Macromechanical Analysis of a Lamina

Transcription

1 3, P. Joyce

2 Macromechanical Analyi of a Lamina

3 Generalized Hooke Law ij Cijklε ij C ijkl i a 9 9 matri! 3, P. Joyce

4 Hooke Law Aume linear elatic behavior mall deformation ε Uniaial loading 3, P. Joyce

5 Triaial tre tate z Uniaial: hear are z z y omitted imilarly δ y εfor dthe δ deformation + δ y + δ in the y- and z-direction δ yz δ ε ε z y δ y δ yy ε δ d d z υ ε δ y y y υ δ y zy z δ υ υ εz z δ δ [ δ y z δ ( ] υ zz y + z δ z y z y z y υε ε d z y d z d υ dy d υ d ( Conider hear + z y eparately [ ] y yz yy υε ε dy dy y z uperimpoe ( three uniaial z zy + υε y dz y z δ υ zz ε z dz dz tree [ υ ] υ dy z y d dy dy dz 3, P. Joyce

6 Triaial tre tate train-tre Relation ε ε ε y z [ υ( + ] [ υ( + ] y [ υ( + ] z y z z y 3, P. Joyce

7 Triaial tre tate tre-train Relation ( + υ( υ y ( + υ( υ z ( + υ( υ [( υ ε + υ( ε + ε ] [( υ ε + υ( ε + ε ] y [( υ ε + υ( ε + ε ] z y z z y 3, P. Joyce

8 imilarly for hear tree 3, P. Joyce

9 hear tre train Relationhip hear tree are independent of each other and all aial tree hear train are independent of each other and all aial train ach obey a imple linear elatic model τ G γ G hear Modulu G ( + υ 3, P. Joyce

10 hear tre-train train Relationhip τ y Gγ y ( + υ γ y τ z Gγ z ( + υ γ z τ yz Gγ yz ( + υ γ yz 3, P. Joyce

11 3, P. Joyce Generalized Generalized Hooke Hooke Law Law train-tre relation for an iotropic material/matri form y z yz z y y z yz z y G G G τ τ τ υ υ υ υ υ υ γ γ γ ε ε ε Compliance matri

12 3, P. Joyce Generalized Generalized Hooke Hooke Law Law tre-train relation for an iotropic material /matri form y z yz z y y z yz z y G G G v v v v v v v v v γ γ γ ε ε ε ν υ ν υ ν υ ν υ ν υ ν υ ν υ ν υ ν υ τ τ τ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( tiffne matri

13 tre-train train Relation in Compoite Material Material 3, P. Joyce No. of independent elatic contant General aniotropic material 8 Aniotropic material (conidering ymmetry of tre and train tenor Aniotropic material with energy conideration General orthotropic material 9 Tranverely orthotropic material 5 Iotropic material 36

14 tre-train train Relation in Compoite Material Orthotropic material ~ ha three mutually perpendicular plan of material ymmetry pecially orthotropic material ~ when the reference ytem of coordinate i elected along principal plane of material ymmetry Tranverely iotropic material ~ one of it principal plane i a plane of iotropy (propertie are the ame in all direction. 3, P. Joyce

15 ε Contracted Notation Thank to ymmetry of the tre and train tenor, the compliance matri reduce to a 3 ε C C matri, introduce a contracted notation.,, 4 ε, ε 4 3 ε,ε C C 3, C, C,, 5 ε, ε ε,ε C ε, C,, C, C 3 6 ε C 3 C, C 3, C 3 3 C 3, P. Joyce 4 C, C 4 3, C 3 C 5 C, C 5, C C 6 C 6

16 tre-train train Relation for Thin UD Lamina Aumed to be under a tate of plane tre τ 6 Q Q Q Q Q 66 ε ε γ 6 Fully characterized by 4 independent contant, Qij ~ reduced tiffnee 3, P. Joyce

17 Reduced tiffne Matri? If the tiffne matri i the invere of the compliance matri, what i the reduced tiffne matri? Q ij Ci3C j3 Cij ( i, j C 33,, 6 3, P. Joyce

