8 Properties of Lamina


 Juliana Hicks
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1 8 Properties of Lamina 8
2 ORTHOTROPIC LAMINA An orthotropic lamina is a sheet with unique and predictable properties and consists of an assemblage of fibers ling in the plane of the sheet and held in place b a matri. The lamina can be composed of continuous or discontinuous fibers. The labeling conventions used here is shown in Fig. 8. T, Y L, Y T, q X X q L, Continuous Fiber Lamina Discontinuous Fiber Lamina Figure 8 Labeling convention for lamina L and T are the principal material directions, also referred to as and. The angle formed b the counter clockwise rotation from an arbitrar direction to L is +. The clockwise rotation produces . Orthotropic material In an orthotropic material there are two unique mutuall orthogonal directions (called the principal material directions) in which tension load causes etension onl parallel and perpendicular to the tensile load direction, and shear load causes shear strains onl. The effects of etensional and shear loads in the principal directions on the strains in an orthotropic material are shown in Fig.83. The unstrained lamina are shown as dotted lines in this figure. The etensional load shown in Fig.83a causes onl etension in the longitudinal direction and contraction in the transverse direction. Loading in shear is shown in Fig.83b which produces onl shear strains. 8
3 T P P L a) Load in principal direction, L b) hear load in LT direction Figure 83 Orthotropic lamina loaded in principal directions or isotropic lamina loaded in an direction Loading, either in shear or in tension in an other direction causes both etensional and shear strains. The effect of load on strain in an orthotropic material loaded in the other than principal directions are shown in Fig.84. In an orthotropic material etension and shear are decoupled onl in the principal material directions. Fig. 84a shows the etensional loads applied in an T L P P a) tensional load in arbitra direction b) hear load in arbitrar direction Figure 84 Orthotropic lamina loaded in an arbitrar direction arbitrar or nonprincipal direction producing both etension and shear. In Fig. 84b shear loads applied in the nonprincipal direction produce etensional strains as well as shear strains. Isotropic material A tensile load in an direction causes onl etension parallel and perpendicular to that direction and shear produces onl shear strains. In an isotropic material etension and shear are alwas decoupled and the strains are represented b Fig
4 Anisotropic material A tensile load in an direction causes shear as well as parallel and perpendicular etensions. A shear load in an direction causes etensional and shear strains. In anisotropic materials etension and shear are never decoupled. trains are alwas represented b Fig HOOK' LAW FOR ANIOTROPIC MATRIAL Hooke s law can be epressed in full tensor notation b ij ijkl kl where ij are the stress components and kl are the strain components. The elastic constants, ijkl are fourth ranked tensors. Their subscripts i,j,k,l,,3,hence there are a total of 3 4 or 8 elastic constants. The number of unique constants can be reduced b using smmetr and thermodnamic arguments. train smmetr produces the result ijkl ijlk hence there are onl 54 unique constants remaining. tress smmetr produces the result ijkl jikl hence there are now onl 36 unique constants remaining. Thermodnamic arguments produce the result ijkl klij hence that leaves onl unique elastic constants as follows, Hooke's law can also be written in contracted notation which replace paired subscripts with a single subscript according to the following: for, for, 3 for 33, 4 for 3, 5 for 3, and 6 for. C i ij j where subscripts i, j can have the values,,3,4,5, and 6. TRANFORMATION OF LATIC CONTANT To transform an elastic constant from the X ais to the X' ais use the following: ' a a aa (8.) where a im, a jn, a kr, a ls are direction cosines. mnrs im jn kr ls ijkl For one plane of smmetr, sa X X as illustrated in Fig.85 then X X', X X', and X 3 X' 3 84
5 3 Figure 85 Plane of smmetr X X in an orthotropic material The direction cosines for transformation through this plane of smmetr are: X' X' X' 3 X a a 0 a 3 0 X a 0 a a 3 0 X 3 a 3 0 a 3 0 a 33  Using the above determined direction cosines in the transformation qn.(8.) produces the following result ' ' ' 3 3 ' The result 3 3 can onl be true if 3 0. amination of the stiffness matri indicates that odd multiples of a 33 will give negative and hence zero values for ijkl. Appling this observation to all the coefficients then the following ijkl are zero: 3, 3, 3, 3, 3, 333, 3, 333 The same technique can be applied to a second plane, sa X X 3 85
6 X' X' X' 3 X a  a 0 a 3 0 X a 0 a a 3 0 X 3 a 3 0 a 3 0 a 33 This time odd multiples of a give negative (hence zero) ijkl T T T T The T'ed ones are new 's eliminated b the second plane of smmetr, unchecked 's fit the criteria but were alread eliminated b the first plane of smmetr. Appling the third plane of smmetr does not eliminate an other coefficients. Hooke's Law for the orthotropic material is C C C C C C C3 C3 C τ C γ 3 τ C55 0 γ 3 τ C66 γ (8.) RDUCD TIFFN MATRIX FOR PLAN TR CONDITION For a two dimensional lamina (sheet) onl the stresses in the plane of the lamina are nonzero. Hence the streses through the thickness are zero under plane stress 3 τ3 τ3 0 Appling these values to qn.(8.) C C C C C C C3 C3 C C γ C55 0 γ 3 τ C γ 66 (8.3) 86
7 then Hooke s Law (qn.(8.3)) for plane stress gives C + C + C 3 3 C + C + C C + C + C τ C γ 66 (8.4) The strain, 3 can be epressed in terms of and C C (8.5) C33 C33 The qns. (8.4) can now be written C C C C + C C33 C33 C C C τ C γ C + C C33 C33 66 (8.6) Using the following notations: C3 C3C3 C, C, C C C3 C and 66 C66 (8.7) C 33 Then Hooke s law for plane stress condition takes the reduced form 0 0 τ 0 0 γ 66 (8.8) For lamina that are transversel isotropic, which is generall the case for most practical composite plies, C3 C and C33 C. Hooke s law for plane stress in terms of compliance is 0 0 γ 0 0 τ 66 (8.9) 87
8 ij 's and ij 's are mutuall inverse [ I] [ ][ ] Inverting [] (8.0) Inverting [] (8.) Compliance and stiffness in terms of engineering constants Using qn. (8.9) and the definitions of Young s ratio and Poisson s ratio the values of the compliance constants can be found to be 66 ν ν G (8.) ubstituting qns. (8.) into qns. (8.0) gives ν ν 66 ν ν ν ν ν ν ν ν G (8.3) 88
9 From qns. (8.) it is also clear that ν ν (8.4) TRTRAIN RLATION FOR OFFAXI LAMINA Hooke s Law is readil epressed b qns. (8.8) and (8.9) for stresses and strains in the principal material direction of a composite. In real composite lamina stresses are often applied in arbitrar direction at an angle θ from the principal directions as seen in Fig.86. s q s X X Figure 86 tress applied to a composite lamina in an arbitrar direction Transforming stress F in the arbitrar direction to the F in the principal material direction [ T ] τ τ (8.5) Transforming strain from the arbitrar to the principle direction is performed b [ T ] γ γ (8.6) where γ is engineering strain, γ is tensorial strain and [T] is second order transformation matri given b 89
10 [ T ] cos θ sin θ cosθsinθ sin θ cos θ cosθsinθ cosθsinθ cosθsinθ cos θ sin θ ( ) (8.7) To transform from principal direction to arbitrar direction, then  and transformation matri becomes [ T ] θ θ θ θ sin θ cos θ cosθsinθ cosθsinθ cosθsinθ ( cos θ sin θ) cos sin cos sin (8.8) Hence, qn. (8.6) is the inverse of qn.(8.7). Transforming stress from principal to arbitrar direction [ T ] τ τ (8.9) Transforming strain from principal to arbitrar direction [ T ] γ γ (8.0) HOOK LAW FOR TR AND TRAIN APPLID IN ARBITRARY DIRCTION To obtain Hooke s Law for loading in a direction other than the principal material directions the stiffness or compliance coefficients which are defined in the principal directions have to be transformed to the arbitrar direction of loading. The sequence of operations to perform this transformation on the stiffness coefficients, given the engineering strain in the arbitrar direction, is as follows: () Convert engineering strain in the arbitrar direction to tensorial strain in the arbitrar direction. 80
11 () Transform tensorial strain in the arbitrar direction to tensorial strain in the principal material direction. (3) Convert tensorial strain in the principal material direction back to the engineering strain in the principal material direction. (4) Find engineering strain in the principal material direction to stress in the principal material direction using Hooke s Law for orthotropic material. (5) Transform stress in the principal material direction to stress in the original arbitrar direction. A convenient wa to perform step () is to multipl the engineering strain b the inverse of the Reuter s matri. [ ] R (8.) γ γ where 0 0 [ R] In step () appl the tensorial strain transformation of qn. (8.6) to find the tensorial strains in the principal material direction. [ T ] γ γ (8.) The stiffness matri [ ] is defined in terms of the engineering strains in the principal material direction, hence in step (3) the tensorial strains in the principal material direction is converted to engineering strain. This can be convenientl performed using the Reuter s matri where [ R] γ γ (8.3) 8
12 [ R] Now step (4) can be performed using qn.(8.8) to find the stresses in the principal material directions [ ] τ γ (8.4) The stress in the principal material directions fron qn (8.9) can be transformed back to the arbitrar direction b in inverse transformation matri [ T ] τ τ (8.5) howing all of these operations in one equation gives [ ] [ ][ ][ ][ ] T R T R τ γ (8.6) Premultipling the conversion, transformation and stiffness matrices gives the transformed stiffness matri It can be shown that qn. (8.7) can be reduced to [ ] [ ][ ][ ][ ] T R T R (8.7) [ ] [ ][ ] T T T (8.8) The transformed stiffness matri epressed in terms of the individual coefficients is (8.9) 8
13 where ( ) ( ) cos θ + + sin θ cos θ + sin θ sin θ + + sin θ cos θ + cos θ ( 4 ) sin θ cos θ ( sin θ cos θ) ( ) ( ) ( 3 ) sinθ cosθ ( + ) sinθ cos θ + + sin θ cosθ ( ) sin θ cos θ ( sin θ cos θ) sinθ cos θ (8.30) B a similar process the tansformed compliance matri can be found for finding the stains in an arbitar direction form the stresses in an arbitrar direction [ ][ ] [ ] R T R [ ][ T] γ τ (8.3) where or [ ][ ] [ ] [ ][ ] R T R T (8.3) T [ ] [ ][ ] T T (8.33) The transformed compliance matri epressed in terms of the individual coefficients is (8.34) where ( ) ( ) cos θ + sin θ + + sin θ cos θ sin θ + cos θ + + sin θ cos θ ( ) sin θ cos θ ( cos θ sin θ) ( ) ( ) 3 ( ) sin θ cosθ ( ) sinθ cos θ sin θ cosθ ( ) θ θ ( θ θ) + 4 sin cos + cos + sin sinθ cos θ (8.35) 83
14 INVARINT FORM OF TIFFN COFFICINT The reduced transformed stiffness coefficients can be epressed in terms of five invariant constants that depend upon the untransformed, ( i.e. independent of direction) stiffness coefficients. These invariant coefficients are U U U U U (8.36) Using these invariants coefficient the transformed stiffness coefficients can be written as U + U cosθ + U 3 U U cosθ + U 3 U U cos4θ cos4θ cos4θ 6 Usinθ U3sin4θ 6 Usinθ + U3sin4θ U U cos4θ (8.37) ach of the terms on the right hand side of qn.(8.33) can be plotted individuall and then summed to produce the transformed stiffness coefficient. This summation is shown schematicall in Fig
15 U U cos q U cos 4 q q q q q Figure 87. Transformed reduced stiffness coefficient constructed form invariant coefficients In this eample the coefficient consists of the summation of a constant, cosine term and a havercosine term. TRANFORMATION OF NGINRING CONTANT From measured elastic engineering constants in the principal material directions, shown in Fig.88, the elastic constants in an arbitrar direction can be determined. n n G Figure 88. lastic engineering constants in the principal material directions This can be performed as follows: Consider the following elastic strains in the principal material direction:, and γ. Transform these strains to the arbitrar direction, let us sa at some angle, θ counterclockwise from the principal direction,. cos θ + sin θ γ cosθsinθ sin θ + cos θ + γ cosθsinθ ( ) γ cosθsinθ cosθsinθ + γ cos θ sin θ (8.38) 85
16 Using Hooke s Law the strains in the principal material directions can be epressed in terms of engineering elastic constants and stresses in the principal material directions. γ ν ν τ G (8.39) If we appl are as the onl nonzero arbitrar stress then the stresses in the principal directions θ θ cos sin τ cosθsinθ (8.40) Combining qns.(8.39) and (8.40) with (8.38) ields the strains in arbitrar direction in terms of the engineering constants for the principal material direction, 4 4 cos θ sin θ ν + + sin θ 4 G (8.4a) ν ν + + sin θ 4 G (8.4b) γ ν θ θ ν sin + cos + + G G (8.4c) The Young s modulus in the arbitrar direction can now be found as follows: From qn, (8.4a) 4 4 cos θ sin θ ν + + sin θ 4 G If qn. (8.4a) is evaluated at θ + 90 (8.4) 4 4 cos θ sin θ ν + + sin θ 4 G (8.43) 86
17 If we appl τ as the onl nonzero stress in the above described procedure the shear modulus in the arbitrar direction is G 4ν sin θ + sin θ + cos θ G G ( ) (8.44) Using the definition of Poisson s ratio and qns. (8.4a) and (8.4b) ν ( ) 4 4 ν sin θ cos θ + sin θ G (8.45) The stress can also produce shear strain γ, and shear stress τ can produce etensional strain, these can be related though a the cross coefficient m b m γ (8.46a) and m τ (8.46b) Likewise for stress m, and and strain γ, and stress τ and strain, using the cross coefficient m γ (8.46a) m τ (8.47b) The cross coefficient m m can be found from qn. (8.4c) as sinθ ν + cos θ + ν + G G (8.48) The cross coefficient m can be similarl found 87
18 m sinθ ν + sin θ + ν + G G (8.49) These transformed engineering elastic constants can be used in the compliance matri for stresses and strains in the arbitrar directions (i.e. offais loading) ν m ν m γ τ m m G (8.50) The offais Young s modulus qns. (8.4 and (8.43) and shear modulus qn. (8.44) for a tpical carbon fiber/polmer matri composite are plotted over the range of orientations in Fig ngineering lastic Constants, Mpsi GLT q Figure 89 Offais Young s modulus and shear modulus for tpical carbon/epo lamina Fig. 80 is a plot of the offais Poisson s ratio qn.(8.45) and offais cross coefficients over the range of orientations qns. (8.46) and (8.47) for a tpical carbon fiber/polmer matri lamina. 88
19 .5 Poisson's Ratio m m q Figure 80 Offais Poisson s ratio and cross coefficients for tpical carbon fiber/polmer matri lamina. RTRICTION ON LATIC CONTANT Isotropic materials Thermodnamic considerations require that Young s modulus and shear modulus are positive values. The work of deformation, and τγ are positive then and > τ > For isotropic materials the relation between the Young s modulus and shear modulus is G 0 0 For G and to be positive ν >. G ( + ν ) The volumetric strain, φ resulting from hdrostatic pressure P is φ + + z P K 89
20 where K the bulk modulus is For K and to be positive ν < K 3( ν ) For all the elastic moduli to be positive then < ν < Orthotropic materials The relation between the elastic constants are more comple for orthotropic materials. Appling thermodnamic restraints to C, C, C33, C44, C55, C 66, > 0 For the plane stress condition,, 66 >0, then for eample ν ν and therefore νν > 0. This is true if νν <. From this it follows that > ν and > ν It is often prudent to verif that the elastic properties used to calculate compliance and stiffness coefficients do not violate these criteria. 80