Trace – a game-changing discovery Stephen W. Tsai Stanford University October 5, 2015
Topics • Why trace? The one and only unifying constant • Theory of trace – simple; easy to understand • C-Ply by Stanford/Chomarat – bi-angle NCF • Homogenization – optimizable; high-speed ATL • Unit circle failure criterion • Trace-based sizing method – power of C-Ply • Conclusions
How Many Data Points? • In netting analysis that is applied to pressure vessels and pipes, fiber strength is the one and only design validation calculated from burst pressure: the answer is one • Carbon fibers are known to be anisotropic but we know only the longitudinal fiber stiffness and strength: the answer is one as before, plus one for stiffness • With trace, laminates become universal
Dominant “11” Component in 2- & 3-D Trace-normalized stiffness: Universal [0] Q 11 * = 0.885; 1.3% E 1 * = 0.880; 1.3% 2D: Plane stress 0.880 0.735 = 1.20 = 3D/2D 2D Trace, GPa C 11 * = 0.752; 2.1% E 1 * = 0.735; 2.1% 3D: Plane strain 3D Trace, GPa
Material Selection for Weight Savings
Relative Cost of Hybrid Laminates Linear correlation with trace, but not with E x or others Trace Hybrid = v 1 Trace 1 + v 2 Trace 2 + …
Black Aluminum vs Trace: 15 CFRP Black aluminum vs Trace E x = 0.88 Trace Ply stiffness Unit circle E y ν x E s X Universal Laminates Ply strength X’ Y (Unit circle) Y’ (Universal) (Trace) (Universal) (e x , e x ’) S So much simpler: 4:1 5:2 15:1 15:1 15:1
Topics • Why trace? The one and only unifying constant • Theory of trace – simple; easy to understand • C-Ply by Stanford/Chomarat – bi-angle NCF • Homogenization – optimizable; high-speed ATL • Unit circle failure criterion • Trace-based sizing method – power of C-Ply • Conclusions
Tensors and Traces for Composites Tr [ σ ] = σ 1 + σ 2 + σ 3 = pressure • Stress components σ i Tr [ ε ] = ε 1 + ε 2 + ε 3 = Δ volume • Strain components ε i • Stiffness: Q ij , A ij , D ij , C ij ,… Tr [Q] = Q 11 + Q 22 + 2Q 66 = Tr [A*] = A 11 * + A 22 * + 2A 66 * • Failure criterion in stress 2D space: F ij , F i = Tr [D*] = D 11 * + D 22 * + 2D 66 * Tr [C] = C 11 + C 22 + C 33 +…+ 2C 66 3D where F ij σ i σ j + F i σ i = 1 • Failure criterion in strain Tr [F]= F 11 + F 22 + F 66 /2 space: G ij , G i Tr {F} = F 1 + F 2 where G ij ε i ε j + G i ε i = 1 Tr [G] = G 11 + G 22 + 2G 66 G ij = Q ik Q jl F kl . G i = Q ij F j Tr {G} = G 1 + G 2
Evolution: Universal Constants [0] CFRP Ply engineering constants Plane-stress stiff matrix Trace Trace-normalized Trace = Q 11 + Q 22 +2Q 66 Average = Universal
E 1 ° master ply, GPa 0.888; 2.1% 0.873; 2.5% 2% CFRP tape tens CFRP tape compr 0.471; 1.2% 0.472; 1.3% CFRP fabric tens CFRP fabric compr 0.433; 1.5% 0.423; 1.3% 2% Multiple CFRP, GFRP: GFRP GFRP RT-dry, cold-dry, fabric tens fabric compr hot-dry, hot-wet - Good correlation No correlation CFRP CFRP braid tens braid compr
Composition of Trace: Carbon, Kev, Glass (88, 6, 3)% (70, 15, 7)% (88, 5, 3)% Q yy Q ss Q xx Carbon/epoxy Kevlar 49/epoxy E-glass/epoxy Q xx * Q yy * Q ss * Q xy *
