Analytical model of strain and strain rate in weld zone

Theoretical Background

3.1.5 Analytical model of strain and strain rate in weld zone

The heat is generated by friction as well as plastic deformation which alters the microstructure and properties. The kinetics depends mainly on the temperature and strain rate and microstructure evolution influence the energy transfer within the system. The phenomenological understanding of heat transfers and material flow demands a strong coupling between thermal, mechanical, and metallurgical aspects. The center of stirred zone (SZ) experiences dynamic recrystallization and is followed by recovered microstructure around the stirred zones at low level of deformation and temperature. A part of the plastic deformation energy is also stored within the thermomechanically affected zone (TMAZ) in the form of increased dislocation densities within the deformed grains. The heat affected zone (HAZ) is affected only by the temperature and the microstructural changes with respect

Chapter 3 to base material are insignificant. In FSW, the flow stress is highly dependent on temperature and the stain rate. Thus, the resulting grain structure in the processed zone mainly depends on the resulting strain rate and temperature distributions. In order to achieve a desired microstructure, the strain rate and temperature distributions must be controlled and relates to the grain structure and process parameters. Therefore, the strain rate distribution in friction stir welding in stir zone is determined in terms of process parameters.

To develop the analytical model for strain and strain rate distribution in weld zone, the following assumptions are considered. The velocity field within the stirred zone are determined by considering the effects of both the shoulder and the pin of the tool on the material flow. The state variables by slip condition are introduced to relate the material velocity within the stirred zone. From the velocity fields, the strain rate distribution in the processed zone can be estimated. In the model two incompressible flow fields are combined to describe the material flow such as forced vortex flow and uniform translation. No material movement in the thickness direction are considered.

The relative velocity of the tool surface and workpiece depends on several factors that control this relation including contact condition at the tool/ workpiece interfaces. There are two interfaces between the tool and the sheet: one is at the shoulder and the other is at the pin as shown in Fig. 3.8(a). Two state variables are used for radial and thickness direction respectively to consider the contact conditions between tool contact surface and workpiece material. The first one is denoted by α which relates the material velocity to shoulder velocity and it is defined as a function of z. The same approach is followed to define the second state variable (β) which relates the material velocity to pin velocity. The state variable β is defined in terms of r. Mathematically, the state variables are given by

𝛼 = 𝐴𝑒𝑥𝑝 {−𝐵 ^𝑧

𝑧₀} (3.49)

𝛽 = 𝐶𝑒𝑥𝑝 {−𝐷 ^{(𝑟−𝑅𝑝)}

(𝑅_𝑠−𝑅_𝑝)} (3.50)

Theoretical Background

where the variables is described by Fig. 3.8(b) and the values of state variables like α, β are functions of A, B, C, D that can be determined from friction condition (between 0 and 1), effect of material properties, pressure, and temperature [266-267].

Figure 3.8(b) depicts the schematic view of linear velocity in r and θ direction corresponding to flat tool shoulder and workpiece interface. Assume the workpiece material is having radius r at an angle θ with respect to the direction of tool transverse speed VT. The material is rotating at an absolute rotational speed of ω. The velocity components on the boundary of the top surface of the tool shoulder is expressed as

𝑈_(𝑠)Ɵ = 𝛼[𝜔𝑟 − 𝑉_𝑇cos Ɵ] (3.51)

𝑈_(𝑠)𝑟 = 𝛼[−𝑉_𝑇sin Ɵ] (3.52)

The velocity at tool pin periphery has been defined as

𝑈_(𝑝)Ɵ = 𝛽[𝜔𝑅_𝑝− 𝑉_𝑇cos Ɵ] (3.53)

𝑈_(𝑝)𝑟 = 𝛽[−𝑉_𝑇sin Ɵ] (3.54)

Figure 3.8: (a) Schematic illustration of shoulder and pin interface with workpiece and (b) Velocity presentation in r and θ direction (top view).

