Understanding Flexoelectric Materials and Their Asymptotic Behavior in Crack Development

Explore the utilization of asymptotic approaches to analyze crack development in flexoelectric materials, considering the influence of intensity of applied stress, limitations, advantages, and the connection to singular perturbation methods. Discover the intriguing property of flexoelectric materials, their impact on crystal symmetry, and the flexoelectric effect resulting from a combination of deformation gradients and electric polarization. Delve into the parameters and equations governing flexoelectric materials, including boundary conditions and nonlocal characteristics in classical elasticity. Discover recent research on outer and inner asymptotic solutions in flexoelectric solids using FEM studies.

• Flexoelectric materials
• Asymptotic behavior
• Crack development
• Singular perturbation methods
• Crystal symmetry

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1. Odhad souinitel asymptotickho rozvoje trhliny ve flexoelektrick m materi lu p i zat en modem I a II Profant T., Sl dek J., Sl dek V., Kotoul M. Slovensk akademie v d, Slovensk republika Vysok u en technick v Brn , esk republika SAV, stav stavebnictva a architekt ry, Bratislava, 10.11.2022

2. Motivace Existuje zp sob vyj d en sou initel asymptotick ho rozvoje p ed elem trhliny ve flexoelektrick m materi lu pomoc sou initel intenzity nap t zaveden ch v line rn elastick lomov mechanice? Jestli e ANO (bude l p?), o jak zp sob a metodu se jedn , jak m omezen a v hody? Co d l? Co je to flexoelektrick efekt? Co je to tzv. vn j a vnit n asymptotick e en pro trhlinu ve flexoelektrick m materi lu? Jak tato e en propojit pomoc singul rn perturba n metody (singular matched asymptotic expansion method) Numerick v sledky SAV, stav stavebnictva a architekt ry, Bratislava, 10.11.2022

3. Flexoelektrick materily ?? ????????,? Gradienty deformace zp sobuj zm nu symetrie krystalov m e a n sledn polarizaci i v centrosymetrick ch krystalech (direct effect). Flexoelektrick jev je d sledkem kombinace velk ch gradient deformace a elektrick polarizace v dielektrick ch materi lech. Flexoelektrick jev je vlastnost materi lu z visl na rozm rech vnit n struktury materi lu. SAV, stav stavebnictva a architekt ry, Bratislava, 10.11.2022

4. Flexoelektrick materily flexoelectrick parametry Rovnice rovnov hy Konstitutuvn vztahy ? ? = 0 ? = ?Tr ? ? + 2?? ?0 2? + ? = 0 ? = ? 1? ?1 Tr ? 2?2 ? ? + ? = 0 ? = ?2 ? + ?1?????+ ????? Boundary conditions +?2????? + ????? +?2?????+ ????? ? = ?, nap t vy ho du ? = ? ? = ? ? = ? ? ? ? ? + ? ? ? ? = ? ? = ? ? ? = ? ? = ? d lkov parameter jako materi lov charakteristika ? = ? ?0? + ? = ? SAV, stav stavebnictva a architekt ry, Bratislava, 10.11.2022

5. local Flexoelektrick materi ly nonlocal classical elasticity ???= ????? ??? ????= ??????? ???,?????,?? flexo SAV, stav stavebnictva a architekt ry, Bratislava, 10.11.2022

6. The outer and inner asymptotic solution in the flexoelectric solid flexoFEM: Sl dek& Sl dek et. al., International Journal of Solids and Structures, 2021 SGET FEM: Kotoul et . al., Fatigue & Fracture of Engineering Materials & Structures, 2020 FEM study: ??= 5.25 nm, ? = 0.25 nm ? ? ?? ?? outer solution inner solution 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

7. The outer and inner asymptotic solution in the flexoelectric solid INPUT ??, ?, ?, ?, ??, ?, ?1, ?2 outer asymptotic solution 4th-order differential equations symmetry OUTPUT crack ??, ??? ??, ??? ???, ???? ?? ???, ??? ?? isolated and traction free crack faces inner asymptotic solution Analytical solution: Mao & Purohit, Journal of the Mechanics and Physics of Solids, 2015 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

