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- 1. Invariant-based Method for accelerating certification testing 𝝈 = 𝑸 𝜺 ⇒ 𝜎𝑥𝑥 𝜎 𝑦𝑦 𝜏 𝑥𝑦 = 𝜎1 𝜎2 𝜏6 = 𝑄11 𝑄12 𝑄16 𝑄21 𝑄22 𝑄26 𝑄61 𝑄62 𝑄66 𝜀1 𝜀2 𝜀6
- 2. 2 • Theory of Linear Elasticity – Basic overview and review of the most important aspects • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool
- 3. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 3 • Cauchy Stress Tensor 𝜎 = 𝜎𝑥𝑥 𝜏 𝑥𝑦 𝜏 𝑥𝑧 𝜏 𝑦𝑥 𝜎 𝑦𝑦 𝜏 𝑦𝑧 𝜏 𝑧𝑥 𝜏 𝑧𝑦 𝜎𝑧𝑧 Valid according to the following assumptions – Continuum Medium (macroscopic analysys) – Homogeneous Material - specific properties are indepedent of the point of evalution, i.e. Mechancial properties of any given point are equal do the specific properties of the solid 𝜏 𝑦𝑥 𝜏 𝑥𝑦 𝑦 𝑥𝜏 𝑦𝑥 𝜏 𝑥𝑦
- 4. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 4 • Cauchy Stress Tensor 𝜎 = 𝜎𝑥𝑥 𝜏 𝑥𝑦 𝜏 𝑥𝑧 … 𝜎 𝑦𝑦 𝜏 𝑦𝑧 … … 𝜎𝑧𝑧 Momentums Equilibrium 𝜏 𝑦𝑧 + 𝜕𝜏 𝑦𝑧 𝜕𝑦 𝑑𝑦 2 + 𝜏 𝑦𝑧 − 𝜕𝜏 𝑦𝑧 𝜕𝑦 𝑑𝑦 2 − 𝜏 𝑧𝑦 + 𝜕𝜏 𝑧𝑦 𝜕𝑧 𝑑𝑧 2 − 𝜏 𝑧𝑦 − 𝜕𝜏 𝑧𝑦 𝜕𝑧 𝑑𝑧 2 𝑑𝑥𝑑𝑦𝑑𝑧 2 = 0 − 𝜏 𝑥𝑧 + 𝜕𝜏 𝑥𝑧 𝜕𝑥 𝑑𝑥 2 − 𝜏 𝑥𝑧 − 𝜕𝜏 𝑥𝑧 𝜕𝑥 𝑑𝑥 2 + 𝜏 𝑧𝑥 + 𝜕𝜏 𝑧𝑥 𝜕𝑧 𝑑𝑧 2 + 𝜏 𝑧𝑥 − 𝜕𝜏 𝑧𝑥 𝜕𝑧 𝑑𝑧 2 𝑑𝑥𝑑𝑦𝑑𝑧 2 = 0 𝜏 𝑥𝑦 + 𝜕𝜏 𝑥𝑦 𝜕𝑥 𝑑𝑥 2 + 𝜏 𝑥𝑦 − 𝜕𝜏 𝑥𝑦 𝜕𝑥 𝑑𝑥 2 − 𝜏 𝑦𝑥 + 𝜕𝜏 𝑦𝑥 𝜕𝑦 𝑑𝑦 2 − 𝜏 𝑦𝑥 − 𝜕𝜏 𝑦𝑥 𝜕𝑦 𝑑𝑦 2 𝑑𝑥𝑑𝑦𝑑𝑧 2 = 0
- 5. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 5 • Cauchy Stress Tensor 𝜎 = 𝜎𝑥𝑥 𝜏 𝑥𝑦 𝜏 𝑥𝑧 … 𝜎 𝑦𝑦 𝜏 𝑦𝑧 … … 𝜎𝑧𝑧 Symmmetry relations: 𝜏 𝑦𝑧 = 𝜏 𝑧𝑦 𝜏 𝑥𝑧 = 𝜏 𝑧𝑥 𝜏 𝑥𝑦 = 𝜏 𝑦𝑥 𝑚 𝑦𝑧 𝜏 𝑦𝑥 𝜏 𝑥𝑦 𝑦 𝑥 𝜏 𝑦𝑥 𝜏 𝑥𝑦 𝑚 𝑦𝑧 𝑚 𝑥𝑧𝑚 𝑥𝑧 Length scale effects (microanalysis) Equilibrium Equations degenerate in the Symetry relations
- 6. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 6 • Infinitesimal Strain Tensor 𝜀 = 𝜀 𝑥𝑥 𝜀 𝑥𝑦 𝜀 𝑥𝑧 𝜀 𝑦𝑥 𝜀 𝑦𝑦 𝜀 𝑦𝑧 𝜀 𝑧𝑥 𝜀 𝑧𝑦 𝜀 𝑧𝑧 = 1 2 2𝜀 𝑥𝑥 𝛾𝑥𝑦 𝛾𝑥𝑧 𝛾𝑦𝑥 2𝜀 𝑦𝑦 𝛾𝑦𝑧 𝛾𝑧𝑥 𝛾𝑧𝑦 2𝜀 𝑧𝑧 Tensorial Strain Engineering Strain Strain Definition – Small Strains and displacments (linear and angular) Theory – Others Strain Definitions: • Finite strains and displacemnts Theory – Elastomers, fluids, biological or not soft tissues • Small Strains and large displacements Theory