Chapter 3 To account for how much each of them is responsible for the material flow, weight functions are introduced. Finally, the two velocity fields are combined using the weight functions to yield the overall velocity field. The weight functions for the shoulder and the pin are defined in both r and θ coordinates as given below:

𝛿_(𝑠)Ɵ = ^𝑈(𝑠)Ɵ

𝑈_(𝑠)Ɵ+𝑈_(𝑝)Ɵ (3.55)

𝛿_(𝑝)Ɵ= ^𝑈(𝑝)Ɵ

𝑈_(𝑠)Ɵ+𝑈(𝑝)Ɵ (3.56)

𝛿_(𝑠)𝑟= ^{𝑈(𝑠)𝑟}

𝑈_(𝑠)𝑟+𝑈_(𝑝)𝑟 (3.57)

𝛿_(𝑝)𝑟 = ^{𝑈(𝑝)𝑟}

𝑈_(𝑠)𝑟+𝑈(𝑝)𝑟 (3.58)

Incorporating the weight function with these velocity fields yields the overall velocity field in the r-Ɵ-z coordinates as given by:

𝑈_Ɵ = 𝛿_(𝑠)Ɵ𝑈_(𝑠)Ɵ+ 𝛿_(𝑝)Ɵ𝑈_(𝑝)Ɵ (3.59)

𝑈_𝑟 = 𝛿_(𝑠)𝑟𝑈_(𝑠)𝑟+ 𝛿_(𝑝)𝑟𝑈_(𝑝)𝑟 (3.60)

𝑈_𝑧 = 0 (3.61)

The velocity-strain rate relation in cylindrical (r-θ-z) coordinates is used to find the strain-rate components. The effective strain rate distribution within the FSW zone estimated by assuming von Mises criteria. The effective strain-rate (𝜀̇_𝑒𝑓𝑓) distribution in the friction stir welded zone can be determined as [267]

𝜀̇_𝑟𝑟 =^𝜕𝑈𝑟 (3.62)

Theoretical Background 𝜀̇_𝜃𝜃 = ¹

𝑟

𝜕𝑈𝜃

𝜕𝜃 −^𝜕𝑈𝑟

𝜕𝑟 (3.63)

𝜀̇_𝑧𝑧= ^𝜕𝑈𝑧

𝜕𝑧 (3.64)

𝜀̇_𝑟𝜃 =¹

2[¹

𝑟

𝜕𝑈𝜃

𝜕𝜃 +^𝜕𝑈𝜃

𝜕𝑟 −^𝑈𝜃

𝑟] (3.65)

𝜀̇_𝜃𝑧 =¹

2[^𝜕𝑈𝜃

𝜕𝑧 +¹

𝑟

𝜕𝑈_𝑧

𝜕𝜃] (3.66)

𝜀̇_𝑧𝑟 =¹

2[^𝜕𝑈𝑧

𝜕𝑟 +^𝜕𝑈𝑟

𝜕𝑧] (3.67)

𝜀̇_𝑒𝑓𝑓 = [²

3{𝜀̇_𝑟𝑟²+ 𝜀̇_ƟƟ²+ 𝜀̇_𝑧𝑧²+ 𝜀̇_𝑟Ɵ²+𝜀̇_Ɵ𝑧²+ 𝜀̇_𝑧𝑟²}]

1

2 (3.68)

𝜀̇_𝑒𝑓𝑓 = (²

3𝜀̇_𝑖𝑗²)

1

2 (3.69)

where, 𝜀̇_𝑟𝑟, 𝜀̇_𝜃𝜃, 𝜀̇_𝑧𝑧, 𝜀̇_𝑟𝜃, 𝜀̇_𝜃𝑧 and 𝜀̇_𝑧𝑟 (Eq. 3.62-3.367) are the strain rate components in the respective plane of r-θ-z coordinate and 𝜀̇_𝑖𝑗 is the strain rate in i direction on the area normal to the j direction.

The strain values are obtained from the strain-rate using rotational speed (N) as,

𝜀_𝑒𝑓𝑓 = ^{𝜀̇𝑒𝑓𝑓}

𝑁 (3.70)

However, the significant microstructural evolution takes place during FSW due to continuous dynamic recrystallization (CDRX) phenomena, resulted in a highly refined grain structure in the weld nugget. The analytical models aimed to the determination of the average grain size due to continuous dynamic recrystallization phenomena in FSW processes. Therefore, the CDRX model takes into account with few material constants to predict average grain size in the weld zone [268].

D_CDRX= C₁ε^kε̇^jD_i^hexp (− ^Q

RT) (3.71)

This initial grain size (D_i), along with the temperature (T), strain-rate 𝜀̇ and strain (ε) values predicted from the analytical models, yields the final grain size distribution in the weld zone using the Eq. (3.71) suitable for dynamic recrystallization. The constants (C1, k, j,

Chapter 3 h) in the equation were obtained by solving the equation at different points of the grain size distribution from Woo et al. [269]. The values of activation energy (Q) and gas constant (R) assumed are 140 kJ/mole and 8.314 J/kg-K.