**2. Finite element simulation on homogeneous SP model**

### **2.1. Homogeneous SP model**

### *2.1.1. Introduction of SOLID164 element*

The explicit dynamic analysis of ANSYS/LS-DYNA program provides a rich element library, including 3D bar element (LINK160), 3D beam element (BEAM161), thin shell element (SHELL163), solid element (SOLID164), spring damping element (COMBI165), mass element (MASS166), cable element (Link167), and ten-node tetrahedral element (Tet-Solid168). Each explicit dynamic element has a corresponded linear displacement function, and the default is set to a single point of integration (one of the reduced integrals). It has been proved that the explicit dynamic element with linear displacement function and the element integration can be used for solving the nonlinear problems effectively, such as the cases of large deformation and material's failure [34]. The element utilized in simulation of SP is the SOLID164 element, which is a 3D explicit solid element consisting of eight nodes. Each node in SOLID164 has the degrees of freedom in *x*, *y*, *z* direction of translation, velocity, and acceleration. The geometry, node locations, and the coordinate system for SOLID164 element are shown in **Figure 1** [35]. This element is only used in the dynamic explicit analysis, which supports all licensed nonlinear characteristics. By default, SOLID164 is the single point integral with the viscous hourglass control to speed up the calculation.

points. So, using the simplified integration of elements can save the data storage capacity and reduce the number of calculations greatly, for instance, the single point integral in SOLID164 element, that is, an element has only one integration point, which is in the center of element. The utilization of single points can save a lot of calculation time, but may lead to the hourglass phenomenon. Mesh deformation with an hourglass effect is called the hourglass phenomenon, and in this situation, the typical feature is that the nonrigid element grid is distorted irregularly, resulting in a mathematically stable state but a physically impossible state. The presence of hourglass phenomenon will distort the results of the solution and even the solution cannot be carried out. Therefore, when using the simplified integration of elements, the hourglass phenomenon should be controlled. In ANSYS/LS-DYNA finite element analysis process, if the result of hourglass energy calculated by the model is less than 10% of the total

Finite Element Dynamic Analysis on Residual Stress Distribution of Titanium Alloy and Titanium…

The choice of material model is not only related to the success of simulation, but also directly related to the rationality and reliability of calculated results. The process of SP causes a high plastic deformation in the surface layer, so Cowper-Symbols in ANSYS is adopted to achieve 3D finite element dynamic analysis [34]. Cowper-Symbols model is a piecewise linear plasticity model, and the yield stress can be obtained via the strain rate, which is shown in Eq. (1).

> ) = *σy*(*εeff p* )[1 + ( *ε*̇ *eff p* \_\_\_ *<sup>C</sup>* ) \_\_1 *p*

original yield stress while the strain rate has not been considered. In this model during simulation, the input data include totally: the density of material *ρ*, the elastic modulus of material *E*, Poisson's ratio *ν*, the yield stress, the tangent modulus, the strain rate parameters *P* and *C*, and the true stress–strain curve. If the load curve is used, the yield stress and tangent modulus

Before establishing SP model for (TiB+TiC)/Ti-6Al-4V, the 3D model for homogeneous matrix Ti-6Al-4V should be established firstly. For reducing the number of elements and the calculation time, the symmetry of SP sample and shot balls should be considered, and a 1/2 model can be set up. The 3D homogeneous model established in this work is shown in **Figure 2** [36], including the top four-layer of shot balls, and the bottom of peened target. The dimension of

of the intensive impacting in the near surface layer, a mesh refinement is adopted and each mesh depth is 0.02 mm. SOLID164 dynamic analysis element is chosen for meshing element,

, in which *R* is the average radius of shot balls. Because

*<sup>ε</sup>*̇ represents the effective strain rate, *P* and *C* are the parameters for strain rate, and *σy*(*εeff*

] (1)

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27

*p* ) is the

energy, this simulation result and the model can be identified as credible.

*<sup>p</sup>* , *ε*̇ *eff p*

are ignored. If *P* and *C* are set to 0, the strain rate effect is omitted.

*2.1.2. Piecewise linear plasticity model*

*σy*(*εeff*

*2.1.3. Establishment of homogeneous SP model*

peened target is 12*R* × 6*R* × 2.1 mm<sup>3</sup>

and total mesh number is 120,000.