Universal ply: Transversely isotropic [0] Fiber C 11 = 75% In-plane PLANE STRESS C 22 C 66 Out-of-plane: 14% Fiber Q 11 = 88% C 44 Q 22 Matrix Transverse In-plane: 86% shear PLANE STRAIN Matrix C 55 Q 66 Transverse C 33 stiffness Sub-trace (out-of-plane) Q 22 + 2Q 66 = 12% = C 33 + 2(C 44 + C 55 ) = matrix controlled = 14% Trace 2D Trace 3D Sub-trace (in-plane) = C 11 + C 22 +2C 66 Q 11 = 88% = 86% = Fiber controlled 13
Trace in GPa E x /Trace E x in GPa 0.880, cv = 1.33% E x of [0] E 1 °/Trace Trace in GPa E 1 ° in GPa 0.337, cv = 0.10% E 1 ° of [π/4]
Ascending 2D and 3D Trace: CFRP 2D/3D trace, GPa
Dominant “11” Component in 2- & 3-D Trace-normalized stiffness: Universal [0] Q 11 * = 0.885; 1.3% E 1 * = 0.880; 1.3% 2D: Plane stress 0.880 0.735 = 1.20 = 3D/2D 2D Trace, GPa C 11 * = 0.752; 2.1% E 1 * = 0.735; 2.1% 3D: Plane strain 3D Trace, GPa
Dominant “11” Component in 2- & 3-D Trace- normalized stiffness: Universal [π/3],[π/4], … E 1 * = E 2 * = 0.336; 0.1% Q 11 * = 0.371; 0.1% 0.336 2D: Plane stress 0.281 = 1.20 = 3D/2D E 6 * = 0.129; 0.1% Q 66 * = 0.129; 0.1% 0.129 0.107 = 1.20 2D Trace, GPa = 3D/2D A 11 * = A 22 * = 0.323; 0.5% E 1 * = E 2 * = 0.281; 0.4% A 66 * = 0.107; 0.1% 3D: Plane strain E 6 * = 0.107; 0.1% A 33 *= 0.057; 0.5% E 3 * = 0.053; 0.5% A 44 * = A 55 * = 0.020; 0.2% 3D Trace, GPa
Universal laminate stiffness Components: 2D Universal Laminates [0/±45] T700 C-Ply 55 [0/90] AS4/MTM45 T700/2510 T4708/MR60H T650/epoxy 15 Individual CFRP AS4/3501 T700 C-Ply 64 T300/F934 T800S/3900 IM7/8552 T800/Cytec IM7/MTM45 T300/5208 2D Trace, GPa IM7/977-3 IM6/epoxy [Quasi-iso] [0/±45/0] T700 C-Ply 55 AS4/MTM45 T700/2510 T4708/MR60H T650/epoxy 15 Individual CFRP AS4/3501 T700 C-Ply 64 T300/F934 T800S/3900 IM7/8552 T800/Cytec IM7/MTM45 T300/5208 2D Trace, GPa IM7/977-3 IM6/epoxy
Universal Laminate Constants Generated from classical laminated plate theory, and universal [0] derived from average of 15 CFRP ν 21 E 1 * E 2 * E 6 * Trace Universal [0] [0/90] Universal laminates [π/4] [0 7 /±45/90] [0/±45 4 /90] [0/±45] [0/±45/0] [0/±30] [0/±30/0] [±12.5]
Components: 3D Universal Laminates C 33 * = 0.057 C 33 * = 0.057 C 33 * = 0.057 Out-plane: 0.14 C 55 * = 0.021 C 55 * = 0.021 C 55 * = 0.022 C 44 * = 0.021 C 44 * = 0.021 C 44 * = 0.019 C 66 * = 0.14 C 66 * = 0.11 C 66 * = 0.026 In-plane: 0.86 C 22 * = 0.41 C 22 * = 0.32 C 22 * = 0.18 C 11 = C 22 = 0.405 C 11 * = 0.41 C 11 * = 0.41 C 11 * = 0.32 [0/90] [π/3], [π/4], … [0/±45] 3D Trace, GPa
Components: 3D Universal Laminates C 33 * = 0.057 C 33 * = 0.057 C 33 * = 0.057 Out-plane: 0.14 C 55 * = 0.024 C 55 * = 0.020 C 55 * = 0.025 C 44 * = 0.017 C 44 * = 0.020 C 44 * = 0.016 C 66 * = 0.059 C 66 * = 0.087 C 66 * = 0.16 C 22 * = 0.16 In-plane: 0.86 C 11 * = 0.58 C 22 * = 0.083 C 11 = C 22 = 0.405 C 22 * = 0.27 C 11 * = 0.604 C 11 * = 0.27 [0 2 /±30] [0 7 /±45/90] [0/±45 4 /90] 3D Trace, GPa
