Lainated oposite Plates Notes y Larry Peel Adarsh Jagannath (grad student) Naveen Kuar Raaiah (grad student) MEEN 4385 1
Introduction What are oposite Materials? They can be deined as a aterial with two (or ore) distinct acroscopical phases. They consist o two or ore aterials cobined in such a way that the individual aterials are easily distinguishable. A coon exaple o a coposite is a iberglass boat. What are Lainated oposites? Lainated coposites can be thought o as sheets o continuous iber coposites lainated such that each layer has the iber oriented in a given direction. MEEN 4385
Lainated oposites MEEN 4385 3
Metal Matrix oposites In boron iber-reinorced aluinu, the ibers are coposed o a thick layer o boron deposited on a sall diaeter tungsten ilaent Silver opper Alloy reinorced with arbon Fibers. MEEN 4385 4
Fiber Properties arbon/graphite The next 5 tables are ro Stress Analysis o Fiber-Reinorced oposite Materials by MW Hyer MEEN 4385 5
Fiber Properties Fiberglass MEEN 4385 6
Fiber Properties Kevlar & Spectra MEEN 4385 7
Selected Properties o Fibers, etc(gibson, oposite Mechanics) MEEN 4385 8
Matrix Properties Thero-set MEEN 4385 9
Matrix Properties Thero-plastic MEEN 4385 1
Shear Modulus and Poisson s Ratios Material Shear Modulus, G (psi) Poisson's Ratio, v Fiberglass (E-glass) 4.3E+6. IM Graphite iber 3.6E+6. AS4 Graphite iber 3.6E+6.3 Kevlar 49 7.4E+6.31 Polyester Resin 1.96E+5.34 Epoxy.3E+5.35 Properties: http://www.alphastarcorp-genoa.co/publications/levonsampepap/strdur.pd, www.iberglast.co., Hyer, Jones. MEEN 4385 11
Typical Orthotropic Properties MEEN 4385
Typical Properties -ontinuous MEEN 4385 13
Typical Properties arbon (Fro MW Hyer) MEEN 4385 14
More Othotropic Props (ro Gibson s Mechanics o oposites) MEEN 4385 15
More Othotropic Props (ro Mil-Handbook 17, volue 3) MEEN 4385 16
MEEN 4385 17 Derivation o Rule o Mixtures Derive the rule o ixtures or the odulus o elasticity o a iber reinorced coposite when a stress (σ) is applied along the axis o the ibers. Solution: The total orce have acting on the coposite is the su o the orces carried by each constituent: c c c c c c A A A A A A A A F F F F : Since,
Derivation (contd...) I the ibers have unior cross - section, the area raction equals Fro Hooke's Law, E. Thereore, c the volue raction, V V V, also, E E c c c c V c E E V V V E V V E V V I the ibers are rigidly bonded to the atrix, both the ibers and the atrix ust stretch equal aounts (iso - strain conditions) : : MEEN 4385 18
What about Shear & Transverse Stiness? Todeterine the copositestiness transverse to the iber, reeber t hat the copositestrain,, is equal to thesu o the iber and atrix strain : Fro Hooke's Law, E. Thereore, / E also, 1/ E 1/ G c ) V ) V ) V V ( (1/ E ) V (1/ G / E I the ibers are rigidly bonded to the atrix, both the ibers and the atrix ust experience equal stresses (iso - stressconditions) : ( / E (1/ E V (1/ G ) V ) V we MEEN 4385 19
Rule o Mixtures The Rule o aounts and The general where property o are R, is the volue raction o A greater volue raction o stiness o 8%, beyond which the ibers surrounded by c R c M ixtures will accurately the propertieso or is : Vi. Ri V 1. the property o the each consituents in the copositev the coposite.the M axiu volue raction is atrix. R 1 the individual V. R the coposite, R each consituent. ibers predict the relative... V increases constituents.. R, R.., R, V are... V the strength and can no longer be copletely 1 n n 1 n the n about MEEN 4385
Rule o Mixtures (contd..) Asor thelainated coposites, the rule o ixtures alwayspredicts thedensity o iber - reinorced coposite: Also, V 1V V c V where the subscripts and reer to the atrix and iber. MEEN 4385 1
Modulus o Elasticity (Axial Stiness) The rule o ixtures is used to predict the Modulus o Elasticity along the iber direction, when the ibers are unidirectional and continuous. I we consider the iber to be isotropic, the Rule o Mixtures can be presented as: E 1 V E V E Here the Modulus o Elasticity is considered to be parallel to the iber. The other or o the above equation is: E (1 V ) E V E 1 MEEN 4385
