THE DISPLACEMENT GRADIENT AND THE LAGRANGIAN STRAIN TENSOR Revision B

Size: px

Start display at page:

Download "THE DISPLACEMENT GRADIENT AND THE LAGRANGIAN STRAIN TENSOR Revision B"

Muriel Shields
6 years ago
Views:

1 HE DISPLACEMEN GRADIEN AND HE LAGRANGIAN SRAIN ENSOR Revision B By om Irvine tom@irvin .org Febrary, 05 Displacement Graient Sppose a boy having a particlar configration at some reference time t o changes to another configration at time t, with both rigi boy motion an elastic eformation possible. Q(t o ) (+) Q(t) P(t o ) () P(t) 0 Figre. A typical material point P nergoes a isplacement so that it arrives at the position,t ()

2 A neighboring point Q at + arrives at ) + ( t, () hs, t - ) + ( (),t ) - + ( t, (4),t ) - + (, t (5) his eqation may be rewritten sing the graient of a vector fiel. (6) is a secon orer tensor known as the isplacement graient with respect to. (7) Note that for rigi boy translation 0.

3 Lagrangian Strain ensor Consier two material vectors an issing from point P in Figre. hrogh the motion, becomes an becomes. (8) (9) A measre of the eformation is given by the ot proct of an. (0) () By the efinition of the transpose, () An () By sbstittion, (4) (5)

4 Define the Lagrangian strain tensor as E* (6) E* (7) E* characteries the eformation of the neighborhoo of the particle P. Note that for rigi boy translation: 0 (8) E* =0 (9) (0) For Cartesian coorinates j E * i k k () j i i j. For small eformations 0 E () he omission of the * symbol inicates small eformations. For small eformations in Cartesian coorinates, j E i () j i 4

5 he infinitesimal strain tensor is E, symmetric (4) he nit elongation in the irection is E e Ee (5) References. Lai, Rbin, Krempl, Introction to Continm Mechanics, Revise Eition in SI/Metric Units, Pergamon Unifie Engineering Series, Volme 7, New York, 98. Hjelmsta, Fnamentals of Strctral Mechanics, Secon Eition, Springer, New York, Chng, General Continm Mechanics, Cambrige University Press,

6 APPENDI A Deformation Graient C is the coorinate in the eforme configration is the coorinate in the neforme configration is the crve is the eformation map B C (s+s) (s+s) - (s) (s+s) (s+s) - (s) s (s) C s (s) C Uneforme Coorinates Deforme Coorinate Figre A-. Measring the istance between two points on a crve 6

7 Noting that the parameterie crve in the eforme configration is etermine by the eformation maps as (s) = (Z(s)), one can apply the chain rle for ifferentiation to relate the vectors tangent to the crves in the eforme an neforme configrations. In components, noting that the map can be written i (s) = i (s),(s), (s) (A-) We can compte the erivative by the chain rle as s (s) = i i j j s s (A-) Note that the components i are simply the components of the tensor. j his tensor plays sch an important role in the sbseqent evelopments that we shall give it a special name an symbol. We call F() the eformation graient becase it characteries that rate of change of eformation with respect to material coorinates. hs F( ) ( ) (A-) F (A-4) s s he eformation graient carries the information abot the stretching in the infinitesimal neighborhoo of the point. It also carries information abot the rotation of the vector /s. he eformation graient F is a tensor with the coorinate representation. 7

8 F( ) i ( ) ei g j (A-5) where e i are the base vectors in the eforme configration g i are the base vectors in the neforme configration Strain in hree-imensional Boies he Green eformation tensor C is C F F (A-6) he stretch of the line oriente in the irection n of the neforme configration can then be compte as ( n) n Cn (A-7) Eqation (A-7) hols for any crve with /s = n. he Lagrangian strain is the ifference between the sqare of the eforme length an the sqare of the original length ivie by twice the sqare of the original length. he strain E in the irection n is ( n) n En E ( n) (A-8) he Lagrangian strain tensor E is efine to be have the ifference between the Green eformation tensor an the ientity tensor I as E C I (A-9) 8

9 APPENDI B Figre B-. Lagrangian Coorinates he neforme state is efine by rectanglar Cartesian coorinates, an the eforme state by arbitrary crvilinear (convecte) coorinates. Strain ensor in Solis s G ij iji j iji j s0 (B-) he metric tensor G ij is Gij Gi G j R i R j m i m j (B-) 9

10 where G i R is a tangent vector is the position vector to the point P in the eforme state at time t=t Note that G i ki i i k k (B-) G m i i i m (B-4) R = R(r, t) (B-5) where r is the position vector from the origin of the rectanglar Cartesian coorinates to a point P o at time t=t o in the neforme state r = i i i (neforme) (B-6) R = i i i (eforme) (B-7) he isplacement vector is =R-r (B-8) 0

11 Jacobian he Jacobian J is j i J (B-9) G ij G J (B-0) he Jacobian for ilatation is o J (B-)

12 APPENDI C Finite Strain ensor H * i j j i k i k j H * k k H * k k H * k k H * k k

13 H * k k H * k k

2.13 Variation and Linearisation of Kinematic Tensors

2.13 Variation and Linearisation of Kinematic Tensors Section.3.3 Variation an Linearisation of Kinematic ensors.3. he Variation of Kinematic ensors he Variation In this section is reviewe the concept of the variation, introce in Part I, 8.5. he variation