Exact Solutions in Finite Compressible Elasticity via the Complementary Energy Function

Transcription

1 Exact Solutions in Finite Comressible Elasticity via the Comlementary Energy Function Francis Rooney Deartment of Mathematics University of Wisconsin Madison, USA Sean Eberhard Mathematical Institute, University of Oxford Oxford, UK Francis Rooney, Deartment of Mathematics, 80 Lincoln Dr., Madison, WI 53706,USA Setember 9, 0 Key Words: Comlementary Energy, exact solutions, comressible elasticity, Legendre transform Abstract The urose of this aer is to establish a method of obtaining closed form solutions in isotroic hyerelasticity using the comlementary energy, the Legendre transform of the strain energy function. Using the comlementary energy, the stress becomes the indeendent variable and the strain the deendent variable. Some of the imlications of this formulation of the equations are exlored and illustrative examles of solutions for sherical and cylindrical ination for several forms of the comlementary energy are resented.

2 Introduction The search for exact solutions of the equations of elasticity initially concentrated on incomressible materials. The simlied kinematics allows a wide variety of analytic solutions to be calculated. Much of the research concerned deformations ossible in any material, the universal deformations, see for examle Ogden []. Early on, though, Ericksen [] showed that in comressible materials that the only universal solutions were homogeneous. The rst inhomogeneous solutions for comressible materials were for the harmonic materials due to John [3]. Another large class of solutions, for strain energies deending on the invariants of the stretch tensor, were discovered by Carroll []. These included the harmonic materials as a secial case. A third class that allows exact solutions are the Blatz-Ko materials [5]. A summary of all these solutions can be found in the aers by Horgan [6], Carroll [7] and Horgan [8]. A more systematic examination of the structure of the equations for sherical and cylindrical equations and their solution is given by Horgan and Murhy [0]. Treating the stress as the indeendent variable, rather than the strain or dislacement, is a common tactic in linear elasticity roblems. Examles include the Prandtl stress function in St. Venant torsion, the Airy function in lane strain or lane stress roblems and the Love stress function in axisymmetric situations. The resent aer utilizes this aroach in nite elasticity. The rst Piola-Kircho stress is taken to be the indeendent variable. It is chosen so that the equilibrium equation is satised identically. The governing equations are obtained by requiring that the tensor obtained from dierentiating the comlementary energy corresonds to the gradient of a deformation eld. Some constitutive restrictions on the comlementary energy are derived from the requirement that the innitesimal shear and bulk moduli be ositive. Several classes of comlementary energy are examined and solutions derived for sherical ination. The Legendre Transform Let B be a homogeneous three-dimensional continuum and let it occuy a xed homogeneous conguration 0 at time t = 0. For any article X B, let X be the osition vector of X in the conguration 0. Let x be the osition vector of X at time t, then a deformation of the body is given by x = (X t): The deformation gradient F is dened as A e i E A where e i and E A are basis vectors in the current and reference conguration and the summation convention is assumed. In the usual method of obtaining analytical solutions in hyerelasticity, one treats F as the indeendent variable, where F is the deformation gradient. The rst Piola-Kirchho stress tensor P is derived using the constitutive relation () where the strain-energy function W (F) is a rescribed function of the deformation gradient. The equilibrium equation Div P = 0 () then yields the dierential equations of deformation.

3 In the alternative aroach, utilized in this aer, one treats the stress P as the indeendent variable. The domain of P is the set of tensor elds satisfying (). The Legendre transform of the strain-energy function W (F) with resect to P is denoted : The deformation gradient F can then be = P (P) = F P W = @W = F: () The dierential equations of deformation are derived from the integrability conditions for F. In cartesian coordinates, these B A i A B = 3: (5) In general these dierential equations dier from those given by the usual method, and this alternative method can lead to new solutions. Isotroy Let F = RU be the right olar decomosition of F. For isotroic materials, the strain-energy W can be written as a symmetric function W y of 3, the eigenvalues of U, or equivalently as a function W? of the rincial invariants i i i 3 of U: where Dene the Biot stress by W = W (F) = W y ( 3 ) = W? (i i i 3 ) (6) i = i = i 3 = 3 : (7) then if i are the eigenvalues of and () imlies Performing the Legendre (8) i i (9) P = R: (0) = W: () The comlementary energy can be written as a symmetric function y of 3, or equivalently as a function? of the invariants j j j 3 of : = (P) = y ( 3 ) =? (j j j 3 ) () where j = j = j 3 = 3 : (3) 3

