Partition of mixed-mode fractures in 2D elastic beams with through-thickness shear force

Christopher Harvey (1256223)

Joseph D. Wood (7119710)

Simon Wang (1250952)

English

Loughborough University Institutional Repository

Partition of Mixed-Mode Fractures in 2D Elastic Beams with Through-Thickness Shear Forces  Joe Wood, Chris Harvey, Simon Wang  J.Wood@lboro.ac.uk Department of Aeronautical & Automotive Engineering Loughborough University, LE11 3TU, UK 19th International Conference of Composite Structures • Cracks tend to propagate along interfaces in laminated materials because they represent a plane of weakness.  • They do not kink in order to propagate under pure mode I opening conditions, as they would tend to in an isotropic material.   • Interfacial cracks therefore propagate in a mixed-mode with a combination of mode I opening, mode II shearing, and/or mode III tearing. Ω Interfacial cracks 2  • Fracture toughness depends on the fracture mode partition.  • Predicting fracture toughness requires the knowledge of the partition of a mixed-mode fracture.   • Essential to have a correct analytical partition theory to predict the fracture toughness.  Fracture toughness 3 One-dimensional fractures 4 Mixed-mode interfacial fracture • 1D fracture of DCB is fundamental case for study – Bending moments 𝑀𝑀1 and 𝑀𝑀2 – Axial forces 𝑁𝑁1 and 𝑁𝑁2 – Shear forces 𝑃𝑃1 and 𝑃𝑃2 – 𝜂𝜂 = 𝐸𝐸2 𝐸𝐸1⁄ , Ν = 𝜈𝜈2 𝜈𝜈1⁄ , 𝛾𝛾 = ℎ2 ℎ1⁄  Double cantilever beam (DCB) 5  Total energy release rate (ERR) • Quadratic form and non-negative definite • Partition total ERR 𝐺𝐺 into its pure mode components, 𝐺𝐺𝐼𝐼 and 𝐺𝐺𝐼𝐼𝐼𝐼 • Use the orthogonal pure fractures modes [ ]=BBBBTBBBBNNMMCNNMMG21212121( )bhhEEfCij ,,,,,, 212121 νν=6  Pure fracture modes • The inner product matrix transforms the 𝑀𝑀1𝐵𝐵 𝑀𝑀2𝐵𝐵   vectors into ERR space • In ERR space, orthogonality between two 𝑀𝑀1𝐵𝐵 𝑀𝑀2𝐵𝐵  vectors means 𝑀𝑀1𝐵𝐵 𝑀𝑀2𝐵𝐵 1 𝐶𝐶 𝑀𝑀1𝐵𝐵 𝑀𝑀2𝐵𝐵 2𝑇𝑇 = 0 • Orthogonal pairs of 𝑀𝑀1𝐵𝐵 𝑀𝑀2𝐵𝐵  vectors exist that represent pure fracture modes • Denote pure mode I as 1 𝑀𝑀2𝐵𝐵/𝑀𝑀1𝐵𝐵 = 1 𝜃𝜃1  • Denote pure mode II as 1 𝑀𝑀2𝐵𝐵/𝑀𝑀1𝐵𝐵 = 1 𝛽𝛽1 , etc. • With 𝜃𝜃𝑖𝑖  and 𝛽𝛽𝑖𝑖 = 𝑓𝑓 𝐸𝐸1,𝐸𝐸2, 𝜈𝜈1, 𝜈𝜈2,ℎ1,ℎ2, 𝑏𝑏  Contours of ERR with 𝐸𝐸 = 1, 𝑏𝑏 = 1, ℎ = 1, 𝛾𝛾 = 1, 𝜂𝜂 = 1 7  