Solid Mechanics Z. Suo

Transcription

1 Sold Mechancs Z Suo Fnte Deformaton: General Theory The notes on fnte deformaton have been dvded nto two parts: specal cases ( and general theory In class I start wth specal cases and then sketch the general theory But the two parts can be read n any order Subect to loads a body deforms We would lke to develop a theory to evolve ths deformaton n tme In contnuum mechancs we model the body by a feld of partcles and update the postons of the partcles by usng an equaton of moton We formulate the equaton of moton by mxng the followng ngredents: knematcs of deformaton conservaton of mass momentum and energy producton of entropy models of materals Each of these ngredents has alternatve but equvalent mathematcal representatons To focus on essental deas we wll frst adopt one set of representatons usng nomnal quanttes At the end of the notes we wll sketch some of the alternatves We wll use the equaton of moton to study several phenomena ncludng wave vbraton and bfurcaton Model a body by a feld of partcles A body s made of atoms each atom s made of electrons protons and neutrons and each proton or neutron s made of Ths knd of descrpton s too detaled We wll not go very far n helpng the engneer by thnkng of a brdge as a ple of atoms Instead we wll develop a contnuum theory Ths theory models the body by a feld of materal partcles or partcles for brevty Each partcle conssts of many atoms As tme progress the clouds of electrons deform and the protons ggle all at a maddenngly hgh frequency The materal partcle however represents the collectve behavor of many atoms over a large sze and a long tme A common method to devse a materal model s to regard each materal partcle as a small specmen undergong homogenous deformaton The body deforms n a three-dmensonal space The space conssts of places labeled by a system of coordnates At a gven tme each materal partcle n the body occupes a place n the space As tme progresses the materal partcle moves from one place to another The moton of the materal partcle s affected by the external loads as well as the by the neghborng partcles When a body deforms does each materal partcle preserve ts dentty? We have tactly assumed that when a body deforms each materal partcle preserves ts dentty Whether ths assumpton s vald can be determned by experments For example we can pant a grd on the body After deformaton f the grd s dstorted but remans ntact then we say that the deformaton preserves the dentty of each partcle If however after the deformaton the grd dsntegrates we should not assume that the deformaton preserves the dentty of each partcle March Fnte Deformaton: General Theory 1

2 Sold Mechancs Z Suo Whether a deformed body preserves the dentty of the partcle s subectve dependng on the sze and tme scale over whch we look at the body A rubber for example conssts of cross-lnked long molecules If our grd s over a sze much larger than the ndvdual molecular chans then deformaton wll not dsntegrate the grd By contrast a lqud conssts of molecules that can change neghbors A grd panted on a body of lqud no matter how coarse the grd s wll dsntegrate over a long enough tme Smlar remarks may be made for metals undergong plastc deformaton Also n many stuatons the body wll grow over tme Examples nclude growth of cells n a tssue and growth of thn flms when atoms dffuse nto the flms The combned growth and deformaton clearly does not preserve the dentty of each materal partcle In these notes we wll assume that the dentty of each materal partcle s preserved as the body deforms ame a materal partcle by the coordnate of the place occuped by the materal partcle when the body s n a reference state Each materal partcle can be named any way we want For example we often name a materal partcle by usng an Englsh letter a Chnese character or a color More systematcally we name each materal partcle by the coordnate of the place occuped by the materal partcle when the body s n a partcular state called the reference state Often we choose the reference state to be the state when the body s unstressed However even wthout external loadng a body may be under a feld of resdual stress Thus we may not be able to always set the reference state as the unstressed state Rather any state of the body may be used as a reference state Indeed the reference state need not be an actual state of the body and can be a hypothetcal state of the body For example we can use a flat plate as a reference state for a shell even f the shell s always curved and s never flat To enable us to use dfferental calculus all that matters s that materal partcles can be mapped from the reference state to any actual state by a 1-to-1 smooth functon We wll use the phrase the partcle as shorthand for the materal partcle that occupes the place wth coordnate when the body s n the reference state Feld of deformaton ow we are gven a body n a three-dmensonal space We have set up a system of coordnates n the space and have chosen a reference state of the body to name materal partcles When the body s n the reference state a materal partcle occupes a place whose coordnate s At tme t the body deforms to a current state and the materal partcle moves to a place whose coordnate s x The tme-dependent feld x x( descrbes the hstory of deformaton of the body A central am of contnuum mechancs s to evolve the feld of deformaton x ( by developng an equaton of moton March Fnte Deformaton: General Theory

3 Sold Mechancs Z Suo x( reference state current state Exercse Gve a pctoral nterpretaton of the followng feld of deformaton: x1 1 + tanγ ( x x 3 Compare the above feld wth another feld of deformaton: x + snγ t x x cosγ () t Dsplacement velocty and acceleraton At tme t the partcle x t At a slghtly later tme t + δt the partcle occupes occupes the place the place ( t + δ x Durng the short tme between t and t + δt the partcle moves by a small dsplacement: δ x x( t + δt ) x( The velocty of the partcle at tme t s x( t + δ x( ( δt The acceleraton of the partcle at tme t s x( The felds of velocty and acceleraton are lnear n the feld of deformaton x t We have ust nterpreted the partal dervatve of the functon x ( wth respect to t We next nterpret the partal dervatve of the functon x ( wth respect to Deformaton gradent For a bar undergong homogenous deformaton we have defned the stretch by March Fnte Deformaton: General Theory 3