4 The equation for F, @P + @? where adj(p) is the adjugate (i.e the matrix of cofactors) of 3 adj(p) () adj(p) = e ijk e ABC P jb P kc e i E A (5) For (), see for examle Steigmann [9]. In the usual formulation of elasticity, the fact that det(f) > 0 imlies that the rotation matrix in the olar decomosition for F is uniquely determined. Since det(p) doesn't have a denite sign there are several dierent rotation matrices and accordingly dierent Biot stresses that can corresond to a given P. For a thorough discussion of this issue see Ogden [] age 358. Materials We will consider the following forms of the comlementary energy A ^ A (j j ) = (f(j ) j ) (6) B ^ B (j j ) = g(aj + j j ) + bj (7) C ^ C ( 3 )) = (h(( + c)( + c)( 3 + c)) + dj ) (8) where a b c d are constants and f g h are arbitrary functions of their arguments. In the strain energy formulation, the innitesimal bulk and shear moduli are given by 3 W (9) (or any cyclic ermutation of 3 ) where the derivatives are evaluated at = = 3 =. Due to the roerties of the Legendre transform, the equivalent exressions for the moduli in terms of the comlementary energy are 3 y y y (0) evaluated at zero stress. In addition the stretches at zero stress should be one. These conditions give the following restrictions on the comlementary energy Material A f 0 (0) = f 00 (0) = Material B a + b = g0 (0) = g 00 (0) = 3a 3 +

5 Material C d c = h 0 (c 3 ) = h 00 (c 3 ) = 3c 3 + Sherical Ination Assume the deformation can be given in sherical olar coordinates by r = ^r(r) = = : () The equilibrium equation () reduces to d dr (P rr) + R (P rr P ) = 0: () For this deformation eld we have R = I: This means that the the rincial values of the Biot stress are given by = P rr = 3 = P : (3) The constitutive equations give F rr = dr The integrability condition in this case is just F = r : () which reduces to where we have written d dr (RF ) = F rr (5) dr + = 0 @ : Material A For model A, = f 0 (j ) = f 0 (j ) ( + ) : (7) Substituting into (6) immediately gives Using the equilibrium equation () dr f 0 (j ) ( + ) + ( ) = 0: (8) f 0 (j ) dr j = 0: (9) 5

6 This can be integrated to give j = constant = 3k 0 : (30) Using the exression for j and the equilibrium equation gives an equation for Integrating leads to Now () gives dr + 3 = 3k 0 : (3) = k 0 + k R 3 = k 0 r = R f 0 (3k 0 ) k 0 k R 3 (3) k : (33) R 3 Material B For model B, let then j B = aj + j j (3) = (a + )g 0 (j B ) + b (35) = (a + )g 0 (j B ) + b : (36) Substituting these into the comatibility equation (6) and using the equilibrium equation (6) leads to (a + ) dr g0 (j B ) = 0: (37) Thus, either a = or j B = constant = 3k 0 : (38) The condition j B = 3k 0 leads to the dierential equation This has solution Using (36), r = R (a ) d dr + 3a 3 = 3k 0 : (39) r = a a k 0 + k R 3 : (0) a + ra k 0 + k R 3! g 0 (3k 0 ) + b! : () The other stress can be found using equilibrium and the condition (38) in the form = a r a k 0 + k R 3 3R q 3 A : () a k 0 + k R 3 6

7 Material C Using the form of stress energy given in (8) we obtain where The comatibility equation (6) becomes since Equation (5) becomes which gives either = c or = ( + c)( 3 + c)h 0 (j C ) + d (3) = ( + c)( 3 + c)h 0 (j C ) + d () j C = ( + c)( + c)( 3 + c): dr ( + c)( + c)h 0 (j C ) + ( )h 0 (j C ) = 0 (5) + c + c = r jc + c : (6) dr jc h 0 (j C ) = 0 (7) j C = constant j C = k 0 or j C = k 0 : (8) Thus + c = k 0 + c or + c = k 0 ( + c) : (9) Using the equilibrium equation () gives dr + + c = k 0 + c or dr + + c = k 0 ( + c) (50) with solutions + c = (k 0 + k R 3 ) =3 or + c = ( k 0 + k R 3 ) =3 (5) where k is an integration constant. From (5) the other stress comonent is + c = k 0 (k 0 + k R 3 ) =3 or + c = k 0 ( k 0 + k R 3 ) =3 : The dislacement is obtained from () as r = R k 0 (k 0 + k R 3 ) =3 h 0 (k 0) + d or r = R k 0 ( k 0 + k R 3 ) =3 h 0 ( k0) + d : (5) 7

8 Cylindrical Ination Assume the deformation can be given in cylindrical olar coordinates by where is a constant. The equilibrium equations () reduce to d dr (P rr) + R (P rr P ) = 0 For this deformation eld we again have R = I: r = ^r(r) = z = Z: (53) dp zz dz This means the the rincial values of the Biot stress are given by The constitutive equations give F rr = dr The comatibility conditions in this case are which reduce to Material A = 0: (5) = P rr = P 3 = P zz : (55) d dr (RF ) = F rr F = r dr + = 0 F zz 3 : (56) d dr (F zz) = 0 (57) d 3 dr = 0: (58) For model A, = f 0 (j ) ( + 3 ) = f 0 (j ) ( + 3 ) 3 = f 0 (j ) ( + ) : Substituting into (58) immediately gives Using the equilibrium equation (5) This together with imlies that (59) dr f 0 (j ) ( + 3 ) + ( ) = 0: (60) dr f 0 (j ) 3 = 0: (6) 3 = = f 0 (j ) ( + ) (6) j = k 0 + = f 0 (k 0 ) 3 = k 0 + f 0 (k 0 ): (63) Integrating (63) using the equilibrium equation (5) gives Now (56) gives = f 0 (k 0 ) + k R = f 0 (k 0 ) + k R : (6) r = R 3 f 0 (k 0 ) k 0 k 3 : (65) R 8