18 Relation Between Mathematical and ngineering Contant Q Q Q Q 66 υ υ G υ υ υ υ υ υ υ υ 3, P. Joyce

19 You aid Four Independent Contant? From ymmetry of the compliance matri υ υ ij i The above can alo be deduced from Betti reciprocal law according to which tranvere deformation due to a tre applied in the longitudinal direction i equal to the longitudinal deformation due to an equal tre applied in the tranvere direction. ji 3, P. Joyce j

20 3, P. Joyce tre tre-train Relation train Relation Alo epreed in term of compliance τ γ ε ε

21 Relation between Mathematical and ngineering Contant 66 υ G υ That Better! 3, P. Joyce

22 ample Calculation For a graphite/epoy UD laminate, find the following:. Compliance matri. Minor Poion ratio 3. Reduced tiffne Matri 4. train in the - coordinate ytem if the applied tree are τ MPa, -3MPa, τ 4 MPa 3, P. Joyce

23 ample Data Property ymbol Unit Fiber volume fraction Long. elatic modulu Tran. elatic modulu Major Poion ratio hear Modulu Gla/ epoy 3, P. Joyce Boron /epoy Graphite /epoy V f GPa GPa ν G GPa

24 ample Calculation The compliance matri element are calculated a follow: 66 ν G 8( 9.8 8(.3(.555( 9.547( 9 7.7(.979( 9.395( 9 (all term have unit of Pa - 3, P. Joyce

25 ample Calculation From Betti reciprocal law: ν ν ν ( 9.8 (.3( 9 8(.593 3, P. Joyce

26 ample Calculation The tiffne matri element are calculated a follow: Q Q Q Q 66 ν ν ν ν ν ν ν G 7.7( 9 8( 8.8( (.8(.593 ν ν ν 9.3(.35( (.8( (.8(.3( (.8( ( 9 (all term have unit of Pa 3, P. Joyce

27 ample Calculation The tiffne matri can alo be calculated by inverting the compliance matri of Part : [ Q].555(.547( 8.8( 9.547(.897(.979( 9 9 [ Q].897(.35( Pa 9 7.7( 9.395( (all term have unit of Pa 9 3, P. Joyce

28 3, P. Joyce ample Calculation ample Calculation ( ( 3( (.395(.979(.547(.547(.555( γ ε ε γ ε ε τ γ ε ε The train in the - coordinate ytem are calculated a follow: ( -6 i microtrain

29 tre-train train Relation for Thin Angle Lamina Generally, a laminate doe not conit only of UD laminae becaue of their tiffne and trength propertie in the tranvere direction. Hence, in mot laminate, ome laminae are placed at an angle. y Global and material ae of an angle lamina. 3, P. Joyce

30 tre-train train Relation for Thin Angle Lamina The ae in the -y coordinate ytem are called the global ae of the off-ae. The ae in the - coordinate ytem are called the material ae or the local ae, where direction i parallel to the fiber (alo called the longitudinal direction and direction i i perpendicular to the fiber (alo called the tranvere direction. The angle between the two ae i denoted by the angle θ. 3, P. Joyce

31 3, P. Joyce tre tre-train Relation train Relation for Thin Angle Lamina for Thin Angle Lamina The tre-train relationhip in the - coordinate ytem ha already been etablihed. From Mechanic of Material, the tree in the global and material ae are related to each other through the angle of the lamina, θ. [ ] ( τ θ τ T y y Where [T(θ] i called the tranformation matri and i defined a [ ] [ ] [ ] T(- ( thu ( θ θ θ c c c c c c c T c c c c c c c T

32 tre-train train Relation for Thin Angle Lamina Uing the tre-train equation in the material ae together with the tranformation equation we obtain: y τ y [ T ] [ Q] ε ε γ 3, P. Joyce

