Mechanics Analysis 2010 Part 8 ppt

Chapter 5. Mechanies of laminates 23 1 a ‘3 Fig. 5.6. Reduction of transverse shear stresses to stress resultants (transverse shear forces). -e -e Because the particular distribution (5.12) of txz and z, does not influence the displace- ments, we can introduce some average stresses having the same resultants as the actual ones, i.e., However, according to Eqs. (5.11), shear strains are linear combinations of shear stresses. So we can use the same law to introduce average shear strains as (5.13) Average shear strains ’yx and yv can be readily expressed in terms of displacements if we substitute Eqs. (5.9) into Eqs. (5.13), i.e., These equations, in contrast to Eqs. (5.9), do not include derivatives with respect to z. So we can substitute Eqs. (5.1) and (5.2) to get the final result aw aw ax aY y.y = e, + - , y), = e, + - . (5.14) 232 Mechanics and analysis of composite materials Consider Eqs. (5.10) and (5.1 1). Integrating them over the layer thickness and using Eqs. (5.12) and (5.13) we get -e -e -e -e Because the actual distribution of stresses and strains according to the foregoing reasoning is not significant, we can change them for the corresponding average stresses and strains: where S 1 sm, = s,, = - J am, d~ . h2 (5.17) (5.18) -e It should be emphasized that Eqs. (5.16) are not the inverse form of Eqs. (5.15). Indeed, solving Eqs. (5.16), using Eqs. (5.18) and taking into account that a55 = A66 I - Am, Am, = a56 = -256 9 a66 = 255 I A55A66 - A:6 ’ we arrive at Eqs. (5.15) in which (5.19) These expressions, in general, do not coincide with Eqs. (5.17). Thus, the constitutive equations for transverse shear are specified by Eqs. (5.15), and there exist two, in general different, approximate forms of stiffnesscoefficients - Eqs. (5.17) and (5.19). The fact that equations obtained in this way are approximate is quite natural because the assumed displacement field, Eqs. (5.1) and (5.2), is also approximate. Chapter 5. Mechanics of laminates 233 To compare two possible forms of constitutive equations for transverse shear, consider for the sake of brevity an orthotropic layer for which For transverse shear in the xz-plane Eqs. (5.15) yield where - 1' in accordance with Eq. (5.17), while Eq. (5.19) yields (5.20) (5.21) (5.22) If the shear modulus does not depend on z, both equations, Eqs. (5.21) and (5.22), give the same result S55 = G,h. Using the energy method applied in Section 3.3 we can show that the Eqs. (5.21) and (5.22) provide the upper and the lower bounds for the actual transverse shear stiffness. Indeed, consider a strip with unit width experiencing transverse shear induced by force Y, as in Fig. 5.7, Assume that Eq. (5.20) links the actual force K. with the actual angle yx = A/l through the actual shear stiffness SSS which we do not know and which we would like to evaluate. To do this, we can use two variational principles described in Section 2.1 1. According to the principle of minimum total potential energy Tact < rtdm 7 (5.23) Y Fig. 5.7. Transverse shear of a strip with unit width. 234 where Mechanics and anatysis of composite materials Tact = u&, - &t, Tadm = u&, - Aadm are the total energies of the actual state and some admissible kinematic state expressed in terms of the strain energy, U, and work A performed by force V, on displacement A (see Fig. 5.7). For both states Aact = Aadm = V,A and condition (5.23) reduces to u& < u:dm . (5.24) For the actual state, with due regard to Eq. (5.20) we get 1 1 2 Uact = - KY, = +Y, . 