2. Plane Elasticity Problems

Transcription

1 S0 Solid Mechanics Fall 009. Plane lasticity Poblems Main Refeence: Theoy of lasticity by S.P. Timoshenko and J.N. Goodie McGaw-Hill New Yok. Chaptes 3..1 The plane-stess poblem A thin sheet of an isotopic mateial is subject to loads in the plane of the sheet. The sheet lies in the (x y) plane. Both top and the bottom sufaces of the sheet ae taction-fee. The edge of the sheet may have two kinds of the bounday conditions: displacement pescibed o taction pescibed. In the latte case we wite τ n n x x τ n n y y t x t y whee t x and t y ae components of the taction vecto pescibed on the edge of the sheet and n x and n y ae the components of the unit vecto nomal to the edge of the sheet. The above two equations povide two conditions fo the components of the stess tenso along the edge. Semi-invese method. We next go into the inteio of the sheet. We aleady have obtained a full set of govening equations fo linea elasticity poblems. No geneal appoach exists to solve these patial diffeential equations analytically although numeical methods ae eadily available to solve most elasticity poblem. In this intoductoy couse in ode to gain insight into solid mechanics we will make easonable guesses of solutions and see if they satisfy all the govening equations. This tial-and-eo appoach has a name: it is called the semi-invese method. It seems easonable to guess that the stess field in the sheet only has nonzeo components in its plane: τ and that the components out of plane vanish: τ τ 0. zz xz yz Futhemoe we guess that the in-plane stess components may vay with x and y but ae independent of z. That is the stess field in the sheet is descibed by thee functions: 9/6/11 Linea lasticity-1

2 S0 Solid Mechanics Fall 009 ( x y) ( x y) τ ( x y). Will these guesses satisfy the govening equations of elasticity? Let us go though the equations one by one. 1. quilibium equations. Using the guessed stess field we educe the thee equilibium equations to two equations: τ τ 0 0. These two equations by themselves ae insufficient to detemine the thee functions.. Stess-stain elations. Given the guessed stess field the 6 components of the stain field ae ν ν γ ( ν ) 1 τ ( ) γ γ 0 ν zz xz yz. 3. Stain-displacement elations. Recall the 6 stain-displacement elations: u v γ u v w u w v w zz γ xz γ yz. z z z It seems easonable to assume that the in-plane displacements u and v vay only with x and y but not with z. Fom these guesses togethe with the conditions that γ γ 0 we find that w w 0. xz yz Thus w is independent of x and y and can only be a function of z. If we insist that zz be independent of z and fom zz w/ z then zz must be a constant zz c and w cz b. On the othe hand we also have zz ν ( ) / which may not be a constant. This inconsistency shows that ou guesses ae geneally incoect. 9/6/11 Linea lasticity-

3 S0 Solid Mechanics Fall 009 Summay of equations of plane elasticity poblems. Instead of abandoning these guesses we will just call ou guesses the plane-stess appoximation. If you neglect the inconsistency between c and ν ( ) at least the following set of equations is selfconsistent: zz zz / τ τ 0 0 ν ( ) ν γ 1 ν τ u v u v γ. These ae 8 equations fo 8 functions. We will focus on these 8 equations. 9/6/11 Linea lasticity-3

4 S0 Solid Mechanics Fall 009. The plane-stain poblem Conside an infinitely long cylinde with axis in the z-diection and a coss section in the ( x y) plane. We assume that the loading is invaiant along the z-diection. Unde these conditions the displacement field takes the fom: u( x y) v( x y) w 0. Fom the stain-displacement elations we find that only the thee in-plane stains ae nonzeo: ( x y) ( x y)γ ( x y). The thee out-of-plane stains vanish: γ γ 0. zz xz yz Because γ xz γ yz 0 the stess-stain elations imply that τ xz τ yz 0. Fom zz 0 and zz ( zz ν ν ) we obtain futhe that zz ν ( ). Futhemoe we have 1 ( ν ν zz ) 1 ν 1 ( ν ν zz ) 1 ν ( ) γ 1 ν τ. ν 1 ν ν 1 ν These thee stess-stain elations look simila to those unde the plane-stess conditions povided we make the following substitutions: 1 ν ν ν 1 ν. The quantity is called the plane stain modulus. Finally we also have the equilibium equations: 9/6/11 Linea lasticity-

