Aerospace Materials&Processes AE 510
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This 39 page Class Notes was uploaded by Miss Sid Klocko on Monday September 7, 2015. The Class Notes belongs to AE 510 at Kansas taught by Richard Hale in Fall. Since its upload, it has received 58 views. For similar materials see /class/186814/ae-510-kansas in Aerospace Engineering (AE) at Kansas.
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Date Created: 09/07/15
Micromechanical Behavior of a Lamina De nition of Micromechanics The study of composite material behavior Where the interaction of constituent material is examined in detail and used to predict and de ne the behavior of the heterogeneous composite material Approaches to the study of Micromechanics Mechanics of Materials Elasticity Bounding Principles Exact Solution Approximate Solutions Mechanics of Materials Approach to Stiffness Determination of E1 6fEf81 6mEm81 P 01A of Af om Am Where 01 Average Stress m m By Substitution Af A But 51 EfT Em A quot Am A Vf Tquot Vm T E1 E Vf Em Vm Rule 0f Mixtures Mechanics of Materials Approach to Stiffness Determination of E2 22 HWH MATRIX 6 c w 8f E g8mE 2 I f m Total Transverse Deformation 82W VfW8f Vm WEm 82Vf fvm 8m 02 02 62 Vf Ef Vm a But 02E2 252 Ef MATRIX Vf 0392 Vm 02 Em FIBER Determination of E2 Solving For E2 E Em 239 Vm EfVf Em EfEm1oo I Ef Em10 Nondnmenstonallzmg E2 1 E m Vm Vf EmEf Mechanics of Materials Approach to Stiffness Determination of 12 WE 2 FIBER A w2 1 39 gt 1 gt x 82 War Aw Amw Afw Aw We2 Wv12 a1 Amw WVm Vm E1 Afw WVf Vf 81 V12Vm Vm Vfo Mechanics of Materials Approach to Stiffness Determination of G12 2 FIEEH m l C FIEER m M I 7 E An I T T Ym Gm I E Total Shear Deformation A 7W Am Af AmVmWYm AfVfWVf L i i G1239Vm Gm Vf Gt 5me G1 Vm GfVme Equations to Approximate Lamina Properties from Constituents E1 Efo Eme 12 fo me E2 EfEmEfVm Eme G12 GmeGfVm Gme Micromechanics of Lamina Examples Your supervisor has given yoll the task of estimating the values of E G12 and D12 for an emerging material The following data is availal Efiber 40M psi Ematrix 05M psi Gfiber 14M psi Gmntrix 02M psi D ber 045 Dmatrix 03 Volume fraction of the fiber is 40 Note M Million Solution E1 40M04 O5MO6 163M psi E2 4OMO5M4OMO6 05M04 20242 083M psi 12 045O4 O3O6 027 G12 14MO2M14M06 02M04 28848 033M psi Micromechanics of Lamina Examples A composite material manufacturer has decided to make a composite material using twg ber systems and one matrix system How would you estimate the value of the longitudinal modulus E of the material Solution E1 Ef1 Vf1 EQVf2 Eme Micromechanics of Lamina Examples Is the transverse modulus E2 of a unidirectional ply stiffer equal or softer than the matrix modulus Em Provide an explanation of your answer Assume that Er E of the ber is much greater than Em Solution E2 must be stiffer than the matrix modulus Em The matrix modulus is the same in any direction and Ef serves to increase E2 according to the equation 139 E2 Ef Em H H1 vm Ef vf Em 0r Em vm vf Em5f If Ef gt Em and knowing that Vf Vm 1 then E2 gt Em Macromechanical Behavior of a Lamina De nition of Macromechanics The study of composite material behavior Where the material is presumed homogeneous and the effects of constituent materials are detected only as averaged apparent properties of the composite material Generalized Hooke s Law Anisotropic Material 6 Generalized Hooke39s Law Relating 3 Stresses to Strains Can Be Written in Contracted Notation as 152 0iCij8j ij16 3 Where ciAre the