Macromechanical Analysis of a Lamina and Stiffness Matrices
Explore the macromechanical analysis of a lamina and dive into the 3D stiffness and compliance matrices, presented by Dr. Autar Kaw from the Department of Mechanical Engineering at the University of South Florida. Discover the complexities of laminates and stiffness matrices in composite materials through detailed illustrations and explanations.
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Presentation Transcript
Chapter 2 MacromechanicalAnalysis of a Lamina Part 3 3D Stiffness and Compliance Matrices Dr. AutarKaw Department of Mechanical Engineering University of South Florida, Tampa, FL 33620 Courtesy of the Textbook Mechanics of Composite Materials by Kaw
FIGURE 2.1 Typical laminate made of three laminas
S S S S S S 1 11 12 13 14 15 16 1 S S S S S S 2 21 22 23 24 25 26 2 S S S S S S 3 3 31 32 33 34 35 36 = S S S S S S 23 23 41 42 43 44 45 46 S S S S S S 31 51 52 53 54 55 56 31 S S S S S S 12 61 62 63 64 65 66 12
C C C C C C 1 11 12 13 14 15 16 1 C C C C C C 2 21 22 23 24 25 26 2 C C C C C C 3 3 31 32 33 34 35 36 = C C C C C C 23 23 41 42 43 44 45 46 C C C C C C 31 51 52 53 54 55 56 31 C C C C C C 12 61 62 63 64 65 66 12 Stiffness matrix [C] has 36 constants
0 0 0 S S S 11 12 12 1 1 0 0 0 S S S 12 11 12 2 2 0 0 0 S S S 12 11 3 12 3 = 0 0 0 ) 0 0 S ( 2 S 12 23 11 23 0 0 0 0 ) 0 S ( 2 S 31 12 31 11 0 0 0 0 0 ) S ( 2 S 12 12 12 11
0 0 0 C C C 11 12 12 1 1 0 0 0 C C C 12 11 12 2 2 0 0 0 C C C 12 11 3 12 3 = 0 0 0 ) 0 0 C ( 2 C 12 23 11 23 0 0 0 0 ) 0 C ( 2 C 31 12 31 11 0 0 0 0 0 ) C ( 2 C 12 12 12 11
1 0 0 0 E E E 1 0 0 0 E E E 1 1 1 0 0 0 2 2 E E E 3 3 = 0 0 0 1 0 0 23 23 G 31 31 0 0 0 0 1 0 12 12 G 0 0 0 0 0 1 G
E(1 - ) E E 0 0 0 (1 - 2 )(1 + ) (1 - 2 )(1 + ) (1 - 2 )(1 + ) x x E E(1 - ) E 0 0 0 y y (1 - 2 )(1 + ) (1 - 2 )(1 + ) (1 - 2 )(1 + ) z z = , E E E(1 - ) 0 0 0 yz yz (1 - 2 )(1 + ) (1 - 2 )(1 + ) (1 - 2 )(1 + ) zx zx 0 0 0 G 0 0 xy xy 0 0 0 0 G 0 0 0 0 0 0 G
S S S S S S 1 11 12 13 14 15 16 1 S S S S S S 2 21 22 23 24 25 26 2 S S S S S S 3 3 31 32 33 34 35 36 = S S S S S S 23 23 41 42 43 44 45 46 S S S S S S 31 51 52 53 54 55 56 31 S S S S S S 12 61 62 63 64 65 66 12
C C C C C C 1 11 12 13 14 15 16 1 C C C C C C 2 21 22 23 24 25 26 2 C C C C C C 3 3 31 32 33 34 35 36 = C C C C C C 23 23 41 42 43 44 45 46 C C C C C C 31 51 52 53 54 55 56 31 C C C C C C 12 61 62 63 64 65 66 12 Stiffness matrix [C] has 36 constants
FIGURE 2.11 Transformation of coordinate axes for 1-2 plane of symmetry for a monoclinic material
FIGURE 2.12 Deformation of a cubic element made of monoclinic material