Topics • Why trace? The one and only unifying constant • Theory of trace – simple; easy to understand • C-Ply by Stanford/Chomarat – bi-angle NCF • Homogenization – optimizable; high-speed ATL • Unit circle failure criterion • Trace-based sizing method – power of C-Ply • Conclusions
[0/25] Unique NCF at Chomarat Shallow angles, wide range ply thicknesses with high quality from tow spreading, Flip noninvasive stitching, hybrid, … over [-25/0] [20 ≤ ϕ ≤ 30]
Wide-range GSM to Meet Requirement 160 120 Laminate thickness 80 in mil 40 0
Thick-thin Combinations of C-Ply Thin [0]; thick [ ϕ ] Thin [0]; thin [ ϕ ] Thick [0]; thin [ ϕ ] [ ϕ ] or [- ϕ ] [ ϕ ] or [- ϕ ] [ ϕ ] or [- ϕ ] [ ϕ ] or [- ϕ ] [0] [0] [0] Black Alum C-Ply laminates [0/±45/90] [0] [0/± ϕ ] [0/± ϕ /90] [0/± ϕ /± ψ /90] 1-axis 4-axis 1-axis 2-axis 2-axis
Layup: 4-axis [0] thk vs 2-axis [0/45] thn/thk Jim Hecht of MAG [45] [-45] [90] [0] More exact estimate: 2.7X faster for thin ply; Time, min 5.4X for thick ply 4-axis Unitape layup 2-axis [0/45] thin-ply layup Laminate length, m
High Speed Layup of Thick-thin C-Ply Starting [0/ϕ 2 ] - Thin-Thick [0/ ϕ ] - Thin-Thin [0 2 / ϕ ] - Thick-Thin C-Ply (33/67/0) – 150 gsm (50/50/0) – 150 gsm (67/33/0) – 150 gsm High shear Low shear Regular 1-axis [0/± ϕ ] 2 [0 2 /± ϕ ] [0 4 /± ϕ ] layup = [π/3] 2 for ϕ = 60 1:0 (33/67/0) (50/50/0) (67/33/0) 5X 2-axis [(0/± ϕ ) 2 /(± ψ /90)] 2 [(0 2 /± ϕ ) 2 /± ψ 2 /90 2 ] [(0 4 /± ϕ ) 2 /± ψ /90 4 ] layup Ψ = 90 - ϕ Ψ = 90 - ϕ Ψ = 90 - ϕ 2:1 (22/67/11) (33/50/17) (44/33/22) 3X 2-axis [0/± ϕ /± ψ /90] 2 [0 2 /± ϕ /± ψ /90 2 ] [0 4 /± ϕ /± ψ /90 4 ] layup = [π/6] 2 for ϕ = 30 = [π/4] 2 for ϕ = 45 Ψ = 90 - ϕ 1:1 (17/66/17) (25/50/25) (33/33/33) 2X 27
Universal Stiffness Components: C-Ply [0 2 /± ϕ ] [0/± ϕ ] 2 [0 4 /± ϕ ] [0/ ϕ ]; [0/- ϕ ] [0/ϕ 2 ]; [0/- ϕ 2 ] [0 2 / ϕ ]; [0 2 /- ϕ ] Thin-thin Thin-thick Thick-thin E 1 * Regular High shear E 1 * Low shear E 1 * 1:0 A;sk 1:0 [0/±30/0] Universal laminate stiffness 1:0 2:1 [0/±30] [0/±45/0] 2:1 2:1 1:1 [0/±45] 1:1 1:1 [π/3] = 0.34 [π/4] [π/6] E 6 * [π/3] = 0.13 E 6 * Off-axis angle ϕ
Topics • Why trace? The one and only unifying constant • Theory of trace – simple; easy to understand • C-Ply by Stanford/Chomarat – bi-angle NCF • Homogenization – optimizable; high-speed ATL • Unit circle failure criterion • Trace-based sizing method – power of C-Ply • Conclusions
Black Aluminum Design: Bricky Cannot be optimized; manufacturing nightmare E 1 * = 0.28 E 1 * = 0.22 [0 5 /±45 3 ] E 1 * = 0.45 E 6 * = 0.16 E 6 * = 0.20 (45/55/0) E 6 * = 0.14 [0/±45 2 ] [0/±45 2 /90] 0.277 0.161 (20/80/0) (17/66/17) 0.466 0.268 0.138 0.187
C-Ply: Black Aluminum Replacement [0/± ϕ ] 2 [0/ϕ 2 ]; [0/- ϕ 2 ] Thin-thick E 1 * High shear 1:0 [0/±35] [0/±45] Trace-normalized stiffness 2:1 Layup ratio 0.45 1:1 [0 5 /±45 3 ] 0.28 0.22 [0/±45 2 ] 0.20 0.16 0.14 [0/±45 2 /90] E 6 * Off-axis angle Off-axis angle
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