2 For the admissible state. we I P should use the following general equation: I and admit some approximation for yxz. The simplest one is yz = yx, so that (5.25) (5.26) -e Then, Eqs. (5.24H5.26) yield S Sssd /G=dz. -e Comparing this inequality with Eq. (5.21) we can conclude that this equation specifies the upper bound for S55. To determine the lower bound, we should apply the principle of minimum strain energy according to which Gc, < U& , (5.27) where Chapter 5. Mechanics qf laminates 235 For the admissible state we should apply and use some admissible distribution for L The simplest approximation is z,, = V;./h so that -e Substitution into condition (5.27) yields Thus, Eq. (5.22) provides the lower bound for S55, and the actual stiffness satisfies the following inequality: So, constitutive equations for the generalized layer under study are specified with Eqs. (5.5) and (5.15). Stiffness coefficients that are given by Eqs. (5.6)-(5.8), and (5.17) or (5.19) can be written in a form more suitable for calculations. To do this, introduce new coordinate t = z + e such that 0 d t d h (see Fig. 5.8). Transforming the integrals to this new variable we get where rnn = 1 1, 12, 22, 14, 24, 44 and (5.28) Fig. 5.8. Coordinates of an arbitrary point A. 236 Mechanics and analysis of composite materials Transverse shear stiffnesses, Eqs. (5.17) and (5.19), acquire the form and where mn = 55,56,66 and (5.29) (5.30) (5.3 1) (5.32) 5.2. Stiffness coefficients of a homogeneous layer Consider a layer whose material stiffness coefficients A,, do not depend on coordinate z. Then (5.33) and Eqs. (5.28), (5.30), and (5.31) yield the following stiffness coefficients of the layer: B,, = A,,h, Cm, =A,, - - (; e>, Dmn = A,, 6 - eh + e2), S,, = A,,h (5.34) Both Eqs. (5.30) and (5.31) give the same result for S,,. As follows from the second of these equations, membrane-bending coupling coefficients C,, become equal to zero if we take e = h/2, Le., if the reference plane coincides with the middle plane of the layer shown in Fig. 5.9. In this case, Eqs. (5.5) and (5.15) acquire the following de-coupIed form: Chapter 5. Mechanics of Inminates 231 X Y Fig. 5.9. Middle plane of a laminate. As can be seen, we have arrived at three independent groups of constitutive equations for in-plane stressed state of the layer, bending and twisting, and transverse shear. Stiffness coefficients, Eqs. (5.34), become For an orthotropic layer, there are no in-plane stretching-shear coupling (B14 = B24 = 0) and transverse shear coupling (s56 = 0). Then, Eqs. (5.35) reduce to In terms of engineering elastic constants material stiffness coefficients of an orthotropic layer can be expressed as (5.38) 238 Mechanics and analysis of composite materials Finally, for an isotropic layer, we have (5.40) where E = E/( I - v’). 5.3. Stiffness coefficients of a laminate Consider a general case, Le., a laminate consisting of an arbitrary number of layers with different thicknesses hi and stiffnesses A!), (i = 1,2,3,. . . ,k). Location of an arbitrary ith layer of the laminate is specified by coordinate ti, which is the distance from the bottom plane of the laminate to the top plane of the ith layer (see Fig. 5.10). Assuming that material stiffness coefficients do not change within the thickness of the layer and using piece-wise integration we can write parameter I,,,,, in Eqs. (5.29) and (5.32) as *k YY L X to =o Fig. 5.10. Structure of the laminate. Chapter 5. Mechanics of laminates 239 (5.41) where I = 0, 1, 2 and to = 0, tk = h (see Fig. 5.10). For thin layers, Eqs. (5.41) can be reduced to the following form, which is more suitable for calculations: k k where hi = ti - ti-l is the thickness of the ith layer. Thus, membrane, coupling, and bending stiffness coefficients of the laminate are specified with Eqs. (5.28) and (5.42). Consider transverse