FIGURE 2.13 A unidirectional lamina as a monoclinic material with fibers arranged in a rectangular array
0 0 S S S S 11 12 13 1 16 1 0 0 S S S S 2 12 22 23 26 2 0 0 S S S S 3 23 33 3 13 36 = 0 0 0 0 S S 23 23 44 45 0 0 0 0 S S 31 31 55 45 0 0 S S S S 12 66 12 16 26 36
0 0 C C C C 11 12 13 1 16 1 0 0 C C C C 2 12 22 23 26 2 0 0 C C C C 3 23 33 3 13 36 = 0 0 0 0 C C 23 23 44 45 0 0 0 0 C C 31 31 55 45 0 0 C C C C 12 66 12 16 26 36
FIGURE 2.14 Deformation of a cubic element made of orthotropic material
0 0 0 S S S 1 11 12 13 1 0 0 0 S S S 2 12 22 23 2 0 0 0 S S S 3 23 33 3 13 = 0 0 0 0 0 S 23 23 44 0 0 0 0 0 S 31 55 31 0 0 0 0 0 S 12 66 12
0 0 0 C C C 1 11 12 13 1 0 0 0 C C C 2 12 22 23 2 0 0 0 C C C 3 23 33 3 13 = 0 0 0 0 0 C 23 23 44 0 0 0 0 0 C 31 55 31 0 0 0 0 0 C 12 66 12
1 0 0 0 12 13 E E E 1 1 1 1 0 0 0 21 23 E E E 2 2 2 1 1 1 0 0 0 2 2 31 32 E E E 3 3 3 3 3 = 0 0 0 1 0 0 23 23 G 23 31 31 0 0 0 0 1 0 12 12 G 31 0 0 0 0 0 1 G 12
+ + 1 0 0 0 23 32 21 23 31 31 21 32 E E E E E E 2 3 2 3 2 3 1 1 + + 1 0 0 0 21 23 31 13 31 32 12 31 2 2 E E E E E E 2 3 1 3 1 3 3 3 = + + 1 0 0 0 31 21 32 32 12 31 12 21 23 23 E E E E E E 1 2 2 3 1 3 31 31 0 0 0 0 0 G 23 12 12 0 0 0 0 0 G 31 0 0 0 0 0 G 12
FIGURE 2.15 A unidirectional lamina as a transversely isotropic material with fibers arranged in a rectangular array
0 0 0 S S S 1 11 12 12 1 0 0 0 S S S 2 12 22 23 2 0 0 0 S S S 3 23 22 12 3 = 0 0 0 0 0 S ( 2 S 23 23 ) 23 22 0 0 0 0 0 S 31 31 55 0 0 0 0 0 S 12 55 12
0 0 0 C C C 11 12 12 1 1 0 0 0 C C C 12 22 23 2 2 0 0 0 C C C 23 22 12 3 3 = 2 0 0 0 0 0 C C 22 23 23 23 31 31 0 0 0 0 0 C 55 12 12 0 0 0 0 0 C 55
0 0 0 S S S 11 12 12 1 1 0 0 0 S S S 12 11 12 2 2 0 0 0 S S S 12 11 3 12 3 = 0 0 0 ) 0 0 S ( 2 S 12 23 11 23 0 0 0 0 ) 0 S ( 2 S 31 12 31 11 0 0 0 0 0 ) S ( 2 S 12 12 12 11
0 0 0 C C C 11 12 12 1 1 0 0 0 C C C 12 11 12 2 2 0 0 0 C C C 12 11 3 12 3 = 0 0 0 ) 0 0 C ( 2 C 12 23 11 23 0 0 0 0 ) 0 C ( 2 C 31 12 31 11 0 0 0 0 0 ) C ( 2 C 12 12 12 11
1 0 0 0 E E E 1 0 0 0 E E E 1 1 1 0 0 0 2 2 E E E 3 3 = 0 0 0 1 0 0 23 23 G 31 31 0 0 0 0 1 0 12 12 G 0 0 0 0 0 1 G
E(1 - ) E E 0 0 0 (1 - 2 )(1 + ) (1 - 2 )(1 + ) (1 - 2 )(1 + ) x x E E(1 - ) E 0 0 0 y y (1 - 2 )(1 + ) (1 - 2 )(1 + ) (1 - 2 )(1 + ) z z = , E E E(1 - ) 0 0 0 yz yz (1 - 2 )(1 + ) (1 - 2 )(1 + ) (1 - 2 )(1 + ) zx zx 0 0 0 G 0 0 xy xy 0 0 0 0 G 0 0 0 0 0 0 G
Independent Elastic Constants Material Type Anisotropic 21 Monoclinic 13 Orthotropic 9 Transversely Isotropic 5 Isotropic 2
= , , 0 0 3 0 = = 31 23 Upper and lower surfaces are free from external loads , 0 = 31 23 3 0 , 0 , = = FIGURE 2.17 Plane stress conditions for a thin plate