Modulus (Transverse Stiness) When the load is applied perpendicular to the ibers, then the odulus o the coposite is now: 1/ E V / E V / E Modulus o Elasticity (Shear Stiness) The rule o ixtures also predicts initial shear stiness (G) or each aterial syste. It is given by : 1/ G V / G (1 V ) / G where G and G are the iber and atrix shear oduli. MEEN 4385 3
Axial, Transverse, and Shear Strengths The rule o ixtures is used to predict the tensile strength along the iber direction, when the ibers are unidirectional and continuous. I we consider the iber to be isotropic, the axial tensile strength can be approxiately calculated as: S 1 V S V There are soe crude approxiations or the laina transverse strength and shear strength, but the ollowing estiates are about as accurate. Testing is required or accurate strengths. S S S MEEN 4385 4
Saple Rule o Mixture alcs AS4 MEEN 4385 5
Saple Rule o Mixture alcs MEEN 4385 6
oparison o Test and Experiental Results or E 1 Silicone & otton MPa (psi) Urethane & otton MPa (psi) Test Results 7 (39) 341 (494) Rule o Mixtures 73 (396) 39 (477) onclusion: Here we can see the Rule o Mixtures predicts accurately the initial axial lainate stiness E 1. MEEN 4385 7
oparison o Test and Experiental Results or E Silicone & otton MPa (psi) Urethane & otton MPa (psi) Test Results 3.76 (546).5 (37) Rule o Mixtures 1.9 (75).1 (37) onclusion: Here we can see that the Rule o Mixtures does not accurately predict Initial Transverse Stiness E. MEEN 4385 8
oparison o Test and Experiental Results or G Silicone & otton MPa (psi) Urethane & otton MPa (psi) Test Results.655 (95).1 (34) Rule o Mixtures.63 (91.7) 1.45 (11) onclusion: The Rule o ixtures does not always give accurate results or G. MEEN 4385 9
Hookes Law We know that Hooke s law is given by E where σ is the stress, ε is the strain and E is a constant called the Young s Modulus In the generalized or it can be written as: ij ijkl kl Where ijkl are stiness coeicients MEEN 4385 3
Deinition (ont ) Hooke s law is a stateent that the stress is proportional to the gradient o the deoration occurring in the aterial. These equations assue that a linear relationship exists between the coponents o stress tensor and strain tensor. Such relations are reerred to as a set o constitutive equations. They relate stress and strain, because they depend on the aterial behavior, whether it be an elastic or plastic solid or a viscous luid. In this presentation we will only consider the constitutive equations or an elastic solid. MEEN 4385 31
onstitutive Relationships They are applicable or aterials exhibiting sall deorations when subjected to external orces. The 81 constants ijkl are called the elastic stiness o the aterial and are the coponents o a artesian tensor o the ourth order. It is the elastic stiness tensor which characterizes the echanical properties o a particular anisotropic Hookean elastic solid. The anisotropy o the aterial is represented by the act that the coponents o ijkl are in general dierent or dierent choices o coordinate axes. I the body is hoogeneous, that is, the echanical properties are the sae or every particle o the body, then ijkl are constants (i.e. independent o position). We shall only study hoogeneous bodies. Due to the syetry o the stress and strain tensor we ind that elastic stiness tensor ust satisy the relation ijkl= jikl= ijlk= jilk and consequently only 36 o the 81 constants are actually independent. MEEN 4385 3
Material Syetries The generalized Hooke s law can be expressed in a or where the 36 independent constants can be exained in ore detail under special aterial syetries. 1 3 4 5 6 11 1 31 41 51 61 3 4 5 6 13 3 33 43 53 63 14 4 34 44 54 64 15 5 35 45 55 65 16 6 36 46 56 66 l l l l l l 1 3 4 5 6 () MEEN 4385 33