In the process of explicit dynamic analysis, the processing of element integral is the most time-consumption, and the processing time by CPU is proportional to the number of integral

**Figure 1.** Geometry properties of SOLID164 [35].

points. So, using the simplified integration of elements can save the data storage capacity and reduce the number of calculations greatly, for instance, the single point integral in SOLID164 element, that is, an element has only one integration point, which is in the center of element. The utilization of single points can save a lot of calculation time, but may lead to the hourglass phenomenon. Mesh deformation with an hourglass effect is called the hourglass phenomenon, and in this situation, the typical feature is that the nonrigid element grid is distorted irregularly, resulting in a mathematically stable state but a physically impossible state. The presence of hourglass phenomenon will distort the results of the solution and even the solution cannot be carried out. Therefore, when using the simplified integration of elements, the hourglass phenomenon should be controlled. In ANSYS/LS-DYNA finite element analysis process, if the result of hourglass energy calculated by the model is less than 10% of the total energy, this simulation result and the model can be identified as credible.

### *2.1.2. Piecewise linear plasticity model*

**2. Finite element simulation on homogeneous SP model**

26 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

The explicit dynamic analysis of ANSYS/LS-DYNA program provides a rich element library, including 3D bar element (LINK160), 3D beam element (BEAM161), thin shell element (SHELL163), solid element (SOLID164), spring damping element (COMBI165), mass element (MASS166), cable element (Link167), and ten-node tetrahedral element (Tet-Solid168). Each explicit dynamic element has a corresponded linear displacement function, and the default is set to a single point of integration (one of the reduced integrals). It has been proved that the explicit dynamic element with linear displacement function and the element integration can be used for solving the nonlinear problems effectively, such as the cases of large deformation and material's failure [34]. The element utilized in simulation of SP is the SOLID164 element, which is a 3D explicit solid element consisting of eight nodes. Each node in SOLID164 has the degrees of freedom in *x*, *y*, *z* direction of translation, velocity, and acceleration. The geometry, node locations, and the coordinate system for SOLID164 element are shown in **Figure 1** [35]. This element is only used in the dynamic explicit analysis, which supports all licensed nonlinear characteristics. By default, SOLID164 is the single point integral with the viscous hourglass control to speed up the

In the process of explicit dynamic analysis, the processing of element integral is the most time-consumption, and the processing time by CPU is proportional to the number of integral

**2.1. Homogeneous SP model**

calculation.

*2.1.1. Introduction of SOLID164 element*

**Figure 1.** Geometry properties of SOLID164 [35].

The choice of material model is not only related to the success of simulation, but also directly related to the rationality and reliability of calculated results. The process of SP causes a high plastic deformation in the surface layer, so Cowper-Symbols in ANSYS is adopted to achieve 3D finite element dynamic analysis [34]. Cowper-Symbols model is a piecewise linear plasticity model, and the yield stress can be obtained via the strain rate, which is shown in Eq. (1).

$$
\sigma\_y \left( \varepsilon\_{gt'}^p, \dot{\varepsilon}\_{gt}^p \right) = \sigma\_y \{ \varepsilon\_{gt}^p \} \left[ 1 + \left( \frac{\dot{\varepsilon}\_{gt}^p}{C} \right)^{\frac{1}{p}} \right] \tag{1}
$$

*<sup>ε</sup>*̇ represents the effective strain rate, *P* and *C* are the parameters for strain rate, and *σy*(*εeff p* ) is the original yield stress while the strain rate has not been considered. In this model during simulation, the input data include totally: the density of material *ρ*, the elastic modulus of material *E*, Poisson's ratio *ν*, the yield stress, the tangent modulus, the strain rate parameters *P* and *C*, and the true stress–strain curve. If the load curve is used, the yield stress and tangent modulus are ignored. If *P* and *C* are set to 0, the strain rate effect is omitted.

### *2.1.3. Establishment of homogeneous SP model*

Before establishing SP model for (TiB+TiC)/Ti-6Al-4V, the 3D model for homogeneous matrix Ti-6Al-4V should be established firstly. For reducing the number of elements and the calculation time, the symmetry of SP sample and shot balls should be considered, and a 1/2 model can be set up. The 3D homogeneous model established in this work is shown in **Figure 2** [36], including the top four-layer of shot balls, and the bottom of peened target. The dimension of peened target is 12*R* × 6*R* × 2.1 mm<sup>3</sup> , in which *R* is the average radius of shot balls. Because of the intensive impacting in the near surface layer, a mesh refinement is adopted and each mesh depth is 0.02 mm. SOLID164 dynamic analysis element is chosen for meshing element, and total mesh number is 120,000.

