is insufficient, making it difficult to reflect the actual flow of particles. With the increase of R_c, the variation of particle velocity curve tends to be gentle, and the particle contact velocity gradually increases. When R_c ≥ 2.5d_dem, due to the limitation of jet diameter d_jet = 1.0 mm, increasing R_c has little effect on the simulation results, so the coincidence degree of the curves is relatively high. The particle contact velocity is about 166 m/s at R_c = 2.0d_dem, which is the closest to the fully resolved model, and the curve is basically consistent. Therefore, the coupling radius R_c of the unresolved SPH-DEM model for AWJ simulation is selected as R_c = 2.0d_dem.

Figure 11

Time history of single abrasive’s velocity (Before colliding with rigid solid wall).

Continuous AWJ impact on ductile materials

Effect of impact time T

In this section, the OFHC copper is selected as the ductile material, and the material parameters are listed in Table 2. The simulation result of ductile surface evolution process by continuous AWJ impact is shown in Fig. 12. The incident velocity v_jet is 500 m/s, and other numerical parameters are listed in Table 4.

Figure 12

Evolution of ductile surface evolution by continuous AWJ impact.

When impact time T = 2.5ms, a series of discontinuous pits appear on the ductile surface, as show in Fig. 12(a). With the increase of T, these smaller pits gradually merge into a bigger crater, and the initial cross-section profile of the crater is V-shaped, as show in Fig. 12(b). When T = 10.0ms, the crater develops to the material interior, and the cross-section profile changes constantly, eventually transforming into U-shape at T = 30.0ms, as show in Fig. 12(c ~ e). Subsequently, the crater width is basically unchanged, while the crater depth continues to increase, as show in Fig. 12(f ~ h). Figure 13 shows the time history of ductile surface erosion process, which can be divided into three stages. In Fig. 13(a), Stage One (0 ≤ T < 3.2ms) is the equivalent plastic strain accumulation stage. When AWJ contacts the ductile surface at T = 0, the plastic deformation occurs, and the equivalent plastic strain begins to accumulate. Stage Two (3.2 ≤ T < 12.5ms) is the erosion crater formation stage. The cross-section profile of the crater is constantly changing, forming the V-shaped first, and then transitioning to the U-shaped. The erosion mass rate of the ductile material is unstable. Stage Three (T$\geq$12.5ms) is the stable erosion stage. The erosion mass rate is constant, and the crater begins to develop inside the material. The cross-section shape of the crater is no longer changed (U-shaped), but the size gradually increases. The erosion mechanism of AWJ on ductile materials is the material failure caused by plastic strain accumulation.

Figure 13

Time history of ductile surface erosion by continuous AWJ impact.

Figure 13(b) shows the relationship between the crater size and the impact time. When T < 10.0ms, the crater width C_w increases sharply, while the crater depth C_d increases slightly. During this period (in Stage One and Two), the ductile surface is basically intact and no significant material removal occurs. When 20.0 ≤ T < 50.0ms, C_w increases less rapidly, which is in the stable erosion stage (Stage Three). When T≥ 50.0ms, C_w is basically unchanged, while C_d increases linearly during the whole erosion process. It can be seen that as the impact time increases, the erosion area no longer extends to both sides, but develops deeper inside the material.

The abrasive moves by the contact of fluid phase, targeted material, and other abrasives, as shown in Fig. 14. Abrasive No.1 located at the center area is embedded into the ductile surface under the squeezing effect of water-jet, and collides with the subsequent abrasives No.2 ~ 4. The initial position of abrasive No.2 is near the center area. After colliding with abrasive No.1, abrasive No.2 deviates and finally contacts with the ductile surface far from the center area, which expands the erosion region. Abrasive No.4 does not touch the ductile surface, but flows to the outlet under the action of fluid phase and other abrasives. It can be seen from Fig. 10(a) that the AWJ has the lowest velocity in the center area, causing the abrasives to gather and collide. This is the main reason for the sharp increase in the crater width during Stage One and Two in Fig. 13(b).

Figure 14

Motion trajectory of abrasives at center area.