shear stiffnesses which have two diflerent forms determined by Eqs. (5.30) and (5.31). Because both equations coincide for a homogeneous layer (see Section 5.2), we can expect that the difference shows itself in laminates consisting of layers with different transverse shear stiffnesses. The laminate for which this difference is the most pronounced is a sandwich structure with metal facings (inner and outer layers) and a foam core (middle layer) that has very low shear stiffness. For such a sandwich, experimentally found transverse shear stiffness is S = 389 kN/m (Aleksandrov et al., 1960), while Eqs. (5.30) and (5.31) yield, respectively, S = 37200 kN/m and S = 383 kN/m. Thus, Eq. (5.31) provides much more accurate result for sandwich structures. This conclusion is also valid for composite laminates (Chen and Tsai, 1996). A particular case, important for applications, is an orthotropic laminate for which Eqs. (5.5) and (5.15) acquire the form: (5.43) where, membrane, coupling, and bending stiffnesses, B,,, C,,,,, and D,,,,, are specified by Eqs. (5.28) and (5.42), while transverse shear stiffnesses are 240 Mechanics and analysis of composite materials (5.44) h2 k hi ' Ci=l (i) A mm smm = Laminates composed of unidirectional plies have special stacking-sequence nota- tions. For example, notation [0;/+45"/-45"/90~] means that the laminate consists of 0" layer having two plies, f45" angle-ply layer, and 90" layer also having two plies. Notation [0"/90"], means that the laminate has five cross-ply layers. 5.4. Quasi-homogeneous laminates Some typical layers considered in Chapter 4 were actually quasi-homogeneous laminates (see Sections 4.4, and 4.5), but being composed of a number of identical plies, they were treated as homogeneous layers. The accuracy of this assumption is evaluated below. 5.4. I. Laminate composed of identical homogeneous Kayers Consider a laminate composed of layers with different thicknesses but the same stiffnesses, Le., such that A:; = A,, for all i = 1,2,3,. . .k. Then, Eqs. (5.29) and (5.32) yield This result coincides with Eqs. (5.33), which means that the laminate consisting of the layers with the same mechanical properties is a homogeneous laminate (layer) studied in Section 5.2. 5.4.2. Laminate composed of inhomogeneous orthotropic layers Let the laminate have the structure [0"/90"],, where p = 1,2,3,. specifies the number of elementary cross-ply couples of 0" and 90" plies. In Section 4.4, this laminate was treated as a homogeneous layer with material stiffness coefficients specified by Eqs. (4.100). Taking 60 = igo = 0.5 in these equations we get In accordance with Eqs. (5.36), stiffness coefficients of this layer should be [...]... 120 90 72 60 135 I 08 90 - - 144 120 - - 150 Table 5.2 Modulus of elasticity and Poisson’s ratio of quasi-isotropic laminates made of typical advanced composites Property Glass-epoxy Carbon-poxy Aramid-epoxy Boron+poxy Boron-AI Modulus, E, GPa Poisson’s ratio, v Specific modulus, 27.0 0.34 1290 54 .8 0.3 1 3530 34 .8 0.33 2640 80 .3 0.33 382 0 183 .1 0. 28 6910 kl; x lo3,m Mechanics and analysis of composite... (5. 28) acquire the form In practical analysis, constitutive equations for the laminates with arbitrary structure are often approximately simplified using the method of reduced or minimum bending stiffnesses described, e.g., by Ashton (1969), Karmishin (1974), and Whitney (1 987 ) To introduce this method, consider the corresponding equation of Eqs (5. 