- 7. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 7 • Infinitesimal Strain Tensor 𝜀 = 𝜀 𝑥𝑥 𝜀 𝑥𝑦 𝜀 𝑥𝑧 … 𝜀 𝑦𝑦 𝜀 𝑦𝑧 … … 𝜀 𝑧𝑧 = 1 2 2𝜀 𝑥𝑥 𝛾𝑥𝑦 𝛾𝑥𝑧 … 2𝜀 𝑦𝑦 𝛾𝑦𝑧 … … 2𝜀 𝑧𝑧 Tensorial Strain Engineering Strain Symmetry due to the Displacement Definition 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝜀 𝑧𝑧 𝛾𝑦𝑧 𝛾𝑥𝑧 𝛾𝑥𝑦 = 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝜀 𝑧𝑧 2𝜀 𝑦𝑧 2𝜀 𝑥𝑧 2𝜀 𝑥𝑦 = 𝜀11 𝜀22 𝜀33 𝜀23 𝜀13 𝜀12 = 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 Voigt Notation
- 8. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 8 • Generalized 3D Hook‘s Law 𝜎 = Ώ 𝜀 ⇒ 𝜎 𝑥𝑥 𝜎 𝑦𝑦 𝜎𝑧𝑧 𝜏 𝑦𝑧 𝜏 𝑥𝑧 𝜏 𝑥𝑦 𝜏 𝑧𝑦 𝜏 𝑧𝑥 𝜏 𝑦𝑥 = 𝐶 𝑥𝑥𝑥𝑥 𝐶 𝑥𝑥 𝑦𝑦 𝐶 𝑥𝑥 𝑧𝑧 𝐶 𝑥𝑥 𝑦𝑧 𝐶 𝑥𝑥 𝑥𝑧 𝐶 𝑥𝑥 𝑥𝑦 𝐶 𝑥𝑥 𝑧𝑦 𝐶 𝑥𝑥 𝑧𝑥 𝐶 𝑥𝑥 𝑦𝑥 𝐶 𝑦𝑦 𝑥𝑥 ⋱ ⋮ 𝐶𝑧𝑧 𝑥𝑥 ⋱ ⋮ 𝐶 𝑦𝑧 𝑥𝑥 ⋱ ⋮ 𝐶 𝑥𝑧 𝑥𝑥 ⋱ ⋮ 𝐶 𝑥𝑦 𝑥𝑥 ⋱ ⋮ 𝐶𝑧𝑦 𝑥𝑥 ⋱ ⋮ 𝐶𝑧𝑥 𝑥𝑥 ⋱ ⋮ 𝐶 𝑦𝑥 𝑥𝑥 … … … … … … … 𝐶 𝑦𝑥 𝑦𝑥 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝜀 𝑧𝑧 ε 𝑦𝑧 ε 𝑥𝑧 ε 𝑥𝑦 ε 𝑧𝑦 ε 𝑧𝑥 ε 𝑦𝑥 4th order Tensor - 9x9 Matrix - 81 Elastic Coefficients
- 9. Theory of Elasticity 9 • Generalized 3D Hook‘s Law Symmetry relations (Stress and strain Tensor) Stiffness Form 𝜎 = 𝐶 𝜀 𝜎 𝑥𝑥 𝜎 𝑦𝑦 𝜎𝑧𝑧 𝜏 𝑦𝑧 𝜏 𝑥𝑧 𝜏 𝑥𝑦 = 𝜎11 𝜎22 𝜎33 𝜏23 𝜏13 𝜏12 = 𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 = 𝐶11 𝐶12 𝐶13 𝐶21 𝐶22 𝐶23 𝐶31 𝐶32 𝐶33 𝐶14 𝐶15 𝐶16 𝐶24 𝐶25 𝐶26 𝐶34 𝐶35 𝐶36 𝐶41 𝐶42 𝐶43 𝐶51 𝐶52 𝐶53 𝐶61 𝐶62 𝐶63 𝐶44 𝐶45 𝐶46 𝐶54 𝐶55 𝐶56 𝑐64 𝐶65 𝐶66 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 Voigt Notation 6x6 Matrix - 36 Elastic Coefficients Unfortunate Notation: [C] ≠ Compliance [C] Stiffness • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool
- 10. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 10 • Generalized 3D Hook‘s Law Symmetry relations (Stress and strain Tensor) Compliance form 𝜀 = 𝑆 𝜎 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝜀 𝑧𝑧 𝛾𝑦𝑧 𝛾𝑥𝑧 𝛾𝑥𝑦 = 𝜀 𝑥𝑥 𝜀 𝑦𝑦 𝜀 𝑧𝑧 2𝜀 𝑦𝑧 2𝜀 𝑥𝑧 2𝜀 𝑥𝑦 = 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 = 𝑆11 𝑆12 𝑆13 𝑆21 𝑆22 𝑆23 𝑆31 𝑆32 𝑆33 𝑆14 𝑆15 𝑆16 𝑆24 𝑆25 𝑆26 𝑆34 𝑆35 𝑆36 𝑆41 𝑆42 𝑆43 𝑆51 𝑆52 𝑆53 𝑆61 𝑆62 𝑆63 𝑆44 𝑆45 𝑆46 𝑆54 𝑆55 𝑆56 𝑆64 𝑆65 𝑆66 𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 Voigt Notation 6x6 Matrix - 36 Elastic Coefficients Unfortunate Notation: [S] ≠ Stiffness [S] Compliance
