journal of computational and applied mechanic1

21
Journal of Com putational and App lied Mechani cs, Vol. 2., No. 2., (2001) pp. 205 221 TIME RATES OF TENSORS IN CONTINUUM MECHANICS UNDER ARBITRARY TIME DEPENDENT  TRANSFORMATIONS PART I. MATERIAL TIME RATES  IMRE KOZAK  Department of Mechanics, University of Miskolc  3515 Miskolc Egyetemvaros, Hungary  mechkozOgold.uni-miskolc.hu  [Received: April 16, 2001]  Abstract. The motion of a body and material time rates of tensors are investigated in three coordinate systems: in a fixed and a relative one the latter is assumed to move arbitrarily wi th respect to the fixed coordinate system, i.e., it is deformable and also in the convected coordinate system. Relations have been derived for the material time rates of arbitrary te nsors in all three coordinate syst ems.  Mathematical Subject Classification: 73A05 Keywords: Kinematics of continua, moving coordinate systems, relative motion of a body,  material time rates of tensors  1. Introduction 1.1. Components of tensors at a point of space can be transformed from on e coordinate system into another by making use of the general transformation rules of tensors. If the coordinate systems move with respect to each other, one speaks about time dependent transformations.  If the motion of the coordinate systems rel ative to each other is arbitrary (o ne of the coordinate systems is deformed with respect to the other) then the transformation is also referred to as arbitrary, otherwise, i.e., for a rigid body motion as relative motion of the coordinate system, the trans formation is an orthogonal one and in both cases time dependent.  From this point of view those tensors (including some time rate of tensors) which can be denned independently of the choice of coordinate systems moving arbitrarily with respect to each other, i.e., which are invariant under any arbitrary and time de pendent transformations, will be referred to as physically (or materially) objective, or, for the sake of brevity, objective tensors or objective rates. (We remark that in the literature criteria of physical objectivity are valid mostly for orthogonal transformation only.)  Fulfillment of physical objectivity is a necessary (but not s ufficie nt) condition for  

Upload: juli-ardita-pribadi

Post on 08-Apr-2018

220 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 1/21

Page 2: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 2/21

Page 3: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 3/21

Page 4: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 4/21

Page 5: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 5/21

Page 6: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 6/21

Page 7: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 7/21

Page 8: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 8/21

Page 9: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 9/21

Page 10: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 10/21

Page 11: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 11/21

Page 12: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 12/21

Page 13: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 13/21

Page 14: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 14/21

Page 15: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 15/21

Page 16: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 16/21

Page 17: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 17/21

Page 18: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 18/21

Page 19: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 19/21

Page 20: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 20/21

Page 21: Journal of Computational and Applied Mechanic1

8/7/2019 Journal of Computational and Applied Mechanic1.....

http://slidepdf.com/reader/full/journal-of-computational-and-applied-mechanic1 21/21