28) for bending stiffnesses, Le Mechanics and analysis. .. integrating we 58) (5 .86 ) where, A,,,,, (mn= 21, 22) are the step-wise functions of z, i.e., A,,,,, = A::: A,,,,, =A::; < hi, for h i < z < h = hl + h2 , for 0 d z and C is the constant of integration Because no pressure is applied to the inner ,z surface of the cylinder, a( = 0) = 0 and C = 0 Substitution of stiffness coefficients, Eqs (5 .80 ), (5 .81 ) and strains, Eqs (5.79) into Eq (5 .86 ) yields (5 .87 ) P z-hl... cylinder under axial compression Mechanics and analysis of composite materials 2 58 My, associated with the change of the curvature of the cross-sectional contour in Eq (5.75) is very small For numerical analysis, we first use Eqs (4.72) to calculate stiffness coefficients for the angle-ply layer, i.e., Ai:) = 25 GPa, A$) = 14.1 GPa, Ai:) = 10 GPa, A:) = 11.5 GPa , (5 .80 ) and for the hoop layer Ai:)... For , the laminate under study, e = 0. 48 mm, i.e., the reference surface is located within the internal angle-ply layer Then, Eqs (5 .88 )-(5.90), and (5.92), upon substitution of strains from Eqs (5.94) and (5.95) can be written as , W IVr=B1~u’+8l2- , R W’ N , = B ~ , U + ~ ~ ~ - ,+ C ~ , ~ ~ R W A & = C I Z - + D I I ~, : R K = SS5(& + w’) , (5.96) (5.97) (5. 98) (5.99) where stiffness coefficients... Fig 5.20) Then, Eqs (5. 28) yield B I I= I/? = 21.2 GPa mm, B22 = 1;;)= 35.6 B12 = 1:;) = 7.7 GPamm, GPamm, CII=Zi:) = 10.1 GPamm2, C12 =Z[i) = 3.3 GPamm’, C22 = Z ; : ) = 27.4 0 1 2 = 1;;’ 5.9 = GPamm2, GPamm3, 022 =I;;) = 94 GPamm3 and in accordance with Eqs (5. 78) for R = 100 mm, B I = 7.7 GPa mm, ~ E 2 = 3.2 1 GPa mm2, 82 2 = 35.3 GPa mm, c = 26.5 2 2 GPa mm2 , 259 Chapter 5 Mechanics of laminates... R and z ? / R in comparison with unity and write &(I) 1' = 0 -E, (5 .83 ) 1 ' Fig 5.21 Experimental composite cylinder in test fixtures P , kN E;,% -0.4 Fig 5.22 Dependence of axial E;,% -0.3 -0.2 -0.1 0 0.1 and circumferential ( E ! ) strains of a composite cylinder on the axial analysis; (oj experiment force: () (E:) Mechanics and analysis of composite materials 260 Applying Eqs (5.71) to calculate... angle-ply layer, a{’) -0.26= 0 P Rh’ P Rh ay) = -0.0 28- , 2) 1 : P ; Rh = 0.023- in the hoop layer, P a?) = 0.073 -, Rh P CY’ = -0. 089 -, Rh T$) = 0 , where h = hl + hz is the total thickness of the laminate To calculate interlaminar stresses acting between the angle-ply and the hoop layers, we apply Eqs (5.73) Using Eqs (5.4) and taking Eqs (5 .82 ) and (5 .83 ) into account, we first find the stresses in the... I , 2, 3 , ,k) we get Mechanics and analysis of composite materials 252 For practical analysis, this result is often used even if Poisson’s ratios of the layers are different In these cases it is approximately assumed that all the layers are characterized with some average value of Poisson’s ratio, i.e., As another example, consider a sandwich structure described in Section 5 .8 In the general case,... Indeed, consider a laminate in Fig 5. 18 and assume that its structure is, in general, not symmetric, Le., that 4 # zi and K # k Using plane z = 0 as the reference plane we can write the membrane-bending coupling coefficients as Z k i’ k t k s s m m s u Fig 5. 18 1;ayer coordinates with respect to the reference plane Chapter 5 Mechanics of laminates 253 + where, z 20 and 4 8 0 Introduce a new layer coordinate . Modulus, E, GPa 27.0 54 .8 34 .8 80.3 183 .1 Poisson’s ratio, v 0.34 0.3 1 0.33 0.33 0. 28 Specific modulus, 1290 3530 2640 382 0 6910 kl; x lo3, m 246 Mechanics and analysis of composite. material stiffness coefficients of an orthotropic layer can be expressed as (5. 38) 2 38 Mechanics and analysis of composite materials Finally, for an isotropic layer, we have (5.40). we get where rnn = 1 1, 12, 22, 14, 24, 44 and (5. 28) Fig. 5 .8. Coordinates of an arbitrary point A. 236 Mechanics and analysis of composite materials Transverse shear stiffnesses,