*σ<sup>t</sup>* = *σε*(1 + *εε*) (2)

*ε<sup>t</sup>* = ln(1 + *εε*) (3)

[37]. The true stress–strain curve is used to provide the corresponding deformation parameters for the shot peened material in the finite element model. Because of the existence of reinforcements in composite and the influences of which on the matrix's residual stress distribution, the strain–stress curves with a single strain rate (10−3 s−1) are utilized during simulation. The dynamic stress–strain curves become flatter with increasing strain rate, which reduces the material strain hardening [38]. Thus, considering the variation of strain rates, based on the references

The true stress–strain curves of Ti-6Al-4V and (TiB+TiC)/Ti-6Al-4V can be obtained from the tensile tests. The dimension of tensile specimens with a gauge section of 4 × 1.8 × 18 mm is shown in **Figure 4(a)**, and the axes direction is parallel to the hot-forging direction. The tensile tests are performed using a Zwick T1-FR020TN test machine in air, and the initial strain rates is 10−3 s−1. The true stress–strain curves are shown in **Figure 4(b)**, which are implemented in the simulation process as the elastic–plastic deformation curves. In addition, the typical

In order to show the deformation of material obviously after SP, the coverage rate of 300% is chosen and the shot velocity is 100 m/s. The simulation results of *x*-direction principal stress (σxx) are shown in **Figure 5**. From the **Figure 5(a)**, it can be found that after SP, there is a compressive stress field with a certain depth in the plastic deformation area below the shot balls, and the

mechanical parameters of shot balls, target, and reinforcements are shown in **Table 1**.

is the true strain, and *εε*

Finite Element Dynamic Analysis on Residual Stress Distribution of Titanium Alloy and Titanium…

is the engineering strain

29

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is the engineering stress, *ε<sup>t</sup>*

[39, 40] and Eq. (1), the parameters *C* and *P* are set as *C* = 1300 and *P* = 5.

*σt*

is the true stress, *σε*

**2.2. Simulated results and discussion**

*2.2.1. Residual stress distribution on surface*

**Figure 3.** Coverage rate of SP with different impacts [36].

**Figure 2.** 3D SP model for homogeneous materials [36].

For improving the computational efficiency, the mesh size gradually increases at the model boundary and the lower half. In order to avoid the influence of reflected stress wave within the target on the distribution of residual stress during the process of impacting, nonreflective boundary conditions are implemented on the bottom and flank of the target. Because the XOZ plane is symmetrical, the symmetrical boundary conditions are applied. Because a small size model (12*R* × 6*R* × 2.1 mm<sup>3</sup> ) may lead to the nonreal oscillation while simulating the process of impacting, in order to eliminate the effect of nonreal oscillation, the alpha damping constraint is applied to the model in the dynamic analysis. The alpha damping is a damping coefficient proportional to the mass and is very effective for the low frequency oscillations.

In 3D SP model, the top four-layers are shot balls made by case steel, and the hardness and strength of matrix is smaller than that of the case steel balls. The deformability of shot balls is very weak, which hardly affects the results of residual stress distribution. Therefore, the shot balls are defined as rigid bodies. During the explicit dynamics analysis, the degrees of freedom of all nodes in the rigid body are coupled to the mass center of rigid body, thus greatly reducing the computational time of the explicit analysis. Moreover, the corresponding mechanical parameters are given to the rigid body center to describe the dynamic characteristics. In addition, the coverage rate is very important to the distribution of residual stresses after SP. In order to keep the simulation accuracy, the coverage rate is defined and the schematic is shown in **Figure 3** [36]. Four impacts with shot balls are performed in current 3D model. On a single impact, the coverage rate is about 25%, and after four impacts, it approaches 100%. If coverage rate increases to 200%, eight impacts should be carried out in turn.

Because the piecewise linear model is adopted during simulation, the true stress–strain curve is needed. Usually, the stress–strain curve obtained by experiment is the engineering stress– strain curve, which is needed to be transformed into the true stress–strain curve, and the formula of transformation is shown in Eqs. (2) and (3).

