ADVANCED MECHANICS OF COMPOSITE MATERIALS Episode 13 pptx

Chapter 7. Environmental, special loading, and manufacturing effects 407 log N , MPa 0 400 800 1200 3456 −1 10 R = 0.1 s 1R Fig. 7.36. Fatigue diagrams for a unidirectional carbon–epoxy composite loaded along the fibers with various stress ratios. Fabric composites are more susceptible to cyclic loading than materials reinforced with straight fibers. This fact is illustrated in Fig. 7.37 showing the experimental results of Schulte et al. (1987). The foregoing discussion deals with high-cycle fatigue. The initial interval 1 ≤ N ≤ 10 3 corresponding to so-called low-cycle fatigue is usually studied separately, because the slope of the approximation in Eq. (7.69) can be different for high stresses. A typical fatigue diagram for this case is shown in Fig. 7.38 (Tamuzh and Protasov, 1986). 0 200 400 600 800 3456 log N 1 2 s R , MPa Fig. 7.37. Tensile fatigue diagrams for a cross-ply (1) and fabric (2) carbon–epoxy composites. 408 Advanced mechanics of composite materials 0 400 800 1200 1600 2000 0123 s 1R , MPa log N Fig. 7.38. Low-cycle fatigue diagram for unidirectional aramid–epoxy composite loaded along the fibers with R = 0.1. Fatigue has also some effect on the stiffness of composite materials. This can be seen in Fig. 7.39 demonstrating a reduction in the elastic modulus for a glass–fabric–epoxy– phenolic composite under low-cycle loading (Tamuzh and Protasov, 1986). This effect should be accounted for in the application of composites to the design of structural members such as automobile leaf-springs that, being subjected to cyclic loading, are designed under stiffness constraints. Stiffness degradation can be used as an indication of material damage to predict fatigue failure. The most sensitive characteristic of the stiffness change is the tangent modulus E t specified by the second equation in Eqs. (1.8). The dependence of E t on the number of cycles, N, normalized to the number of cycles that cause material fatigue fracture under the preassigned stress, is presented in Fig. 7.40 corresponding to a ±45 ◦ angle-ply carbon–epoxy laminate studied by Murakami et al. (1991). 7.3.4. Impact loading Thin-walled composite laminates possessing high in-plane strength and stiffness are rather susceptible to damage initiated by transverse impact loads that can cause fiber breakage, cracks in the matrix, delamination, and even material penetration by the impactor. Depending on the impact energy determined by the impactor mass and veloc- ity and the properties of laminate, impact loading can result in considerable reduction Chapter 7. Environmental, special loading, and manufacturing effects 409 0 10 20 30 0123 E, GPa log N Fig. 7.39. Dependence of elastic modulus of glass fabric–epoxy–phenolic composite on the number of cycles at stress σ = 0.5¯σ (¯σ is the static ultimate stress). 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 E t N Fig. 7.40. Dependence of the tangent modulus normalized to its initial value on the number of cycles related to the ultimate number corresponding to fatigue failure under stress σ max = 120 MPa and R =−1 for ±45 ◦ angle-ply carbon–epoxy laminate. 410 Advanced mechanics of composite materials 0 0.2 0.4 0.6 0.8 1 0 5 10 15 20 −− s 0 s i / E i , J/m m 1 2 3 Fig. 7.41. Dependence of compression strength after impact normalized to the initial compressive strength on the impact energy related to the plate thickness for glass fabric–epoxy (1), unidirectional glass–epoxy (2), and carbon–epoxy composite plates (3). in material strength under tension, compression, and shear. One of the most dangerous consequences of an impact loading is an internal delamination in laminates, which can sometimes be hardly noticed by visual examination. This type of defect causes a dra- matic reduction in the laminate compressive strength and results in unexpected failure of thin-walled composite structures due to microbuckling of fibers or local buckling of plies. As follows from Fig. 7.41, showing the experimental results of Verpoest et al. (1989) for unidirectional and fabric composite plates, impact can reduce material strength in compression by a factor of 5 or more. To study the mechanism of material interlaminar delamination, consider the problem of wave propagation through the thickness of the laminate shown in Fig. 7.42. The motion equation has the following well-known form ∂ ∂ z  E z ∂ u z ∂ z  = ρ ∂ 2 u z ∂ t 2 (7.70) Here, u z is the displacement in the z-direction, E z is material modulus in the same direction depending, in the general case on z, and ρ is the material density. For the laminate in Fig. 7.42, the solution of Eq. (7.70) should satisfy the following boundary and Chapter 7. Environmental, special loading, and manufacturing effects 411 h 1 i k h i z z i z i−1 P(t) Fig. 7.42. Laminate under impact load. initial conditions σ z (z = 0,t)=−p(t), σ z (z = h, t) = 0 (7.71) u z (z, t = 0) = 0, ∂ u z ∂ t (z > 0,t= 0) = 0 (7.72) in which σ z = E z ∂ u z ∂ z (7.73) is the interlaminar normal stress. Consider first a homogeneous layer such that E z and ρ do not depend on z. Then, Eq. (7.70) takes the form c 2 ∂ 2 u z ∂ z 2 = ∂ 2 u z ∂ t 2 where c 2 = E z /ρ. Transform this equation introducing new variables, i.e., x 1 = z +ct and x 2 = z −ct. Performing conventional transformation and rearrangement, we arrive at ∂ 2 u z ∂ x 1 ∂ x 2 = 0 412 Advanced mechanics of composite materials The solution for this equation can be readily found and presented as u z = φ 1 (x 1 ) +φ 2 (x 2 ) = φ 1 (z +ct) +φ 2 (z −ct) Here, φ 1 and φ 2 are some arbitrary functions. Using Eq. (7.73), we get σ z = E z [f 1 (x +ct) +f 2 (x −ct)] where f 1 = ∂ φ 1 ∂ z ,f 2 = ∂ φ 2 ∂ z Applying the boundary and initial conditions, Eqs. (7.71) and (7.72), we arrive at the following final result σ z = E[f(x+ ct) − f(x−ct)] (7.74) in which the form of function f is governed by the shape of the applied pulse. As can be seen, the stress wave is composed of two components having opposite signs and moving in opposite directions with one and the same speed c, which is the speed of sound in the material. The first term in Eq. (7.74) corresponds to the applied pulse that propagates to the free surface z = h (see Fig. 7.43, demonstrating the propagation of a rectangular pulse), whereas the second term corresponds to the pulse reflected from the free surface z = h. It is important that for a compressive direct pulse (which is usually the case), the reflected pulse is tensile and can cause material delamination since the strength of laminated composites under tension across the layers is very low. P z h P P s C C Fig. 7.43. Propagation of direct and reflected pulses through the layer thickness. Chapter 7. Environmental, special loading, and manufacturing effects 413 Note that the speed of sound in a homogeneous material, i.e., c =  E z ρ (7.75) is the same for the tensile and compressive waves in Fig. 7.43. This means that the elastic modulus in Eq. (7.75) must be the same for both tension and compression. For compos- ite materials, tensile and compressive tests sometimes produce modulus values that are slightly different. Usually, the reason for such a difference is that the different specimens and experimental techniques are used for tensile and compression tests. Testing of fiber- glass fabric coupons (for which the difference in the experimental values of tensile and compressive moduli is sometimes observed) involving continuous loading from compres- sion to tension through zero load does not show any ‘kink’ in the stress–strain diagram at zero stress. Naturally, for heterogeneous materials, the apparent (effective) stiffness can be different for tension and compression as, for example, in materials with cracks that propagate under tension and close under compression. Sometimes stress–strain diagrams with a ‘kink’ at the origin are used to approximate nonlinear experimental diagrams that, actually, do not have a ‘kink’ at the zero stress level at all. For laminates, such as in Fig. 7.42, the boundary conditions, Eqs. (7.71), should be supplemented with the interlaminar conditions u (i) z = u (i−1) z and σ (i) z = σ (i−1) z . Omitting the rather cumbersome solution that can be found elsewhere (Vasiliev and Sibiryakov, 1985), we present some numerical results. Consider the two-layered structure: the first layer of which has thickness 15 mm and is made of aramid–epoxy composite material with E (1) z =4.2 GPa, ρ 1 = 1.4 g/cm 3 , and the second layer is made of boron–epoxy composite material and has E (2) z = 4.55 GPa, ρ 2 = 2 g/cm 3 , and h 2 = 12 mm. The duration of a rectangular pulse of external pressure p acting on the surface of the first layer is t p = 5×10 −6 s. The dependence of the interlaminar (z = 15 mm) stress on time is shown in Fig. 7.44. As can be seen, at t ≈ 3t p the