- 11. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 11 • Coordinate system Transformation 𝑇 = 𝑙 𝑥′ 𝑙 𝑦′ 𝑙 𝑧′ 𝑚 𝑥′ 𝑚 𝑦′ 𝑚 𝑧′ 𝑛 𝑥′ 𝑛 𝑦′ 𝑛 𝑧′ = cos 𝛼 𝑥′ 𝑥 cos 𝛼 𝑥′ 𝑦 cos 𝛼 𝑥′ 𝑧 cos 𝛼 𝑦′ 𝑥 cos 𝛼 𝑦′ 𝑦 cos 𝛼 𝑦′ 𝑧 cos 𝛼 𝑧′ 𝑥 cos 𝛼 𝑧′ 𝑦 cos 𝛼 𝑧′ 𝑧 [ 𝑇 ] = − cos 𝜓 sin 𝜓 0 sin 𝜓 cos 𝜓 0 0 0 1 𝑧 1 0 0 0 cos 𝜃 sin 𝜃 0 − sin 𝜃 cos 𝜃 𝑥 − cos 𝜑 sin 𝜑 0 sin 𝜑 cos 𝜑 0 0 0 1 𝑧 Euler Angles
- 12. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 12 • Coordinate system Transformation 𝜎 = 𝑇 𝜎 𝑇 𝑇 𝜀 = 𝑇 𝜀 𝑇 𝑇 𝑧 𝑥 𝑦 𝑥′ 𝛼 𝑥′ 𝑥 𝛼 𝑥′ 𝑦 𝛼 𝑥′ 𝑧
- 13. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 13 • Generalized 3D Hook‘s Law Anisotropic and Conservative Material 𝜎𝑥𝑥 𝜎 𝑦𝑦 𝜎𝑧𝑧 𝜏 𝑦𝑧 𝜏 𝑥𝑧 𝜏 𝑥𝑦 = 𝜎11 𝜎22 𝜎33 𝜏23 𝜏13 𝜏12 = 𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 = 𝐶11 𝐶12 𝐶13 𝐶14 𝐶15 𝐶16 … 𝐶22 𝐶23 𝐶24 𝐶25 𝐶26 … … 𝐶33 𝐶34 𝐶35 𝐶36 … … … 𝐶44 𝐶45 𝐶46 … … … … 𝐶55 𝐶56 … … … … … 𝐶66 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 ; 𝜀11 𝜀22 𝜀33 𝛾12 𝛾23 𝛾13 = 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 Voigt Notation 6x6 Matrix 21 Independent elastic coefficients - Conservative Material - Hyperelastic materials for infinitesimal strains Stiffness and Compliance Matrix Symmetry
- 14. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 14 • Orthotropic Materials Material Symmetry 𝑒3 𝑒2 𝑒1 𝑒3 𝑒2 𝑒1 𝑒3 𝑒2 𝑒1
- 15. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 15 • Hook‘s Law for Orthotropic Materials – Stiffnes form 𝜎1 𝜎2 𝜎3 𝜏4 𝜏5 𝜏6 = 𝐸1 1 − 𝑣23 𝑣32 𝑘 𝐸1 𝑣13 𝑣32 + 𝑣12 𝑘 𝐸1 𝑣12 𝑣23 + 𝑣13 𝑘 0 0 0 𝐸2 𝑣23 𝑣31 + 𝑣21 𝑘 𝐸2 1 − 𝑣13 𝑣31 𝑘 𝐸2 𝑣21 𝑣13 + 𝑣23 𝑘 0 0 0 𝐸3 𝑣32 𝑣21 + 𝑣31 𝑘 𝐸3 𝑣31 𝑣12 + 𝑣32 𝑘 𝐸3 1 − 𝑣12 𝑣21 𝑘 0 0 0 0 0 0 𝐺44 0 0 0 0 0 0 𝐺55 0 0 0 0 0 0 𝐺66 𝜀1 𝜀2 𝜀3 𝛾4 𝛾5 𝛾6 9 Independent elastic coefficients Material Symmetry