5 S0 Solid Mechanics Fall 009 τ τ Solution of plane poblems and the Aiy stess function Fom the fogoing it is clea that plane stess and plane stain poblems ae descibed by the same equations as long as one uses the appopiate elastic constants. This also means that the solution technique fo both types of poblems is the same. We make use of the following calculus theoem: A theoem in calculus. If two functions f ( x y) and g( x y) satisfy the following elationship f g then thee exists a function ( x y) A such that f A g A. The Aiy stess function. We now apply the above theoem to the equilibium equations. Fom the equation τ 0 we deduce that thee exists a function ( x y) A τ A. Fom the equation A such that τ 0 we futhe deduce that that thee exists a function ( x y) B τ B. B such that 9/6/11 Linea lasticity-5

6 S0 Solid Mechanics Fall 009 Finally fom A B we deduce that that thee exists a function φ ( x y) such that A φ B φ. The function ( x y) φ is known as the Aiy stess function. The thee components of the stess field can now be epesented by the stess function: φ ; φ ; φ. Using the stess-stain elations we can also expess the thee components of stain field in tems of the Aiy stess function: ( ) 1 φ ν φ 1 φ ν φ γ 1 ν Compatibility equation. Recall the stain-displacement elations. u v γ u v. φ. We deived the compatibility equation by eliminating the two displacements in the thee stain displacement elations to obtain the compatibility equation γ. Bihamonic equation. Inseting the expessions of the stains in tems of ( x y) compatibility equation we obtain that φ φ φ 0. φ into the 9/6/11 Linea lasticity-6

7 S0 Solid Mechanics Fall 009 This equations can also be witten as φ φ 0. Because of its obvious similaity to the hamonic equation it is called the bihamonic equation. Thus a pocedue to solve a plane stess poblem is to solve fo ( x y) φ fom the above PD and then calculate stesses and stains. Afte the stains ae obtained the displacement field can be obtained by integating the stain-displacement elations. Dependence on elastic constants. Fo a plane poblem with taction-pescibed bounday conditions both the govening equation and the bounday conditions can be expessed in tems of φ. All these equations ae independent of the elastic constants of the mateial. Consequently the stess field in such a bounday value poblem is independent of the elastic constants. Once we go ove specific examples we will find that the above statement is only coect fo bounday value poblems in simply connected egions. Fo multiply connected egions the above equations in tems of φ do not guaantee that the displacement field is continuous. When we insist that displacement field be continuous elastic constants may ente the stess field..3.1 Solution of D poblems in Catesian coodinates: A half space subject to peiodic taction on the suface An elastic mateial occupies a half space x > 0. On the suface of the mateial x 0 taction vecto is pescibed ( 0 yz) 0 cosky τ ( 0yz ) 0 τ xz ( 0yz ) 0. Detemine the stess field inside the mateial. Solution: The mateial clealy defoms unde the plane stain conditions. It is easonable to guess that the Aiy stess function should take the fom φ( x y) f ( x)cos ky. The bihamonic equation then becomes the d f dx k d f dx k f 0. 9/6/11 Linea lasticity-7

8 S0 Solid Mechanics Fall 009 This is a homogenous OD with constant coefficients. The solution must be of the fom f αx ( x) e. Inset this fom into the OD and we obtain that ( α k ) 0. The algebaic equation has double oots of the geneal solution is of the fom α k and double oots of α k. Consequently f kx kx ( x) Ae Be Cxe Dxe whee A B C and D ae constants of integation. We expect that the stess field vanish as x so that the stess function should be of the fom f ( x) Be Dxe. We next detemine the constants B and D by using the taction bounday conditions. The stess fields ae τ φ φ Be φ Be ( Be Dxe ) k D e k D e k Recall the bounday conditions cosky Dxe Dxe ( 0 y) 0 cos ky τ ( 0 y) 0. Consequently we find that B k D / k. 0 / 0 The stess field inside the mateial is k k sin ky cosky 9/6/11 Linea lasticity-8