In the same incident condition (v_jet = 500 m/s and T = 30.0ms), the simulation results of OFHC copper surface erosion by pure water-jet and abrasive water-jet impact are shown in Fig. 15. Under the continuous impact of pure water-jet, local plastic deformation and a small amount of material removal occur on the ductile surface, resulting in a wide and shallow crater with 6.89 mm in width and 0.54 mm in depth. Under the continuous impact of AWJ, a large amount of material removal occurs, and the crater is narrow and deep with a U-shaped of 6.18 mm in width and 1.16 mm in depth. Therefore, the erosion ability of AWJ to the targeted material is much greater than that of pure water-jet. As can be seen from Fig. 15(b), the water-jet not only accelerates the abrasives, but also cleans the material surface. The abrasives and the material debris located at the crater bottom flow to outlet on both sides by the continuous water-jet, which reduces the obstruction of the accumulation of abrasives and materials on impact, and provides space for subsequent abrasives.

Figure 15

Simulation results of OFHC copper surface erosion by water-jet impact in two types.

Effect of incident velocity v _jet

The incident velocity v_jet is defined as the flow velocity of AWJ at the inlet, which is one of the key factors affecting jet erosion ability. In the actual AWJ cutting process, the v_jet value cannot be directly set, and it is determined by adjusting the working pressure P of the booster cylinder. The relationship between P and v_jet can be solved by Bernoullis’s equation, expressed as⁵⁸:

$${v_{jet}}=\kappa \sqrt {\frac{{2{P_{}}}}{\rho }},$$

(52)

where ρ = 1000 kg/m³ is the density of liquid medium. κ is the volume efficiency ranging from 0.7 to 0.85⁵⁸. In this study, κ = 0.7, and Eq. (52) can be rewritten as:

$${v_{jet}}=31.3\sqrt P.$$

(53)

The SPH-DEM model proposed in this study is verified by AWJ impact experiment, which is carried out by a gantry CNC abrasive water-jet cutting machine, as shown in Fig. 16(a). The OFHC Copper and Al6061-T6 specimens are selected as the targeted ductile materials with a dimension of 120*40*8 mm and 100*40*8 mm, as shown in Fig. 16(b) and 16(c). Several cases are conducted through the impact experiment and the numerical model, and the specific experimental parameters are shown in Table 5.

Figure 16

AWJ erosion experimental system and ductile specimens.

Table 5 Experimental parameters with different impact velocities on ductile surface.

The crater profiles obtained from impact experiment and simulation results under different incident velocities are shown in Fig. 17. Among them, the targeted material of Cases 1 ~ 3 (Fig. 17(a ~ c)) are OFHC copper, and Cases 4 ~ 6 (Fig. 17(d ~ f)) are Al6061-T6. The crater profile images are taken with a microscope (maximum magnification is 1500X). Under the continuous AWJ impact, craters with different sizes are produced on the surface of two ductile materials. Due to the lower hardness of OFHC copper (35 ~ 45HB) compared to the material (90 ~ 95HB), material accumulation occurs on both sides of the crater for copper, while there is no obvious material accumulation for Al6061-T6. These phenomena are also reflected in the simulation process. Besides, the experiment and simulation results are basically the same, indicating that the unresolved SPH-DEM coupled model proposed in this study can accurately simulate the plastic deformation and material removal phenomena of the ductile materials, which can be used for AWJ study.

Figure 17

Experiment and simulation results of crater profile on ductile surface with different incident velocities.

The crater profiles obtained from simulation results for copper and Al under different incident velocities are shown in Fig. 18. In these cases, the nozzle transverse velocity v_tran = 0, the incidence angle α = 90°, and the impact time T = 30.0ms. With the increase of v_jet, the crater depth C_d and crater width C_w increase obviously. The relationship between C_d and v_jet is basically linear, which is similar to the result of Ti6Al4V material erosion process by AWJ impact in Ref⁵⁸. , as shown in Fig. 18(c). While the grow of C_w gradually slows down and tends to a stable value, the crater length-diameter ratio C_d / C_w increases. In Fig. 18(a) and (b), the erosion effect of AWJ is concentrated at the crater bottom, and the crater profile gradually develops from a V-shaped to a U-shaped with the increase of v_jet. Due to the lower density of Al6061-T6 material, although it has higher hardness, the erosion resistance is still weaker than OFHC copper, so the crater size of Al6061-T6 is significantly larger with the same incident velocity.