- 16. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 16 • Hook‘s Law for Orthotropic Materials – Compliance form 𝜀11 𝜀22 𝜀33 𝛾12 𝛾23 𝛾13 = 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 = 1 𝐸11 − 𝑣12 𝐸22 − 𝑣13 𝐸33 0 0 0 − 𝑣21 𝐸11 1 𝐸22 − 𝑣23 𝐸33 0 0 0 − 𝑣31 𝐸11 − 𝑣32 𝐸22 1 𝐸33 0 0 0 0 0 0 1 𝐺44 0 0 0 0 0 0 1 𝐺55 0 0 0 0 0 0 1 𝐺66 𝜎11 𝜎22 𝜎33 𝜏23 𝜏13 𝜏12 Material Symmetry 9 Independent elastic coefficients
- 17. Theory of Elasticity • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 17 • Hook‘s Law for Orthotropic Materials – 12 Elastic Coefficients – 9 Independent Elastic Coefficients 𝐺44 𝐺55 𝐺66 𝐸11 𝐸22 𝐸33 𝑣23 𝑣32 𝑣13 𝑣31 𝑣12 𝑣21 Symmetry Relations: 𝒗 𝟏𝟐 𝒗 𝟐𝟏 = 𝑬 𝟐𝟐 𝑬 𝟏𝟏 𝒗 𝟐𝟑 𝒗 𝟑𝟐 = 𝑬 𝟑𝟑 𝑬 𝟐𝟐 𝒗 𝟏𝟑 𝒗 𝟑𝟏 = 𝑬 𝟑𝟑 𝑬 𝟏𝟏𝐸11 𝐸22 𝐸33 𝐺44 𝐺55 𝐺66 𝑣23 = 𝑓 𝑣32 𝑣13 = 𝑓 𝑣31 𝑣12 = 𝑓(𝑣21)
- 18. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 18 • Laminate – Series of superimposed plies, fabrics or mats, building a thin layered body – Stacking process (draping or lay-up) • Ply Concept – Semi-finished product with reinforcement(fiber) and matrix – Can be understood as a thin quasi 2D geometry
- 19. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 19 • Laminate – Building Blocks – Ply Fabric Mat
- 20. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 20 • Laminate – Ply Fabric Mat
- 21. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 21 • Model of UD Ply 2D Orthotropic Governing Equations 𝜎 = 𝑄 𝜀 𝜎1 𝜎2 𝜏6 = 𝑄11 𝑄12 𝑄16 𝑄21 𝑄22 𝑄26 𝑄61 𝑄62 𝑄66 𝜀1 𝜀2 𝜀6 𝜎1 𝜎2 𝜏6 = 𝐸1 1 − 𝑣12 𝑣21 𝐸1 𝑣12 1 − 𝑣12 𝑣21 0 𝐸2 𝑣21 1 − 𝑣12 𝑣21 𝐸2 1 − 𝑣12 𝑣21 0 0 0 𝐺66 𝜀1 𝜀2 𝜀6 4 Independent elastic coefficients Symmetry Relations: 𝒗 𝟏𝟐 𝒗 𝟐𝟏 = 𝑬 𝟐𝟐 𝑬 𝟏𝟏 Engineering Notation – Based on the angular distortion
- 22. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 22 • Model of UD Ply 2D Orthotropic Stress and Strains tensors σ = 𝜎11 𝜏12 𝜏21 𝜎22 ; ε = 𝜀1 𝜀12 𝜀21 𝜀22 • Coordinate system Transformation 𝑇 = cos 𝛼 𝑥′ 𝑥 cos 𝛼 𝑥′ 𝑦 cos 𝛼 𝑥′ 𝑧 cos 𝛼 𝑦′ 𝑥 cos 𝛼 𝑦′ 𝑦 cos 𝛼 𝑦′ 𝑧 cos 𝛼 𝑧′ 𝑥 cos 𝛼 𝑧′ 𝑦 cos 𝛼 𝑧′ 𝑧 [ 𝑡 ] = cos 𝜓 sin 𝜓 −sin 𝜓 cos 𝜓 𝑧 In ply plane Rotation