9 S0 Solid Mechanics Fall 009 τ 0 ( 1 kx) kxe 0 0 e sin ky cosky ( 1 kx) e cosky The stess field decays exponentially. We have solved the poblem whee the taction on the bounday of a half space is given by a simple cosine function. Though application of the supeposition pinciple which is valid fo linea elastic mateials it is now staightfowad to extend this analysis to any peiodic taction distibution. Indeed a peiodic taction distibution can be witten as a Fouie seies each tem of which is of the fom found in the pevious poblem. Application: de Saint-Venant s pinciple When a load is applied in a small egion and the load has a vanishing esultant foce and esultant moment then the stess field is localized. We used this pinciple in discussing the laminate poblem whee we have neglected the edge effects. While Saint-Venant s pinciple cannot be poved in such a loose fom the foegoing is a nice example of the pinciple: the taction applied to the bounday is self-balancing and hence the stess field associated with the tactions die out. If we had imposed an additional constant taction tem the stess field would quickly decay to a constant stess. 9/6/11 Linea lasticity-9

10 S0 Solid Mechanics Fall Solution of D poblems in pola coodinates 1. Tansfomation of stess components due to change of coodinates. A mateial paticle is in a state of plane stess. If we epesent the mateial paticle by a squae in the ( x y) coodinate system the components of the stess state ae paticle unde the same state of stess by a squae in the ( ) components of the stess state ae τ. If we epesent the same mateial coodinate system the τ. Fom the tansfomation ules we know that the two sets of the stess components ae elated as τ cos τ cos τ sin τ cos sin sin. quations in pola coodinates. The Aiy stess function is a function of the pola coodinates φ ( ). The stesses ae expessed in tems of the Aiy stess function: φ 1 φ φ φ τ The bihamonic equation is φ φ φ 0. The stess-stain elations in pola coodinates ae simila to those in the ectangula coodinate system: ν The stain-displacement elations ae ν γ ( ν ) 1 τ u u u u u u γ. 9/6/11 Linea lasticity-10

11 S0 Solid Mechanics Fall A stess field symmetic about an axis. Let the Aiy stess function be φ ( ) equation becomes. The bihamonic d d 1 d d d φ 1 dφ 0 d d. ach tem in this equation has the same dimension in the independent vaiable. Such an OD is known as an equi-dimensional equation. A solution to an equi-dimensional equation is of the fom m φ. Inseting into the bihamonic equation we obtain that ( m ) m. The fouth ode algebaic equation has a double oot of 0 and a double oot of. Consequently the geneal solution to the OD is ( ) Alog B log C D φ. whee A B C and D ae constants of integation. The components of the stess field ae φ 1 φ A B( 1 log) C φ A B( 3 log) C φ 0 τ. The stess field is linea in A B and C. The contibutions due to A and C ae familia: they ae the same as the cylindical Lamé poblem. Fo example fo a hole of adius a in an infinite sheet subject to a emote biaxial stess S the stess field in the sheet is 9/6/11 Linea lasticity-11

12 S0 Solid Mechanics Fall 009 a a S 1 S 1. The stess concentation facto of this hole is. We may compae this poblem with that of a spheical cavity in an infinite elastic solid unde emote tension: 3 3 a 1 a S 1 S 1. A cut-and-weld opeation. How about the contibutions due to B? Let us study the stess field (Timoshenko and Goodie pp ) B( 1 log) B( 3 log) The stain field is τ 0. 1 B ( ν ) [( 1 3ν ) ( 1 ν ) log] 1 B ( ν ) [( 3 ν ) ( 1ν ) log] γ 0 To obtain the displacement field ecall the stain-displacement elations u u u u u u γ. Integating we obtain that whee ( ) B u [ ( 1ν ) log ( 1 ν ) ] f ( ) f is a function still undetemined. Integating we obtain that B u f ( ) d g( ) 9/6/11 Linea lasticity-1