33 3, P. Joyce tre tre-train Relation train Relation for Thin Angle Lamina for Thin Angle Lamina imilarly, the train in the global and material coordinate ae are related through the tranformation matri [ ] y y T γ ε ε γ ε ε which can be rewritten a [ ][ ][ ] y y R T R γ ε ε γ ε ε where [R] i the Reuter matri and i defined a [ ] R

34 tre-train train Relation for Thin Angle Lamina Multiplying out the firt five matrice on the RH of the previou equation we obtain the tranformed reduced tiffne matri, [ Q y ] Thu, [ ], y [ Q ], y [ ε ], y ummarizing, Q Q Q Q Q Q yy y y c c c c c Q Q 3 Q Q Q Q + + c 4 4 c Q Q + c c 3 + c 3 Q Q + c + c Q Q + ( c 3 + ( c + ( c 3 Q Q c 4 c + 4c + 4c + 3 c 3 4 Q Q Q Q + ( c Q Q c + ( c + ( c 3 3 Q 66 c Q 3 c 66 Q 3 Q The ubcript correpond to hear tre or train component in the -y ytem, i.e., τ τ y and γ γ y 3, P. Joyce

35 train-tre tre Relation for Thin Angle Lamina imilarly, the tranformed compliance can be obtained: yy y y c c 4 4 c 3 c 4c 3 ε + + c c c c 3 + 4c + c + c 3 Thu, [] y [ ], y [ ], y, + ( c 4 + ( c + ( c 3 8c + 4c + 4c c 3 c c + ( c + ( c + ( c c 3 c , P. Joyce

36 3, P. Joyce train train-tre Relation for Thin tre Relation for Thin Angle Lamina Angle Lamina How about in term of ngineering Contant? If we imagine a erie of imple eperiment on an element with ide parallel to the -and y-ae, we obtain: y y y y y y y y y y y y G G G τ η η η ν η ν γ ε ε

37 train-tre tre Relation for Thin Angle Lamina What i η? hear coupling coefficient η, the firt ubcript denote normal loading in the -direction; the econd ubcript denote hear train. η η η η y y γ γ ε ε y ε ε γ γ y 3, P. Joyce

38 train-tre tre Relation for Thin Angle Lamina Comparion of equivalent train-tre relation yield the following relationhip: yy y y y G y y y ν η η y y y ν η G y ηy G y y y G ν η η 3, P. Joyce y y y y yy y yy y ; ; ; ν η η y y y y yy

39 ample Calculation Find the following for a 6 angle lamina of graphite/epoy. Tranformed compliance matri Tranformed reduced tiffne matri Global train Local train If the applied tree are MPa, y -3MPa, τ y 4 MPa 3, P. Joyce

40 ample Calculation From the previou eample: (.547(.979(.395( 9 (all term have unit of Pa - 3, P. Joyce

41 3, P. Joyce ample Calculation ample Calculation The tranformed compliance matri element are calculated a follow:.4( ( ( ( (.3475( ( (.334( 4 (.7878( 4.853( c c c c c c c c c c c c c c c c c c c c c c c c c c y y yy (All term are in Pa -

42 ample Calculation Net, invert the tranformed compliance matri [] to obtain the tranformed reduced tiffne matri [Q]. [ Q] [ ].853(.7878(.7878(.3475(.334(.4696( [ Q] (.334(.4696(.4( 9 (all term in Pa 3, P. Joyce

43 ample Calculation The global train in the -y plane are given by [] ε, y [], y [ ], y ε ε y γ y ε ε y γ y.853(.7878(.334(.5534(.378(.538( (.3475(.4696(.334(.4696(.4( 9 3 ( 4 6 3, P. Joyce

44 ample Calculation The local train in the lamina can be calculated uing the Tranformation equation. ε ε γ ε ε γ [ T ] ε ε y γ y co 6 in 6 co6in 6 4 ε.367( 3 ε.66( 3 γ.589( in co 6 6 co6in 6 4 co6in ( 3 co6in 6.378( 3 co 6 in 6.538( / 3, P. Joyce

45 Tranformation of ngineering Contant Flow chart for determination of tranformed elatic contant of UD lamina. [], [],y [], θ [],y [Q], [Q],y 3, P. Joyce