- 23. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 23 • Coordinate system Transformation 𝜎 = 𝑡 𝜎 𝑡 𝑇 Basic Algebraic Manipulation 𝜎 = 𝑡σ 𝜎 𝜎1′ 𝜎2′ 𝜏6′ = cos2 𝜃 sin2 𝜃 −2 sin 𝜃 cos 𝜃 sin2 𝜃 cos2 𝜃 2 sin 𝜃 cos 𝜃 sin 𝜃 cos 𝜃 −sin 𝜃 cos 𝜃 cos2 𝜃 − sin2 𝜃 𝜎1 𝜎2 𝜏6
- 24. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 24 • Coordinate system Transformation Basic Algebraic Manipulation 𝜀 = 𝑡 𝜀 𝑡 𝑇 ε = 𝑡 𝜀 ε 𝜀1′ 𝜀2′ 𝛾6′ = cos2 𝜃 sin2 𝜃 − sin 𝜃 cos 𝜃 sin2 𝜃 cos2 𝜃 sin 𝜃 cos 𝜃 2 sin 𝜃 cos 𝜃 −2 sin 𝜃 cos 𝜃 cos2 𝜃 − sin2 𝜃 𝜀1 𝜀2 𝛾6
- 25. Classical Laminated Plate Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 25 • Coordinate system Transformation 𝜎 = 𝑄 𝜀 ⇒ 𝑡 𝜎 𝜎 = 𝑡 𝜎 𝑄 𝜀 ⇒ 𝒕 𝝈 𝝈 = 𝒕 𝝈 𝑸 𝒕 𝜺 −𝟏 𝒕 𝜺 𝜺 𝝈 = 𝑡 𝜎 𝑄 𝑡 𝜀 −1 𝜺 ⇒ 𝑸 = 𝒕 𝝈 𝑸 𝒕 𝜺 −𝟏 𝑄11 𝑄12 𝑄16 𝑄21 𝑄22 𝑄26 𝑄61 𝑄62 𝑄66 = 𝑡 𝜎 𝑄11 𝑄12 𝑄16 𝑄21 𝑄22 𝑄26 𝑄61 𝑄62 𝑄66 𝑡 𝜀 −1
- 26. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 26 • Trace = 1st Invariant or Linear Invariant 𝑇𝑟 = 𝑄11 + 𝑄22 + 2𝑄66 Trace of the on-axis ply plane Stiffness Matrix Trace is a material property Trace is Independent from the axis (by definition of 1st invariant) 𝑥1 𝑧3 𝑦2
- 27. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 27 • Trace = 1st Invariant or Linear Invariant 𝑇𝑟 = 𝑄11 + 𝑄22 + 2𝑄66 Only the stress-strain relation in terms of tensorial stress and strain has the invariant tensor properties 𝑥1 𝑧3 𝑦2
- 28. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 28 • Trace = 1st Invariant 𝑇𝑟 = 𝑄11 + 𝑄22 + 2𝑄66 𝑥1 𝑧1𝜎1 𝜎2 𝜏6 = 𝐸1 1 − 𝑣12 𝑣21 𝐸1 𝑣12 1 − 𝑣12 𝑣21 0 𝐸2 𝑣21 1 − 𝑣12 𝑣21 𝐸2 1 − 𝑣12 𝑣21 0 0 0 𝐺66 𝜀1 𝜀2 𝜀6 𝜎1 𝜎2 𝜏6 = 𝐸1 1 − 𝑣12 𝑣21 𝐸1 𝑣12 1 − 𝑣12 𝑣21 0 𝐸2 𝑣21 1 − 𝑣12 𝑣21 𝐸2 1 − 𝑣12 𝑣21 0 0 0 𝟐𝑮 𝟔𝟔 𝜀1 𝜀2 𝜀12 𝜺 𝟔 = 𝟐𝜺 𝟏𝟐 𝑦2
- 29. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 29 • Trace = 1st Invariant 𝑇𝑟 = 𝑄11 + 𝑄22 + 2𝑄66 𝑇𝑟 = 𝐸1 1 − 𝑣12 𝑣21 + 𝐸2 1 − 𝑣12 𝑣21 + 2𝐺66 = 𝐸1 + 𝐸2 1 − 𝑣12 𝑣21 + 2𝐺66 Composite System 𝐸1 𝐸 𝟐 𝑣 𝟐𝟏 𝐺66 Tr(Q) M55j/epoxy 340 6.40 0.30 3.90 355 M55j/954-6 321 6.20 0.30 4.70 337 … … … … … …