13 S0 Solid Mechanics Fall 009 whee g ( ) is anothe function still undetemined. Inseting the two displacements into the expession u u u γ 0 and we obtain that f ' ( ) f ( ) d g( ) g' ( ). In the equation the left side is a function of and the ight side is a function of. Consequently the both sides must equal a constant independent of and namely f g '( ) f ( ) d ( ) g' ( ) G G Solving these equations we obtain that f g ( ) H sin ( ) F G K cos Substituting back into the displacement field we obtain that u u B [ ( 1ν ) log ( 1 ν ) ] H sin K cos. B F H cos K sin Consequently F epesents a igid-body otation and H and K epesent a igid-body tanslation. Now we can give an intepetation of B. Imagine a ing with a wedge of angle α cut off. The ing with the missing wedge was then welded togethe. This opeation equies that afte a otation of a cicle the displacement is u ( π ) u 0 This condition gives α B. 8π ( ) α 9/6/11 Linea lasticity-13

14 S0 Solid Mechanics Fall 009 This cut-and-weld opeation clealy intoduces a stess field in the ing. The stess field is axisymmetic as given above.. A cicula hole in an infinite sheet unde emote shea. Remote fom the hole the sheet is in a state of pue shea: τ S 0. The emote stesses in the pola coodinates ae Recall that S sin S sin τ S cos. φ 1 φ φ φ τ. We guess that the stess function must be in the fom ( ) f ( ) sin φ. The bihamonic equation becomes d d d d f f f 0. A solution to this equi-dimensional OD takes the fom ( ) m OD we obtain that (( m ) ) ( m ) 0. f. Inseting this fom into the The algebaic equation has fou oots: - 0. Consequently the stess function is C φ( ) A B D sin. The stess components inside the sheet ae 9/6/11 Linea lasticity-1

15 S0 Solid Mechanics Fall 009 9/6/11 Linea lasticity-15 φ φ sin 6 1 D C A φ sin 6 1 C B A φ τ cos 6 6 D C B A. To detemine the constants A B C D we invoke the bounday conditions: 1. Remote fom the hole namely τ cos sin S S giving 0 / B S A.. On the suface of the hole namely a 0 0 τ giving Sa D and / C Sa. The stess field inside the sheet is sin 3 1 a a S sin 3 1 a S τ cos 3 1 a a S 5. A hole in an infinite sheet subject to a emote uniaxial stess. Use this as an example to illustate linea supeposition. A state of uniaxial stess is a linea supeposition of a state of pue shea and a state of biaxial tension. The latte is the Lame poblem. When the sheet is subject to emote tension of magnitude S the stess field in the sheet is given by 1 1 a S a S.

16 S0 Solid Mechanics Fall 009 Illustate the supeposition in figues. Show that unde uniaxial tensile stess the stess aound the hole has a concentation facto of 3. Unde uniaxial compession the mateial may split in the loading diection. 6. A line foce acting on the suface of a half space. A half space of an elastic mateial is subject to a line foce on its suface. Let P be the foce pe unit length. The half space lies in x > 0 and the foce points in the diection of x. This poblem has no length scale. Lineaity and dimensional consideations equies that the stess field take the fom ij whee ( ) ij P ij ( ) g ( ) g ae dimensionless functions of. We guess that the stess function takes the fom ( ) Pf ( ) φ whee f ( ) is a dimensionless function of. (A homewok poblem will show that this guess is not completely coect but it suffices fo the pesent poblem.) Inseting this fom into the bihamonic equation we obtain an OD fo f ( ): d f d f f 0. d d The geneal solution is φ Obseve that ( ) P( Asin B cos C sin D cos ). sin y and cos x do not contibute to any stess so we dop these two tems. By the symmety of the poblem we look fo stess field symmetic about 0 so that we will dop the tem ( ) PC sin φ. cos. Consequently the stess function takes the fom We can calculate the components of the stess field: CP cos τ 0. 9/6/11 Linea lasticity-16

