The Deformation Gradient in Curvilinear Coordinates

Transcription

1 The Deformation Gradient in Curvilinear Coordinates St. Hartmann Technical Report Series Fac

2 Impressum Publisher: Fakultät für Mathematik/Informatik und Maschinenbau, Technische Universität Clausthal Am Regenbogen 15, Clausthal-Zellerfeld, Germany Editor-in-chief: Alfons Esderts Technical editor: Martina Wächter Contact: URL: ISSN: The Faculty of Mathematics/Computer Science and Mechanical Engineering Review Board Prof. Dr. Frank Endres Prof. Dr. Alfons Esderts Prof. Dr. Stefan Hartmann apl. Prof. Dr. Günter Kemnitz Prof. Dr. Armin Lohrengel Prof. Dr. Norbert Müller Prof. Dr. Hubert Schwarze Prof. Dr. Volker Wesling

3 The Deformation Gradient in Curvilinear Coordinates Stefan Hartmann Institute of Applied Mechanics, Clausthal University of Technology, Adolph-Roemer-Str. 2a, Clausthal-Zellerfeld, Germany Abstract This short article offers an overview of the deformation gradient and its erminant in the case of curvilinear coordinates. In particular, the relation to a pure Cartesian description and the representation relative to a curvilinear basis is treated. This brief exercise is known, but hard to be found in the basic literature on Continuum Mechanics. Further, most of the literature on constitutive modeling refers to the symbolj, butj is not generally equivalent tof. The difference is derived in ail. 1

4 1 Introduction The basic problem arises from the fact that the Jacobian of the deformation gradient is not identical to the erminant of the deformation gradient in the case of curvilinear coordinates. In order to show this, we recall some relations provided by Marsden and Hughes (1983). (X 1,X 2,X 3 ) ˆX i (Θ 1,Θ 2,Θ 3 ) (Θ 1,Θ 2,Θ 3 ) Φ i R (X1,X 2,X 3,t) (x 1,x 2,x 3 ) ˆx i (ϑ 1,ϑ 2,ϑ 3 ) χ i R (Θ1,Θ 2,Θ 3,t) (ϑ 1,ϑ 2,ϑ 3 ) (1) represents the occurring coordinate transformations. (X 1,X 2,X 3 ) and (x 1,x 2,x 3 ) are Cartesian coordinates in the reference and the current configuration, respectively. Accordingly,x i = Φ i R (X1,X 2,X 3,t) defines the motion of a particle (X 1,X 2,X 3 ) if Cartesian coordinates are chosen. In the reference configuration, it is possible to choose curvilinear coordinates (Θ 1,Θ 2,Θ 3 ) as well. X i = ˆX i (Θ 1,Θ 2,Θ 3 ) defines the coordinate transformation between the curvilinear and the Cartesian coordinates in the reference configuration. In the current configuration other curvilinear coordinates might be chosen, e.g. (ϑ 1,ϑ 2,ϑ 3 ), which are given by the coordinate transformation between the curvilinear and the Cartesian coordinates in the current configuration x i = ˆx i (ϑ 1,ϑ 2,ϑ 3 ). Thus, the motion can also be written by ϑ i = χ i R (Θ1,Θ 2,Θ 3,t). In the following, tangent and gradient vectors are required. They are defined as follows: g i = ˆxk ϑ i e k, G i = ˆX k Θ i e k, g i = ˆϑ i x k e k (2) G i = ˆΘ i X k e k. (3) e k are the basis vectors in a Cartesian coordinate system. Furthermore, co- and contravariant metric coefficients relative to the reference and current configuration must be defined: g ij = g i g j = ˆxk ϑ i ˆx m ϑ j δ km = ˆxk ϑ i ˆx k ϑ j (4) g ij = g i g j = ˆϑ i x k ˆϑ j x mδ km = ˆϑ i x k ˆϑ j G ij = G i G j = ˆX k Θ i ˆX m Θ j δ km = ˆX k Θ i ˆX k G ij = G i G j = ˆΘ i X k ˆΘ j X mδ km = ˆΘ i X k ˆΘ j Without a recapitulation of the theory of tensor calculus, the following properties are stated, x k (5) Θ j (6) X k (7) [g ij ] = [g ij ] 1 and [G ij ] = [G ij ] 1, (8) see (Malvern, 1969; Ogden, 1984). In other words, the inverse of the matrix of covariant metric coefficients is identical to the matrix of contravariant metric coefficients. 2 Representation of the deformation gradient The deformation gradient F( X,t) = Grad χ R ( X,t) is defined as the gradient of the motion of a particle X occupying the place x at timet. In curvilinear coordinates, this reads F( X,t) = χi R Θ k g i G k. (9) If we insert the tangent vector g i and the gradient vector G k by the definitions (2) 1 and (3) 2, this yields F( X,t) = ˆxm ϑ i χ i R Θ k ˆΘ k X n e m e n. (10) FACULTY 3 2

5 Obviously, this represents the chain-rule Φ m R X n = ˆxm χ i R ˆΘ k ϑ i Θ k X n, (11) which implies the equivalence of (9) with the Cartesian representation of the deformation gradient F( X,t) = Φm R X n e m e n. (12) 3 Jacobian and erminant of deformation gradient In the following, it will be shown that the Jacobian J = [ χ i ] R Θ k is not equivalent to the erminant of the deformation gradient, F. This can be shown by insering relation G k = G kl Gl as well as the Eqns.(2) 1 and (3) 1 into the deformation gradient in curvilinear representation (9), which leads to F( X,t) = χi R Θ k g i G k = χi R ˆx m ˆX n Θ k ϑ i Θ l Glk e m e n. (14) The erminant of the matrix of covariant metric coefficients (4) reads ( k])( k]) [g ki ] = ϑ i ϑ j = ( k])2 where the multiplication theorem of matrices,(ab) = ( A)( B), is exploited. Or, to put it differently, k] ϑ i = [g ki ], (16) which holds also for the metric coefficients in the reference configuration [ ˆX ] n Θ l = [G ln ]. (17) Accordingly, the erminant of the deformation gradient reads ( [ χ i ])( [ F = R ˆx m]) ( [ Θ }{{ k ϑ i ˆX ]) n ( [G lk Θ }}{{} l ] ) = (18) }{{} }{{} 1/[G J [gmi] lk ] [Gln ] [gmi ] = J [Glk ], (19) see property (8) as well. In other words, F = J holds only if the erminants of the covariant metric coefficientsg ij andg ij are identical. This holds for Cartesian coordinates, but even for the most common cylindrical or spherical coordinates this is violated. In particular, J is used in the development of constitutive models to indicate the volumetric behavior of the material, which is misleading due to the aforementioned property. Its real application are in analytical solutions for compressible materials, see (Ogden, 1984), or in beam and shell theories, where curvilinear coordinates and compressible constitutive equations are used. Thus, J as an abbreviation, instead of F, has to be carefully applied. References Malvern, L. E. (1969). Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs. Marsden, J. E. and Hughes, T. J. R. (1983). Mathematical Foundations of Elasticity. Prentice-Hall, New Jersey. Ogden, R. W. (1984). Non-Linear Elastic Deformations. Ellis Horwood, Chichester. ϑ i (13) (15) 3 Technical Report