ERR partitions general theory • Euler beam partitions:    • Timoshenko beam partitions:   • 2D elasticity partitions:    ′−′−′−−−−=32211213221121 ββββββBBBBBBBBIEIENNMMNNMMcG−−−−−−='32'21'1213221121 θθθθθθBBBBBBBBIIEIIENNMMNNMMcG252413221121 −−−−−=βββββBBBBBBITITPPNNMMcG252413221121 −−−−−=θθθθθBBBBBBIITIITPPNNMMcG22D-522D-412D-322D-212D-1212D-2D- −−−−−=βββββBBBBBBIIPPNNMMcG22D-522D-412D-322D-212D-1212D-2D- −−−−−=θθθθθBBBBBBIIIIPPNNMMcG8 • Bending moments 𝑀𝑀1𝐵𝐵 and 𝑀𝑀2𝐵𝐵 and axial forces 𝑁𝑁1𝐵𝐵 and 𝑁𝑁2𝐵𝐵 • Revisit the orthogonal pure fracture modes 𝜃𝜃𝑖𝑖 ,𝛽𝛽𝑖𝑖  – Condition using beam theories does not produce the same stress distribution in 2D elasticity theory  – Apply a correction factor for 2D elasticity to the part of the condition that represents the intact portion of the beam   – Calibrate correction factor for 𝜃𝜃1−2𝐷𝐷 using 𝜃𝜃1 ≤ 𝜃𝜃1−2𝐷𝐷 ≤ 𝜃𝜃1′   – Obtain other pure modes (𝜃𝜃2−2𝐷𝐷 ,𝛽𝛽1−2𝐷𝐷 ,𝛽𝛽2−2𝐷𝐷 , etc.) using orthogonality     General 2D elasticity partition theory 9  10 Timoshenko beam partition theory • Crack tip through-thickness shear forces 𝑃𝑃1𝐵𝐵 and 𝑃𝑃2𝐵𝐵 only – 𝑀𝑀1𝐵𝐵 = 𝑀𝑀2𝐵𝐵 = 𝑁𝑁1𝐵𝐵 = 𝑁𝑁2𝐵𝐵 = 0    • 𝜃𝜃𝑃𝑃−𝑇𝑇  ,𝛽𝛽𝑃𝑃−𝑇𝑇 = −1 , 𝛾𝛾     ∴     𝐺𝐺𝐼𝐼𝐼𝐼 = 0 • Shear correction factor 𝜅𝜅 = 5 6⁄   𝐺𝐺𝜃𝜃𝑃𝑃−𝑇𝑇 = 12𝑏𝑏2ℎ1𝜅𝜅𝜅𝜅 1 + 𝜃𝜃𝑃𝑃−𝑇𝑇2𝛾𝛾  𝐺𝐺𝛽𝛽𝑃𝑃−𝑇𝑇 = 12𝑏𝑏2ℎ1𝜅𝜅𝜅𝜅 1 + 𝛽𝛽𝑃𝑃−𝑇𝑇2𝛾𝛾 − 1 + 𝛽𝛽𝑃𝑃−𝑇𝑇 21 + 𝛾𝛾   • Crack tip through-thickness shear forces 𝑃𝑃1𝐵𝐵 and 𝑃𝑃2𝐵𝐵 only – 𝑀𝑀1𝐵𝐵 = 𝑀𝑀2𝐵𝐵 = 𝑁𝑁1𝐵𝐵 = 𝑁𝑁2𝐵𝐵 = 0    • 𝜃𝜃𝑃𝑃−2𝐷𝐷 ,𝛽𝛽𝑃𝑃−2𝐷𝐷 = ? ?,  ? ?  • Shear correction factor now 𝛾𝛾 dependent 𝜅𝜅 𝛾𝛾  • 𝐺𝐺𝐼𝐼𝐼𝐼 ≠ 0 and introduce pure-mode-II correction factor 𝑐𝑐 𝛾𝛾     𝐺𝐺𝛽𝛽𝑃𝑃−2𝐷𝐷 = 12𝑏𝑏2ℎ1𝜅𝜅 𝛾𝛾 𝜅𝜅 1 + 𝛽𝛽𝑃𝑃−2𝐷𝐷2𝛾𝛾 − 1 + 𝛽𝛽𝑃𝑃−2𝐷𝐷 21 + 𝛾𝛾 𝑐𝑐 𝛾𝛾  𝐺𝐺𝜃𝜃𝑃𝑃−2𝐷𝐷 = 12𝑏𝑏2ℎ1𝜅𝜅 𝛾𝛾 𝜅𝜅 1 + 𝜃𝜃𝑃𝑃−2𝐷𝐷2𝛾𝛾  , 11 2D elasticity partition theory  Shear Force Pure Modes • 𝜃𝜃𝑃𝑃−2𝐷𝐷,𝛽𝛽𝑃𝑃−2𝐷𝐷  • FEM simulations  • −1.7 ≤ log10 1 𝛾𝛾⁄ ≤ 1.7  • Pure mode I 𝜃𝜃𝑃𝑃−2𝐷𝐷 – 𝐺𝐺𝐼𝐼𝐼𝐼 = 0,  𝜃𝜃𝑃𝑃−2𝐷𝐷 = −1 – ∴ 𝑃𝑃2𝐵𝐵 = −𝑃𝑃1𝐵𝐵  • Pure mode II 𝛽𝛽𝑃𝑃−2𝐷𝐷 – 𝐺𝐺𝐼𝐼 = 0 – 𝛽𝛽𝑃𝑃−2𝐷𝐷 = 𝛾𝛾exp(−1.986060 atanh(0.563483𝛾𝛾𝑖𝑖))  12 Shear & Pure Mode II Correction Factors • FEM Simulations • 𝑀𝑀1𝐵𝐵 = 𝑀𝑀2𝐵𝐵 = 𝑁𝑁1𝐵𝐵 = 𝑁𝑁2𝐵𝐵 = 0 • −1.7 ≤ log10 1 𝛾𝛾⁄ ≤ 1.7  • Shear Correction Factor – 𝜅𝜅(𝛾𝛾) – 𝑃𝑃2𝐵𝐵 𝑃𝑃1𝐵𝐵⁄ = 𝜃𝜃𝑃𝑃−2𝐷𝐷 = −1 • Pure-mode-II ERR Correction Factor – 𝑐𝑐(𝛾𝛾) – 𝑃𝑃2𝐵𝐵 𝑃𝑃1𝐵𝐵⁄ = 𝛽𝛽𝑃𝑃−2𝐷𝐷 13 Numerical Verification 110 ≤ 𝛾𝛾 ≤ 10 14 Numerical Verification 110 ≤ 𝛾𝛾 ≤ 10 15 Blister Test Image from Koenig (2011) • Interface fracture toughness 16  Thick substrate Adhesion of graphene membranes 𝛾𝛾 = ℎ2 ℎ1⁄ → ∞ 𝐺𝐺𝐼𝐼 = 6𝑀𝑀𝐵𝐵𝐵𝐵2𝐸𝐸ℎ3 1 − 𝜈𝜈2 1 − 𝑁𝑁𝐵𝐵𝐵𝐵ℎ4.450𝑀𝑀𝐵𝐵𝐵𝐵 − 𝜆𝜆 2 0.6227 𝐺𝐺𝐼𝐼𝐼𝐼 = 6𝑀𝑀𝐵𝐵𝐵𝐵2𝐸𝐸ℎ3 1 − 𝜈𝜈2 𝑁𝑁𝐵𝐵𝐵𝐵ℎ2.697𝑀𝑀𝐵𝐵𝐵𝐵 2 0.3773 17 Adhesion of graphene membranes • Pressure loaded blister test – Linear failure critierion  – 𝐺𝐺𝐼𝐼𝐼𝐼 = 0.226 𝐽𝐽 𝑚𝑚2⁄   – 𝐺𝐺𝐼𝐼𝐼𝐼𝐼𝐼 = 0.683 𝐽𝐽 𝑚𝑚2⁄     18  Adhesion of graphene membranes • Pressure loaded blister test – Linear failure critierion  – 𝐺𝐺𝐼𝐼𝐼𝐼 = 0.226 𝐽𝐽 𝑚𝑚2⁄   – 𝐺𝐺𝐼𝐼𝐼𝐼𝐼𝐼 = 0.683 𝐽𝐽 𝑚𝑚2⁄   – 𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚 = 𝐺𝐺𝐼𝐼 𝐺𝐺𝐼𝐼𝐼𝐼⁄ = 0.431 – 𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑖𝑖 = 𝐺𝐺𝐼𝐼 𝐺𝐺𝐼𝐼𝐼𝐼⁄ = 0.764     19  Experimental validation • Pressure loaded blister test – Koenig et al. (2011) – Linear failure critierion  – 𝐺𝐺𝐼𝐼𝐼𝐼 = 0.226 𝐽𝐽 𝑚𝑚2⁄  and 𝐺𝐺𝐼𝐼𝐼𝐼𝐼𝐼 = 0.683 𝐽𝐽 𝑚𝑚2⁄   • Point loaded blister – Zong et al. (2012) – Experimental Results – 𝛿𝛿 𝑅𝑅𝐵𝐵⁄ = 0.2309 , 𝐸𝐸 = 1TPa , 𝑛𝑛𝑛𝑛 = 1.7nm and 𝑛𝑛 = 5. – 𝐺𝐺𝐵𝐵𝑒𝑒𝑒𝑒 = 0.438 𝐽𝐽 𝑚𝑚2⁄  – Mode mixity 𝜌𝜌𝑚𝑚𝑡 = 𝐺𝐺𝐼𝐼 𝐺𝐺𝐼𝐼𝐼𝐼⁄ = 0.381 – Linear failure criterion 𝐺𝐺𝑚𝑚𝑡 = 0.438 𝐽𝐽 𝑚𝑚2⁄      20  Conclusion • 2D elasticity partition theory – Developed for general loading conditions (bending moments, axial forces and shear forces).  – Numerically verified for a number of loading conditions • Application to: – Adhesion of graphene membranes – Adhesion energy has been explained and well-predicted   21  Thank you very much for your attention • Submitted for publication at Composite Structures – Partition of mixed-mode fractures in 2D elastic orthotropic laminated beams under general loading (2016). Questions are now welcome 22  

Partition of mixed-mode fractures in 2D elastic beams with through-thickness shear forces

https://repository.lboro.ac.uk/articles/Partition_of_mixed-mode_fractures_in_2D_elastic_beams_with_through-thickness_shear_forces/9221624/files/16799084.pdf

Partition of mixed-mode fractures in 2D elastic beams with through-thickness shear forces

Abstract

Similar works

Full text

Available Versions

Loughborough University Institutional Repository