Finite Element Dynamic Analysis on Residual Stress Distribution of Titanium Alloy and Titanium… http://dx.doi.org/10.5772/intechopen.73120 29

$$
\sigma\_i = \sigma\_i (1 + \varepsilon\_i) \tag{2}
$$

$$
\varepsilon\_t = \ln(1 + \varepsilon\_t) \tag{3}
$$

*σt* is the true stress, *σε* is the engineering stress, *ε<sup>t</sup>* is the true strain, and *εε* is the engineering strain [37]. The true stress–strain curve is used to provide the corresponding deformation parameters for the shot peened material in the finite element model. Because of the existence of reinforcements in composite and the influences of which on the matrix's residual stress distribution, the strain–stress curves with a single strain rate (10−3 s−1) are utilized during simulation. The dynamic stress–strain curves become flatter with increasing strain rate, which reduces the material strain hardening [38]. Thus, considering the variation of strain rates, based on the references [39, 40] and Eq. (1), the parameters *C* and *P* are set as *C* = 1300 and *P* = 5.

The true stress–strain curves of Ti-6Al-4V and (TiB+TiC)/Ti-6Al-4V can be obtained from the tensile tests. The dimension of tensile specimens with a gauge section of 4 × 1.8 × 18 mm is shown in **Figure 4(a)**, and the axes direction is parallel to the hot-forging direction. The tensile tests are performed using a Zwick T1-FR020TN test machine in air, and the initial strain rates is 10−3 s−1. The true stress–strain curves are shown in **Figure 4(b)**, which are implemented in the simulation process as the elastic–plastic deformation curves. In addition, the typical mechanical parameters of shot balls, target, and reinforcements are shown in **Table 1**.

### **2.2. Simulated results and discussion**

For improving the computational efficiency, the mesh size gradually increases at the model boundary and the lower half. In order to avoid the influence of reflected stress wave within the target on the distribution of residual stress during the process of impacting, nonreflective boundary conditions are implemented on the bottom and flank of the target. Because the XOZ plane is symmetrical, the symmetrical boundary conditions are applied. Because a small size

impacting, in order to eliminate the effect of nonreal oscillation, the alpha damping constraint is applied to the model in the dynamic analysis. The alpha damping is a damping coefficient

In 3D SP model, the top four-layers are shot balls made by case steel, and the hardness and strength of matrix is smaller than that of the case steel balls. The deformability of shot balls is very weak, which hardly affects the results of residual stress distribution. Therefore, the shot balls are defined as rigid bodies. During the explicit dynamics analysis, the degrees of freedom of all nodes in the rigid body are coupled to the mass center of rigid body, thus greatly reducing the computational time of the explicit analysis. Moreover, the corresponding mechanical parameters are given to the rigid body center to describe the dynamic characteristics. In addition, the coverage rate is very important to the distribution of residual stresses after SP. In order to keep the simulation accuracy, the coverage rate is defined and the schematic is shown in **Figure 3** [36]. Four impacts with shot balls are performed in current 3D model. On a single impact, the coverage rate is about 25%, and after four impacts, it approaches 100%. If coverage

Because the piecewise linear model is adopted during simulation, the true stress–strain curve is needed. Usually, the stress–strain curve obtained by experiment is the engineering stress– strain curve, which is needed to be transformed into the true stress–strain curve, and the

proportional to the mass and is very effective for the low frequency oscillations.

rate increases to 200%, eight impacts should be carried out in turn.

formula of transformation is shown in Eqs. (2) and (3).

) may lead to the nonreal oscillation while simulating the process of

model (12*R* × 6*R* × 2.1 mm<sup>3</sup>

**Figure 2.** 3D SP model for homogeneous materials [36].

28 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

### *2.2.1. Residual stress distribution on surface*

In order to show the deformation of material obviously after SP, the coverage rate of 300% is chosen and the shot velocity is 100 m/s. The simulation results of *x*-direction principal stress (σxx) are shown in **Figure 5**. From the **Figure 5(a)**, it can be found that after SP, there is a compressive stress field with a certain depth in the plastic deformation area below the shot balls, and the

**Figure 3.** Coverage rate of SP with different impacts [36].

**Figure 4.** (a) Specimen size for room temperature tests (unit: mm); (b) stress-strain curves of the matrix and composite (strain rate: 10−3 s−1).


on the simulation results. So, the tensile stress field of surface boundary has no effect on the residual stress distribution in the plastic deformation zone. Moreover, in the actual SP experiment, the whole surface of material is subjected to SP process, and the uniform CRS can be

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The effect of SP on the residual stress distribution is mainly described by the residual stress variation with the increase of depth and the four characteristic parameters (the surface residual stress, the max residual stress, the depth of residual stress layer, and the depth of max residual stress). Usually, in experiment, after SP, the surface layer of samples are subjected to electrochemical etching and removed layer by layer (each layer is about 15–20 μm), and then the stress value of each layer can be measured by XRD method, and the curves of residual stress variation along the depth are obtained. In order to avoid measurement error, the irradia-

simulation, for obtaining the distribution of residual stress along the depth, the stress value of all nodes at a certain depth along Z direction are selected and averaged, which can represent the average stress in a certain depth. Using this method to average the stress values of all nodes, the curve of residual stress distribution with the increase of depth can be obtained.