- 30. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 30 • Master Ply Concept 𝑸𝒊𝒋 𝑪𝒐𝒆𝒇𝒇. 𝑽𝒂𝒓. [%] 𝑄11 ∗ = 𝑄11 𝑇𝑟(𝑄) ≈ 0.8815 ; 1.5 % 𝑄22 ∗ = 𝑄22 𝑇𝑟(𝑄) ≈ 0.0499 ; 10.9 % 𝑄12 ∗ = 𝑄12 𝑇𝑟(𝑄) ≈ 0.0164 ; 1.09 % 𝑄66 ∗ = 𝑄66 𝑇𝑟(𝑄) ≈ 0.0342 ; 16.4 % The master ply is defined by the mean values of the normalized plane stiffness matrix coefficients from a variety of CFRP composite systems. Normalized On-Axis Stiffness Matrix Coefficients
- 31. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 31 • Master Ply Concept Normalized On-Axis Stiffness Matrix Coefficients 𝑸 𝟏𝟏 ∗ = 𝑸 𝟏𝟏 𝑻𝒓(𝑸) ≈ 𝟎. 𝟖𝟖𝟏𝟓 𝑄22 ∗ = 𝑄22 𝑇𝑟(𝑄) ≈ 0.0499 𝑄12 ∗ = 𝑄12 𝑇𝑟(𝑄) ≈ 0.0164 𝑄66 ∗ = 𝑄66 𝑇𝑟(𝑄) ≈ 0.0342 • The normalized Coefficients depend of the Axis!! • Fiber dominated Stiffness Coefficient • Less Matrix/process dependent • More accuracy in the Trace evaluation
- 32. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 32 • Master Ply Concept 𝐸11 ∗ ≈ 0.8796 𝐸22 ∗ ≈ 0.0522 𝑣12 ∗ ≈ 0.3181 𝐺66 ∗ ≈ 0.0313 [𝑆] = 𝑄 −1 𝑆 = 1 𝐸11 − 𝑣12 𝐸22 0 − 𝑣21 𝐸11 1 𝐸22 0 0 0 1 𝐺66 Engineering Elastic Coefficients 𝐸11 ∗ = 1 𝑆11 𝐸22 ∗ = 1 𝑆22 𝑣12 ∗ = − 𝑆12 𝑆22 𝐺66 ∗ = 1 𝑆66
- 33. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 33 • Master Ply Concept Laminate testing - Properties as-built, allowing to take in account for: • Lay-up technique and associate flaws and typical human variation • Interface • Particular Matrix Material • Curing conditions 𝑬 𝟏𝟏 ∗ ≈ 0.8796 • Fiber dominated Stiffness Coefficient • Less Matrix/process dependent • More accuracy in the Trace evaluation
- 34. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 34 • Application of Master Ply Concept Ply Characterization Coupon Testing (UD Ply) Laminate Testing Laminate Design
- 35. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 35 • Application of Master Ply Concept – Coupon Testing 𝑬 𝟏𝟏 ∗ = 𝑬 𝟏𝟏 𝑻𝒓 ⇒ 𝑻𝒓 = 𝑬 𝟏𝟏 𝑬 𝟏𝟏 ∗ = 𝑬 𝟏𝟏 𝟎.𝟖𝟕𝟗𝟔 𝑬 𝟏𝟏 empirical evaluation
- 36. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 36 • Application of Master Ply Concept – Coupon Testing 𝑸 𝟏𝟏 = 𝑸 𝟏𝟏 ∗ ∙ 𝑻𝒓 𝑸 =𝟎.𝟖𝟖𝟏𝟓 ∙ 𝑻𝒓 𝑸 𝑸 𝟐𝟐 = 𝑸 𝟐𝟐 ∗ ∙ 𝑻𝒓 𝑸 = 𝟎. 𝟎𝟒𝟗𝟗 ∙ 𝑻𝒓 𝑸 𝑸 𝟏𝟐 = 𝑸 𝟏𝟐 ∗ ∙ 𝑻𝒓 𝑸 = 𝟎. 𝟎𝟏𝟔𝟒 ∙ 𝑻𝒓 𝑸 𝑸 𝟔𝟔 = 𝑸 𝟔𝟔 ∗ ∙ 𝑻𝒓 𝑸 = 𝟎. 𝟎𝟑𝟒𝟐 ∙ 𝑻𝒓 𝑸 𝑄 = 𝑄11 𝑄12 0 𝑄21 𝑄22 0 0 0 𝑄66 [𝑆] = 𝑄 −1