4 Sold Mechancs Z Suo length n current state stretch length n reference state We now extend ths defnton to a body undergong nhomogeneous deformaton n three dmensons Consder two nearby materal partcles n the body When the body s n the reference state frst partcle occupes the place wth the coordnate and the second partcle occupes the place wth the coordnate + d When the body s n the current state at tme t the frst partcle occupes the place wth the coordnate x ( and the second partcle occupes the place wth the coordnate x ( + d The stretch s generalzed to x ( + d x ( d Ths obect represents 9 ratos and s gven a symbol ( F and s called the deformaton gradent That s we have generalzed the stretch to a tme-dependent feld of tensor F ( The deformaton gradent ( lnear n the feld of deformaton x ( F s +d x( x(+d reference state current state As a body deforms each materal element of lne rotates and stretches ow focus on the vector connectng the two nearby materal partcles and + d We call the vector a materal element of lne When the body s n the reference state the materal element of lne s represented by the vector d When the body s n the current state at tme t ths materal element of lne s represented by the vector d x x( + d x( Thus March Fnte Deformaton: General Theory 4

5 Sold Mechancs Z Suo dx ( d d We have defned the partal dervatve as the deformaton gradent so that dx F ( d or d x F t d Thus the deformaton gradent ( F s a lnear operator that maps the vector between two nearby materal partcles from the reference state d to the vector between the same two materal partcles n the current state d x x( d dx reference state current state When the body deforms the materal element of lne both stretches and rotates In the reference state let dl be the length of the element and M be the unt vector n the drecton of the element namely d MdL In the current state let dl be the length of the element and m be the unt vector n the drecton of the element namely dx mdl Thus the deformaton of the body stretches the length of the materal element of lne from dl to dl and rotates the drecton of the materal element from M to m Recall that the stretch of the element s defned by dl λ dl Combnng the above four equatons we obtan that λ m F( M In the current state at tme t once the feld of deformaton x ( s known the deformaton gradent F ( can be calculated Once the drecton M of a materal element of lne s gven when the body s n the reference state the above m t M and expresson gves n the current state the drecton of the element March Fnte Deformaton: General Theory 5

6 Sold Mechancs Z Suo the stretch of the element ( tm) λ Green deformaton tensor We can also obtan an explct expresson for the stretch λ ( tm) Takng the nner product of the vector λ m F( M we obtan that λ M T CM where T C F F The tme-dependent feld C ( known as the Green deformaton tensor s symmetrc and postve-defnte x t s In the current state at tme t once the feld of deformaton known the Green deformaton tensor C ( can be calculated and so can the stretch of any materal element of lne λ ( tm) Exercse Gven a feld of deformaton: x1 1 + snγ ( t ) x cosγ () t x 3 Calculate the tensors F ( and C ( a materal element of lne s n the drecton [ 010] T 3 When the body s n the reference state M Calculate the stretch and the drecton of the element when the body s n the current state at tme t Exercse Any symmetrc and postve-defnte matrx has three orthogonal egenvectors along wth three real and postve egenvalues Interpret the geometrc sgnfcance of the egenvectors and the egenvalues of C ( Exercse For the above feld of deformaton determne the egenvectors C t Interpret your result and egenvalues of the tensor Conservaton of mass In our prevous work we have tactly assumed that mass s conserved as a body deforms We now make ths assumpton explct When a body s n the reference state a materal partcle occupes a place wth coordnate and a materal element of volume around the partcle s denoted by dv When the body s n a current state at t the element deforms to some other volume Let ρ be the nomnal densty of mass namely mass n current state ρ volume n reference state Durng deformaton we assume that a materal partcle does not gan or lose mass so that the nomnal densty of mass ρ s tme-ndependent If the body n the reference state s nhomogeneous the nomnal densty of mass n general vares from one materal partcle to another Combnng these two consderatons we wrte the nomnal densty of mass as a functon of materal partcle: ρ ρ March Fnte Deformaton: General Theory 6

7 Sold Mechancs Z Suo Ths functon s often gven as an nput to our theory Consder agan a materal element of volume When the body s n the reference state the volume of the materal element s dv When the body s n the current state at tme t the materal element may deform to some other volume The mass of the materal element of volume at all tme s ρ dv Consequently at all tme the mass of any part of the body s ρ dv The ntegral extends over the volume of the part n the reference state Thus the doman of ntegraton remans fxed ndependent of tme even though the body deforms dv be a materal element of volume In the current state the materal element of volume has the ρ dv and the lnear momentum Conservaton of lnear momentum Let mass ( ) x t ρ dv The lnear momentum n any part of the body s ( ρ dv The ntegral extends over the volume of the part n the reference state The rate of change of the lnear momentum of the part s d ( x ( ρ dv ρ dv dt The ntegrals extend over the volume of the part n the reference state Conservaton of lnear momentum requres that the rate of change of the lnear momentum n any part of a body should equal the force actng on the part We next express ths prncple n useful terms Body force Consder a materal element of volume around materal partcle When the body n the reference state the volume of the element s dv element s denoted by B( dv namely force n current state B ( volume n reference state The force B( dv s called the body force and the vector ( When the body s n the current state at tme t the force actng on the B the nomnal densty of the body force The body force s appled by an agent external to the body and s often taken as an nput to our theory Tracton Consder a materal element of area at materal partcle When the body s n the reference state the area of the element s da and the unt vector normal to the element s When the body s n the current state at tme t the force actng on the element of area s denoted as T( da March Fnte Deformaton: General Theory 7