46 Macromechanical trength Parameter From a macromechanical POV, the trength of a lamina i an aniotropic property. It i deirable, for eample, to correlate the trength along an arbitrary direction to ome baic trength parameter (analogou to micromechanic definition before. 3, P. Joyce

47 trength Failure Theorie of an Angle Lamina Variou theorie have been developed for tudying the failure of an angle lamina. Generally baed on the normal and hear trength of a UD lamina. Need to conider tenion and compreion UD lamina ha material ae, -direction parallel to the fiber and -direction which i perpendicular to the fiber. Hence there are 4 normal trength parameter for UD lamina. Tenile trength in fiber direction Tranvere tenile trength Compreive trength in fiber direction Tranvere compreive trength The fifth trength parameter i the hear trength 3, P. Joyce

48 trength Failure Theorie of an Angle Lamina Unlike the tiffne parameter, thee trength parameter cannot be tranformed directly for an angle lamina. Hence, the failure theorie are baed on firt finding the tree in the material ae and then uing thee five trength parameter of a UD lamina to find whether the lamina ha failed. 3, P. Joyce

49 Macromechanical trength Parameter Alo predict tranvere compreive trength and in-plane hear trength uing micromechanic... Failure mechanim vary greatly with material propertie and type of loading. ven when prediction are accurate with regard to failure initiation at critical point, they are only approimate a far a global failure of the lamina i concerned. Furthermore, the poible interaction of failure mechanim make it difficult to obtain reliable trength prediction under a general type of loading. A macromechanical or phenomological approach to failure analyi may be preferable. 3, P. Joyce

50 Macromechanical trength Parameter Thi characterization recognize the fact that mot compoite material have different trength in tenion and compreion. By convention the ign of the hear tre i immaterial, a long a the hear trength i referred to the principal material direction. ception, refer to the cae when the hear tre i applied at an angle wrt the principal material direction. ince mot compoite have different tenile and compreive trength and they are weaket in tranvere tenion, it follow that in thi cae the lamina would be tronger under poitive hear. 3, P. Joyce

51 Macromechanical trength Parameter y τ 6 τ 6 Poitive hear tre -τ 6 hear tre acting along principal material ae. 3, P. Joyce

52 Macromechanical trength Parameter y τ 6 τ 6 Negative hear tre -τ 6 hear tre acting along principal material ae. 3, P. Joyce

53 Macromechanical trength Parameter τ 6 τ Poitive hear tre -τ hear tre acting at 45 wrt principal material ae. 3, P. Joyce

54 Macromechanical trength Parameter -τ τ Negative hear tre τ hear tre acting at 45 wrt principal material ae. 3, P. Joyce

55 Macromechanical Failure Theorie Given a tate of tre, the principal tree and their direction are obtained by tre tranformation (independent of material propertie. The principal train and their direction are obtained by uing the appropriate aniotropic tre-train relation and train tranformation. In general, the principal tre, principal train, and material ymmetry direction do not coincide. ince trength varie with orientation, maimum tre alone i not the critical factor in failure. 3, P. Joyce

56 Macromechanical Failure Theorie An aniotropic failure theory i needed. Failure criteria for homogeneou iotropic material, uch a Maimum normal tre (Rankine, Maimum hear tre (Treca, Maimum ditortional energy (von Mie, and o forth are well etablihed. More than 4 aniotropic theorie have been propoed look at the four mot widely ued. 3, P. Joyce

57 Maimum tre Failure Theory Related to the Maimum Normal tre theory by Rankine and the Maimum hear tre theory by Treca. The tree acting on a lamina are reolved into the normal and hear tree in the material ae. Failure i predicted in a lamina, if any of the normal or hear tree in the material ae are equal to or greater than the correponding ultimate trength of a UD lamina. C T C T ( < ( ( ( ult ( ult ( ult <, ult < ult <, τ < τ < τ ult ach component of tre i compared with the correponding trength and hence doe not have an interaction with the other. 3, P. Joyce