- 37. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 37 • Application of Master Ply Concept – Coupon Testing 𝑬 𝟏𝟏 = 𝟏 𝑺 𝟏𝟏 𝑬 𝟐𝟐 = 𝟏 𝑺 𝟐𝟐 𝒗 𝟏𝟐 = − 𝑺 𝟐𝟏 𝑺 𝟐𝟐 𝑮 𝟔𝟔 = 𝟏 𝑺 𝟔𝟔 𝑄 = 𝑄11 𝑄12 0 𝑄21 𝑄22 0 0 0 𝑄66 [𝑆] = 𝑄 −1 UD Ply Elastic Properties
- 38. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 38 • Application of Master Ply Concept – Laminate Testing Master Ply Plane Stiffness Matrix 𝑄∗ = 𝑄11 ∗ 𝑄12 ∗ 0 𝑄21 ∗ 𝑄22 ∗ 0 0 0 𝑄66 ∗ = 0.8815 0.0164 0 0.0164 0.0499 0 0 0 0.0342 Normalized Plane Stiffness Matrix for each Ply (Laminate Global System) 𝑄∗ = 𝑄 𝑥𝑥 ∗ 𝑄 𝑥𝑦 ∗ 𝑄 𝑥𝑠 ∗ 𝑄 𝑦𝑥 ∗ 𝑄 𝑦𝑦 ∗ 𝑄 𝑦𝑠 ∗ 𝑄 𝑠𝑥 ∗ 𝑄𝑠𝑦 ∗ 𝑄𝑠𝑠 ∗ = 𝑡 𝜎 𝑄∗ 𝑡 𝜀 −1
- 39. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 39 • Application of Master Ply Concept – Laminate Testing 𝑨∗𝟎 = 𝑘 𝑁 𝑸 𝒌 𝑧 𝑘 − 𝑧 𝑘−1 Normalized in- plane Stiffnes Matrix from the laminate Normalized in- plane Stiffnes Matrix from each Ply
- 40. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 40 • Application of Master Ply Concept – Laminate Testing 𝐴∗0 = 𝑘 𝑁 𝑄 𝑘 𝑧 𝑘 − 𝑧 𝑘−1 𝑥1𝑧3 𝑦2
- 41. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 41 • Application of Master Ply Concept – Laminate Testing [𝑎∗ ] = 𝐴∗ −1 𝑎∗ = 𝑎 𝑥𝑥 ∗ 𝑎 𝑥𝑦 ∗ 𝑎 𝑥𝑠 ∗ 𝑎 𝑦𝑥 ∗ 𝑎 𝑦𝑦 ∗ 𝑎 𝑦𝑠 ∗ 𝑎 𝑠𝑥 ∗ 𝑎 𝑠𝑦 ∗ 𝑎 𝑠𝑠 ∗ 𝑬 𝒙𝒙 ∗ = 𝟏 𝒂 𝒙𝒙 ∗ 𝑬 𝒚𝒚 ∗ = 𝟏 𝒂 𝒚𝒚 ∗ 𝒗 𝒙𝒚 ∗ = − 𝒂 𝒚𝒙 ∗ 𝒂 𝒚𝒚 ∗ 𝑮 𝒔𝒔 ∗ = 𝟏 𝒂 𝒔𝒔 ∗
- 42. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 42 • Application of Master Ply Concept – Laminate Testing 𝑬 𝒙𝒙 ∗ = 𝑬 𝒙𝒙 𝟎 𝑻𝒓 ⇒ 𝑻𝒓 = 𝑬 𝒙𝒙 𝟎 𝑬 𝒙𝒙 ∗ 𝑬 𝒙𝒙 𝟎 empirical evaluation
- 43. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 43 • Application of Master Ply Concept – Laminate Testing 𝑸 𝟏𝟏 = 𝑸 𝟏𝟏 ∗ ∙ 𝑻𝒓 𝑸 =𝟎.𝟖𝟖𝟏𝟓 ∙ 𝑻𝒓 𝑸 𝑸 𝟐𝟐 = 𝑸 𝟐𝟐 ∗ ∙ 𝑻𝒓 𝑸 = 𝟎. 𝟎𝟒𝟗𝟗 ∙ 𝑻𝒓 𝑸 𝑸 𝟏𝟐 = 𝑸 𝟏𝟐 ∗ ∙ 𝑻𝒓 𝑸 = 𝟎. 𝟎𝟏𝟔𝟒 ∙ 𝑻𝒓 𝑸 𝑸 𝟔𝟔 = 𝑸 𝟔𝟔 ∗ ∙ 𝑻𝒓 𝑸 = 𝟎. 𝟎𝟑𝟒𝟐 ∙ 𝑻𝒓 𝑸 𝑄 = 𝑄11 𝑄12 0 𝑄21 𝑄22 0 0 0 𝑄66 [𝑆] = 𝑄 −1
- 44. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 44 • Application of Master Ply Concept – Laminate Testing 𝑬 𝟏𝟏 = 𝟏 𝑺 𝟏𝟏 𝑬 𝟐𝟐 = 𝟏 𝑺 𝟐𝟐 𝒗 𝟏𝟐 = 𝑺 𝟐𝟏 𝑺 𝟏𝟐 𝑮 𝟔𝟔 = 𝟏 𝑺 𝟔𝟔 𝑄 = 𝑄11 𝑄12 0 𝑄21 𝑄22 0 0 0 𝑄66 [𝑆] = 𝑄 −1 UD Ply Elastic Properties
