Generalized Hooke’s eco-law for unidirectional ply

Generalized Hooke's eco-law for unidirectional ply

In high-performance composite structures, structural components are manufactured mainly from unidirectional fibre-reinforced plies, where the ply axes are identified in the principal coordinate system with numbers 1, 2 and 3. Whereas, the laminate axes are identified in the global coordinate system (Cartesian coordinates) with letters x, y and z (Jones, 1999 & Saarela, 1994). The 1-axis and z-axis are combined into a single axis (Figure 8).

The Young’s moduli £i, E2, E3 and the shear moduli G12, G23, G13 of the unidirectional ply

determined from experimental tests will become E1, E2 , E3 , G12 , G23 , G13, respectively. In relation to this orientation and for a unidirectional fibre-reinforced ply, the eco-efficiency factors related to the engineering constants for the linear-elastic mechanical behaviour may be defined in the principal coordinate system as (Attaf, 2008):

E G’ E E-

J = —L, J j with —L = – І – (i, j=1,2,3 and i^j) (9)

Ei Gij Vij Vji

where Ei, Ei are classical and sustainable Young’s moduli in i-direction (i=1, 2 and 3),

Gij, Gij are classical and sustainable shear moduli in the i-j plane (i-j=2-3, 3-1 and 1-2),

Ji, Jij are the eco-efficiency factors of Young’s and shear moduli, respectively and, vij is the Poisson’s ratio for transverse strain in the j-direction when stressed in the i-

Generalized Hooke's eco-law for unidirectional ply

direction. For this value, it is important to note that no attempt was made to investigate the sustainability of Poisson’s ratios. Their influence is beyond the scope of this analysis. According to these assumptions and generalized Hooke’s law, the strain-stress eco-relations for an orthotropic material in the principal coordinate system (1,2,3) may be written in compact matrix form as:

6×6 (6 rows by 6 columns). The components Sij = Sji are defined as:

"r—^

II

S12 = – V12/

E1

S13 = ~V13I E1

S14

= 0

S15

= 0

S16

= 0

S21 = – У21І

E2

S22 = 1E2

S23 = – V2^l E2

S24

= 0

S25

= 0

S26

= 0

S31 = – V31 /

/E3

S32 = – V321

^3

S33 = 1E3

S34

= 0

S35

= 0

S36

= 0

S41 = 0

S42 = 0

S43 = 0

S44

=V G23

S45

= 0

S46

= 0

Ul. •

II

0

S52 = 0

О

II

CO

. • LO

S54

= 0

S55

=V G13

S56

= 0

S61 = 0

S62 = 0

О

II

CO

. • 40

чл

S64

= 0

S65

= 0

S66

= V G1

Also, the eco-stiffness matrix of the ply may be obtained by the inverse of the eco­compliance matrix. Thus, we can write:

Generalized Hooke's eco-law for unidirectional ply Generalized Hooke's eco-law for unidirectional ply Generalized Hooke's eco-law for unidirectional ply

When using the eco-stiffness matrix, the stress-strain relations can be obtained by the inverse of Equation (10). Thus, we obtain:

Подпись: Mi Подпись: Qu Подпись: (4 12 ( i,j= 1,2,6) Подпись: (12b)

Since orthotropic fibres have almost the same characteristics along 2- and 3-axis, the 2-3 plane is considered to be a plane of symmetry and the three-dimensional 123-coordinate system will be reduced to two-dimensional 12-coordinate system. Thus, Equation (12a) becomes:

Подпись: coordinate system, and Подпись: Qu Подпись: is the eco-stiffness matrix of order 3x3 (3 rows by 3

where (M12 = {m m2 t12}T, {s}12 = {e1 s2 y12}T are the stress and strain vectors in 12-

12

columns).

Generalized Hooke's eco-law for unidirectional ply

Using the transformation matrix, Equation (12b) may be written, after some rearrangements, in the global xy-coordinate system as:

by 3 columns). The components Qij = Qji (i, j = 1, 2, 6) are functions of the fibre orientation

angle, 0 and the orthotropic elastic eco-moduli of the ply. By letting C=cos 0 and S=sin 0, the eco-stiffness components are defined in the global xy-coordinate system as:

Подпись: Q11 = Q11C4 + 2(Q12 + 2 Q66)S2C2 + Q22 S4 Q12 = (Q11 + Q22 - 4 Q66)S2C2 + Q12(S4 + C4) Q22 = Q11S4 + 2(Q12 + 2 Q66)S2C2 + Q,, C4Q16 = (Q11 – Q12 – 2 Q66 )SC3 + (Q12 – Q22 + 2 Q66 )S3C Q26 = (Q11 – Q12 – 2 Q66)S3C + (Q12 – Q22 + 2 Q66 )SC3 Q66 = (Q11 + Q22- 2Q12- 2Q66)S2C2 + Q66(S4 + C4)

With these ecological considerations, the constituent equations for laminated composite structures can be derived using the classical lamination theory (Jones, 1999).

1.6 Constitutive eco-equations of laminates

When the environment and health impacts besides quality are taken into consideration in the structural analysis, the constitutive relations for an unsymmetrically и-layered laminated composite plate (к=1,2,..и), and without transverse shear deformations can, after integration through each ply thickness and summation, be written in matrix form as:

Подпись:(14a)

Generalized Hooke's eco-law for unidirectional ply
Generalized Hooke's eco-law for unidirectional ply
Подпись: (14b)

Or in compacted matrix form:

where N and M are the resultant in-plane forces and bending/ torsional moments, respectively. Whereas, є0 and к are the asso ciated strains and curvatures.

Подпись: (Ay , Bij
Подпись: azk JQiV)k (1, z, z2)dz
Подпись: (i, j = 1, 2, 6)
Подпись: These sub-matrices may be assembled in a single matrix Подпись: C Generalized Hooke's eco-law for unidirectional ply

The eco-components of the sub-matrices Ay (extensional eco-stiffnesses), By (coupling eco­stiffnesses) and Dy (bending eco-stiffnesses) are expressed as:

Подпись: C Подпись: A I B B I D Подпись: (16)

bending eco-stiffness matrix. Thus,

Generalized Hooke's eco-law for unidirectional ply

and Equation (14b) may be written as: