3. Stress-strain relationships of a composite layer

Transcription

1 OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman.. tress-stran reatonshps of a composte ayer Introducton In the frst chapter t was shown that the propertes of the basc consttuents of a composte matera fbre and matrx- are extremey dfferent. hs mpes that a composte made of these two consttuents s heterogeneous propertes vary from pont to pont when consdered at a fbre matrx scae as shown n fg... When comng to a composte ayer t can be assumed that the ansotropc fg..: Fbre-matrx scae py scae. propertes are ndependent of the poston from whch they are ooed at. It maes t possbe to assume a composte as a pate havng averaged ansotropc matera propertes. he stress-stran reatons of a composte ayer n genera w be deveoped next. tress-stran reatonshp he stresses at a pont n a matera can be descrbed by 9 components as shown n fg.. where. Accordng to ths notaton are norma stresses when shear stresses when aso noted τ. It can be shown that ony of these stress components are ndependent.e.. mary there are stran components to descrbe the deformaton at ths pont. he fg..: tate of stress n a matera prncpe s smar as for the stress components athough the engneerng shear components γ are twce the shear deformaton tensor components. ensor notaton ontracted Notaton τ τ τ γ γ γ tabe.: onverson tensor notaton-contracted notaton. Assumng sma near eastc deformatons the stran components are reated to the deformatons by the foowng reatons:

2 OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman.. where u u and u are the dspacement aong the axs respectvey. he reaton between stress and stran components descrbes the behavour of the matera when oadng s apped. We w consder near eastc materas and n ths case each stran component s a near functon of the stress components : or n matrx form: { } [ ] { }.. where [] s the compance matrx. here are x compance constants. he nverse of the compance matrx s caed the stffness matrx [][] -. he stress-stran reatonshp ust descrbed s vad for any ansotropc eastc matera. It w be shown that the amount of compance or stffness constants can be reduced. Frsty t can be shown that the compance and stffness matrx s symmetrc. he demonstraton of ths symmetry s based on the fact that the energy necessary to nduce deformaton to the eastc and hence reversbe matera s ndependent from the way the energy s produced. hs eads to equatons of the form. hs means that the compance matrx s symmetrc and therefore the stffness matrx.e.:. hs eaves of the compance constants to descrbe an ansotropc matera. Further smpfcatons can be made by consderng geometrca symmetres n the composte ayer consdered. ae as an exampe a ayer of undrectona matera as shown n fg... Let us defne a artesan coordnate system where the frst axs concdes wth the drecton aong the fbres. he two other drectons are then perpendcuar to the fbres & wth the thrd axs perpendcuar to the ayer. hs coordnate system s caed the matera coordnate fg..: A undrectona ayer and ts matera coordnate system

3 ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman. OM PO I O U P U N I V I Y O F W N system. It can be seen that there are panes of matera property symmetry.e. the panes defned by the coordnate system axs & & &. A matera agreeng to these symmetry assumptons s caed orthotropc. It can be shown that ths eaves a compance matrx wth 9 ndependent components. ym. As an exampe appy a stress aong the frst axs a other stress components are zero. hs means that the reaton for the n-pane shear reduces to.7 It can be seen from fg.. that a stress apped aong the fbres cannot generate any n-pane shear deformaton. herefore the compance matrx component. uch smpe experments can be apped for the other zerocomponents. It s mportant to note that the concept of orthotropy s dependent on the coordnate system chosen. For exampe a norma stress apped n an arbtrary drecton w nduce shear strans. hs w be shown ater on when rotatng the compance or stffness matrx. A partcuar case of orthotropy s transverse sotropy. hs assumpton s taen for undrectona ayers where the propertes are assumed sotropc n the - pane.e. perpendcuar to the fbres. he compance matrx contans now ndependent constants and can be further reduced to the foowng set of reatons: ym.8 Isotropy reduces the amount of ndependent constant to whch gves the foowng set of equatons: ym.9

4 OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman. he reducton n the amount of ndependent constants n the compance and stffness matrx s aso due to the reaton between the engneerng constants and :. he reaton between the compance coeffcents and the engneerng constants w be deveoped n the next anea. ngneerng constants It s convenent to express the compance or stffness matrx components nto engneerng constants such as the Young s moduus. hese engneerng constants are defned by consderng smpe experments. A stress stuaton s defned where ony one stress component s dfferent from zero.e.: for.... If we consder the -stran component reated to ths -stress component t s possbe to defne a moduus as: / where the superscrpt n the stran component n a -stress. For : For :. If we now consder the. remnds that t fnds ts orgn s the moduus of eastcty aong the -axs and w be gven the notaton. s the shear moduus reatng respectvey to the - - and - panes and aong the -axs and w be gven the notaton -stran component a deformaton rato s defned as: ρ /. For and ρ are the Possons ratos wth a negatve sgn. In the case of an orthotropc ayer a other deformaton ratos are zero whch s not true for an ansotropc matera n genera. In ths case t s worth mentonng the shear-extenson coupng coeffcents η Lehnts and the shear-shear coupng coeffcents µ hentsov. he earer appes for and or and. he ater one concerns and. hese defntons mae t possbe to f the compance matrx. For an orthotropc ayer ths gves: / for... for.. he equatons are:

5 ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman. OM PO I O U P U N I V I Y O F W N. From the symmetry of the []-matrx we fnd: for... Pane stress assumpton A smpfcaton of the stress-stran reatons for an orthotropc ayer can be made n case of pane state of stress.e.. If ony the n-pane deformatons are consdered equaton. reduces to: {} []{} or.7 he other stran components can be non-zero but we w focus on the behavour of the n-pane strans. Invertng the compance matrx gves: {} [] {} or.8 In-pane rotaton Up to now the stress-stran reatonshps for a composte ayer wth the matera coordnate correspondng wth the pate coordnate system were consdered. he expressons for a rotated coordnate system w be derved here for a rotaton n the - pane axs s fxed. fg..: otaton from x x x to x x x x x x x x x

6 OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman. In genera the transformaton of the coordnates of a pont from the x x x system to the x x x system fg.. can be wrtten as:.e. x x t wth x x.9 he transformaton of the artesan components of any tensor t s then gven by: t In matrx form the stress transformaton gves:. m n mn n m mn m n n m mn mn m n wth m n. hs transformaton matrx w be caed [] n the foowng. eturnng to the composte ayer probem : ayer : matera at Ө fg.. shows the chosen notatons for the coordnate system: for the matera as n fg.. for the of the ayer. he propertes n the coordnate system are nown and need to be transformed nto the ayer. ompared to the stuaton n fg.. t therefore mpes to rotate wth an ange - whch means ug [] -. In case of pane stress the stress transformaton can be wrtten: or fg..: matera and ayer { } [] {}. he transformaton of the stran components s performed smary but the transformaton descrbed above based on tensors s vad for and therefore / see tabe..e:

7 ompostes ourse 8-9 Unversty of wente ng. &ech..7. tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman. OM PO I O U P U N I V I Y O F W N []. hs s usuay corrected by ntroducng the euter matrx []. In pane stress form t can be wrtten as: []. In matrx form the transformaton of the stran components n pane stress becomes: [] [][] [][] []..e.:. It s now possbe to formuate a transformed stffness matrx []: [] [] [] [] [][][][].7 hs gves for an orthotropc matera wth ts stffness matrx wrtten n ts matera coordnate system.e. :.8 mary t s possbe to formuate an expresson for the transformed compance matrx [ ]: [][] [] [][] [] [] [][] [] [][].9 [ ] [ ]

8 OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech..8. tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman. From ths transformaton an expresson for the engneerng constants of a ayer n any coordnate system can be wrtten. For exampe appy a stress aong the axs. he stran aong the same axs s then:. Whch means that /. xpandng.9 gves:. tress-stran reatonshp of a ayer ncudng hygrotherma effects he ansotropc behavour of the deformaton of a ayer s not ony evdent when a mechanca stress s apped but s aso subect to changes n envronmenta condtons. he most mportant are change n temperature and mosture content. hs can affect the composte both durng fabrcaton as durng ts end-use. It s therefore mportant to ncude them n the anayss. H he stran nduced by a change n temperature or n mosture content can be expressed as: H βc for for. where: s the coeffcent of therma expanson s the change n temperature - the temperature for whch. β s the coeffcent of hygroscopc expanson H c s the mosture concentraton wth a mosture free-state as a reference. hese hygrotherma stran terms are vad for temperature and mosture content ranges where the coeffcent of therma expanson and the coeffcent of hygroscopc expanson can reasonaby be assumed constant. For an orthotropc matera the stress-stran reatonshp n the matera coordnate system can be wrtten as: β β c. β or: { } [ ] { } { } { β} c. he transformaton of the hygrotherma stran from the matera coordnate system to an arbtrary coordnate system s performed as for the stran components.e. for the therma expanson components: [] [][] [][] [].

9 OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech..9. tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman. Probems. Wrte expressons for. Produce a graph of η and η. η and η as a functon of for a arbon-pi havng orthotropc propertes n ts matera coordnate system wth: Pa 8Pa.Pa... A fament wound cyndrca pressure vesse of mean dameter dm and wa thcness t.m s subected to an nterna pressure p. he vesse s wound at an ange of o from ts ongtudna axs and the gass-epoxy matera used has the foowng mechanca propertes: Pa Pa.Pa.. By the use of a stran gauge the stran aong the fbre drecton s measured and has at the pressure a vaue of.m/m. Determne the nterna pressure n the vesse.