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Variational principles for Cosserat body
International Journal of Non-Linear Mechanics 37 (2002) 565}569
Note
Variational principles for Cosserat body Ion Nistor Department of Mathematics, Technical University of Ias7 i, Ias7 i 6600 Romania Received 15 July 2000; accepted 12 October 2000
Abstract Two variational principles are derived for the mixed boundary value problem of Cosserat solid. These principles are a generalization of the stationary principle of potential energy and the stationary principle of complementary energy from non-linear theory of elasticity. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cosserat solid; Principle of stationary potential energy; Principle of stationary complementary energy
1. Basic equations In the following discussion, all quantities are referred to the reference con"guration which is an undistorted stress-free con"guration. They are therefore functions of the coordinates of the place occupied by a particle X in reference con"guration. The deformation of a Cosserat body is described [1] by y "y (x ), R "R (x ), (1.1) G G H GH GH I where x and y are the coordinates of place occuG G pied by the particle X in the reference con"guration and in the present con"guration respectively, R are the components of a proper orthogonal GH tensor R which characterizes the rotation undergone by the material particles. The components of the tensor R can be represented [2] in the form R "[(1! ) # #e ]/ GH I I GH G H HGN N (1# ), F F E-mail address: i } [email protected] (I. Nistor).
where are the components of the microrotation H vector , e is the alternating symbol. GHN We consider the strain tensors c and de"ned by c "y R , "e R R , (1.3) GH I G IH GH HKL NL NK G where the symbol `,ia denotes partial derivation with respect to the variable x . G The tensor can be written [2,3], as "H , GH HI I G where
1 R FK H " e R HI 2 HKL FL I "(2 #e )/(2# ). (1.5) HI HIN N G G It is easy to verify that the matrix H"(H ) is GH non-singular and satis"ed the relations RH"HR"HR
(1.2)
(1.4)
(1.6)
where HR is the transpose of the matrix H. Substituting (1.5) into (1.6) and using the identity e R R "e R , (1.7) LHK FL GH FGN NK
0020-7462/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 1 1 3 - X
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I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
we obtain 1 R NK . H " e R (1.8) IG 2 GHN HK I Multiplying both sides of relation (1.8) by e adding and using the identities GLO e e " ! , GHI GNO HN IO HO IN R R "R R " , (1.9) LN LK NI KI NK we have R R ON "e R H . OK , (1.10) H e "R OGL LN IG IG GLO LK I I We can easily show, with the help of relations (1.8), (1.9) and (1.10) , that H H GH ! IH "e H H . (1.11) HKL GL IK I G From (1.6) we "nd, on di!erentiating H H R GH "R HO #H OG . (1.12) OG HO I I I Eq. (1.12) can be written, after using relations (1.10) and (1.11), as H H GH "R IN . (1.13) NG I H The equilibrium equations of Cosserat media are [1,4] (R t ) #F "0, GH IH I G (R m ) #e y R t #G "0, (1.14) GH IH I GHI H N IO NO G =(c, ) =(c, ) t " , m " . (1.15) IH IH c IH IH In these relations we have used the following notations: t , m * components of stress tensor GH GH T and couple stress tensor M; F , G * compoG G nents of body force vector F and body couple vector G per unit volume; = * strain energy per unit volume. The boundary < of the reference con"guration consists of regular surfaces S , 1)i)4 G
(S S "S S ", S S "S S "<) for which the following holds: u(x)"u(x) on S , R t n "f (x) on S , GH IH I G (x)"(x) on S , R m n "g (x) on S , GH IH I G (1.16) where u(x), (x) are given displacement and microrotation vectors, f( f ), g(g ) are given surface G G tractions and couples, n(n ) is the unit outward G normal on <.