After SP, the surface coverage rate refers to the ratio of the area occupied by the shot craters to the area of surface required for SP. During the process of SP, the coverage rate is usually required for reaching or exceeding 100%. Moreover, in the experiment, the coverage rate more than 100% can be expressed as a multiple of the time required for a full coverage rate of 100%, for example, the coverage rate of 200% means that the SP time is two times of full coverage.

, so actually, the measured stress by

. In this numerical

obtained. The surface tensile stress on the boundary disappears at the same time.

**Figure 5.** Simulation results of residual stress field of SP (σxx), (a) 3D result; (b) 2D result of surface.

experiment is the average stress under the statistical of surface area in 1 mm<sup>2</sup>

tion area of X-ray on the surface is generally about 1 mm2

*2.2.2. Influence of coverage rate on residual stress distribution*

**Table 1.** Typical mechanical parameters of shot balls and target.

distribution of CRS in the plastic deformation zone is uniform, which is about −900 to −1000 MPa. However, in the 2D result of surface in **Figure 5(b)**, the distribution of residual stress is not uniform, and the max CRS is about −1453 MPa, which results from the stress concentration on the impact point of shot ball. In addition, there is tensile residual stress in some area between two craters. As the impacting time of shot ball on material's surface is different, the size of crater on the surface is not consistent, and the area of craters formed firstly is significantly reduced by the extrusion of follow-up craters. In these extruded area, the tensile residual stress is formed, but after repeated SP, the distribution of residual stress is gradually tended to uniform.

Due to the repeated impact and extrusion of shot balls, the plastic deformation appears and many craters are formed on surface. The formation of craters results in the increase of surface area and the appearance of tensile residual stress, which is also the reason that the max CRS does not appear on surface. On surface, the residual stress includes both the compressive stress formed by the plastic deformation and the tensile stress formed by the increase of surface area. With the increase of depth, the tensile residual stress decreases gradually but the CRS is gradually increased, then the max CRS is reached at a certain depth. At last, due to the reduction of plastic deformation, the CRS is gradually reduced to the level of stress before SP. In **Figure 5(b)**, it can be found that there is tensile stress on the surface boundary. The formation of such tensile stress field is mainly due to the limitation of finite element model size (only 12*R* × 6*R*), and the SP area only 8*R* × 4*R* in order to avoid the effect of model boundary Finite Element Dynamic Analysis on Residual Stress Distribution of Titanium Alloy and Titanium… http://dx.doi.org/10.5772/intechopen.73120 31

**Figure 5.** Simulation results of residual stress field of SP (σxx), (a) 3D result; (b) 2D result of surface.

on the simulation results. So, the tensile stress field of surface boundary has no effect on the residual stress distribution in the plastic deformation zone. Moreover, in the actual SP experiment, the whole surface of material is subjected to SP process, and the uniform CRS can be obtained. The surface tensile stress on the boundary disappears at the same time.

The effect of SP on the residual stress distribution is mainly described by the residual stress variation with the increase of depth and the four characteristic parameters (the surface residual stress, the max residual stress, the depth of residual stress layer, and the depth of max residual stress). Usually, in experiment, after SP, the surface layer of samples are subjected to electrochemical etching and removed layer by layer (each layer is about 15–20 μm), and then the stress value of each layer can be measured by XRD method, and the curves of residual stress variation along the depth are obtained. In order to avoid measurement error, the irradiation area of X-ray on the surface is generally about 1 mm2 , so actually, the measured stress by experiment is the average stress under the statistical of surface area in 1 mm<sup>2</sup> . In this numerical simulation, for obtaining the distribution of residual stress along the depth, the stress value of all nodes at a certain depth along Z direction are selected and averaged, which can represent the average stress in a certain depth. Using this method to average the stress values of all nodes, the curve of residual stress distribution with the increase of depth can be obtained.

