Parameters of interest based on prior models

The following parameters are important when considering those which are relevant to the model to be developed. Based on the previous models, the heat generated on each surface is looked at in more detail as well as what previous models have used and assumed on the convection and friction coefficients. The contact condition is also discussed further since it has been found that this incorporates not only if the weld is of a good quality but the friction coefficient as well as tool geometry. Since friction is created by the two surfaces interacting with one another and this generates heat, the slip condition is then an important parameter to investigate since it can also be related to the temperature the weld produces.

2.3.5.1 Heat generation

The amount of heat generated in the model is predominantly caused by the tools properties and the friction between the tool and work piece surfaces. The total amount of energy is the sum of the heat generated from the tools probe and shoulder surface.

𝑄 = 𝑄1+ 𝑄2+ 𝑄3 (2.4)

Q1= Heat generated from shoulder [34] Q2= Heat generated from probe [34]

46 The final result after integration of above with respect to the radii of each, the following total heat generation can be calculated as follows according to [34]

𝑄𝑇𝑜𝑡𝑎𝑙 = 2

3𝜋𝜏𝑐𝑜𝑛𝑡𝑎𝑐𝑡𝜔(𝑅𝑠ℎ𝑜𝑢𝑙𝑑𝑒𝑟 3 _{+ 3𝑅}

𝑝𝑟𝑜𝑏𝑒2 𝐻𝑝𝑟𝑜𝑏𝑒) (2.5)

Based on Chao-Qi-Tang, [23] concludes that 5% of the heat generated by friction flows to the tool and the remaining 95% flows through the work piece

The following heat generation continues from the above and describes the surfaces of interest that need to be accounted for during the FSW process. These heat contributions on the surfaces are shown in Figure 2.20.

From Figure 2.20, the following heat generation contribution is as follows and is according to Chao-Qi-Tang [23]

𝑄3= 𝑄4+ 𝑞1 (2.6)

𝑄1 = 𝑄2+ 𝑞2+ 𝑄 (2.7)

Q1 = Heat flux into the model due to the friction between tool and work piece surface Q2 = Heat conducted from the bottom of the work piece to the backing plate

Q3 = Heat flux from the friction between tool and work piece surface Q4 = Heat transferred from the clamped tool to the machine bushing q1 = Convection due to heat lost from the tool surface to the air

Q

Q

Q

Q

q

q

47 q2 = Convection due to heat lost from work piece surface to the air

Q = Increase of heat content in the work piece

Since Q3 is the heat generation coming from the tool and Q1 is the heat generation into the work piece from the tool, the assumption is;

𝑄3= 𝑄1= 𝑄𝑡𝑜𝑡𝑎𝑙 (2.8)

We can describe these two as the total heat generated by each of the contributing factors as described in the above equations.

Each of the definitions above can be represented in their own form. For the work piece Q2 and q2 are described in terms of heat flux loss, according to Muhsin et al [17]

𝑞2= ℎ𝑞2(𝑇 − 𝑇0) + 𝜀𝐹1𝜎𝑆𝐵(𝑇4− 𝑇04) (2.9)

𝑄2= ℎ𝑄2(𝑇 − 𝑇0) (2.10)

Then, based on convection, the heat transfer and heat flux [13] for the tool pieces can be defined as;

𝑄4= ℎ4𝐴𝑐(𝑇 − 𝑇0) (2.11)

𝑞1= 𝜇𝐹𝜔𝑅𝑠ℎ𝑜𝑢𝑙𝑑𝑒𝑟2𝜋 (2.12)

Where Ac and As are the surface areas for the clamped tool surface and shoulder surface respectively.

Additional heat transfer into the work piece is described as

𝑄 = ℎ𝑄(𝑇 − 𝑇0) (2.13)

Heat flux along the tool into the work piece is expressed as follows with reference to Chao-Qi- Tang [20]

2.3.5.2 Contact states for slip condition

The contact condition between surfaces is important as it affects the friction coefficient variable. Three contact conditions exist;

48  Sticking

 Sticking and sliding combination  Sliding

Contact condition variable is expressed as δ and the slip condition as γ [34].

𝛿 =𝑣𝑚𝑎𝑡𝑟𝑖𝑥 𝑣_{𝑡𝑜𝑜𝑙} = 1 − 𝛾̇ 𝑣_{𝑡𝑜𝑜𝑙} (2.15) Where; 𝑣𝑡𝑜𝑜𝑙= 𝜔𝑟 𝑎𝑛𝑑 𝛾̇ = 𝑣𝑡𝑜𝑜𝑙− 𝑣𝑚𝑎𝑡𝑟𝑖𝑥 (2.16)

The following Table 2.13 [34] outlines the contact conditions and the results of each for the state variable value and the shear stress and matrix velocity conditions.

Based on the heat generated in equation 2.3, each of the above conditions change with contact shear conditions [34]

For sticking conditions, the contact shear is equal to the yield stress divided by the root of three. Yield stress is temperature dependent and the heat generated for pure sticking conditions is as follows;

(2.17) For pure sliding, the contact shear stress is the friction coefficient multiplied by the contact pressures [34] and the heat generated is expressed as

Combination of sliding and sticking condition takes into account the contact condition variable (δ)

(2.18) Table 2.13: Contact condition

49 𝑄𝑡𝑜𝑡𝑎𝑙= 𝛿𝑄𝑡𝑜𝑡𝑎𝑙,𝑠𝑡𝑖𝑐𝑘𝑖𝑛𝑔+ (1 − 𝛿)𝑄𝑡𝑜𝑡𝑎𝑙,𝑠𝑙𝑖𝑑𝑖𝑛𝑔 (2.19)

2.3.5.3 Friction coefficient

Friction coefficient is affected by the changes in temperature. An example appears on waxed- wood skis, where the friction coefficient differs at 0.02 for temperatures above -10°C but 0.4 for temperatures below this amount [45].

Based on [28] it is concluded that for sliding conditions, friction coefficient lies within known values between surface conditions and for sticking conditions, yield shear stress relating to work piece material for varying temperatures is used.

M. Mijajlovic et al [44] predicts the friction coefficient in terms of the momentum of friction and axial force which is dominantly expressed for the plunge phase

𝜇 =

𝐹𝑧𝑑

(2.20)

Where;

 d is the diameter of the tool probe  Ft is the tangential force

 L is the length of the force/friction pole  Fz is the axial force

For the slip condition to be taken into account which is affected by the temperature and strain rate conditions, the friction coefficient is expressed as follows with reference to A.H Kheireddine [26]

2.3.5.4 Convection coefficient

The convection coefficient is expressed as follows;

ℎ =

𝐴∆𝑇 (2.22)

Where;

A=Surface area

ΔT = Difference in temperature between surface temperature and fluid temperature (In our case is air)

50 Q=Heat transfer