1. Frictional Heating During an Earthquake
By Yao Yu & Yue Du

2. Coseismic Temperature Increases During a Crack-like Rupture
v/2
- v/2
(Fialko, 2004)
Heat transfer: 1-D diffusion equation 𝜕𝑇 𝜕𝑡 =𝜅 𝜕 2 𝑇 𝜕 𝑦 2 + 𝑄 𝑐𝜌

3. Solve for 1-D diffusion equation
Approach: Fourier Transform ℱ 𝜕∆𝑇 𝜕𝑡 =ℱ 𝜅 𝜕 2 ∆𝑇 𝜕 𝑡 2 + 𝑄 𝑐𝜌 =ℱ 𝜅 𝜕 2 ∆𝑇 𝜕 𝑡 2 +ℱ 𝑄 𝑐𝜌 ℱ ∆𝑇 =𝐹, ℱ 𝑄 𝑐𝜌 =𝐺 ℱ 𝜕∆𝑇 𝜕𝑡 = 𝜕 𝜕𝑡 ℱ ∆𝑇 = 𝜕 𝜕𝑡 𝐹 ℱ 𝜅 𝜕 2 ∆𝑇 𝜕 𝑦 2 = (𝑖2𝜋𝑘) 2 𝜅𝐹=−4 𝜋 2 𝑘 2 𝜅𝐹

4. 𝜕 𝜕𝑡 𝐹=−4 𝜋 2 𝑘 2 𝜅𝐹+𝐺 Integrating factor: 𝑒 4 𝜋 2 𝑘 2 𝜅𝑑𝑡 = 𝑒 4 𝜋 2 𝑘 2 𝜅𝑡 𝐹= 𝑒 −4 𝜋 2 𝑘 2 𝜅𝑡 0 𝑡 𝑒 4 𝜋 2 𝑘 2 𝜅𝜏 𝐺𝑑𝜏 ∆ 𝑇=𝑓 𝐹 = −∞ ∞ 𝑒 −4 𝜋 2 𝑘 2 𝜅𝑡 0 𝑡 𝑒 4 𝜋 2 𝑘 2 𝜅𝜏 𝐺𝑑𝜏 𝑒 𝑖2𝜋𝑘𝑦 𝑑𝑘 = −∞ ∞ 0 𝑡 𝐺 𝑒 −4 𝜋 2 𝑘 2 𝜅(𝑡−𝜏) 𝑒 𝑖2𝜋𝑘𝑦 𝑑𝜏𝑑𝑘
(1st order ODE)

5. Convolution theorem
𝑓∗𝑔= ℱ −1 ℱ 𝑓 ∙ℱ 𝑔 ∆ 𝑇= −∞ ∞ 0 𝑡 𝐺 𝑒 −4 𝜋 2 𝑘 2 𝜅(𝑡−𝜏) 𝑒 𝑖2𝜋𝑘𝑦 𝑑𝜏𝑑𝑘 Since ℱ 𝑄 𝑐𝜌 =𝐺,ℱ 1 2 𝜋𝜅(𝑡−𝜏) 𝑒 − 𝑥 2 4𝜅(𝑡−𝜏) = 𝑒 −4 𝜋 2 𝑘 2 𝜅(𝑡−𝜏) ∆𝑇(𝑦,𝑡)= 1 2𝑐𝜌 𝜋𝜅 0 𝑡 −∞ ∞ exp⁡ (𝑦−𝜁) 2 4𝜅(𝜏−𝑡) 𝑄(𝜁,𝜏) 𝑡−𝜏 𝑑𝜁𝑑𝜏

6. Q - Rate of Frictional Heat Generation
Finite thickness: 2𝜔
Velocity varies linearly across the fault zone: 𝜕𝑉 𝜕𝑦 =𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑄= 𝜎 𝑠 2𝜔 𝜕𝐷 𝜕𝑡 = 𝜇𝜎 𝑛 2𝜔 𝑣
𝑄= 𝜇𝜎 𝑛 𝑣 2𝜔 , 𝑡>0, 𝑦 <𝜔 0, 𝑦 >𝜔

7. Solution
∆𝑇 𝑦,𝑡 = 1 2𝑐𝜌 𝜋𝜅 0 𝑡 −∞ ∞ exp 𝑦−𝜁 2 4𝜅(𝜏−𝑡) 𝑄(𝜁,𝜏) 𝑡−𝜏 𝑑𝜁𝑑𝜏 Substitute 𝑄= 𝜇𝜎 𝑛 𝑣 2𝜔 , 𝑡>0, 𝑦 <𝜔 0, 𝑦 >𝜔 ∆𝑇 𝑦,𝑡 = 𝜇𝜎 𝑛 𝑣 4𝑐𝜌𝜔 0 𝑡 erf 𝑦+𝜔 2 𝜅 𝑡−𝜏 − erf 𝑦−𝜔 2 𝜅 𝑡−𝜏 𝑑𝜏 0<𝑡< 𝑡 𝑚 𝜇𝜎 𝑛 𝑣 4𝑐𝜌𝜔 0 𝑡 𝑚 erf 𝑦+𝜔 2 𝜅 𝑡−𝜏 − erf 𝑦−𝜔 2 𝜅 𝑡−𝜏 𝑑𝜏 𝑡≥ 𝑡 𝑚
(in unit of J/m3/s)

8. Example Density 𝜌 2700 kg/m3 Heat capacity 𝑐 1 kJ/kg/K
Thermal diffusivity 𝜅
1 mm2/s
Coefficient of friction 𝜇
0.6
Fault normal stress
𝜎 𝑛
100 MPa
Slip velocity 𝑣
1 m/s
Slip duration 𝑡
5 s

9. Finite thickness: 2𝜔 = 1 mm & 2𝜔 = 10 cm, at t = 5 s

10. i). Finite thickness: 2𝜔 = 1 mm, t = 0-5 s

11. i). Finite thickness: 2𝜔 = 1 mm, t = 0-20 s

12. ii). Finite thickness: 2𝜔 = 10 cm, t = 0-5 s

13. ii). Finite thickness: 2𝜔 = 10 cm, t = 0-5 s

14. Characteristic length: 2𝜅𝜏 ≈3.2 𝑚𝑚
2𝑤 2𝜅𝜏 = 1𝑚𝑚 3.2𝑚𝑚 ≈ 0.31< 𝑤 2𝜅𝜏 =0.5, 1, 2, 5
2𝑤 2𝜅𝜏 = 10 𝑐𝑚 3.2𝑚𝑚 ≈31≫1 2
Non-dimensional temperature increasement
Non-dimensional distance
Cardwell et al., 1978