### *2.2.2. Influence of coverage rate on residual stress distribution*

distribution of CRS in the plastic deformation zone is uniform, which is about −900 to −1000 MPa. However, in the 2D result of surface in **Figure 5(b)**, the distribution of residual stress is not uniform, and the max CRS is about −1453 MPa, which results from the stress concentration on the impact point of shot ball. In addition, there is tensile residual stress in some area between two craters. As the impacting time of shot ball on material's surface is different, the size of crater on the surface is not consistent, and the area of craters formed firstly is significantly reduced by the extrusion of follow-up craters. In these extruded area, the tensile residual stress is formed, but

**Figure 4.** (a) Specimen size for room temperature tests (unit: mm); (b) stress-strain curves of the matrix and composite

**Shot balls Ti-6Al-4V TiB TiC**

Due to the repeated impact and extrusion of shot balls, the plastic deformation appears and many craters are formed on surface. The formation of craters results in the increase of surface area and the appearance of tensile residual stress, which is also the reason that the max CRS does not appear on surface. On surface, the residual stress includes both the compressive stress formed by the plastic deformation and the tensile stress formed by the increase of surface area. With the increase of depth, the tensile residual stress decreases gradually but the CRS is gradually increased, then the max CRS is reached at a certain depth. At last, due to the reduction of plastic deformation, the CRS is gradually reduced to the level of stress before SP. In **Figure 5(b)**, it can be found that there is tensile stress on the surface boundary. The formation of such tensile stress field is mainly due to the limitation of finite element model size (only 12*R* × 6*R*), and the SP area only 8*R* × 4*R* in order to avoid the effect of model boundary

after repeated SP, the distribution of residual stress is gradually tended to uniform.

Density ρ (Kg·m−3) 7800 4500 4520 4930 Poisson's ratio ν 0.30 0.34 0.14–0.15 0.18–0.2 Young's modulus *E* (GPa) 210 112 371–485 400–460 Constraint of the part Rigid Elastic–plastic Elastic–plastic Elastic–plastic

**Table 1.** Typical mechanical parameters of shot balls and target.

30 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

(strain rate: 10−3 s−1).

After SP, the surface coverage rate refers to the ratio of the area occupied by the shot craters to the area of surface required for SP. During the process of SP, the coverage rate is usually required for reaching or exceeding 100%. Moreover, in the experiment, the coverage rate more than 100% can be expressed as a multiple of the time required for a full coverage rate of 100%, for example, the coverage rate of 200% means that the SP time is two times of full coverage. In order to simplify the model, the collision between two shot balls are not considered, and SP position of each ball can be precisely controlled. Therefore, the different coverage rate can be simulated by the multi-layer shot balls and the coverage increases linearity with the number of layers. In this work, the characteristics of 100% coverage rate (4-layer shot balls), 200% coverage rate (8-layer shot balls), and 300% coverage rate (12-layer shot balls) are simulated. The shot velocity are set as 50 and 100 m/s based on experiment (mentioned in Section 2.2.4), and the average radius of shot balls are chosen as 0.15, 0.3, and 0.6 mm. The influence of coverage rate, radius of shot balls (*r*), and shot velocity (*v*) on residual stress distribution are investigated and discussed.

**Figure 6** shows the distribution of residual stress along depth with different coverage rates. While the radius and speed of shot balls are constant, the similar variation trend of residual stress can be obtained. With the increase of coverage rate, the CRS and the depth of stress layer are improved. In addition, the depth of max CRS decreases with the increase of coverage rate, which is more obvious in **Figure 6(e)** and **(f)**. Comparing the results under coverage rate of 200 and 300%, it can be found that the increment of surface residual stresses are not obvious with increasing the coverage rate, since the surface of almost all area is covered by craters and the stress field reaches saturation. Though there is a little bit increment of the max CRS while coverage rate increases from 200 to 300%, there is not obvious comparing the coverage variation from 100 to 200%.

The simulated pictures about residual stress distribution under different coverage rates are shown in **Figure 7**, while *r* = 0.3 mm and *v* = 100 m/s. With increasing coverage, the number of craters on surface increases, and the surface residual stresses becomes more uniform. As well, the uniformity of residual stress distribution on subsurface is also improved while viewed from the cross section, and slight tensile stresses in the deep surface decrease gradually and disappear with the increase of coverage rate.

