1. EM 388F Term Paper: Discussion of Fracture Criterions under Impermeable and Permeable Crack Surface of Piezoelectric Materials RONG JIAO April 27, 2008

2. Outline of this topic Fundamental equations for piezoelectric material Impermeable crack surface J-integral or Total Potential Energy Release Rate (TPERR ) Mechanical Strain Energy Release Rate ( MSERR ) Permeable crack surface Total Potential Energy Release Rate (TPERR ) Mechanical Strain Energy Release Rate ( MSERR )

3. Fundamental equations for piezoelectric materials Field equations: Boundary equations: Constitutive equations: Crack surface Impermeable or permeable? ~stress tensor ~body force ~ electric displacement ~body charge ~traction vector ~surface change ~elastic module ~dielectric constants ~piezoelectric constants ~electric field

4. Stroh’s complex potential theorem (Suo et al.,1992)

5. Impermeable crack surface: Boundary condition: Traction free condition: Charge free condition: Stress intensity factors (SIF) : Electric displacement intensity factor (EDIF):  SIF is not a good fracture criterion for the impermeable crack since they do not consider the influence of the electric field on fracture.

6. J-integral (TPERR: Total Potential Energy Release Rate) (Park and Sun,1995, International Journal of Fracture 70, 203~216) Why TPERR is not a good fracture criterion here? 1. Negative last term always block the crack growth 2. Do not agree with the experiment Mechanical Coupling Electric, Negative? ( Mode I )

7. MSERR (Mechanical Strain Energy Release Rate)  Fracture is mechanical process and it’s more reasonable to only use the mechanical energy release rate as the fracture criterion. (Mode I) (Park and Sun,1995,International Journal of Fracture 70, 203~216)  The negative electric term vanishes.  Positive electric field decreases the critical stress and hence accelerate crack growth while negative electric field increases the fracture stress hence blocks crack growth.  These conclusions match experimental results very well.

8. Permeable crack surface Boundary condition: Traction free condition: Charge continuous condition:  Stress intensity factors (SIF) :

9. Energy calculations for permeable crack surface: ( Xu and Rajapakse, 2001, International Journal of Solids and Structures, 38: 7643) ~distributed normal eclectic displacement ~complex coefficient in the complex potentials  Electric displacement intensity factor (EDIF):

10. Energy analysis for impermeable and permeable crack ( Liu and Chen, 2002, International Journal of Fracture, 116: 15-20 ) Mechanical part of TPERR (MSERR) 1. It’s constant for permeable crack under different electric loading. 2. It’s always bigger than that for impermeable crack, which implies the permeable boundary enhance the crack growth. 3. For the impermeable crack, MSERR is influenced by the applied electric loading: Positive electric load increases MSERR while negative decreases MSERR.

11. Energy analysis for impermeable and permeable crack (Liu and Chen,2002) (Continue) Electric part of TPERR 1. It’s almost zero for permeable crack under different electric loading and this is the reason why TPERR could be used as a fracture criterion for the permeable crack. 2. For the impermeable crack, it is affected by the applied electric loading and this is the reason why TPERR could not be used as a fracture criterion for the impermeable crack.

12. Comments? End and thank You!