58 ample Calculation Find the off-ai hear trength of a 6 graphite/epoy lamina uing the Maimum tre failure criteria. Aume the following tre tate,, τ τ, y y Find the tree along the principal material ae, uing the Tranformation quation τ τ.866τ τ.5τ , P. Joyce τ

59 ample Calculation Applying the Maimum tre Failure Criteria together with trength data for graphite/epoy compoite from the Data heet,we have 5 46 <.866τ < 4 68 < 46.9 <.866τ < 5.5τ < or < τ < 73 < τ < < τ < 36. 3, P. Joyce

60 ample Calculation The off-ai hear trength of a lamina i defined a the minimum of the poitive and negative hear tre which can be applied to an angle lamina before failure. Calculation how that τ y 46.9 MPa i the larget magnitude of hear tre one can apply to the 6 graphite/epoy compoite. However, the larget poitive hear tre one could apply i 36. MPa, and the larget negative hear tre one could apply i 46.9 MPa. Thi how that the maimum magnitude of allowable hear tre in other than the material ae direction depend on the ign of the hear tre. Thi i becaue the tenile trength perpendicular to the fiber direction i much lower than the compreive trength perpendicular to the fiber direction. 3, P. Joyce

61 Failure nvelope A failure envelope i a 3D plot of the combination of normal and hear tree which can be applied to an angle lamina before failure. Drawing 3D graph i time conuming... One may develop failure envelope for contant hear tre, τ y, and then ue the normal tree and y a the ae. If the applied tre i within the failure envelope, the lamina i afe; otherwie it ha failed. 3, P. Joyce

62 Failure nvelope For a UD lamina at a given hear tre loading, the failure envelope take the form of a rectangle a hown. For a 6 lamina at a given hear tre loading, the failure envelope take the form of a rectangle a hown. τ y T ( ult T ( ult C T ( ult ( ult C T ( ult ( ult C ( ult C ( ult 3, P. Joyce

63 Maimum train Failure Theory Baed on the Maimum Normal train Theory by t. Venant and the Maimum hear tre Theory by Treca. The train applied to a lamina are reolved into the normal and hear tree in the material ae. Failure i predicted in a lamina, if any of the normal or hear train in the material ae are equal to or greater than the correponding ultimate train of a UD lamina. C T C T ( ε < ε ( ( ( ult ( ult ( ult < ε, ε ult < ε ult < ε, γ < γ < γ ult The ultimate train can be found directly from the ultimate trength parameter and the elatic moduli, auming the tre-train repone i linear until failure. ach component of train i compared with the correponding ultimate train and hence doe not have an interaction with the other. Yield different reult from Maimum tre Failure Theory, becaue the local train in a lamina include the Poion ratio effect (allow ome interaction of tre component. 3, P. Joyce

64 Maimum train Failure Theory Aume a general biaial tate of tre on an angle lamina. Obtain the tre component along the principal material ae by tre tranformation. y τ y [ T ( θ ] τ 3, P. Joyce

65 Maimum train Failure Theory Then the correponding train component can be calculated by mean of the lamina tre-train relation: ε Net calculate the ultimate train a follow γ 6 ε τ 6 G ν ν T ( T ( C ( ε C ( T ( ε T ( C ( ε C ( ( γ ult ult ult ult ε,,,, ult ult ult ult ult ( τ G ult 3, P. Joyce

66 Maimum train Failure Theory Failure ubcriteria retated in term of the tree: ν ν T ( C ( T ( C ( ult ult ult ult ( τ ult < τ < ( τ ult when ε > when ε < when ε when ε > < 3, P. Joyce

67 Maimum train Failure Theory For a D tate of tre with τ 6, the failure envelope take the form of a parallelogram with it center off the origin. y T ( ult T ( ν C T ( ult ( ult C ( ν C ( ult C ( ν T ( ν 3, P. Joyce