### *2.2.3. Influence of shot balls' radius on residual stress distribution*

According to the formula of *<sup>m</sup>* <sup>=</sup> \_\_4 <sup>3</sup> *<sup>R</sup>*<sup>3</sup> , the mass of shot ball is not only linearly related to the density, but also is proportional to *R*<sup>3</sup> . In most of the industrial production, the criterion for choosing the size of shot ball is based on the surface roughness after SP. If it requires high quality of surface (small roughness), the small radius of shot ball is chosen. If there is no high quality requirement of surface, the big radius of shot ball is determined, because the cost of shot balls with small radius are much higher than that of big shot balls. Besides the influence of shot ball's radius on surface roughness, the shot ball also influences the residual stress distribution. So the influence of three kinds of different radius (*r* = 0.15, 0.3, and 0.6 mm) on the residual stress distribution of Ti-6Al-4V is simulated by the model, while the shot velocity is 100 m/s. The results of the residual stress versus depth are shown in **Figure 8**. From the simulation results, it can be found that the surface CRS and max CRS are higher while using small shot balls, but the layer depth of CRS is smaller and the CRS decreases rapidly with increasing depth. When using the big shot balls, both the surface and max CRS are smaller, however, the layer depth of CRS is deeper and CRS decreases slowly with increasing depth.

Some work have shown that when the coverage rate is 100%, the depth of CRS layer has the following relationship with the crater diameter and the shot ball's diameter.

*Z*<sup>0</sup> = *a* × *D* − *c* × *R* (4)

**Figure 6.** Residual stress distribution in depth with different coverage rates, (a) *r* = 0.15 mm, *v* = 50 m/s; (b) *r* = 0.15 mm, *v* = 100 m/s; (c) *r* = 0.3 mm, *v* = 50 m/s; (d) *r* = 0.3 mm, *v* = 100 m/s; (e) *r* = 0.6 mm, *v* = 50 m/s; (f) *r* = 0.6 mm, *v* = 100 m/s.

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constant coefficients. In general, the value of *a* is between 1 and 1.5, and the value of *c* is in

is the depth of CRS layer, R is the shot ball's diameter, and *a, c* are

*D* is the crater diameter, Z<sup>0</sup>

Finite Element Dynamic Analysis on Residual Stress Distribution of Titanium Alloy and Titanium… http://dx.doi.org/10.5772/intechopen.73120 33

In order to simplify the model, the collision between two shot balls are not considered, and SP position of each ball can be precisely controlled. Therefore, the different coverage rate can be simulated by the multi-layer shot balls and the coverage increases linearity with the number of layers. In this work, the characteristics of 100% coverage rate (4-layer shot balls), 200% coverage rate (8-layer shot balls), and 300% coverage rate (12-layer shot balls) are simulated. The shot velocity are set as 50 and 100 m/s based on experiment (mentioned in Section 2.2.4), and the average radius of shot balls are chosen as 0.15, 0.3, and 0.6 mm. The influence of coverage rate, radius of shot balls (*r*), and shot velocity (*v*) on residual stress distribution are

**Figure 6** shows the distribution of residual stress along depth with different coverage rates. While the radius and speed of shot balls are constant, the similar variation trend of residual stress can be obtained. With the increase of coverage rate, the CRS and the depth of stress layer are improved. In addition, the depth of max CRS decreases with the increase of coverage rate, which is more obvious in **Figure 6(e)** and **(f)**. Comparing the results under coverage rate of 200 and 300%, it can be found that the increment of surface residual stresses are not obvious with increasing the coverage rate, since the surface of almost all area is covered by craters and the stress field reaches saturation. Though there is a little bit increment of the max CRS while coverage rate increases from 200 to 300%, there is not obvious comparing the coverage variation from 100 to 200%.

The simulated pictures about residual stress distribution under different coverage rates are shown in **Figure 7**, while *r* = 0.3 mm and *v* = 100 m/s. With increasing coverage, the number of craters on surface increases, and the surface residual stresses becomes more uniform. As well, the uniformity of residual stress distribution on subsurface is also improved while viewed from the cross section, and slight tensile stresses in the deep surface decrease gradually and

choosing the size of shot ball is based on the surface roughness after SP. If it requires high quality of surface (small roughness), the small radius of shot ball is chosen. If there is no high quality requirement of surface, the big radius of shot ball is determined, because the cost of shot balls with small radius are much higher than that of big shot balls. Besides the influence of shot ball's radius on surface roughness, the shot ball also influences the residual stress distribution. So the influence of three kinds of different radius (*r* = 0.15, 0.3, and 0.6 mm) on the residual stress distribution of Ti-6Al-4V is simulated by the model, while the shot velocity is 100 m/s. The results of the residual stress versus depth are shown in **Figure 8**. From the simulation results, it can be found that the surface CRS and max CRS are higher while using small shot balls, but the layer depth of CRS is smaller and the CRS decreases rapidly with increasing depth. When using the big shot balls, both the surface and max CRS are smaller, however, the

Some work have shown that when the coverage rate is 100%, the depth of CRS layer has the

, the mass of shot ball is not only linearly related to the

. In most of the industrial production, the criterion for

investigated and discussed.

disappear with the increase of coverage rate.