Stress Components 2 Cij is the Stiffness Matrix 1 Three Dimensional Ej Are the Strain Components State of Stress c 51 011012013014015015 1 02 C12 C22 c23024025026 L2 03 013 C23 033034035036 E3 123 2 CM C24 034 C44C4504e 23 T13 015 C25 035C45C55056 quot13 016 026 CSGC4BC56065 y T 12 12 36 constants 6X6 matrix 21 independent constants symmetry Derivation of Compliance and Stiffness Symmetry W 1 C C 8 alcuel 5 U I l dW 0 8 d8 1 39l l I W 2 S 0quotO39J From the Elastic Potentials One Can Derive the Generalized Hooke39s Law aw TE rquot CUEI and 32 gc 3305 i Similarly 82W C aelas Since the Order of differentiation is immaterial then Cl CJI in a similar manner Si 5 One Plane of Material Symmetry z 0 Monoclinic Material 01 C11 C12 C13 0 0 C15 81 02 C12 C22 C23 0 0 C25 52 a3 C13 C23 C33 0 0 C35 53 123 0 0 0 C44 C45 0 723 731 0 0 0 C45 C 55 0 731 T12 Cl C26 C35 0 0 C55 712 13 independent constants Two Orthogonal Planes of Symmetry Orthotropic Material 01 C11 C12 C13 0 0 0 81 0392 C12 C22 C23 0 0 0 2 03 C13 C23 C33 0 0 0 83 123 o o 0 344 o o m 131 o o o 0 C55 0 7 31 112 O 0 0 0 0 066 912 9 independent constants StressStrain Relations for Plane Stress in an Orthotropic Lamina Material 030 172320 I13 0 Therefore 723 O Y13 0 7 independent and constants 83 313 01S23 CS2 where Hm4 We Neglect 3 for Laminate Analysis 01 011 012 0 1 lt32 012 Q22 0 82 I12 0 0 Q66 712 Where Now Q C 4 independent constants The Q Matrix 0 Q Q 0 8 I12 0 O 065 312 The Components of the Lamina Stiffness Matrix Q Are Given by E D E 1 E Q11 1 39Q121 21 1 quot 150 1 012D21 i 12021 L 4012 21 E Q 2 Q go Engineering Constants for Orthotropic Materials Engineering constants are E Young39s moduli in 1 direction E2 Young39s moduli in 2 direction D12 2 2 2 Poisson39s ratio for strain in 2 direction 1 when stressed in 1 direction only D21 2 51 Poisson39s ratio for strain in 1 direction 62 when stressed in 2vdirection only G12 Shear modulus in 12 plane where 1k g must be satisfied for orthotropic material E1 H E2 Macromechanics of a Lamina 3Unidirectional Fiber and Matrix Behave Homogeneousiy 3 Woven Cloth r Fill Q 39 Direction IWarp r raquot Direction Lamina Principal Axis Lamina Coordinate System Coordinates 1 2 3 Are Called PrincipalrMateriai Directions Coordinates X Y Z Are Caiied Transformed or Laminate Axes 32 2 Transverse 90 Lamina Properties E17 Ezv G121 12 1 21 Are Determined for Principal Material Directions by Testing StressStrain Relations for a Lamina of Arbitrary Orientation or 6X 8X 6y Q 8y M W 00326 sinze 25in0c050 T1 sinze 00320 25inecoso sinOcose sinOcose c0320 7 sin20 00520 sinzo sinOcosO Q12 Q22 026 T2 sinzo 00320 sinOcosO 515 526 556 25inecoso 2sin8coso 00320 sin20 Expression for the General Case Becomes Ox a 8 x 11 12 gm 8x 0y Q E y I 912 922 926 8y Txy ny Q16 Q26 Q66 ny in which 611011cos4e2Q12 2055sin2000520022 sin4e 612 0110224066sin20cos20012sin40cos40 622 Q11sin4 2Q12 265sin20cos2 0 022 cos4 0 616 011 012 2066sin0cos30012 022 2056sin3 ecoso 625 Q11 Q12 2065sin30coso012 7022 2056sinecos30 655 Q11022 ZQ12 2055sin2 0c0520Q 36sin4 0 cos4 0 Invariant Properties of An Orthotropic Lamina 611 U1 U2c0s20 U3cos4e 212 U4 U3cos48 222 U1 U200820 U3cos40 616 12Uzsin29 U3Sin49 626 12Uzsin20 U3sin49 666 U5 U3cos46 Invariants U1 3Q11 3sz 2le 4Q668 U2 Q11 Q222 U3 Q11 Q22 2Q12 4Q668 U4 Q11 Q22 6Q12 4Q668 U5 Q11 f Q22 39 2le 4Q663 Macromechanical Behavior of a Laminate Laminate Mechanical Behavior Derived From Lamina Building Blocks Principal Material Directions 123 01 Q11012 0 8 1 102 C212sz 0 12 O 0 Gas