MEEN 4385 34 Anisotropic Material (ont ) (3) 6 5 4 3 1 66 56 55 46 45 44 36 35 34 33 6 5 4 3 16 15 14 13 11 6 5 4 3 1 l l l l l l sy
Anisotropic Material (ont ) Elastic deoration under anisotropic conditions is described by the elastic constants ij, whose nuber can vary ro 1 or the ost anisotropic solid to 3 or one exhibiting cubic syetry. As we are going to see next, or isotropic solids, the nuber o independent elastic constants is. There are two sources o anisotropy: Texture, in which the grains are not randoly oriented, but have one or ore preerred orientations. Texturing is oten introduced by deoration processes, such as cold rolling, wire drawing, and extrusion. Alignent o inclusions or second-phase particles along speciic directions. When steel is produced, the inclusions existing in the ingot take the shape and orientation o the rolling. These inclusions produce echanical eects called ibering. Anisotropy can strongly aect the yield stress and also inluence racture. Soe anisotropic aterials, such as wood and iber-reinorced coposites, ay have low strength in the radial direction. MEEN 4385 35
Orthotropic Material Materials such as wood, lainated plastics, cold rolled steels, reinorced concrete, various coposite aterials such as lainated coposites ade by the consolidation o pre-ipregnated sheets, with individual plies having dierent iber orientation, and even orgings can be treated as orthotropic. They possess 3 orthogonal planes o aterial syetry and three corresponding orthogonal axes called orthotropic axes. In soe aterials (orgings) these axes ay vary ro point to point. In other aterials (iber-reinorced plastics, reinorced concrete), orthotropic directions reain constant as long as the ibers and steel reinorcing bars aintain constant directions. In any case, or an elastic orthotropic aterial, independent constants ij reain unchanged at a point under a rotation o 18 about any o the orthotropic axes. Then, the original 36 constants ij reduce to and the generalized Hooke s law (constitutive equation) has the or o equation (4) MEEN 4385 36
MEEN 4385 37 Orthotropic Material (ont ) (4) 6 5 4 3 1 66 55 44 33 3 31 3 1 13 11 6 5 4 3 1
Orthotropic Material (ont ) Where: E x = E 1 E y = E G yz = G 3 G zx = G 31 G xy = G xx = 11, etc MEEN 4385 38
Resources - Orthotropic Materials heck out this web site http://www.eunda.co/orulae/solid _echanics/at_echanics/hooke_orth otropic.c MEEN 4385 39
Isotropic Materials Fro: www.eunda.co/orulae/solid_echanics/at_echanics/hooke_isotropic.c MEEN 4385 4
Overview o lassical Laination Theory The theory uses the orthotropic aterial properties E 1,E,G and ν to describe the various properties. For each layer: Q Q Q Q 11 66 / E 1 E 1 E G /(1 E These Q s or the reduced stiness can be cobined in atrix or to ind the stresses in an orthotropic layer: 1 /(1 / /(1 E 1 ) 1 ) 1 ) MEEN 4385 41
MEEN 4385 4 lassical Laination Theory (contd...) 1 1 66 11 1 sin cos cos sin cos sin cos sin cos sin cos sin sin cos : get In the transored atrix we xy y x Q Q Q Q Q
Transored Stiness E 1 turns into E MEEN 4385 43
lassical Laination Theory (contd...) It can be written as : x y xy where Q Q Q Q ij 11 16 are Q Q Q 6 Q Q Q 16 6 66 x y xy the transored stiness The lainate stiness can be assebled by suing the contributions ro all the layers. Each layer k is t k thick, and its id-plane is a distance z k ro the idplane o the total lainate. MEEN 4385 44
Qbars or transored Stinesses MEEN 4385 45
Stiness (psi) Saple Plotted Qbars Plot o IM7 gr/epoxy @ V=.65, Qbar vs 3,, 5,,,, Q11AR QAR QAR Q16AR Q6AR Q66AR 15,, 1,, 5,, 15 3 45 6 75 9 Angle (degrees) MEEN 4385 46
MEEN 4385 47 lassical Laination Theory (contd...) n k k k k ij ij n k k k k ij ij n k k k ij ij z z Q D z z Q t Q A 1 3 1 3 1 1 1 ] [ ) ( 3 1 ] [ ) ( 1 ) (