Figure 18

Simulation results of crater on ductile surface with different incident velocities.

Effect of abrasive mass flow rate M

Abrasive mass flow rate M is also one of the key factors affecting material removal and crater size. Under a certain working pressure, M is indirectly controlled by the opening of the sand tank outlet, so the abrasive mass flow rate cannot be set precisely during AWJ impact experiment. In the SPH-DEM model proposed in this study, a liquid column with length y_dis and width d_jet is generated at the inlet every y_dis/v_jet seconds, which is composed of SPH particles. Each liquid column contains a randomly distributed abrasive (DEM particle), as shown in Fig. 9. The relationship between abrasive volume fraction ζ and liquid column length y_dis can be obtained as follows:

$$\zeta =\frac{{{V_a}}}{{{y_{dis}} \cdot {d_{jet}}}} \times 100\%,$$

(54)

where V_a is the volume occupied by a single abrasive particle. The expression of abrasive mass flow rate M is given by:

$$M={\rho _{dem}}{v_{jet}}{d_{jet}}\zeta ={\rho _{dem}}{v_{jet}} \cdot \frac{{{V_a}}}{{{y_{dis}}}},$$

(55)

The relationship between M and y_dis is shown in Fig. 19. Therefore, M can be accurately determined from the value of y_dis in the simulation process.

Figure 19

Relationship between abrasive mass flow rate and fluid column length.

The simulation results of the profiles and sizes of the crater on OFHC copper surface with different abrasive mass flow rates are shown in Fig. 20. In these cases, v_tran = 0, α = 90°, v_jet = 350 m/s, and T = 60.0ms, the crater size increases with the increase of M. Among them, C_w is more affected, while C_d is less affected. When M$\:\geq$0.07 kg/s, C_d is basically a constant value, resulting in a decrease in C_d / C_w, as shown in Fig. 20(b). The crater profile gradually develops from a U-shaped to a V-shaped with the increase of M. The collision of abrasive particles may be the main reason, as shown in Fig. 21. When M$\:\geq$0.07 kg/s, the abrasives accumulate at the crater bottom and collide each other, making the subsequent abrasives contact the ductile surface far from the center area, or unable to make contact. It indicates that in the practical application of AWJ cutting ductile materials, it is not possible to infinitely increase the material erosion rate and crater size by increasing the abrasive mass flow rate. Excessive abrasive mass flow rate will greatly increase the crater width, affect the kerf surface quality, and cause blockage of the abrasive pipe, which will reduce the service life of the equipment.

Figure 20

Simulation results of crater on OFHC copper surface with different abrasive mass flow rates.

Figure 21

Accumulation of abrasive particles at the crater bottom (M = 0.07 kg/s).

Continuous AWJ impact on brittle materials

Effect of impact time T

In this section, the red sandstone is selected as the brittle material, and the material parameters are listed in Table 3. The simulation result of brittle surface evolution process by continuous AWJ impact is shown in Fig. 22. The incident velocity v_jet is 400 m/s, and other numerical parameters are listed in Table 4.

Figure 22

Evolution of brittle surface evolution by continuous AWJ impact.

At the instant of contact between water-jet and brittle surface (T = 0.125ms), radial cracks appear on both sides of the water-jet edge. These cracks are generated by the action of water hammer pressure before the abrasive particles contact the brittle surface. With the continuous impact of water-jet, the water hammer pressure at the center area rapidly decreases to the stagnation pressure (P_s = $\:\frac{\text{1}}{\text{2}}\rho{\text{v}}_{\text{jet}}^{\text{2}}$), so the radial crack number no longer increases, but gradually spreads inside the brittle material. The abrasive particles contact with the brittle surface at T = 0.25ms, resulting in a semi-circular pit, and the radial cracks gradually develop to both sides, forming new lateral cracks. With the continuous impact of abrasive particles, several surface pits gradually merge into a large V-shaped crater at T = 1.0ms, and new radial cracks appear at the crater bottom. The V-shaped crater expands rapidly into the targeted material along direction of the bottom radial cracks at T > 2.0ms. The crater section profile changed constantly, and the lateral cracks on crater sides continue to increase and cross each other. The crater changes from V-shaped to U-shaped at T = 4.0ms. The network cracks are generated at the same time, which greatly increases the damage area. When T > 4.0ms, the crater depth continues to expand, while the crater width no longer increases. The lateral cracks on both sides begin to spread towards the brittle surface. When the lateral cracks extend to the surface, the brittle material is fragmented, forming large fragments and material removal.