2. The potential of body and surface couples The body couples G are conservative if there exists a function such that ()"G H , (2.1) G RG R where is the total di!erential of the function . The function is the potential of body couples G. Relation (2.1) is equivalent to the three partial di!erential equations "H G (2.2) HG G H for the dependent variable . We de"ne as a function of the by means of implicit equation of the form <( ,)"0. H From (2.3) we "nd, di!erentiating
(2.3)
< < # "0 (2.4) H H and if we solve for / and substitute the values H so obtained in (2.2) we see that < satis"es a system of three linear homogeneous partial di!erential equations < < #H G "0. HG G H The conditions of integrability of (2.2) are (H G )" (H G ). HG G IG G I H
(2.5)
(2.6)
I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
It is easy to show that (2.6) are necessary and su$cient conditions in order that system (2.5) be a Jacobean system [5]. The general integral of the system will be a function of one variable. We shall consider only the case in which the function G depends only of microrotation tensor (G"G(R)). Theorem. Conditions (2.6) are fulxlled if and only if the functions G satisfy the relations I G G G. H !R (2.7) G "R GO R I IN R IO HN Proof. Conditions (2.6) can be written as
H G H R R HG ! IG " H G . NO !H NO G G IG HG R I NO H H I (2.8) Making use of the relations (1.10) and (1.11) we obtain e
G P. G H H "e R (H H !H H ) KLG G NK RL HIO OF NP RI NI RP R HI (2.9)
Multiplying both sides of relation (2.9) by e H\H\ summing and using (1.9) and the HIJ IN JR identity e e "2 (2.10) GHI NHI GN we obtain condition (2.7). Suzciency: The right hand side of the relation (2.7) can be written as G G H !R G R IN R GO R HN IO G G G !R G "R IN GH R FO GF R HN IO G G. "R ( ! ) (2.11) FN GH IF GF IH R HN Substituting relation (2.11) into (2.7) we "nd that G G, G "e e R I GIO HFO FQ R HQ
(2.12)
567
where, to obtain Eq. (2.12) we have used relation (1.9) . Multiplying (2.12) by e H H and using IKL RL NK (1.9) , (1.10) and (1.11) we obtain conditions (2.6). Let us "nd the potential in the case in which the function G is of the form G "A R , (2.13) G GHI HI where A are real constants. GHI Substituting these relations into (2.7) we obtain (A !A !A )R "0. (2.14) GGO NR NRO RNO RO Eq. (2.14) will be satis"ed for arbitrary values of tensor R if and only if are veri"ed the conditions A !A !A "0. NR GGO NRO RNO Taking t"p we obtain
(2.15)
A !2A "0. RRO NNO The solution of system (2.16) is
(2.16)
A "A "A "0. O O O Substituting (2.17) into (2.15) we "nd
(2.17)
A "!A . (2.18) NRO RNO With (2.17) and (2.18) the solution of system (2.15) can be written as A "e B , GHI GHN IN where B are arbitrary constants. IN Relation (2.13) becomes
(2.19)
G "e R B . (2.20) G GHN HI IN Now the system of equations (2.5) has the form < < #e H R B "0. (2.21) GON HG OI IN H The adjoint system [5] corresponding to (2.21) consists of the single equation H R B d . (2.22) GON HG OI IN H If (1.10) is taken into account, then from (2.22) we deduce
d"e
d"B dR . IN NI
(2.23)
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I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
The general integral of system (2.21) must be of the form <"(!B R ), IN NI where is an arbitrary function. The general integral of Eq. (2.1) is to be found by setting (!B R )"0. We take the function IN IN < to coincide with the identity function, in other words the potential of the body couple is of the form "B R . (2.24) IN NI Proceeding exactly as we did in obtaining relation (2.24) from Eq. (2.1) we "nd the potential of couple stress vector "b R , NI NI where b are arbitrary constants. IN
(2.25)
3. Variational principles With the help of relation (1.10) we "nd that the "rst variation of tensor c may be written in the form c "R y #e y R H . GH NH N G NKL N G LH RK R From relation (1.4) we obtain
(3.1)
H (3.2) "H # HI . GH HI I G I G O O By employing (1.6) and (1.13), the following form of relation (3.2) is obtained: R " (H )! NH H . GH x HI I x IN I G G Consider the integral
(3.3)
J(u, )" (=(c, )!F u !B R ) d< G G IN NI T
(f u #b R ) d. (3.4) G G IN NI /4 First Variational Theorem. The deformation (u, ) is a solution of the boundary-value problem specixed by (1.14) and (1.16) if and only if J(u, ) of (3.4) vanishes for all variation u and such that u"0 when x3S and "0 when x3S . !
To prove this theorem, we compute the variation J(u, ) when u"0 on S and "0 on S . An easy calculation, making use of relations (3.1), (3.3), (1.6) , (2.1), (2.24), (2.25) and the divergence theorem, gives the result
J(u, )"! [(R t ) #F ]u #[(R m ) NH GH G N N NH GH G T #e y R t #G ]H d< NKL K G LH GH N IN I
#
Q
(R t n !f )u NH GH G N N
(R m n !g )H d. (3.5) NH GH G N IN I Q It is clear that (3.5) vanishes for arbitrary u and if and only if (1.14) and (1.16) are valid, which proves the theorem. Assume now that relation (1.15) is invertible, so that the second Piola}Kirchho! tensors t and GH m determine the strain tensors NO c "c (t , m ), " (t , m ). (3.6) GH GH FI NO GH GH FI NO The complementary strain-energy is de"ned by #
Z(t , m )"c t # m !=(c , ), (3.7) GH NO GH GH GH GH GH NO where c and are given by (3.6). GH NO We can easily show, with the help of Eq. (1.15) that Z Z . (3.8) c " , " GH m GH t GH GH For any given deformation (u, ) and any given "eld T, M of second Piola}Kirchho! tensors in <, consider the integral
K(u, , T, M)" (t c #m !Z(t , m ) GH GH GH GH GH NO T !F u !B R ) d< N N IN NI
!