lassical Laination Theory (contd...) y assuing the Kircho reain planar when a treating u o x y xy u x v y u y,v w z x w z y v x and w as plateis a lainated plate, the linear strains at a point (x,y,z) are; w z xy hypothesisthat planes will under bending, and the id - plane displaceents Thesestrains can be put in the or : x x x y y z y xy xy xy MEEN 4385 48
lassical Laination Theory (contd...) Now orces N i and oents M i (per unit length) can be obtained or the entire lainate using the lainate sti nesses, id-plane lainates strains and id-plane lainate curvatures. N N N M M M x y xy x Y xy A A A 11 16 11 16 A A A 6 6 A A A 16 6 66 16 6 66 D D D 11 16 11 16 D D D 6 6 D D D 16 6 66 16 6 66 x y xy kx k y kxy MEEN 4385 49
MEEN 4385 5 lassical Laination Theory (contd...) Likewise, id-plane strains and curvatures can be obtained by: xy y x xy y x xy y x xy y x M M M N N N D D D D D D D D D A A A A A A A A A k k k 1 66 6 16 66 6 16 6 6 16 11 16 11 66 6 16 66 6 16 6 6 16 11 16 11
Reerences Hyer, M.W., Stress Analysis o Fiber-Reinorced oposite Materials, DEStech Pubs., 9 Tsai, S.W., Hahn, H.T., Introduction to oposite Materials, Technoic, 198 Jones, R.M., Mechanics O oposite Materials, R Press, nd ed, 1998 Askeland, D., Phule, P.P., The Science and Engineering o Materials, Thoson, 6 Peel, L.D., Jensen, D.W., Fabrication and Mechanics o Fiber-Reinorced Elastoers, Ph.D. dissertation, righa Young University, 1998. MEEN 4385 51
Hoework A onstitutive Properties & Unit onversions 1) In an Excel spreadsheet, ake a table and enter the Tensile, Shear strengths, Tensile and Shear Modulii, Poisson s ratios or Fibers: AS-4 arbon, E-glass, S-glass, Kevlar 49, and Matrices: Polyester, Generic Epoxy, and 351-6 Epoxy. onvert the values to SI & English units using the Excel Spreadsheet. Each property value should have a reerence nuber on it s row, stating where it cae ro. You can collect iber and atrix property data ro the lecture notes, Matweb.co, your textbook, or any other place, but they ust be published values, not generic or average ones. ) Docuent all your unit conversion calculations and collected constituent data in a report. The report ust include discussion, tables and charts o data, and reerences. Indicate where each data set cae ro. Must be proessional and coplete. The report ust stand alone, and not need the Excel Spreadsheet, hence it ust include tables o all data in the English and SI units. 3) Eail e the Excel ile & typed Suary and also hand in the typed suary, copare with others in the class but don t copy. Nae the eailed report MEEN4385_unit_conversion_nae.xls or *.doc. 4) Due Monday, Feb 19 th, 9 a MEEN 4385 5
Hoework 1) Using your Spreadsheet ro Hwk A, and assuing V = 55% or starters, copute laina orthotropic properties S 1, S, E 1, E, v,, and G or each iber cobined with polyester, and with epoxy atrices, using the previously collected isotropic properties or each aterial syste. alculate in English Units only. Make sure that you can enter the V in a separate place or each aterial syste, and that it can be changed as needed (not hard-coded in). ) Reuse as any equations as possible to save tie, ake one table or each aterial cobination, copy table, and change values or others. See what happens to the orthotropic properties as you vary the V. Discuss that in your report. 3) opare your epoxy atrix results only, with Mil-Handbook 17 published values in a typed 3+-page report that discusses unit conversion, V, equations used, and dierences between published & RM results, conclusions. Report ust include tables, charts o data, discussion, and reerences. Indicate where data cae ro. Must be proessional and coplete. The report ust stand alone, and not need the Excel Spreadsheet. Please note that you will need to adjust the V or each aterial cobination. 4) Eail e the Excel ile & Suary and also hand in the typed suary, copare with others in the class but don t copy. 5) Graduate students should ind the relationships or Qbars, and plot Q11, Q, Q, and Q66 vs ro to 9, or IM7/Epoxy at V=.6. 6) Due Monday, Feb 6, 18 MEEN 4385 53