15 INTERLAMINAR STRESSES
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1 15 INTERLAMINAR STRESSES 15-1
2 OUT-OF-PLANE STRESSES Classical laminate plate theor predicts the stresses in the plane of the lamina,, and τ but does not account for out-of-plane stresses, τ and τ. It assumes these stresses are (ie. onl plane stress is considered). These out-of-plane stresses however, are not at discontinuities such as holes or free edged. The non-ero stress, tends to peel the plies apart while the non-eroτ andτ tends to slide the plies over each other, thus causing them to delaminate. Classical laminate plate theor also predicts τ are non ero even at a free edges of the laminate where it cannot eist. An accurate model or theor would predict that interlaminar shear stress are ver high at holes and free edges of a laminate. Laer-stacking sequence significantl affects transverse tensile strength due to interlaminar normal stresses, and interlaminar shear stresses at pl boundaries. These out-of-plane stresses can be determined b finite element or finite difference methods. The origin of these stresses depends on pl laup such as angle pl, [±θ], cross pl, [/9] and combinations of angle and cross-pl, [/±θ/9]. Interlaminar stresses in angle-pl laminates Consider a laminate that consists onl of plies of off-ais la-up [(±θ) n ] S The shear strain on each pl if loaded separatel is illustrated Fig q -q +q -q Figure Shear strain in off-ais plies. 15-2
3 For these two plies Q Q Q 16 θ 16+ θ Q 26 θ 26+ θ (15.1) Therefore for each [±θ] pair A Q + Q (15.2) θ θ The strains from Hooke s law are 1 ν m E E E 1 ν 1 m E E E1 γ τ m m 1 E1 E1 G (15.3) Appling onl (i.e.,andτ ) then γ E ν E 1 m E (15.4) For plies with the fibers at -θ For plies with the fibers at +θ S 11 S 22 γ S 16 τ S 11 S 22 γ - S 16 τ 15-3
4 The shear stresses, τ on the interior surfaces of the plies are shown in Fig θ θ θ +θ τ τ +q -q Figure Shear stress on surface of plies between ( ± θ ) pairs At the free surface the shear force can not be sustained but to maintain static equilibrium of the moments a shear stress can be induced on the interior plane surface of a pl as seen in Fig.15-3 τ Free surface Figure Shear stresses at the edge of a lamina 15-4
5 Shear stress distribution along width of plate for a [±θ] s laminate is shown in Fig t t edge center edge Figure 15-4 Shear stress distribution along width of plate. For L>>h, the average shear stress, τ, is close to the prediction based on C. The effect of pl angle, θ is shown in Fig t Figure 15-5 Effect of pl angle on interlaminar shear stress τ. q 15-5
6 Interlaminar Stresses in Cross Pl Laminates For E lamina Sij Sij 1 ν ν 1 γ 1 G TL ET ET τ (15.5) For the 9E lamina 1 ν ET ν 1 γ 1 G TL ET τ 9 9 (15.6) Appling stresses in the direction onl τ.hence 1 1 E 9 9 T (15.7) where and E ν ν T TL 9 (15.8) ν ν ν ν TL TL TL 9 ν (15.9) 15-6
7 The strain mismatch in the direction between the two plies, ( ) 9 TL E L is ν ν (15.1) which give rise to shear stresses on the faces of the plies as shown in Fig τ τ O 9 O Figure 15-6 Interlaminar shear stresses on the lamina in a cross-pl composite It can be seen from this figure that τ causes epansion in the º laer and contraction in the 9º laers. These shear stresses are maimum at the free edges and fall to near ero in the center of the pl. This interlaminar shear stress, τ must be equilibrated b the interlaminar normal stress,. This is demonstrated in the free bod diagram of in cross section of a quarter of the [/9] S laminate shown in Fig The normal stress, is maimum at the free edge and drops rapidl, reversing its sign awa from the free edge. A plot of the normal and shear stresses from the edge of the laminate to the center is shown in Fig
8 Z s o τ s s 9 o Figure 15-7 Free bod diagram of -9º adjacent plies in cross pl laminate. s + - t Center /b Free edge Figure 15-8 Stress distribution at interface of -9 plies in cross pl laminate Failure criterion for delamination A prediction of the initiation of delamination has been proposed b Kim and Soni. The criterion proposed is that when the average normal stress, over a characteristic distance, b from the free edge eceeds the interlaminar strength Z int, failure occurs. This criterion can be epressed as 15-8
9 1 (,) b b d Zint (15.11) b b Delamination can also occur when the average interlaminar shear stress, τ eceeds the shear strength, S over another characteristic distance, d, thus τ 1 b τ (,) d S (15.12) d b d These characteristics distances are shown in Fig A more general criterion that addresses both angle pl and cross-pl laminates and accounts for the stress interactions is the Quadratic Delamination Criterion (QDC) proposed b Brewer and Lagace τ τ S S Zint Z int (15.13) where the primed quantities denote compression. The problems with these criteria are that interlaminar strengths S, S, Zint and Z int are not easil measured and the characteristic distances are found empiricall. These problems can be avoided b using the fracture mechanics approach discussed in the net chapter. s + Z int - b t S Center /b d Free edge Figure 15-9 Characteristic distances for delamination failure criteria 15-9
10 Pl sequence effects The interlaminar stresses can be significantl modified b pl stacking sequence. Figure 15-1 shows the interlaminar normal stress, resulting from three smmetric arrangements of ±45º and ±15º pair. Both the sign and the magnitude of the stresses are modified b pl arrangement. Compression /t Tension s Stacking Sequence [15/-15/45/-45] S [15/45/-45/-15] S [45/-45/15/-15] S Figure 15-1 Effect of stacking sequence on through thickness normal stress Eamples of interlaminar stresses produced in various pl sequences for quasiiostropic laminates are shown in Table Table Interlaminar stresses in quasiisotropic laminates Pl Sequence Stress, ksi Location Stress, ksi Location [9/45//-45] S -6.8 middle -6.9 /45 [/-45/9/45] S 6.2 9º 6.6 9/45 [45/9//-45] S 6.6 9º 5.9 /-45 [45//9/-45] S 1.4 middle / [9/-45/45/] S º /45 [45/-45//9] S 1.9 middle /-45 [/45/-45/9] S 1. 45º /-45 It can be seen from this table that the interlaminar stresses, and τ can be reduced b separating the 45º and -45º plies. τ 15-1
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