Q
!
Q
!
Q
(u !u)R t n d N N NH GH G ( !)H m n d I I HI GH G
f u d! G G
Q
b R d. (3.9) IN NI
I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
The Second Variational Theorem. The deformation (u, ), the stress-tensor T and couple-stress tensor M give a solution of the boundary value problem specixed by (1.14), (1.16) and (3.8) if and only if K of (3.9) vanishes for all variations u, , T, M. Proceeding exactly as we did in obtaining relation (3.5) from (3.4) we "nd
Z c ! t GH t GH T GH Z m # ! GH m GH GH ![(R t ) #F ]u NH GH G N N ! [(R m ) #e y R t NH GH G NOL O G LH GH #G ]H d< N IN I
K(u, , T, M)"
!
Q
#
Q
(u !u)n (R t ) d N N G NH GH (t R n !f )u d GH NH G N N
!
Q
569
( !)n (m H ) d I I G GH HI
(R m n !g )H d. NH GH G N IN I Q This variation vanishes for arbitrary u, , T and M if and only if (3.8), (1.14) and (1.16) hold which proves the theorem. #
References [1] B.C. Kafadar, A.C. Eringen, Micropolar media * I. The classical theory, Int. J. Engng. Sci. 9 (1971) 271. [2] I. Nistor, On "nite deformation of elastic Cosserat continuum, Bull. Inst. Politehnic Iasi, s. IV XXII (1976) 20. [3] E. Reissner, Note on the equations of "nite strain force and moment stress elasticity, Stud. Appl. Math. 54 (1975) 1. [4] I. Nistor, Systematic methods of approximation in nonlinear theory of Cosserat media I, Bull. Inst. Politehnic Iasi 38 (s.I) (1982) 53. [5] G.F.D. Du!, Partial Di!erential Equations, The University of Toronto Press, Toronto, 1956.
Note
Variational principles for Cosserat body Ion Nistor Department of Mathematics, Technical University of Ias7 i, Ias7 i 6600 Romania Received 15 July 2000; accepted 12 October 2000
Abstract Two variational principles are derived for the mixed boundary value problem of Cosserat solid. These principles are a generalization of the stationary principle of potential energy and the stationary principle of complementary energy from non-linear theory of elasticity. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Cosserat solid; Principle of stationary potential energy; Principle of stationary complementary energy
1. Basic equations In the following discussion, all quantities are referred to the reference con"guration which is an undistorted stress-free con"guration. They are therefore functions of the coordinates of the place occupied by a particle X in reference con"guration. The deformation of a Cosserat body is described [1] by y "y (x ), R "R (x ), (1.1) G G H GH GH I where x and y are the coordinates of place occuG G pied by the particle X in the reference con"guration and in the present con"guration respectively, R are the components of a proper orthogonal GH tensor R which characterizes the rotation undergone by the material particles. The components of the tensor R can be represented [2] in the form R "[(1! ) # #e ]/ GH I I GH G H HGN N (1# ), F F E-mail address: i } [email protected] (I. Nistor).