- 45. Invariant-based Theory • Theory of Elasticity (Review) • Classical Laminated Plate Theory • Invariant-based Theory • MATLAB Tool 45 • MATLAB Tool Input • Coupon / Laminate Testing • Longitudinal Modulus Tested • Stacking Sequence Output 𝑬 𝒙𝒙 𝟎 ; 𝑬 𝒚𝒚 𝟎 ; 𝒗 𝒙𝒚 𝟎 ; 𝒗 𝒚𝒙 𝟎 ; 𝑮 𝒙𝒚 𝟎
- 46. References 46 • Tsai, S.W. and J.D.D. Melo, An invariant-based theory of composites. Composites Science and Technology, 2014. 100: p. 237-243. • Nettles, A.T., Basic mechanics of laminated composite plates. 1994. • Roylance, D., Laminated composite plates. Massachusetts Institute of Technology Cambridge, 2000. • Tavakoldavani, K., Composite materials equivalent properties in lamina, laminate, and structure levels. 2014: The University of Texas at Arlington. • Liu, S. and W. Su, Effective couple-stress continuum model of cellular solids and size effects analysis. International Journal of Solids and Structures, 2009. 46(14): p. 2787-2799. • Irgens, F., Continuum Mechanics. 2008: Springer Berlin Heidelberg. • Seth, B., Generalized strain measure with applications to physical problems. 1961, WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER.
- 47. References 47 • Slaughter, W.S., The Linearized Theory of Elasticity. 2002: Birkhäuser Boston. • Peck, S. Invariant-Based Design of Laminated Composite Materials. in 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference 17th AIAA/ASME/AHS Adaptive Structures Conference 11th AIAA No. • Theory of elasticity. 2001: McGraw-Hill. • Moakher, M., The algebra of fourth-order tensors with application to diffusion MRI. Visualization and Processing of Tensor Fields, 2009: p. 57-80. • Sih, G.C., A. Carpinteri, and G. Surace, Advanced Technology for Design and Fabrication of Composite Materials and Structures: Applications to the Automotive, Marine, Aerospace and Construction Industry. 1995: Springer Netherlands. • Gay, D., Composite Materials: Design and Applications, Third Edition. 2014: Taylor & Francis.