18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS STUDY ON THE OUT-OF-PLANE SHEAR PROPERTIES OF SUPERALLOY HONEYCOMB CORES Z. H. Xie*, W. Li, J. Zhao, J. Tian College of Astronautics, Northwestern Polytechnical University, Xi’an, China * Corresponding author(xzhae@nwpu.edu.cn) Keywords : honeycomb core, superalloy, shear modulus, shear strength higher than the theoretical ones [6]. Lee et al. 1 Introduction investigated the compressive and shear deformation Honeycomb sandwich panels are used extensively behavior and failure mechanism of honeycomb due to their high specific stiffness and specific composites consisting of Nomex honeycomb cores strength. These panels usually consist of two thin, and 2024Al alloy face sheets at room and elevated stiff and strong face-sheets connected by a temperatures [7], However, experimental shear lightweight honeycomb core. Today these panels are strength results obtained were significantly higher used in aerospace industry (such as satellite than the theoretical ones too. The mechanical structures, airfoil, tail unit, rotor blades, …) as well behavior, especially the shear strength and failure as in the marine industry (such as submarine hull, process of honeycomb cores under shear loads still yacht, …). need to be further investigated. The primary function of honeycomb sandwich 2 Analysis panels is to carry bending loads, so the honeycomb core (Fig. 1) must be stiff and strong enough to 2.1 Prediction on the Equivalent Out-of-plane ensure the face sheets do not slide over each other. Shear Moduli of Metallic Honeycomb Cores These properties are mainly related to the out-of- Typical regular honeycomb core is periodic, so a plane shear moduli and strength of the honeycomb simple unit-element can be isolated which can be cores. Characterization of these out-of-plane repeated exactly to build up the entire honeycomb parameters is quite critical for honeycomb cores. core, and the analysis could be conducted on the Kelsey et al. [1] and Gibson et al. [2] derived the unit-element instead of the entire honeycomb core. upper and lower bonds of mathematical expressions The unit-elements of “Kelsey” model [1] and of the equivalent out-of-plane shear moduli of “Gibson” model [2] are showed in fig. 2. honeycomb cores using the theorems of minimum Honeycomb cores built up by repeating the unit- potential energy and minimum complementary element of “Gibson” model are seldom used in energy. Grediac [3] used the finite element method engineering while honeycomb cores with double to calculate the out-of-plane shear modulus as well thickness horizontal walls are more familiar which as the state of stress in the cell walls of honeycomb can be built up by repeating the unit-element of core. Meraghni, Desrumaux and Benzeggagh “Kelsey” model. This paper proposed a more presented a new analytical method to analyze the simpler unit-element as shown in fig. 3. out-of-plane stiffness of honeycomb cores based on The stress distribution of cell walls of honeycomb the modified laminate theory [4]. core under out-of-plane shear loads is not simple; The out-of-plane shear strength is a significant stress each cell wall suffers a non-uniform deformation. of honeycomb cores. To present day, few studies Exact calculations are possible only by using have been conducted on the out-of-plane shear numerical methods. But upper and lower bounds of strength and shear failure models of honeycomb the out-of-plane shear moduli can be obtained by cores. Zhang and Ashby analyzed the buckling using the theorems of minimum potential energy and strength of a wide range of Nomex honeycomb cores minimum complementary energy [1]. [5]. Pan experimentally investigated the longitudinal The first theorem gives an upper bound. It states that shear deformation behavior and failure process of the strain energy calculated from any postulated set 5056Al alloy honeycomb cores using single block of displacements which are compatible with the shear test and compared the results with the elastic external boundary conditions and with themselves buckling strength model, however the experimental shear strength results obtained were significantly
will be a minimum for the exact displacement Considering, first, loading in the 13 direction. We postulate that an external stress τ 13 induces a set of distribution. A uniform strain, γ 13 , is acting on the honeycomb in shear stresses τ a and τ b in the two cell walls. As the the 13 direction. The shear strains in walls a and b wall a is loaded in simple bending, it carries no significant load, so τ a equal to zero. ( as shown in fig. 3)are: Equilibrium requires that: γ = (1) 0 a ( ) τ + θ θ = τ θ l l l tl sin cos cos (10) c i i b i 13 γ = γ θ (2) 13 cos Then b ( ) τ = τ + θ The unit-element volume is: l l t sin / (11) b c i 13 ( ) = + θ θ V * l l l b (3) sin cos c i i Then the inequality equation (7) give a lower bound: Where b is the height of the honeycomb core. ≥ θ + θ G * t G l l cos /( sin ) (12) s c i 13 Almost all the elastic strain energy is stored in the shear displacements in the cell walls; the bending For loading in the 23 direction, we again postulate that the external shear stress τ 23 induces a set of stiffness and energies associated with the bending are much smaller. Ignoring the energies associated shear stresses τ a , τ b and τ c in the three cell walls( as with the bending, The theorems can then be shown in fig. 4). expressed as an inequality which, for shear in the 13 the equilibrium with the external stress gives: direction, has the form: τ + θ θ = τ + τ θ l l l l t l t 23 ( sin )2 cos 2 2 sin (13) c i i a c b i ≤ ∑ ( ) 1 1 γ γ G V G V * 2 * 2 (4) s i i The equilibrium with the stress at the nodes require: 13 13 2 2 i τ = τ + τ t t t (2 ) (14) Where G s is the shear modulus of the cell wall a b c material, γ i are the shear strains in the two cell wall. Symmetry require τ b = τ c , so τ a = τ b = τ c ,and then Evaluating the sum gives: ( ) + θ tG l l sin ( ) ≥ s c i ≤ θ + θ G G t G l l * * cos / sin (15) (5) ( ) 23 + θ s c i l l l 13 cos c i i Repeating the calculation for a shear γ 23 in 23 Combining the equations ( Eqn5, 8, 12, 15) gives: direction, the strains in cell wall a and b are: ( ) = θ + θ G * t G l l cos / sin (16) γ = γ s c i 13 (6) a 23 ( ) ( ) + θ + θ tG l l tG l l 2 sin sin γ = γ θ s c i s c i ≤ ≤ G 23 sin * (7) (17) ( ) ( ) b + θ + θ θ l l l 23 l l l cos sin cos c i i c i i And For the regular hexagon honeycomb cores, giving ： ( ) + θ tG l l 2 l c =l i =l, θ =30 ° , these equations reduced to: sin s c i ≤ G * (8) ( ) 23 + θ θ l l l sin cos c i i t 3 = G G * (18) s 13 l The minimum complementary theorem gives a 3 lower bound for the moduli. It state, that, among the t t stress distributions that satisfy equilibrium at each 3 5 3 ≤ ≤ G G G * (19) s s 23 point and are in equilibrium with the external loads, l l 2 9 the strain energy is a minimum for the exact stress distribution. Expressed as an inequality, for shear in 2.2 Prediction on the Out-of-plane Shear Elastic the 13 direction: Buckling Strength of Metallic Honeycomb Cores τ ⎛ τ ⎞ The cell walls of a metallic honeycomb core may 2 2 1 1 ∑ ≤ V i V * ⎜ ⎟ 13 (9) buckle elastically under out-of-plane shear loads. i G * G ⎝ ⎠ 2 2 i s 13 The buckling load for a cell wall [9] is determined
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