Macromechanical Analysis of Laminate Modulus - Dr. Autar Kaw
Explore the macromechanical analysis of laminate modulus as discussed by Dr. Autar Kaw, a professor at the University of South Florida. This chapter delves into laminate properties, compliance matrices, modulus calculations, and effective moduli for various planes. Gain insights into the extensional, coupling, and bending compliance matrices for symmetric laminates. Understand the effective longitudinal and transverse moduli, shear modulus, and Poisson's ratio in plane. Delve into the intricate mechanics of composite materials through this comprehensive analysis.
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Presentation Transcript
Chapter 4 MacromechanicalAnalysis of a Laminate Laminate Modulus Dr. Autar Kaw Department of Mechanical Engineering University of South Florida, Tampa, FL 33620 Courtesy of the Textbook Mechanics of Composite Materials by Kaw
0 x N A A A B B B x 11 12 16 11 12 16 0 y N A A A B B B y 12 22 26 12 22 26 N 0 xy A A A B B B xy 16 26 66 16 26 66 = N B B B D D D M 0 11 12 16 11 12 16 x x x x [ = ] N N B B B D D D M 0] 0 [ = 2 12 22 6 12 22 26 y y y y B B B D D D N M 0 16 26 66 16 26 66 xy xy xy xy = N M A B 0 x x [ = ] M [ ] = B D M M y y M xy xy
0 A B N = M C D 1 = A B A B B D C D T * [ ] = [ * ] C B The [A*], [B*], and [D*] matrices are called the extensional compliance matrix, coupling compliance matrix, and bending compliance matrix respectively.
For a symmetric laminate: M 0 N A B A B 0 0 N = = B D M C D n [( ] ) ij ( ) 1 2 6 1 2 6 = - , i = , , ; j = , , Q h h A = [ ] , 0 B ij 1 k k - k 1 k = n 1 = A 1 2 k 2 k - [ *] [ ] , [( )] ( ) 1 2 6 1 2 6 A = - , i = , , ; j = , , Q h h B ij 1 k ij 2 1 k = n 1 3 3 1 - k [( ] ) ij ( ) = i 6 6 = ij h - k 1, 2, ; = j 1, 2, Q = D h D 1 [ *] [ ] D k 3 1 = k
0 x * 11 * 12 * 16 A A A N x = 0 y * 12 * 22 * 26 N A A A y * 16 * 26 * 66 N A A A 0 xy xy
Effective in longitudin plane modulus al E x x 0, = y 0, 0 = xy N N N 0 x * 11 * 12 * 16 A A A N x = 0 y * 12 * 22 * 26 0 A A A 0 * 16 * 26 * 66 A A A 0 xy 0 * 11 = x N A x /h 1 N x x = 0 x = x E x * 11 * 11 N hA A
Effective in plane transvers modulus e E y 0, = x 0, 0 = xy N N N y 0 x * 11 * 12 * 16 A A A 0 = 0 y * 12 * 22 * 26 N A A A y * 16 * 26 * 66 0 A A A 0 xy 0 * 22 = y N A y h / y 1 N y = y = y E y 0 * 22 * 22 h N A A
Effective in plane shear modulus G xy 0 = x 0, = y 0, N 0 x N N xy * 11 * 12 * 16 A A A 0 = 0 y * 12 * 22 * 26 0 A A A N * 16 * 26 * 66 A A A 0 xy xy 0 = xy * 66 N A xy h / xy 1 N xy = = xy G xy 0 xy * 66 * 66 h N A A
Effective in plane Poisson' ratio s xy x 0, = y 0, 0 = xy N N N 0 x * 11 * 12 * 16 A A A N x = 0 y * 12 * 22 * 26 0 A A A 0 * 16 * 26 * 66 A A A 0 xy 0 * 12 = = y N N A A x 0 * 11 x x 0 y * 12 * 12 N A A - x - = 0 x = x - xy * 11 * 11 N A A
Effective in plane Poisson' ratio s yx 0, = x 0, 0 = xy N N N y 0 x * 11 * 12 * 16 A A A 0 = 0 y * 12 * 22 * 26 N A A A 0 x y yx 0 y * 16 * 26 * 66 0 A A A 0 xy * 12 N A y = * 22 N A y 0 * 12 = x N A y * 12 A = 0 * 22 = y * 22 N A A y
symmetric a For laminate : [B] = 0 * 11 * 12 * 16 D D D M x x = * 12 * 22 * 26 D D D M y y * 16 * 26 * 66 M D D D xy xy
f x Effective flexural longitudin modulus al E x 0, = y 0, 0 = xy M M M * 11 * 12 * 16 D D D M x x = * 12 * 22 * 26 0 D D D y 0 * 16 * 26 * 66 D D D xy * = x M 11 D x 12 12 M x f x = E 3 3 * 11 h h D x
Other flexural elastic moduli : * 12 D f - = xy * 11 D 12 3 f = y E * 22 h D * 12 D f - = yx * 22 D 12 3 f = xy G * 66 h D f xy f yx = f x f y E E