8 Sold Mechancs Z Suo namely force n current state T ( area n reference state T the nomnal tracton For a materal element of area on the external surface the tracton may be prescrbed by an agent external to the body and s taken as an nput to our theory For a materal element of area nsde the body the tracton represents the force exerted by one materal partcle on ts neghborng materal partcle In ths case the tracton s nternal to the body and need be determned by the theory Conservaton of lnear momentum stpulates that for any part of a body and at any tme the force actng on the part equals the rate of change n the lnear momentum namely The force T( da s called the surface force and the vector ( ( d T( da + B( dv ρ dv dt The ntegrals extend over the surface and the volume of any part of the body Inertal force From the above equaton t s evdent that we can also regard ( x ρ as a specal type of the body force called the nertal force Usng ths nterpretaton conservaton of lnear momentum s vewed as a balance of forces on any part of the body ncludng the surface force body force and nertal force namely ( x ( ) + T t da B( ρ 0 dv We next derve from the conservaton of lnear momentum two consequences Stress For a bar pulled by a force the nomnal stress s defned as the force n the current state dvded by the cross-sectonal area of the bar n the reference state We now generalze ths defnton to three dmensons Consder a materal tetrahedron around a materal partcle When the body s n the reference state the materal tetrahedron has three faces on the coordnate planes and the fourth face s nclned to the axes When the body s n the current state at tme t the faces deform Consder one face of the tetrahedron When the body s n the reference state the face s a flat trangle normal to the axs When the body s n the current state at tme t the face deforms nto a curved trangle Denote by s ( the component of the force actng on ths face n the current state dvded by the area of the face n the reference state We adopt the followng sgn conventon When the outward normal vector of the area ponts n the postve drecton of axs we take s to be postve f the component of the force ponts n the postve drecton of axs x When the outward normal vector of the area ponts n the negatve drecton of the axs March Fnte Deformaton: General Theory 8

9 Sold Mechancs Z Suo we take s to be postve f the component of the force ponts n the negatve drecton of axs x Stress-tracton relaton Consder the materal tetrahedron agan When the body s n the reference state the four faces of the tetrahedron are trangles: the three trangles on the coordnate planes and one trangle on the plane normal to the unt vector Let the areas of the three trangles on the coordnate planes be A and the area of the trangle normal to be A The geometry dctates that A A When the body s n the current state the tetrahedron deforms to a shape of four curved faces Regard ths deformed tetrahedron n the current state as a free-body dagram ow apply conservaton of momentum to ths deformed tetrahedron Actng on each of the four faces s a surface force As the volume of the tetrahedron decreases the rato of area over volume becomes large so that the surface forces preval over the body force and the change n the lnear momentum Consequently the surface forces on the four faces of the tetrahedron must balance gvng s A T A A combnaton of the above two equatons gve s T Consequently the nomnal stress s a lnear operator that maps the unt vector normal to a materal plane n the reference state to the tracton actng on the materal plane n the current state 3 T 3 A A A reference state 1 s A 3 current state Conservaton of lnear momentum n dfferental form Consder any part of the body The force due to tracton s s T da s da dv The frst equalty comes from the stress-tracton relaton and the second equalty nvokes the dvergence theorem Consequently conservaton of lnear momentum requres that March Fnte Deformaton: General Theory 9

10 Sold Mechancs Z Suo ( ( s x dv + B ( dv ρ dv Ths equaton holds for arbtrary part of the body so that the ntegrands must equal: s ( x ( + B t ρ Ths equaton expresses conservaton of lnear momentum n a dfferental form The equaton s lnear n the feld of stress and the feld of deformaton s 31 Cd 3 s 11 Bd s 1 Ad 1 reference state current state An alternatve dervaton of the above equaton goes as follows Consder a block of materal ABC When the body s n the reference state the block s rectangular When the body s n the current state the block deforms to some other shape Conservng lnear momentum of ths free-body dagram we obtan that BCs + A t BCs t + CAs 1 + ABs 3 + ABCB ( 1 3 ) 1( 1 3 ) ( 1 + B 3 CAs( 1 3 ( C ABs3( 1 3 ( x ( ABCρ Dvde ths equaton be the volume ABC and we obtan that s ( x ( + B t ρ Conservaton of angular momentum Ths prncple stpulates that for any part of a body and at any tme the moment actng on the part equals the rate of change n the angular momentum namely March Fnte Deformaton: General Theory 10

11 Sold Mechancs Z Suo ( d x( T( da + x( B( dv x( ρ dv dt The frst term can be converted nto a volume ntegral as follows: ( x s p ) ε px TpdA ε px sp da ε p dv s p ε psp + ε px dv where ε k s the symbol for permutaton Combnng the above two equatons and usng the equaton for conservaton of lnear momentum we obtan that ε psp dv 0 Ths equaton holds for arbtrary part of the body so that the ntegrand must vansh: ε psp 0 Ths equaton s equvalent to T T sf Fs T That s conservaton of angular momentum requres that the product sf be a symmetrc tensor In general nether the deformaton gradent F nor the nomnal stress s s a symmetrc tensor An alternatve dervaton of the above equaton goes as follows Consder a small block n a body In the reference state the block s rectangular of lengths A B C In the current state the block deforms to some other shape Consder the free-body dagram of the block n the current state: a par of forces s 1 BC act on the par of the materal elements of area BC a par of forces s CA act on the par of the materal elements of area CA and a par of forces s 3 AB act on the par of the materal elements of area AB In the current state n the coordnate plane ( ) we balance the moments of the forces actng on the block: BCs x + A t x t + CAs + ABs BCs + CAs + ABs 1[ ( 1 3 ) ( 1 3 )] [ x ( 1 + B 3 x ( 1 3 ] 3[ x ( C x ( 1 3 ] 1[ x ( 1 + A 3 x ( 1 3 ] [ x ( 1 + B 3 x ( 1 3 ] [ x ( + C x ( ] 3 Dvde the equaton by the volume ABC and we obtan that March Fnte Deformaton: General Theory 11