where are the components of the microrotation H vector , e is the alternating symbol. GHN We consider the strain tensors c and de"ned by c "y R , "e R R , (1.3) GH I G IH GH HKL NL NK G where the symbol `,ia denotes partial derivation with respect to the variable x . G The tensor can be written [2,3], as "H , GH HI I G where
1 R FK H " e R HI 2 HKL FL I "(2 #e )/(2# ). (1.5) HI HIN N G G It is easy to verify that the matrix H"(H ) is GH non-singular and satis"ed the relations RH"HR"HR
(1.2)
(1.4)
(1.6)
where HR is the transpose of the matrix H. Substituting (1.5) into (1.6) and using the identity e R R "e R , (1.7) LHK FL GH FGN NK
0020-7462/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 1 1 3 - X
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I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
we obtain 1 R NK . H " e R (1.8) IG 2 GHN HK I Multiplying both sides of relation (1.8) by e adding and using the identities GLO e e " ! , GHI GNO HN IO HO IN R R "R R " , (1.9) LN LK NI KI NK we have R R ON "e R H . OK , (1.10) H e "R OGL LN IG IG GLO LK I I We can easily show, with the help of relations (1.8), (1.9) and (1.10) , that H H GH ! IH "e H H . (1.11) HKL GL IK I G From (1.6) we "nd, on di!erentiating H H R GH "R HO #H OG . (1.12) OG HO I I I Eq. (1.12) can be written, after using relations (1.10) and (1.11), as H H GH "R IN . (1.13) NG I H The equilibrium equations of Cosserat media are [1,4] (R t ) #F "0, GH IH I G (R m ) #e y R t #G "0, (1.14) GH IH I GHI H N IO NO G =(c, ) =(c, ) t " , m " . (1.15) IH IH c IH IH In these relations we have used the following notations: t , m * components of stress tensor GH GH T and couple stress tensor M; F , G * compoG G nents of body force vector F and body couple vector G per unit volume; = * strain energy per unit volume. The boundary < of the reference con"guration consists of regular surfaces S , 1)i)4 G
(S S "S S ", S S "S S "<) for which the following holds: u(x)"u(x) on S , R t n "f (x) on S , GH IH I G (x)"(x) on S , R m n "g (x) on S , GH IH I G (1.16) where u(x), (x) are given displacement and microrotation vectors, f( f ), g(g ) are given surface G G tractions and couples, n(n ) is the unit outward G normal on <.
2. The potential of body and surface couples The body couples G are conservative if there exists a function such that ()"G H , (2.1) G RG R where is the total di!erential of the function . The function is the potential of body couples G. Relation (2.1) is equivalent to the three partial di!erential equations "H G (2.2) HG G H for the dependent variable . We de"ne as a function of the by means of implicit equation of the form <( ,)"0. H From (2.3) we "nd, di!erentiating
(2.3)
< < # "0 (2.4) H H and if we solve for / and substitute the values H so obtained in (2.2) we see that < satis"es a system of three linear homogeneous partial di!erential equations < < #H G "0. HG G H The conditions of integrability of (2.2) are (H G )" (H G ). HG G IG G I H
(2.5)
(2.6)
I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
It is easy to show that (2.6) are necessary and su$cient conditions in order that system (2.5) be a Jacobean system [5]. The general integral of the system will be a function of one variable. We shall consider only the case in which the function G depends only of microrotation tensor (G"G(R)). Theorem. Conditions (2.6) are fulxlled if and only if the functions G satisfy the relations I G G G. H !R (2.7) G "R GO R I IN R IO HN Proof. Conditions (2.6) can be written as
H G H R R HG ! IG " H G . NO !H NO G G IG HG R I NO H H I (2.8) Making use of the relations (1.10) and (1.11) we obtain e
G P. G H H "e R (H H !H H ) KLG G NK RL HIO OF NP RI NI RP R HI (2.9)
Multiplying both sides of relation (2.9) by e H\H\ summing and using (1.9) and the HIJ IN JR identity e e "2 (2.10) GHI NHI GN we obtain condition (2.7). Suzciency: The right hand side of the relation (2.7) can be written as G G H !R G R IN R GO R HN IO G G G !R G "R IN GH R FO GF R HN IO G G. "R ( ! ) (2.11) FN GH IF GF IH R HN Substituting relation (2.11) into (2.7) we "nd that G G, G "e e R I GIO HFO FQ R HQ
(2.12)
567
where, to obtain Eq. (2.12) we have used relation (1.9) . Multiplying (2.12) by e H H and using IKL RL NK (1.9) , (1.10) and (1.11) we obtain conditions (2.6). Let us "nd the potential in the case in which the function G is of the form G "A R , (2.13) G GHI HI where A are real constants. GHI Substituting these relations into (2.7) we obtain (A !A !A )R "0. (2.14) GGO NR NRO RNO RO Eq. (2.14) will be satis"ed for arbitrary values of tensor R if and only if are veri"ed the conditions A !A !A "0. NR GGO NRO RNO Taking t"p we obtain
(2.15)
A !2A "0. RRO NNO The solution of system (2.16) is
(2.16)
A "A "A "0. O O O Substituting (2.17) into (2.15) we "nd
(2.17)
A "!A . (2.18) NRO RNO With (2.17) and (2.18) the solution of system (2.15) can be written as A "e B , GHI GHN IN where B are arbitrary constants. IN Relation (2.13) becomes
(2.19)
G "e R B . (2.20) G GHN HI IN Now the system of equations (2.5) has the form < < #e H R B "0. (2.21) GON HG OI IN H The adjoint system [5] corresponding to (2.21) consists of the single equation H R B d . (2.22) GON HG OI IN H If (1.10) is taken into account, then from (2.22) we deduce
d"e
d"B dR . IN NI
(2.23)
568
I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
The general integral of system (2.21) must be of the form <"(!B R ), IN NI where is an arbitrary function. The general integral of Eq. (2.1) is to be found by setting (!B R )"0. We take the function IN IN < to coincide with the identity function, in other words the potential of the body couple is of the form "B R . (2.24) IN NI Proceeding exactly as we did in obtaining relation (2.24) from Eq. (2.1) we "nd the potential of couple stress vector "b R , NI NI where b are arbitrary constants. IN
(2.25)
3. Variational principles With the help of relation (1.10) we "nd that the "rst variation of tensor c may be written in the form c "R y #e y R H . GH NH N G NKL N G LH RK R From relation (1.4) we obtain
(3.1)
H (3.2) "H # HI . GH HI I G I G O O By employing (1.6) and (1.13), the following form of relation (3.2) is obtained: R " (H )! NH H . GH x HI I x IN I G G Consider the integral
(3.3)
J(u, )" (=(c, )!F u !B R ) d< G G IN NI T
(f u #b R ) d. (3.4) G G IN NI /4 First Variational Theorem. The deformation (u, ) is a solution of the boundary-value problem specixed by (1.14) and (1.16) if and only if J(u, ) of (3.4) vanishes for all variation u and such that u"0 when x3S and "0 when x3S . !