8. The outer and inner asymptotic solution in the flexoelectric solid ?0 2? + ? = 0, ? ? = 0, ? + ? = 0 Outer solution: ?0 ? ?? 1 2?? 5 8? cos1 2? cos3 4?2? 1 ???,?= ? 2? , ?? 2 1 2?? 2 7 8? sin1 2? + sin3 4?2? 1 ???,?= ? 2? , ?? ????,?= 0. ????= ????= ? gauge ? = ? ?? ?0 2? + ? = 0, ? ? = 0, ? + ? = 0 Inner solution: ?0? 1< ?? 1 ??,? = ?? 1,? 1 3 2 ?1cos1 2? + ?2cos5 3 2? ??,??= ?? ?1+ ?2cos2? + ? ?? 2? , 3 2 ?4sin1 2? ?2sin5 3 2 ? ??,??= ???2sin2? + ? ?? 2? , ???= ???= ? 1 2?1?1,?2,?4cos1 2? ? 1? 1 2? ???,??= ?0+ ? ?? ???,??. 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

9. The singular matched asymptotic expansion method ??? ??? ?? ?? ?0?for ? ?. ?0?and ?? ? ? =?0 ??? ?? ? ? =?0 ??? ?= ?= ???? ??? ?0?for ? ?. ? =?0 ?= in out 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

10. The singular matched asymptotic expansion method 1 2 1 4??2??0 2 5cos1 2? cos3 5 2? + 3? + ?4 1 2? cos1 2? + 3?2?cos5 1 2? ?0 ?1 2? 1 21 1 4??2??0 2 sin1 2? + sin3 2? ?1+ ?4 sin1 2? 3?2?sin5 1 2? ?0 2? cos5 2? cos3 2? and sin5 2? sin3 2? for ?,? < 1 ?? ? 5? ? + ?= ?? 1 212?2+ 7? 5 ?1 1 2 12??02? 6??02? ?? ?? ?2 1 2= 1 2? + 1 , 12??02? 6??02? ?? 5? + 11? ? + ? ?? 1 212?2+ ? 11 ?4 = 1 2 12??02? 6??02? 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

11. The singular matched asymptotic expansion method ?0=??? 5 4? 7 6? ?2 1 ?2 ??? 2 ???= 2+ 8?12 3?11 2?12 16 1 ? Mao & Purohit, Journal of the Mechanics and Physics of Solids, 2015 3 2 3 2 ? ? ?2 ???,? , ???,? , ?11= lim ?? ?12= lim ? = ?? ?? ? ?? ? 0 ? 0 ???=??2 1 ?2 ? 1 2 ?? 9 8 ?3 6?2 7? + 6 1 ?2 +3?22? 1 + 2?2 6? + 1 1 ?2 1 ?2 ?0 =1 ? 8 1 ? 1 ? 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

12. Numerical results The influence of the flecoelectricity? and Poisson ratio ? on the parameter ?0 ? = 0.01, ? ?0= 10 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

13. Numerical results The influence of the component ?2on the inner asymptotic solution 0.1 ?, ? = 0.01, ? = 0.2, ? ?0= 10, ? =0.2 ? = 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

14. Numerical results The crack opening displacement ?? ???, ? ? = 0.01, ? = 0.1, ? ?0= 10 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

15. Numerical results The Cauchy stress ??? ????,0 in front of the crack tip ? = 0.01, ? = 0.1, ? ?0= 10 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

16. Numerical results The electric potential ? ? ?,0 in front of the crack tip ? = 0.01, ? = 0.1, ? = 0, ? ?0= 10 27/28th International Conference ENGINEERING MECHANICS 2022, Milovy, Czech Republic, May 9 12, 2022

17. Thank you for your attention The authors acknowledge the supports by the Slovak Science and Technology Assistance Agency registered under number APVV-18-0004, VEGA-2/0061/20 CCTF: 5thIJFatigue & FFEMS Joint Workshop, Characterisation of Crack Tip Fields, Heidelberg April 08 10, 2019