9 Material B For model B, let then j B = aj + j j (66) = (a + 3 )g 0 (j B ) + b (67) = (a + 3 )g 0 (j B ) + b (68) 3 = (a + 3 )g 0 (j B ) + b : (69) 3 = ) ( 3 )g 0 (j B ) = (a )g 0 (j B ) = b : (70) Substituting this into the comatibility equation (58) reduces to Thus, either dr (a )g 0 (j B ) + b + ( )g 0 (j B ) = 0: (7) = a or j B = constant = k 0 : (7) The condition 3 = leads to 3 = + + K K = b g 0 (j B ) a (73) Using this, the condition j B = k 0, and the equilibrium equation gives the dierential equation with solution (a ) dr + K = a + ak = k 0 (7) = a r (K + ak + a k 0 ) + k R : (75) The stress can be found from the condition j B = k 0, and 3 from (73) 3 = a + K The dislacement is q = a K + ak + a k 0 (76) (K + ak + a k 0 ) + k R r (K + ak + a k 0 ) + k R + r = R b r (K + ak + a k 0 ) + k R! q K + ak + a k 0 A : (77) (K + ak + a k 0 ) + k R : (78) 9

10 Material C Using the form of stress energy given in (8) we obtain where The comatibility equation (58) becomes This gives = ( + c)( 3 + c)h 0 (j C ) + d (79) = ( + c)( 3 + c)h 0 (j C ) + d (80) = ( + c)( 3 + c)h 0 (j C ) + d : (8) j C = ( + c)( + c)( 3 + c): dr ( + c)( 3 + c)h 0 (j C ) + ( )( 3 + c)h 0 (j C ) = 0: (8) The other comatibility equation 3 = gives Together (83) and (8) imly ( 3 + c)h 0 (j C ) = constant: (83) ( + c)( + c)h 0 (j C ) = : (8) j C = k 0 ( + c)( + c) = h 0 (k 0 ) : (85) The second equation together with the equilibrium equation gives the dierential equation This has the solution The other stresses are dr + + c = = c The dislacement is obtained from (56 ) as s ( + c)h 0 (k 0 ) : (86) h 0 (k 0 ) + k R : (87) = c = h 0 (k 0 ) h 0 (k 0 ) + k R (88) r = R k 0(h 0 (k 0 )) 3 = c + k 0h 0 (k 0 ) : (89) s h 0 (k 0 ) + k R + d! : (90) 0

11 Conclusion A new method for obtaining closed form solutions for sherical and cylindrical ination of isotroic comressible materials is roosed in this aer. Exloiting the duality relation between the comlementary energy and strain energy, taking the stress to be the indeendent variable, gives rise to a dierent set of equations than the usual aroach using the strains. In some cases they are more tractable and allow new solutions to be found. Postscrit This research received no secic grant from any funding agency in the ublic, commercial, or not-for-rot sectors References [] Ogden R.W., Non-Linear Elastic Deformations, Ellis Horwood, Chichester (98). [] Ericksen J.L., Deformations ossible in every comressible isotroic erfectly elastic material, J. Math. Physics 955 : 6 { 8. [3] John F., Plane Strain Problems for a Perfectly Elastic Material of Harmonic Tye, Comm. Pure Al. Math : 39 { 96. [] Carroll M.M., Finite Strain Solutions in Comressible Finite Elasticity, J. Elasticity : 65 { 9. [5] Blatz P.J. and Ko. W.L., Alication of nite elasticity to the deformation of rubbery materials, Trans. Soc. Rheol :3 { 5. [6] Horgan C.O., On axisymmetric solutions for comressible nonlinearly elastic solids, Zeit. Angew. Math. Phys : S07 { S5. [7] Carroll M.M., On obtaining closed form solutions for comressible nonlinearly elastic materials, Zeit. Angew. Math. Phys : S6 { S5. [8] Horgan C.O., Equilibrium solutions for comressible nonlinearly elastic materials, Nonlinear Elasticity : Theory and Alications, Fu Y.B. and Ogden R.W. eds, London Mathematical Society { 59. [9] Steigmann D.J., Invariants of the Stretch Tensors and their Alications to Finite Elasticity Theory, Mathematics and Mechanics of Solids 007: 393{0 [0] Horgan C.O. and Murhy J.G., A Lie Grou Analysis of the Axisymmetric Equations of Finite Elastostatics for Comressible Materials, Mathematics and Mechanics of Solids 005 0: 3{333.