To prove this theorem, we compute the variation J(u, ) when u"0 on S and "0 on S . An easy calculation, making use of relations (3.1), (3.3), (1.6) , (2.1), (2.24), (2.25) and the divergence theorem, gives the result
J(u, )"! [(R t ) #F ]u #[(R m ) NH GH G N N NH GH G T #e y R t #G ]H d< NKL K G LH GH N IN I
#
Q
(R t n !f )u NH GH G N N
(R m n !g )H d. (3.5) NH GH G N IN I Q It is clear that (3.5) vanishes for arbitrary u and if and only if (1.14) and (1.16) are valid, which proves the theorem. Assume now that relation (1.15) is invertible, so that the second Piola}Kirchho! tensors t and GH m determine the strain tensors NO c "c (t , m ), " (t , m ). (3.6) GH GH FI NO GH GH FI NO The complementary strain-energy is de"ned by #
Z(t , m )"c t # m !=(c , ), (3.7) GH NO GH GH GH GH GH NO where c and are given by (3.6). GH NO We can easily show, with the help of Eq. (1.15) that Z Z . (3.8) c " , " GH m GH t GH GH For any given deformation (u, ) and any given "eld T, M of second Piola}Kirchho! tensors in <, consider the integral
K(u, , T, M)" (t c #m !Z(t , m ) GH GH GH GH GH NO T !F u !B R ) d< N N IN NI
!
Q
!
Q
!
Q
(u !u)R t n d N N NH GH G ( !)H m n d I I HI GH G
f u d! G G
Q
b R d. (3.9) IN NI
I. Nistor / International Journal of Non-Linear Mechanics 37 (2002) 565}569
The Second Variational Theorem. The deformation (u, ), the stress-tensor T and couple-stress tensor M give a solution of the boundary value problem specixed by (1.14), (1.16) and (3.8) if and only if K of (3.9) vanishes for all variations u, , T, M. Proceeding exactly as we did in obtaining relation (3.5) from (3.4) we "nd
Z c ! t GH t GH T GH Z m # ! GH m GH GH ![(R t ) #F ]u NH GH G N N ! [(R m ) #e y R t NH GH G NOL O G LH GH #G ]H d< N IN I
K(u, , T, M)"
!
Q
#
Q
(u !u)n (R t ) d N N G NH GH (t R n !f )u d GH NH G N N
!
Q
569
( !)n (m H ) d I I G GH HI
(R m n !g )H d. NH GH G N IN I Q This variation vanishes for arbitrary u, , T and M if and only if (3.8), (1.14) and (1.16) hold which proves the theorem. #
References [1] B.C. Kafadar, A.C. Eringen, Micropolar media * I. The classical theory, Int. J. Engng. Sci. 9 (1971) 271. [2] I. Nistor, On "nite deformation of elastic Cosserat continuum, Bull. Inst. Politehnic Iasi, s. IV XXII (1976) 20. [3] E. Reissner, Note on the equations of "nite strain force and moment stress elasticity, Stud. Appl. Math. 54 (1975) 1. [4] I. Nistor, Systematic methods of approximation in nonlinear theory of Cosserat media I, Bull. Inst. Politehnic Iasi 38 (s.I) (1982) 53. [5] G.F.D. Du!, Partial Di!erential Equations, The University of Toronto Press, Toronto, 1956.