68 Tai-Hill Failure Theory Baed on the deviatoric or ditortional energy failure theory of von Mie. Adapted to aniotropic material by Hill. Then adapted to a UD lamina by Tai. T T ( T ( ( ( τ + + < ult ult Given the global tree in a lamina, one can find the local tree in a lamina and apply the above failure theory to determine whether or not the lamina ha failed. ult τ ult 3, P. Joyce

69 Tai-Hill Failure Theory The failure envelope decribed by the Tai-Hill criterion i a cloed urface in the,, τ pace. Failure envelope for contant value of have the form Where: F F + k F F F T ( C ( T ( C ( ult ult ult ult when > when < when > when < k τ ( τ ult Graphically repreent 4 different elliptical arc joined at the, ae. (Modified Tai-Hill Criterion 3, P. Joyce

70 Tai-Hill Failure Theory Conider the interaction between the 3 UD lamina trength parameter, unlike the Maimum tre and Maimum train Theorie. Tai-Hill Failure Theory i a Unified Theory and hence doe not give the mode of failure like the Maimum tre and Maimum train Theorie. 3, P. Joyce

71 Tai-Wu Failure Theory Baed on a general failure theory for aniotropic material firt propoed by Gol denblat and Kopnov (965. Capable of predicting trength under general tate of tre for which no eperimental data are available. Ue the concept of trength tenor. Ha the form of an invariant formed from tre and train tenor component Ha the capability to account for the difference between tenile and compreive trength 3, P. Joyce

72 Tai-Wu Failure Theory Tai and Wu (97 propoed a modified tenor polynomial theory by auming the eitence of a failure urface in the tre pace of the form H + H + H 6τ + H + H + H 66τ + H < The coefficient are obtained by applying elementary loading condition to the lamina. Thu H H T C ( ( ult T C ( ( ult ult ult H H T C ( ( ult T C ( ( ult ult ult H H 6 66 ( τ ult 3, P. Joyce

73 Tai-Wu Failure Theory The remaining coefficient H mut be obtained by ome type of biaial teting. Direct biaial teting i not eay or practical to perform. An eaier tet producing a biaial tate of tre i the off-ai tenile tet. For θ 45 we can meaure the off-ai tenile trength,. ( H + H Then, H ( H + H + H Again produce an elliptical failure envelope. 66 3, P. Joyce

74 Comparion with perimental Reult 3, P. Joyce

75 Comparion with perimental Reult 3, P. Joyce

76 Comparion with perimental Reult 3, P. Joyce

77 Comparion with perimental Reult 3, P. Joyce

78 Comparion with perimental Reult Obervation The difference between the Maimum tre and Maimum train Failure Theorie and the eperimental reult i quite pronounced. Tai-Hill and Tai-Wu Failure Theorie are in good agreement with eperimentally obtained reult. The cup oberved in the Maimum tre and Maimum train Failure Theorie correpond to the change in failure mode. The variation of the trength a a function of angle i mooth in the Tai-Hill and Tai-Wu Failure Theorie. 3, P. Joyce

79 Macromechanical Failure Theorie Theory Phyical Bai Operational Convenience Maimum tre Maimum train Deviatoric train energy (Tai-Hill Interactive tenor polynomial Tenile behavior of brittle material No tre interaction Tenile behavior of brittle material ome tre interaction Ductile behavior of aniotropic material Curve fitting for heterogeneou brittle compoite Mathematically conitent Inconvenient Inconvenient Can be programmed Different function required for tenile and compreive trength General and comprehenive; operationally imple Req d eperimental characterization Few parameter By imple teting Few parameter by imple teting Biaial teting i needed in addition to uniaial teting Numerou parameter Tai-Wu Reliable curve fitting Comprehenive eperimental program 3, P. Joyce needed.

80 Reference ngineering Mechanic of Compoite Material, Daniel, I.M. and Ihai, O., 994. Mechanic of Compoite Material, Kaw, A.K., 997. Introduction to Compoite Material, Tai,. W. and Hahn, H. T., 98. Application of Advanced Compoite in Mechanical ngineering Deign, Zweben, C., Proceeding of the 3 t International AMP Technical Conference, , P. Joyce