12 Sold Mechancs Z Suo s F s F Conservaton of energy We now vew any materal part of the body as a system In the current state the part s subect to the body force surface force and nertal force The forces do work to the part when the body deforms The part s also n thermal contact wth reservors of energy and wth neghborng parts of the body so that the part receves heat from the rest of the world Conservaton of energy requres that the change n the energy of the part should equal the sum of the work done to the part and heat receved by the part We next express ths prncple n useful terms Work Between a tme t and a slghtly later tme t + δt a materal partcle moves by a small dsplacement: x x( t + δ x( δ Assocated wth ths feld of small dsplacement the surface force the body force and the nertal force together do work to a part of the body: x work T + δxda B ρ δxdv The ntegrals extend over the surface and the volume of the part of the body We can express the work n terms of the stress Assocated wth the small dsplacement δ x the deformaton gradent of a materal partcle changes by ( t + δt ) F ( δ F F δx Recall that conservaton of lnear momentum results n two equatons: T s x s + B ρ Insert the two equatons nto the expresson for work and apply the dvergence theorem ( sδx ) s δ xda dv We reduce the work done by all the forces to the followng expresson: work sδfdv Ths expresson holds for any part of the body Consequently assocated wth a feld of small dsplacement δ x all the forces together do ths amount work per unt volume: work n the current state s δf volume n the reference state That s the nomnal stress s work-conugate to the deformaton gradent The above result generalzes the one dmensonal result for a bar pulled by a force: March Fnte Deformaton: General Theory 1

13 Sold Mechancs Z Suo work n the current state volume n the reference state sδλ To apprecate the general result t mght be nstructve for you to revew how ths one-dmensonal result arses from elementary consderatons ( Heat The body s n thermal contact wth a feld of reservors of energy Consder a materal element of volume around the partcle When the body s n the reference state the volume of the element s dv When the body s n the current state at tme t the element deforms to some other volume Between a reference tme and the current tme t the materal element receves from the Q t dv namely reservors an amount of energy ( energy receved up to current state Q volume n reference state We adopt the sgn conventon that Q > 0 when the materal element of volume receves energy from the reservors Consder a materal element of area at the partcle When the body s n the reference state the area of the element s da and the unt vector normal to the element s When the body s n the current state at tme t the element deforms to some other shape Between a reference tme and the current tme the energy across the materal element of area s denoted by q ( da namely energy across up to current state q( area n reference state We adopt the sgn conventon that q > 0 when energy transfers n the drecton of Consder a materal tetrahedron around a materal partcle When the body s n the reference state the tetrahedron has three faces on the coordnate planes and the fourth face s nclned to the coordnate axes When the body s n a current state at tme t the faces deform Consder one face of the tetrahedron When the body s n the reference state the fact s normal to the axs Between a reference tme and the current tme the energy per area across the face s denoted by I ( We adopt the sgn conventon that I > 0 f the energy goes out the tetrahedron Conservaton of energy In the current state at tme t denote the nomnal u t namely densty of nternal energy by ( nternal energy n current state u volume n reference state Thus when the body s n the current state at tme t the nternal energy of any part of the body s u( dv March Fnte Deformaton: General Theory 13

14 Sold Mechancs Z Suo The ntegral extends over the volume of the part Between tme t and t + δt the nomnal densty of nternal energy changes by δ u ( u( t + δ u( Smlarly defne varous amounts of heat transfer durng ths short tme: δ Q Q( t + δ Q( δ q q( t + δ q( δ I I ( t + δ I ( Consder any part of the body Conservaton of energy requres that the work done by the forces upon the part and the heat transferred nto the part equal the change n the nternal energy of the part Thus δ udv sδfdv + δqdv δqda The ntegrals extend over the volume and the surface of the part We next reduce conservaton of energy nto a dfferental form 3 I 1 A 1 qa A 1 A A 3 A I A reference state 1 I A 3 3 current state q I relaton Consder a materal element of tetrahedron When the body s n the reference state the four faces of the tetrahedron are trangles: the three trangles on the coordnate planes and one trangle on the plane normal to the unt vector Let the areas of the three trangles on the coordnate planes be A and the area of the trangle normal to be A The geometry dctates that A A When the body s n the current state the tetrahedron deforms to a shape of four curved faces ow apply conservaton of energy to ths deformed tetrahedron As the volume of the tetrahedron decreases the rato of area over volume becomes large so that conducton across the surface prevals over terms due to volumetrc changes Consequently the conducton through all four faces must balance: I A qa A combnaton of the above two equatons gve March Fnte Deformaton: General Theory 14

15 Sold Mechancs Z Suo I q Consequently I s a lnear operator that maps the unt vector normal to an area to the energy across per unt area q Applyng the dvergence theorem we obtan that I qda I da dv Insertng ths expresson nto the expresson for conservaton of energy we obtan that δ udv sδf + δq δi dv Ths equaton holds for any part of the body so that the ntegrands must equal: δu sδf + δq δi Ths equaton expresses conservaton of energy n a dfferental form Producton of entropy We regard a materal partcle as a thermodynamc system When the body s n the current state at tme t denote η t namely the nomnal densty of entropy by ( entropy n current state η volume n reference state Between tme t and tme t + δt the nomnal densty of entropy changes by δη η( t + δ η( and the entropy of any part of the body changes by δη dv The ntegral extends over the volume of the part Each materal partcle s n thermal contact wth a reservor of energy held at temperature θ ( ) R t Upon losng energy δ Q to the materal partcle the reservor changes ts entropy by δ Q / θ R Consequently the change n the entropy of the reservors n thermal contact wth the part of the body s δq dv θ R When the body s n the current state at tme t the feld of temperature n the body s denoted by θ ( The part of the body s also n thermal contact wth neghborng parts of the body We may as well thnk the materal partcles on the surface of the part as reservors of energy When energy flows from the part to the reservors the entropy of these hypothetcal reservors changes by δq da θ The ntegral extends over the surface of the part Recall that conservaton of March Fnte Deformaton: General Theory 15

16 Sold Mechancs Z Suo energy results n two equatons: I q δi δ u sδf + δq Usng these two equatons and applyng the dvergence theorem δi δi da dv θ θ we obtan that δ q δu s 1/ θ δq da + δf + δi + dv θ θ θ θ We form a composte from the followng thngs: a part of the body the reservors of energy n thermal contact wth the part the partcles on the surface of the body represented as hypothetcal reservors of energy all the mechancal forces Between tme t and tme t + δt the entropy of the composte changes by δu s 1/ θ 1 1 change n entropy δη + δf + I + δ δq dv θ θ θ θ R The mechancal forces do not contrbute to the entropy By constructon the composte s an solated system Consequently all the changes must proceed n the drecton to ncrease the entropy of the composte Ths prncple apples to any part of the body so that all the changes must proceed n the drecton δu s ( 1/ ) 1 1 θ δη + δf + I + δ δq 0 θ θ θ θ R Ths solated system has a large number of nternal varables ncludng η u F I Q Thermodynamc model of a materal partcle We regard each materal partcle as a thermodynamc system characterzed by the entropy as a functon of the nternal energy and deformaton gradent: η η( uf) Assocated wth the varaton δ u and δ F the entropy vares by η( uf ) η( uf) δη δu + δf u F Assume that the partcle s n local equlbrum so that η ( u F) 1 u θ March Fnte Deformaton: General Theory 16

17 Sold Mechancs Z Suo ( u F) η s F θ The change n entropy of the solated system reduces to ( 1/ θ ) 1 1 δi + δq 0 θ θ R Ths nequalty can be satsfed by nvokng Fourer s law as dscussed n the notes on heat conducton ( Isothermal deformaton In partcular we wll be mostly nterested n stuatons where the body s n thermal equlbrum so that the temperature s unform n the body and s an nput to the theory Let W be the nomnal densty of the Helmholtz free-energy free energy n current state W volume n reference state so that W u θη As a materal model we assume that the free-energy densty s a functon of the deformaton gradent W W ( F) The temperature s unform n the body and we wll not ndcate temperature explctly A combnaton of the above equatons gves that W ( F ) s F Ths equaton relates the stress to the deformaton gradent once the functon W(F) s prescrbed We may as well regard ths equaton as the defnng equaton for a materal model known as the hyperelastc model The free-energy densty s unchanged f the body undergoes a rgd-body rotaton The free energy s nvarant f the partcle undergoes a rgd body rotaton Thus W depends on F only through the product C L F FL or T C F F We wrte W ( F) f ( C) Recall that the stress s work-conugate to the deformaton gradent Usng the chan rule n dfferental calculus we obtan that W ( F) f ( C) C M s F C M F or March Fnte Deformaton: General Theory 17

18 Sold Mechancs Z Suo f s FL C L ote that ths equaton reproduces the equaton resultng from conservaton of angular momentum: s F s F Tensor of tangent modulus for a hyperelastc materal The tensor of tangent modulus s defned by s ( F ) L F L or δ s LδF L For a hyperelastc materal the tensor of tangent modulus relates to the free-energy densty as W ( F ) L F F L Consequently for a hyperelastc materal the tensor of tangent modulus possesses the maor symmetry: L L Isotropc materal To specfy a hyperelastc materal model we need to specfy the free energy densty as a functon of the Green deformaton tensor C That s we need to specfy the functon W f ( C) The tensor C s postve-defnte and symmetrc In three dmensons ths tensor has 6 ndependent components Thus to specfy a hyperelastc materal model we need to specfy the free energy densty as a functon of the 6 varables For a gven materal such a functon s specfed by a combnaton of expermental measurements and theoretcal consderatons Trade off s made between the amount of effort and the need for accuracy For an sotropc materal the free energy densty s nvarant when the coordnates rotate n the reference state That s W depends on C through 3 scalars: α C β C LC L γ C LC LJC J These scalars are known as the nvarants of the tensor C That s to specfy an sotropc hyperelastc materal model we need to prescrbe a functon of three varables: W f ( α β γ ) Once ths functon s specfed we can derve the stress by usng the chan rule: ( C) March Fnte Deformaton: General Theory 18

19 Sold Mechancs Z Suo ( F) W f α f β f γ s + + F α F β F γ F Algebra of ths knd s recorded n many textbooks on contnuum mechancs Incompressble materal Rubbers and metals can undergo large change n shape but very small change n volume A commonly used dealzaton s to assume that such materals are ncompressble namely det F 1 s W F / F we have tactly assumed that each component of In arrvng at F can vary ndependently The condton of ncompressblty places a constrant among the components To enforce ths constrant we replace the free-energy functon W ( F) by a functon W ( F) Π( det F 1) wth Π as a Lagrange multpler We then allow each component of F to vary ndependently and obtan the stress from s [ W ( F) Π( det F 1) ] F Recall an dentty n the calculus of matrx: det F H det F F where H s defned by H F L δ L and H F δ Thus for an ncompressble materal the stress relates to the deformaton gradent as W ( F) s ΠH det F F The Lagrange multpler Π s not a materal parameter For a gven boundaryvalue problem Π s to be determned as part of the soluton eo-hookean materal For a neo-hookean materal the free-energy functon takes the form W µ ( F ) ( 3) F F The materal s also taken to be ncompressble namely det F 1 The stress relates to the deformaton gradent as s µ F H Π Equaton of moton We now state an ntal and boundary value problem to evolve the feld of deformaton x ( Gven a body we choose a reference state to name a feld of materal partcles The body s prescrbed wth a March Fnte Deformaton: General Theory 19

20 Sold Mechancs Z Suo feld of mass densty ρ The materal s modeled by an energy functon ( F) W whch depends on F through the product F T F The body s subect to a feld of body force B ( The feld equatons are ( F ( s W F F ( s x + B ( ρ A combnaton of the above three equatons gves the equaton of moton Ths equaton evolves the feld of deformaton x ( n tme subect to the followng ntal and boundary condtons Intal condtons are gven by prescrbng at tme t 0 the places of all the partcles x ( t 0 ) and the veloctes of all the partcles V ( t 0 ) For every materal partcle on the surface of the body we prescrbe ether one of the followng two boundary condtons On part of the surface of the body S t the tracton s prescrbed so that s ( prescrbed for St On the other part of the surface of the body S the poston s prescrbed so that ( prescrbed x for Su Perturb an equlbrum state of fnte deformaton A body s made of a materal characterzed by a relaton between the stress and the deformaton gradent: s g ( F) The body s subect to a dead load B n the volume One part of the surface of the body s subect to dead load T and the other part of the surface s held at prescrbed place Let x 0 be a state of equlbrum compatble wth the above prescrptons Assocated wth ths state s the deformaton gradent 0 0 F and the stress 0 0 s F g u The stress feld balances the forces 0 s ( ) 0 n the volume of the body and + B March Fnte Deformaton: General Theory 0

21 Sold Mechancs Z Suo T 0 s on the part of surface where the tracton s prescrbed On the part of the surface x 0 s held at the prescrbed place where the place s prescrbed When the state of equlbrum x 0 s now perturbed by an nfntesmal deformaton u ( the feld of deformaton becomes x ( t ) x 0 + u( Assocated wth ths feld of deformaton s the deformaton gradent 0 u ( F t F + and the stress u ( 0 0 s ( F ) s ( F ) + L ( F ) L Here we have expanded the stress nto the Taylor seres up to terms lnear n the perturbaton Recall the equaton of force balance s x + B ρ Insertng the stress-stran relaton nto the equaton of force balance we obtan a homogeneous partal dfferental equaton: u 0 t u ( L F ρ L The boundary condtons also homogeneous: u ( ) 0 t L ( F ) 0 L T s prescrbed and on the part of the surface where the dead force u ( 0 on the part of the surface where the place s prescrbed These equatons are solved subect to an ntal condton The equaton of moton for the perturbaton u ( looks smlar to that n lnear elastcty Thus solutons n lnear elastcty may be renterpreted for phenomena of ths type We gve examples below Lnear vbraton around a state of equlbrum We have already analyzed the vbraton of beam subect to an axal force In general we look for soluton of the form u ( a snωt where ω s the frequency and a s the feld of ampltude The equaton of moton becomes that March Fnte Deformaton: General Theory 1

22 Sold Mechancs Z Suo a 0 L ( F ) ρ ω a L Ths equaton n conuncton wth the homogeneous boundary condtons forms an egenvalue problem The soluton to ths egenvalue problem determnes ω a for a set of normal modes and Plane waves supermposed on a homogeneous deformaton n an nfnte body When a bar s subect to unaxal tenson homogenous deformaton satsfes all governng equatons However the state of homogeneous soluton may not be the only soluton eckng s an alternatve soluton a bfurcaton from the homogenous deformaton In the prevous analyss we have ust consdered the homogenous deformaton and have dentfed the onset of neckng wth the axal force n the force-stran curve peaks ow we wsh to push the analyss a step further: we wsh to consder the possble small perturbaton from a homogenous deformaton Consder an nfnte body Assume a plane wave of small ampltude: u ( af ( c where a s the unt vector n the drecton of dsplacement the unt vector n the drecton of propagaton c the wave speed and f the wave form The equaton of moton becomes 0 L ( F ) La ρc a Ths s an egenvalue problem When the modulus of tangent modulus possesses the maor symmetry L L the acoustc tensor kl L s also symmetrc Accordng to a theorem n lnear algebra ths symmetrc acoustc tensor wll have three real egenvalues and the three egenvectors are orthogonal to one another Furthermore when the acoustc tensor s postve-defnte the three egenvalues are postve leadng to three dstnct plane waves Exercse An nfnte body of an ncompressble materal cannot support longtudnal wave but can support shear wave Assume that the body s n a homogeneous state of unaxal stress A small dsturbance propagates along the axs of stress as a shear wave Determne the speed of ths shear wave Bfurcaton The tensor of tangent modulus depends on the state of 0 homogeneous deformaton L ( F ) and may no longer be postve-defnte When the tangent modulus reaches the crtcal condton 0 det[ L ( F ) L ] 0 n some drecton we can fnd a nonvanshng dsplacement a at c 0 Ths equaton may be regarded as a 3D generalzaton of the Consdère condton Readng J R Rce "The Localzaton of Plastc Deformaton" n March Fnte Deformaton: General Theory