tensile interface stress exceeds the intensity of the pulse of pressure by the factor of 1.27. This stress is a result of interaction of the direct stress wave with the waves reflected from the laminate’s inner, outer, and interface surfaces. Thus, in a laminate, each interface surface generates elastic waves. For laminates consisting of more than two layers, the wave interaction becomes more complicated and, what is more important, can be controlled by the appropriate stacking sequence of layers. As an example, consider a sandwich structure shown in Fig. 7.45a. The first (loaded) layer is made of aluminum and has h 1 = 1 mm, E (1) z = 72 GPa, ρ 1 = 2.7 g/cm 3 , the second layer is a foam core with h 2 = 10 mm, E (2) z = 0.28 GPa, ρ 2 = 0.25 g/cm 3 , and the third (load-carrying) composite layer has h 3 = 12 mm, E (3) z = 10 GPa, ρ 3 = 1.4 g/cm 3 . The duration of a rectangular pulse of external pressure is 10 −6 s. The maximum tensile stress occurs in the middle plane of the load-carrying layer (plane a–a in Fig. 7.45). The normal stress induced in this plane is presented in Fig. 7.46a. As can be seen, at the moment of time t equal to about 1.75 × 10 −5 s, this stress is tensile and can cause delamination of the structure. 414 Advanced mechanics of composite materials −1.5 −1 −0.5 0 0.5 1 1.5 4121620 s z / P 10 6 t, s Fig. 7.44. Dependence of the interlaminar stress referred to the acting pressure on time. aaa P PP aa h 1 h 2 h 3 (a) (b) (c) Fig. 7.45. Structure of the laminates under study. Now introduce an additional aluminum layer in the foam core as shown in Fig. 7.45b. As follows from Fig. 7.46b, this layer suppresses the tensile stress in section a–a.Two intermediate aluminum layers (Fig. 7.45c) working as generators of compressive stress waves eliminate the appearance of tensile stress in this section. Naturally, the effect under discussion can be achieved for a limited period of time. However, in reality, the impact- generated tensile stress is dangerous soon after the application of the pulse. The damping capacity of real structural materials (which is not taken into account in the foregoing analysis) dramatically reduces the stress amplitude in time. A flying projectile with relatively high kinetic energy can penetrate through the laminate. As is known, composite materials, particularly, high-strength aramid fabrics, are widely Chapter 7. Environmental, special loading, and manufacturing effects 415 −2 −1 0 1 1.1 1.2 1.3 1.4 1.5 1.6 1.8 10 2 s z / p 10 5 t, s (a) −6 −4 −2 0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 10 5 t, s 10 2 s z / p (b) −1.5 −1 −0.5 0 2.7 2.8 2.9 3 3.1 3.2 3.3 10 5 t, s 10 2 s z / p (c) Fig. 7.46. Normal stress related to external pressure acting in section a–a of the laminates in Fig. 7.45 (a)–(c), respectively. used for protection against flying objects. To demonstrate the mechanism of this protection, consider a square composite plate clamped in the steel frame shown in Fig. 7.47 and subjected to impact by a rectangular plane projectile (see Fig. 7.47) simulating the blade of the turbojet engine compressor. The plate consists of layers of thin aramid fabric impregnated with epoxy resin at a distance from the window in the frame (see Fig. 7.47) and co-cured together as shown in Fig. 7.48. The front (loaded) surface of the plate has a 1-mm-thick cover sheet made of glass fabric–epoxy composite. The results of ballistic tests are presented in Table 7.2. Front and back views of plate No. 2 are shown in Fig. 7.47, and the back view of plate No. 3 can be seen in Fig. 7.48. Since the mechanical properties of the aramid fabric used to make the plates are different in the warp and fill directions, the plates consist of couples of mutually orthogonal layers of fabric that are subsequently referred to as 0 ◦ /90 ◦ layers. All the plates listed in Table 7.2 have n = 32 of such couples. 416 Advanced mechanics of composite materials (a) (b) Fig. 7.47. Plate no. 2 (see Table 7.2) after the impact test: (a) – front view; (b) – back view. Fig. 7.48. Back view of plate no. 3 (see Table 7.2) after the impact test. [...]