Figure 23(a) shows the time history of brittle material mass loss during AWJ impact process. Material failure and removal occurs at the moment of AWJ impact, without undergoing the equivalent plastic strain accumulation stage (Stage One). Instead, it directly enters the erosion crater formation stage (Stage Two), which is different from the ductile materials. When T > 0.9ms, the erosion process enters the stable erosion stage (Stage Three), and the brittle material mass loss increases linearly with the impact time. Figure 23(b) shows the time history of the crater size. When T≤ 1.0ms, the crater profile is a series of discontinuous surface pits. The crater width C_w increases rapidly, while the crater depth C_d increases by steps. When T > 1.0ms, the increasing trend of C_w and C_d gradually decreases. When T > 4.0ms, C_w is generally unchanged, and C_d increases linearly with the impact time. For brittle materials, it can be concluded that the relationship between the crater width and the impact time is similar to that of the ductile materials. However, the variation of crater depth is mainly reflected in the initial erosion stage (Stage One and Two). The erosion mechanism of AWJ on brittle materials is the propagation and diffusion of cracks inside the material caused by repeated impact of abrasive particles.

Figure 23

Time history of brittle surface erosion by continuous AWJ impact.

Effect of incident velocity v _jet

In this section, the red sandstone and granite specimens are selected as the targeted brittle materials with sizes of 100*100*100 mm and 80*80*80 mm, respectively, as shown in Fig. 24. Several impact experiment cases are conducted to verify the numerical model, and the specific experimental parameters are shown in Table 6.

Figure 24

Brittle specimens for AWJ impact experiment.

Table 6 Experimental parameters with different impact velocities on brittle surface.

The impact experiment and numerical simulation results of the crater profiles with different incident velocities are shown in Fig. 25. Among them, the targeted material of Cases 1 ~ 3 (Fig. 25(a ~ c)) are red sandstone, and Cases 4 ~ 6 (Fig. 25(d ~ f)) are granite. U-shaped craters with different depths appear on the sandstone surface, while the craters on the granite surface gradually transit from the initial V-shaped to U-shaped with the increase of incident velocity v_jet. When v_jet.= 313 m/s (Fig. 25(a, d)), the water-jet produces only two radial cracks on both sides, which are generated by the action of water hammer pressure at the contact moment. None of new radial cracks can be observed during the subsequent impact process. When v_jet.= 343 m/s (Fig. 25(b, e)), more radial and lateral cracks are produced on both sides. When v_jet.= 370 m/s (Fig. 25(c, f)), new radial and lateral cracks appear at the crater bottom, and these cracks continue to expand to the material interior, greatly increasing the damage area. In these experiment cases, the shapes of the crater profile of two brittle materials are generally consistent with the simulation results, indicating that the numerical model proposed in this study can accurately simulate the crack propagation and surface fragmentation of the brittle materials by AWJ impact.

Figure 25

Experiment and simulation results of crater profile on brittle surface with different incident velocities.

The simulation results of the profiles and sizes of the crater with different incident velocities are shown in Fig. 26. In these cases, the nozzle transverse velocity v_tran = 0, the incidence angle α = 90°, and the impact time T = 30.0ms. When v_jet > 325 m/s, the profile at the crater bottom gradually expands from a sharp point to a straight line segment, as shown in Fig. 26(a) and (b), indicating that the cross section shape of the crater changes from V-shaped to U-shaped with the increase of v_jet. The crater depth C_d and crater width C_w increase linearly with v_jet. Due to the absence of material accumulation on both sides, the growth rate of C_d is significantly greater than that of C_w, which is different from the ductile materials. Although the density of red sandstone and granite is similar, the compressive strength of red sandstone is obviously smaller than that of granite, so the erosion resistance of red sandstone is weaker. Besides, v_jet as well as crater size of sandstone is obviously larger with the same incident velocity, as shown in Fig. 26(c).

