HYGROTHERMALLY STABLE LAMINATES WITH EXTENSION-TWIST AND BEND-TWIST - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS HYGROTHERMALLY STABLE LAMINATES WITH EXTENSION-TWIST AND BEND-TWIST COUPLINGS R. Haynes*, E. Armanios Mechanical and Aerospace Engineering, University of Texas at Arlington, Arlington, TX, USA * Corresponding author (rahaynes@uta.edu) Keywords : extension-twist coupling, bend-twist coupling, hygrothermal stability, elastic tailoring, optimization, classical lamination theory 1 Summary tilt rotor as well as variable speed rotor aircraft could be achieved by designing blades with both The material-independent hygrothermal-stability extension-twist and bend-twist couplings conditions are derived from Classical Lamination Some couplings, such as extension-twist coupling, Theory (CLT) [1] and used in conjunction with require the use of asymmetric stacking sequences, specific compliance coefficients in an optimization which are prone to hygrothermal instabilities. routine to identify hygrothermally stable stacking Hygrothermally unstable laminates will warp out-of- sequences with optimal coupling [2]. Of particular plane with changes in temperature or moisture. interest is the combined effect of extension-twist and Since this behavior is typically undesirable, this bend-twist couplings. The objective is to achieve work aims at identifying only hygrothermally stable the highest twist distribution possible among flat laminates with the couplings mentioned previously. laminates of a given number of plies. Optimal Families of hygrothermally stable asymmetric stacking sequences are presented for flat laminates stacking sequences have been published previously with up to ten plies. by Winckler [4] but have not been shown to be unique or optimal. This work makes use of the necessary and sufficient conditions for hygrothermal 2 Introduction stability, derived previously [5] to ensure the entire Laminated composite materials have the potential domain of hygrothermally stable laminates are for coupling of deformation modes in ways not considered. achievable with conventional materials. For A laminate with both extension-twist and bend-twist example, a rotor blade constructed with a laminate couplings has the potential for greater twist exhibiting extension-twist coupling can passively distribution changes than either coupling alone. The change its twist distribution with increasing rotor present work analyzes all families of hygrothermally speed, and thereby change its angle of attack. This stable laminates for optimal combined extension- effect has been shown to achieve maximum bend-twist couplings. Next, a constrained efficiency in tilt rotor aircraft in both the vertical and optimization is performed to identify stacking forward flight regimes, with measurable horsepower sequences that produce the highest twist rate savings [3]. achievable from coupling of deformation modes. Lifting surfaces are also subject to transverse 3 Hygrothermal Stability Conditions aerodynamic lift forces, which often induce bending moments in the structure. A lifting surface made The necessary and sufficient conditions for from a laminate with bend-twist coupling can hygrothermal stability have been derived in previous passively change its twist distribution upon work [5]. A summary will be provided in the experiencing a change in lift. This has application in following for convenience. The relationship aircraft with swept-forward wings; in this case an between force and moment resultants and the mid- increase in lift can produce a change in twist plane strains and curvatures are expressed in CLT as distribution that reduces the angle of attack, which in turn can delay the onset of aerodynamic instabilities such as divergence. Further increase in efficiency in