... M.Ya., Tamuzh, V.P and Tarashuch I.V (1991) Fatigue failure of laminated carbon-fiber-reinforced plastic Mechanics of Composite Materials, 27(1), 58–62 434 Advanced mechanics of composite materials Apinis, R.P., Mikelsons, M.Ya and Khonichev, V.I (1991) Fatigue resistance of carbon material in symmetric tension-compression Mechanics of Composite Materials, 5, 928–930, (in Russian) Barnes, J.A., Simms,... The damping and dynamic moduli of symmetric laminated composite beams – theoretical and experimental results Mechanics of Composite Materials, 18(2), pp 104–121 Rabotnov, Yu N (1980) Elements of Hereditary Solid Mechanics Mir Publishers, Moscow Rach, V.A and Ivanovskii, V.S (1986) On the effect of fiber length variation in filament wound structures Mechanics of Composite Materials, Riga, 67–72 (in Russian)... Environmental Effects on Composite Materials Vol 1 Technomic Publ Springer, G.S (ed.) (1984) Environmental Effects on Composite Materials Vol 2 Technomic Publ Springer, G.S (ed.) (1988) Environmental Effects on Composite Materials Vol 3 Technomic Publ Strife, J.R and Prevo, K.M (1979) The thermal expansion behavior of unidirectional kevlar/epoxy composites Mechanics of Composite Materials, 13, 264–276 Sukhanov,... Anderson, Ya.A (1991) Effect of stress ratio on the fatigue strength of organic plastics Mechanics of Composite Materials, 27(3), 276–283 Milyutin, G.I., Bulmanis, V.N., Grakova, T.S., Popov, N.S and Zakrzhevskii, A.M (1989) Study and prediction of the strength characteristics of a wound epoxy organic-fiber plastic under different environmental effects Mechanics of Composite Materials, 25(2), 183–189 Morozov,... Experimental dependence of carbon–epoxy composite longitudinal compression strength related to the corresponding strength of material without ply waviness on the ratio of the waviness amplitude to the ply thickness 426 Advanced mechanics of composite materials 7.4.2 Warping and bending of laminates in fabrication process There exist also some manufacturing operations that are specific for composites that cause... Rogers, E.F., Phillips, L.M., Kingston-Lee, D.M et al (1977) The thermal expansion of carbon fiber-reinforced plastics Journal of Material Science, 1(12), 718–734 Schapery, R.A (1974) Viscoelastic behavior and analysis of composite materials In Composite Materials (L.J Broutman and R.H Krock eds.), Vol 2, Mechanics of Composite Materials, (G.P Sendeckyj ed.) Academic Press Inc., New York, pp 85–168 Schulte,... Popkova, L.K (1987) Combined theoretical and experimental method of determining residual stresses in wound composite shells Mechanics of Composite Materials, 23(6), 802–807 Murakami, S., Kanagawa, Y., Ishida, T and Tsushima, E (1991) Inelastic deformation and fatigue damage of composite under multiaxial loading In Inelastic Deformation of Composite Materials (G.J Dvorak ed.) Springer Verlag, New York, pp 675–694... (1990) Thermal deformation of composites for dimensionally stable structures Mechanics of Composite Materials, 26(4), 432–436 Survey (1984) Application of Composite Materials in Aircraft Technology Central Aero-hydrodynamics Institute (in Russian) Chapter 7 Environmental, special loading, and manufacturing effects 435 Tamuzh, V.P and Protasov V.D (eds.) (1986) Fracture of Composite Structures Zinatne,... projectile velocities before and after the failure of the kth couple of fabric layers, W is, as earlier, the fracture work consumed by the kth couple of 2 layers, Tk−1 = 0.0006 Vk−1 , and the last term in the right-hand side of Eq (7.77) means that we account for the kinetic energy of only those fabric layers that have been already 418 Advanced mechanics of composite materials P P, kN 7 fm 6 5 4 3 2 1 0 5 10... woven fabric and hybrid fabric composites In Proc 6th Int Conf on Composite Materials and 2nd European Conf on Composite Mater., (ICCM and ECCM), Vol 4, (F.L Matthews, N.C.R Buskel, J.M Hodgkinson, and J Morton eds.) Elsevier Science Ltd, London, pp 89–99 Shen, S.H and Springer, G.S (1976) Moisture absorption and desorption of composite materials Journal of Composite Materials, 10, 2–20 Skudra, A.M., . carbon–epoxy composites. 408 Advanced mechanics of composite materials 0 400 800 1200 1600 2000 0123 s 1R , MPa log N Fig. 7.38. Low-cycle fatigue diagram for unidirectional aramid–epoxy composite. can be seen, at the moment of time t equal to about 1.75 × 10 −5 s, this stress is tensile and can cause delamination of the structure. 414 Advanced mechanics of composite materials −1.5 −1 −0.5 0 0.5 1 1.5 4121620 s z . couples of mutually orthogonal layers of fabric that are subsequently referred to as 0 ◦ /90 ◦ layers. All the plates listed in Table 7.2 have n = 32 of such couples. 416 Advanced mechanics of composite