Figure 26

Simulation results of crater on brittle surface with different incident velocities.

Effect of abrasive mass flow rate M

The simulation results of the crater profiles on red sandstone surface with different abrasive mass flow rates are shown in Fig. 27(a). In these cases, v_tran = 0, α = 90°, v_jet = 350 m/s, and T = 30.0ms. When the abrasive mass flow rate M increases from 0.02 kg/s to 0.06 kg/s, the crater depth C_d increases from 6.58 mm to 9.51 mm. By comparison, the crater width C_w fluctuates around 1.73 mm with no significant change. Compared with C_w, C_d is more affected by M, leading to an increase in C_d / C_w. The crater profile gradually develops from V-shaped to U-shaped with the increase of M, which is different from the erosion process of ductile materials. Figure 27(b) and (c) show the relationship between the brittle material mass loss, the crater size and the abrasive mass flow rate. With the increase of M, the variation law of the mass loss depends on the crater depth C_d. When M ≤ 0.055 kg/s, the brittle material mass loss and C_d increase linearly with the increase of M. However, there is a significant increase in above two parameters when M > 0.055 kg/s, which can be analyzed through the damage distribution inside the brittle material, as shown in Fig. 28.

Figure 27

Simulation results of crater on red sandstone surface with different abrasive mass flow rater.

Figure 28

Damage distribution inside red sandstone with different abrasive mass flow rates.

It can be seen from Fig. 28(a, b) that with the increase of M, the length of radial cracks on both sides of the crater increases significantly. When M = 0.055 kg/s, a radial crack appears at the crater bottom, which spreads to the interior along the incident direction of AWJ, as shown in Fig. 28(c). When M increases to 0.060 kg/s, both the crack number and the crater depth at the bottom increase significantly, resulting in a substantial increase in the damage area, as shown in Fig. 28(d). The above phenomenon indicates that there is a threshold of abrasive mass flow rate M_cri during the AWJ impact process on brittle materials. If M ≤ M_cri, no radial cracks are generated at the crater bottom, and the crater depth shows a linear increasing relationship with M. If M > M_cri, radial cracks appear at the crater bottom, the material mass loss and crater depth increase greatly, even causing the brittle material to fracture. Therefore, in the process of AWJ cutting, the abrasive mass flow rate should be controlled below M_cri. While for the process of rock breaking, increasing the abrasive mass flow rate above M_cri can significantly increase the drilling speed. In the simulation cases involved in this section, the M_cri value is 0.055 kg/s.

Conclusion

In this study, an unresolved SPH-DEM coupled model is proposed for AWJ study. The water-jet and targeted materials are modelled by SPH method, and the abrasive particles are modelled by DEM method. The capability of the coupled model in predicting ductile and brittle surface erosion behaviour is verified by comparing with the impact experiment results. The conclusions are summarized as follows:

• 1. For ductile materials, the erosion mechanism of AWJ is the material failure caused by plastic strain accumulation. The erosion process is divided into three stages: plastic strain accumulation, erosion crater formation and stable erosion. The crater depth C_d increases linearly with the incident velocity v_jet, while the crater width C_w tends to a stable value. The abrasive mass flow rate M has a great influence on C_w. The crater profile changes from U-shaped to V-shaped with the increase of M, so the erosion rate and the crater size cannot be infinite improved by increasing M.

• 2. For brittle materials, the erosion mechanism of AWJ is the propagation and diffusion of cracks inside the material caused by repeated impact of abrasive particles. The erosion process is divided into two stages: erosion crater formation and stable erosion. The crater depth C_d and crater width C_w basically increase linearly with the incident velocity v_jet., and the number and size of cracks inside the material also increase significantly. The abrasive mass flow rate M has a great influence on C_d. The crater profile changes from V-shaped to U-shaped with the increase of M. There is a threshold M_cri. When M < M_cri, C_d increases linearly with M. When M > M_cri, radial cracks appear at the crater bottom. The material mass loss and C_d increase significantly, resulting in the material fragmentation.