17 S0 Solid Mechanics Fall 009 This field satisfies the taction bounday conditions τ 0 at π / and π /. To detemine C we equie that the esultant foce acting on a cylindical suface of adius balance the line foce P. On each element d of the suface the adial stess povides a vetical component of foce cos d. The foce balance of the half cylinde equies that π / P cosd 0. π / Integating we obtain that C 1/ π. The stess components in the x-y coodinates ae P cos πx P sin πx cos τ P sin cos πx 3 The displacement field is u u ( 1ν ) P P cos log sin π π ν P P sin sin log π π π ( 1ν ) P ( 1ν ) cos π P sin 7. Sepaation of vaiable. One can obtain many solutions by using the pocedue of sepaation of vaiable assuming that ( ) R( ) Θ( ) φ. Fomulas fo stesses and displacements can be found on p. 05 Defomation of lastic Solids by A.K. Mal and S.J. Singh. A eal-life example. Fom: S. Ho C. Hillman F.F. Lange and Z. Suo " Suface cacking in layes unde biaxial esidual compessive stess" J. Am. Ceam. Soc (1995). In pevious teatment of laminates we have ignoed edge effect. Howeve we also know that edges ae often the site fo failue to initiate. Hee is a phenomenon discoveed in the lab of Fed Lange at UCSB. A thin laye of mateial 1 was sandwiched in two thick blocks of mateial. Mateial 1 has a smalle coefficient of themal expansion than mateial so that upon cooling 9/6/11 Linea lasticity-17

18 S0 Solid Mechanics Fall 009 mateial 1 develops a biaxial compession in the plane of the laminate. The two blocks ae nealy stess fee. Of couse these statements ae only valid at a distance lage than the thickness of the thin laye. It was obseved in expeiment that the thin laye cacked as shown in Fig. 1. 9/6/11 Linea lasticity-18

19 S0 Solid Mechanics Fall 009 It is clea fom Fig. that a tensile stess can develop nea the edge. We would like to know its magnitude and how fast it decays as we go into the laye. We analyze this poblem by a linie supeposition shown in Fig. 3. Let M be the magnitude of the biaxial stess in the thin laye fa fom the edge. In Poblem A we apply a compessive taction of magnitude M on the edge of the thin laye so that the stess field in thin laye in Poblem A is the unifom biaxial stess in the thin laye with no othe stess components. In poblem B we emove themal expansion misfit but applied a tensile taction on the edge of the thin laye. The oiginal poblem is the supeposition of Poblem A and Poblem B. Thus the esidual stess field in the oiginal poblem is the same as the stess in Poblem B. 9/6/11 Linea lasticity-19

20 S0 Solid Mechanics Fall 009 With efeence to Fig. let us calculate the stess distibution ( x0) space is subject to a line foce P the stess is given by. Recall that when a half P sin cos. πx We now conside a line-foce acting at taction applied the line foce field in the laye: dη t / M ( x0) sin cos. πx t / Note that η x tan and let tan β t / x. Consequently x η d cos d and the integal becomes π β M M ( x0) sin d β Integating we obtain that y η. On an element of the edge d η the tensile P dη. Summing up ove all elements we obtain the stess M π M 1 ( x 0) β sin β. π β β 1 cos d. 9/6/11 Linea lasticity-0

21 S0 Solid Mechanics Fall 009 At the edge of the laye x / t 0 and π / t / x 0 M t ( x 0). 6π x Thus this stess decays as 3 3 x. β so that ( 0) M 0. Fa fom the edge 9/6/11 Linea lasticity-1