23 Sold Mechancs Z Suo Theoretcal and Appled Mechancs (Proceedngs of the 14th Internatonal Congress on Theoretcal and Appled Mechancs Delft 1976 ed WT oter) Vol 1 orth-holland Publshng Co Surface nstablty of rubber n compresson M A Bot Appl Sc Res A (1963) A sem-nfnte block of a neo-hookean materal s n a homogeneous state of deformaton wth λ 1 and λ beng the stretches n the drectons parallel to the surface of the block and λ 3 beng the stretch n the drecton normal to the surface The materal s taken to be ncompressble so that λ 1 λλ3 1 The compresson n drecton 1 s taken to be more severe than that n drecton so that when the surface wrnkles n one orentaton leavng λ unchanged That s the wrnkled block s n a state of generalzed plane stran Bot found the crtcal condton for ths nstablty: λ 3 / λ 1 34 The analyss of ths problem s analogous to that of the Raylegh wave Weak statement of conservaton of momentum We next consder a number of refnements of the theory Recall that conservaton of lnear momentum results n two equatons: T s s x + B ρ Ths par of equatons may be called the strong statement of conservaton of momentum We now express conservaton of lnear momentum n a dfferent form Denote an arbtrary feld by In the lterature on the fnte element method s also called a test functon Consder any part of the body Multplyng the two equatons n the strong statement by ntegratng over the area of the part and the volume of the part respectvely and then addng the two we obtan that x s + dv T da B ρ dv The ntegrals are taken over the volume and the surface of the part In reachng the above equaton we have used the dvergence theorem Consequently conservaton of lnear momentum mples that the above equaton holds for any Ths statement s known as the weak statement of conservaton of lnear momentum Conversely we can start wth the weak statement and show that the weak statement mples the strong statement test functon March Fnte Deformaton: General Theory 3

24 Sold Mechancs Z Suo An alternatve formulaton of sothermal deformaton of a hyperelastc materal A hyperelastc materal s modeled by a free-energy densty as a functon of the deformaton gradent W ( F) To ensure that the free energy s nvarant when the body undergoes a rgd-body rotaton we requre T that W depend on F only through the Green deformaton tensor C F F As an alternatve startng pont of the theory we stpulate that the vrtual work done by all the forces equals the vrtual change n the energy namely W ( F) x dv T + da B ρ dv F Ths equaton holds for arbtrary test functon Exercse The above equaton s the bass for the fnte element method ( In that context the above volume ntegrals extend over the entre body The test functon s set to vansh on part of the surface where dsplacement s prescrbed and on the remanng part of the surface the surface force T s prescrbed Sketch the fnte element method by usng the above equaton Exercse Ths formulaton s so swft that t does not even menton the stress Recover all the prevous equatons by dentfyng the stress by the relaton W ( F ) s F Exercse You can go back to verfy the followng statements Ths alternatve formulaton conserves mass lnear momentum angular momentum and energy Consder the composte of a part of the body n thermal contact wth varous reservors Under the sothermal condton the entropy of the composte does not change Polar decomposton By defnton the deformaton gradent s a lnear operator that maps one vector to another vector For any lnear operator there s a theorem n lnear algebra called polar decomposton Any lnear operator F can be wrtten as a product: F RU T where R s an orthogonal operator satsfyng R R I and U s calculated from T F F U Thus U s symmetrc and postve-defnte Exercse For a gven F show that the polar decomposton yelds unque U and R Exercse Gve pctoral nterpretaton of the polar decomposton Lagrange stran Defne the Lagrange stran E L by March Fnte Deformaton: General Theory 4

25 Sold Mechancs Z Suo Recall and dl dl E L d dl d d dl F F L d d d L L We obtan that E L 1 ( F F δ ) where δ L 1 when L and δ L 0 when L The E L tensor s symmetrc The three tensors are related as 1 E ( C I) C U Exercse The Lagrange stran forms a lnk between fnte deformaton and the nfntesmal deformaton approxmaton Let the dsplacement feld be U( t ) x( Show that the Lagrange stran s 1 U U L U U E + + L L L Thus the Lagrange stran concdes wth the stran obtaned n the nfntesmal stran formulaton f U << 1 Ths n turn requres that all components of lnear stran and rotaton be small However even when all strans and rotatons are small we stll need to balance force n the deformed state as remarked before Exercse We can relate the general defnton of the Lagrange stran to that ntroduced n the begnnng of the notes for a tensle bar Consder a materal element of lne n a three dmensonal body In the reference state the element s n the drecton specfed by a unt vector M In the current state the stretch of the element s λ We have defned the Lagrange stran n one dmenson by 1 η λ Show that the Lagrange stran of the element n drecton M s gven by η E L M M L Exercse The tensor of Lagrange stran can also be used to calculate the engneerng shear stran Consder two materal elements of lne In the reference state the two elements are n two orthogonal drectons M and In the current state the stretches of the two elements are λ M and λ and the angle L L March Fnte Deformaton: General Theory 5

26 Sold Mechancs Z Suo between the two elements become shear stran Show that π γ We have defned γ as the engneerng EL M L sn γ λ M λ Deformaton of a materal element of volume Let dv d1d d 3 be an element of volume n the reference state After 1 deformaton the lne element d 1 becomes d x wth components gven by 1 dx F 1d1 3 We can wrte smlar relatons for d x and d x In the current state the volume of the element s 1 3 dv ( dx dx ) dx We can confrm that ths expresson s the same as dv det( F)dV Deformaton of a materal element of area Let an element of area be da n the reference state where s the unt vector normal to the element and da s the area of the element After deformaton the element of area becomes n da n the current state where n s the unt vector normal to the element and da s the area of the element An element of length d n the reference state becomes d x n the current state The relaton between the volume n the reference state and the volume n the current state gves dx nda det ( F) d da or d Fnda det( F) d da Ths relaton holds for arbtrary d so that Fnda det( F) da or n vector form T F da det F d A The second Pola-rchhoff stress Defne the tensor of the second Pola-rchhoff stress S by L work n the curent state S δe L L volume n the reference state S Ths expresson defnes a new measure of stress L symmetrc tensor S L should also be symmetrc Recall that we have also expressed the same work by two expressons for work Because E L s a s δ F Equatng the March Fnte Deformaton: General Theory 6