Y Coordinates x y z x 611 oy Txy 912 Q 16 Where Qij Are Functions ofg Engineering Constants E1 E2112 and G12 12 1a 8x 922 926 026 Q and ij Are Functions of Qij Thus E1 E2 D12 and G12 and Angie 6 This Is the Starting Point for Building Laminates The Building Block For Arbitrary Coordinates the StressStrain Relations for the Km Layer of a Multilayered Laminate Are 5X 611 612 515 ggtlt 6y 6112 Cgt22 626 8y Txy k 616626 666 k 7X k or in Reduced Form 1 H otk ajk stk GT Classical Lamination Theory For Thin Laminates a Line Originally X U Straight and Perpendicular to the Middle Surface of the Laminate is Assumed to Remain Straight and Perpendicular to the Middle Surface hen the Laminate is Extended and Bent Also This Normal ls Assumed to Have Constant Length yr V U0 EXL L 7 C Theor i D lt ZC B Geometry of deformation in the zx plane Undeformed Deformed Cross Section Cross Section Displacements At Point O UC UO 205 Where aw BWQ At Any Point 3W0 039 W and BW VV Z Q 0 BY small angle plane sections remain plane is slope of mid surface Strains Linear Elasticity 8 7U av Sx ax EV ay ny E Small strains Substitute in the Displacements EX 8 xx 5y 8 z Ky ny y ny s BUDax Where 8 3V0 ay qu Buo ay avoax KX a 2 W0 BX 2 Ky a 82 W0 8y 2 ny 2 92 wO exay Stresses 0x 311 612 616 8x0 0y 912 022 926 eye 2 K TXy K 016 026 066 K YXYO KXy Note that the Equation is valid for values of 2 within the limits of the Kth layer thickness While it should be obvious most textbooks fail to point this fact out Force and Moment Resultants lnPlane Forces on a Flat Laminate lnPlane Forces 71 74quot Nx Running Loads unit Width Moments Running Moments unit Width y Force and Moment Resultants The Force and Moment Resultants for an N Layered Laminate ls Given as I 1 g dz i N351 dz ny 42 LTny k1zk 1 1ny and W 1 ail zdz 2quot fig 1 zdz LMXVJ t2 ny k1zk 1 1ny Layer Number Equation Manipulation The stiffness matrix 6 U is constant within each lamina Therefore the stiffness matrix can go outside the integration over each layer but is within the summation Also we recall thatthe strains and curvatures x e y y xy Kx Ky KX are middle surface values and are not functions of 2 Therefore they can be removed from under both the integration and summation signs E K N Zr X N Zr X N 2 lok J dz 8 zlolk de lg k1 k1 2m 7 A ny 8 K N Zk x N 7 Zk X M Zlolk lzdz 5 Z lok fzzdz Ky k1 k1 2m Yx y Zkt ny The AB D Matrices The Integrations Are Straightfonrvard Equations Can Now Be Written in the Form Z K Idbzk metk A1TA12A16 N 5 B11 B12 B15 KK X 2 RV A12 A22 A26 EV B12 B22 B26 Xy 7y Y Izdzlzfzi1j A16A26A66 B16B26 Bee 2 2m Zk M 311812316 5 DT1D12D16 K X X I22d21ziz1j My B12822 i325 Ev 012ng Dee 3 gt W y Y Zk1 B16 826 Bee D16 DEG D66 Where Using lndicial Notation N n Z A 2 Q39JJK Zk k4 Extensiunal S ffnesscs K1 N s 2 2 3 95 Q39ij 2k 2 Coupling Sti nesses K21 N a a DU Q39ij 2k 2H Bending Stiffnesses K1 Determination of Laminate EX Ey GXy Xy Generally the determination of elastic properties for the laminate involves using the extensional A matrix We shall use this approach below Considering the extensional matrix A only is the same as assuming that the coupling matrix B 0 We can further extend this to implying that the strains are uniform through the laminate thickness or equal to mid surface values through the thickness Therefore we can write Nx A11 A12 A16 8x Ny A12 A22 A26 8y ny A16 A26 A66 lxy The stresses are related to the resultant forces by 1 a TtN Where t Laminate total thickness
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