According to the formula of *<sup>m</sup>* <sup>=</sup> \_\_4

density, but also is proportional to *R*<sup>3</sup>

*2.2.3. Influence of shot balls' radius on residual stress distribution*

32 Finite Element Method - Simulation, Numerical Analysis and Solution Techniques

<sup>3</sup> *<sup>R</sup>*<sup>3</sup>

layer depth of CRS is deeper and CRS decreases slowly with increasing depth.

following relationship with the crater diameter and the shot ball's diameter.

**Figure 6.** Residual stress distribution in depth with different coverage rates, (a) *r* = 0.15 mm, *v* = 50 m/s; (b) *r* = 0.15 mm, *v* = 100 m/s; (c) *r* = 0.3 mm, *v* = 50 m/s; (d) *r* = 0.3 mm, *v* = 100 m/s; (e) *r* = 0.6 mm, *v* = 50 m/s; (f) *r* = 0.6 mm, *v* = 100 m/s.

$$Z\_{\mathbf{o}} = \mathbf{a} \times \mathbf{D} - \mathbf{c} \times \mathbf{R} \tag{4}$$

*D* is the crater diameter, Z<sup>0</sup> is the depth of CRS layer, R is the shot ball's diameter, and *a, c* are constant coefficients. In general, the value of *a* is between 1 and 1.5, and the value of *c* is in

the range from 0 to 0.1. In order to obtain the values of *a* and *c*, the SP experiment should be carried out. Firstly, the same material as the workpiece is chosen and then the surface of this material is subjected to SP treatment. Secondly, the diameter of crater and the depth of CRS are measured. And at last, the values of *a* and *c* are calculated by linear fitting. Based on the values of *a* and *c*, the depth of CRS layer can be estimated. At the same shot velocity, the diameter of crater increases with increasing shot diameter, and thus the depth of CRS layer increases.

Finite Element Dynamic Analysis on Residual Stress Distribution of Titanium Alloy and Titanium…

SP is the process of consuming shot balls' kinetic energy and transfer the kinetic energy to the deformation energy of target material. So, after SP, the elastic and plastic deformations are introduced in the surface layer of target material. The shot balls' mass and velocity directly affect the value of kinetic energy. When the material of shot balls is same, the kinetic energy increases with the improvement of shot velocity. In SP experiment by using an air blast machine, the shot velocity can be varied and obtained by adjusting the air pressure. During the flight of shot balls, the velocity will be decreased because of the collision between them and the effect of air resistance, and the attenuation is related to the distance between the nozzle and the material. The smaller diameter of shot ball, the velocity attenuation is more obvi-

shot distances are shown in **Figure 9** [41]. From this figure, it can be found that when the shot distance is less than 2 m, the attenuation rates are proportional to the distance increment.

The shot velocity is also affected by the shot angle in addition to the attenuation with the distance. When the shot balls impact on the surface of workpiece at a certain angle, the velocity can be decomposed into two directions. One is perpendicular to the surface (normal velocity) and the other is parallel to the surface (tangential velocity). The former velocity contributes to the plastic deformation of the surface layer, but the latter only promotes the friction effect. Based on the above analysis, in the experiment of this work, the distance between nozzle and samples is 100 mm (0.1 m),

) with different diameters at different

http://dx.doi.org/10.5772/intechopen.73120

35

*2.2.4. Influence of shot velocity on residual stress distribution*

ous. The attenuation rates of cast shot balls (ρ = 7.8 g/cm<sup>3</sup>

**Figure 9.** Relationship between decrement of shot velocity and shooting distance [41].

**Figure 7.** Residual stress distribution after SP with different coverage rates: (a) 100%; (b) 200%; (c) 300%. [36].

**Figure 8.** Influence of shot radius on depth distribution of residual stress (*v* = 100 m/s and coverage = 100%).

the range from 0 to 0.1. In order to obtain the values of *a* and *c*, the SP experiment should be carried out. Firstly, the same material as the workpiece is chosen and then the surface of this material is subjected to SP treatment. Secondly, the diameter of crater and the depth of CRS are measured. And at last, the values of *a* and *c* are calculated by linear fitting. Based on the values of *a* and *c*, the depth of CRS layer can be estimated. At the same shot velocity, the diameter of crater increases with increasing shot diameter, and thus the depth of CRS layer increases.