( ) T , H n n ε ⎧ N ⎫ ⎡ ⎤ ⎧ ⎫ ⎧ N ⎫ A A A B B B ∑ ∑ ( ) ( ) − − θ = − − θ = xx xx xx 2 k n 1 cos 2 0 2 k n 1 cos 2 0 11 12 16 11 12 16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ k k ε N N A A A B B B ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ = = k 1 k 1 yy yy yy 12 22 26 12 22 26 ⎪ ⎪ ⎪ γ ⎪ ⎪ ⎪ ⎢ ⎥ N A A A B B B N n n ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∑ ∑ ( ) ( ) xy xy xy = 16 26 66 16 26 66 + − − θ = − − θ = ⎨ ⎬ ⎢ ⎥ ⎨ ⎬ ⎨ ⎬ 2 k n 1 sin 2 0 2 k n 1 sin 2 0 k k κ M M B B B D D D ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ xx 11 12 16 11 12 16 xx xx = or = k 1 k 1 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ κ M M B B B D D D n n ∑ ∑ ( ) yy yy yy ⎪ ⎪ ⎢ 12 22 26 12 22 26 ⎥ ⎪ ⎪ ⎪ ⎪ cos θ = − − θ = 2 0 2 k n 1 cos 4 0 κ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ M ⎢ B B B D D D ⎥ M k k ⎩ ⎭ ⎣ ⎦ ⎩ ⎭ ⎩ ⎭ xy xy xy 16 26 66 16 26 66 = = k 1 k 1 (1) n n ∑ ∑ ( ) sin θ = − − θ = where N xx , N yy , N xy , ε xx , ε yy , γ xy are the in-plane 2 0 2 k n 1 sin 4 0 k k forces and strains respectively, M xx , M yy , M xy , κ xx , = = k 1 k 1 (4a,b) κ yy , and κ xy are the out-of-plane moments and respectively. Since the equations that satisfy either curvatures respectively, A ij , B ij , and D ij , are the in- Condition A or Condition B are only functions of k plane, coupling, and bending stiffness coefficients respectively, and ( ) ( T , H ) indicates non-mechanical and θ k , they are material independent. This benefits the designer by allowing for choice of stacking quantities, i.e. hygral or thermal effects. Since sequence before a material system and provides hygrothermal stability can be defined as having the robustness to the hygrothermal stability from out-of-plane curvatures equal to zero for any change variations in material properties. in temperature or moisture, by considering no mechanical resultants, this can be expressed as 4 Optimization Procedure ( ) T , H ⎧ ⎫ N ⎡ ⎤ A A A The stacking-sequence-dependent equations that xx 11 12 16 ⎪ ⎪ ⎢ ⎥ N A A A ensure hygrothermal stability are used as constraints ⎪ ⎪ ⎢ ⎥ yy 12 22 26 ⎧ ε ⎫ to the optimization procedure. The objective ⎪ ⎪ ⎢ ⎥ xx N ⎪ ⎪ A A A ⎪ ⎪ xy = 16 26 66 ε function arises from the compliance coefficients in ⎨ ⎬ ⎢ ⎥ ⎨ ⎬ yy M B B B CLT. To define the objective function, invert ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ xx 11 12 16 γ ⎩ ⎭ ⎢ ⎥ ⎪ ⎪ Equation (1) to arrive at the mid-plane strains and xy M B B B yy ⎪ ⎪ ⎢ 12 22 26 ⎥ curvatures as a function of the resultants, given as ⎪ M ⎪ ⎢ B B B ⎥ ⎣ ⎦ ( ) ⎩ ⎭ T , H ε ε xy ⎧ ⎫ α α α β β β ⎧ ⎫ ⎧ ⎫ 16 26 66 (2) ⎡ ⎤ N xx xx xx 11 12 16 11 12 16 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ For a laminate where all plies have the same ε α α α β β β ε N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎥ yy 12 22 26 12 22 26 yy yy mechanical and hygrothermal properties in their ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ γ ⎢ ⎥ γ α α α β β β N ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ principal material directions, solving Equation (2) xy xy xy = 16 26 66 16 26 66 + ⎢ ⎥ ⎨ ⎬ ⎨ ⎬ ⎨ ⎬ κ κ β β β δ δ δ M produces the necessary and sufficient conditions to ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ xx xx xx 11 12 16 11 12 16 ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ ensure hygrothermal stability, given as κ β β β δ δ δ κ M yy 12 22 26 12 22 26 yy yy ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎪ ⎪ (T,H) =N yy (T,H) and N xx κ β β β δ δ δ κ ⎪ ⎪ ⎢ ⎥ ⎪ M ⎪ ⎪ ⎪ ⎣ ⎦ ⎩ ⎭ ⎩ ⎭ ⎩ ⎭ (T,H) =M xx (T,H) =M yy (T,H) =M xy (T,H) = 0 N xy (3a) xy xy xy 16 26 66 16 26 66 (5) or Of interest is the twist rate, φ , and for a B ij =0. (3b) hygrothermally stable laminate, the relationship meaning that either (Condition A) the laminate has between the twist rate and the extension and bending equal axial in-plane non-mechanical stress resultants deformation modes can be found as and all other resultants are zero or (Condition B) the (6) ϕ = κ = β + δ 2 N M laminate has a coupling stiffness matrix identically xy 16 xx 16 xx equal to zero. The equations that satisfy these Previous work [6,7] has presented optimal conditions can be cast in terms of the fiber hygrothermally stable stacking sequences for orientation angle, θ k , and location of each ply within extension-twist coupling and bend-twist coupling the stacking sequence as separately. Specifically, to optimize extension-twist 2 , coupling, the objective function is taken to be - β 16 because the optimizer minimizes the objective function. The results of the optimization are