27 Sold Mechancs Z Suo SLδ E L sδf Recall defnton of the Lagrange stran 1 E F F δ L We obtan that 1 δ EL ( FLδF + FδFL ) and s SLFL For a hypoelastc materal the free energy s a functon of the Lagrange stran Ω ( E) The second Pola-rchhoff stress s work-conugate to the Lagrange stran: Ω( E ) SL E L L L Lagrangan vs Euleran formulatons The above formulaton uses the materal coordnate as the ndependent varable The functon x ( gves the place occuped by the materal partcle at tme t Ths formulaton s known as Lagrangan The nverse functon ( x tells us whch materal partcle s at place x and tme t The formulaton usng the spatal coordnate x s known as Euleran In the above we have used the Euleran formulaton to study cavtaton because we know the current confguraton In general the current confguraton s unknown and we have used the Lagrangan formulaton to state the general problem In the followng we lst a few results related to the Euleran formulaton Tme dervatve of a functon of materal partcle At tme t a materal partcle moves to poston x ( The velocty of the materal partcle s ( V t If we would lke to use x as the ndependent varable we change the varable xt and then wrte from to x by usng the functon ( x t ) V( Let G ( v be a functon of materal partcle and tme For example G can be the temperature of materal partcle at tme t The rate of change n temperature of the materal partcle s G ( t ) Ths rate s known as the materal tme dervatve We can calculate the materal tme dervatve by an alternatve approach Change the varable from to x by March Fnte Deformaton: General Theory 7

28 Sold Mechancs Z Suo usng the functon ( x and wrte g ( x G( Usng chan rule we obtan that G( g( x g( x ( + Thus we can calculate the substantal tme rate from G( g( x g( x + v ( x Let us apples the dea to the acceleraton of a materal partcle In the Lagrangean formulaton the acceleraton s V ( x( A t In the Euleran formulaton the acceleraton of a materal partcle s v ( x v ( x a x t + v ( x Conservaton of mass In the Lagrange formulaton the nomnal mass densty s defned by mass n current state ρ R volume n reference state That s ρ RdV s the mass of a materal element of volume A subscrpt s added here to remnd us that the volume s n the reference state In the Eularan formulaton the true mass densty s defned by mass n current state ρ volume n current state That s ρ dv s the mass of a spatal element of volume The two defntons of densty are related as ρ RdV ρdv or ρ R ρ detf Conservaton of mass requres that the mass of the materal element of volume be tme-ndependent Thus the nomnal densty can only vary wth materal partcle ρ R and s tme-ndependent By contrast the true densty ρ xt Conservaton of mass requres that s a functon of both place and tme ρ( x + [ ρ( x v ( x ] 0 When the materal s ncompressble det F 1 we obtan that ρ R ρ( x March Fnte Deformaton: General Theory 8

29 Sold Mechancs Z Suo True stran Recall the defnton of the true stran n one dmenson When the length of a bar changes by a small amount from l ( to l + δl the ncrement n the true stran s defned as δl δε l x t Imagne a small ow consder a body undergong a deformaton dsplacement u( x) δ n the current state Consder two nearby materal partcles n the body When the body s n the reference state the coordnates of the two partcles are and + d The vector d s a materal element of lne When the body s n the current state at tme t the two materal partcles occupy places x ( and x ( + d so that the vector between the two partcles s d x x( + d x( Consder a materal element of lne In the reference state the element s d MdL where dl s the length of the element and M s the unt vector n the drecton of the element In the current state the element becomes dx mdl where dl s the length of the element and m s the unt vector n the drecton of the element The length of a materal element of lne at tme t s dl dx dx Subect to the small dsplacement the above equaton becomes dlδ ( dl) d( δu ) dx ote that ( u ) d δ u δ dx and δ ( dl) δε dl Thus ( u ) δε δ mm Ths s the ncrement of the stran of the materal element of lne Only the symmetrc part ( δu ) ( δu ) 1 + wll affect the ncrement of the stran of the materal element of lne Ths tensor generalzes the noton of the true stran and s gven the symbol: ( δu ) ( δu ) 1 δε ( x + March Fnte Deformaton: General Theory 9

30 Sold Mechancs Z Suo v be the feld of true velocty n the current state The same lne of reasonng shows that the tensor 1 v x t v x t d x t + allows us to calculate the rate of change the length of materal element of lne namely 1 [ dl( ] dmm dl( The tensor d s known as the rate of deformaton Rate of deformaton Let ( x Force balance The true stress obeys that σ x t v x t v x t + b x t ρ x t + v ( x n the volume of the body and σ n t on the surface of the body These are famlar equatons used n flud mechancs Relaton between true stress and nomnal stress Recall the defnton of the nomnal stress work n the curent state sδf volume n the reference state and the defnton of the true stress work n the curent state σ δε volume n the current state Equatng the work on an element of materal we obtan that σ δε dv sδfdv Because the true stress s a symmetrc tensor we obtan that δu ( x) σ δε σ Also note that δu x ( δu x δu ( x) δ F F ( and that dv / dv det( F) Insstng that the two expressons for the work be the same for all moton we obtan that March Fnte Deformaton: General Theory 30