abaqus经典声学分析例题a1-finite-deations

1、Appendix 1Finite DeformationsCopyright 2006 ABAQUS, Inc.A1.2OverviewMotions and Displacements Extension of a Material Line Element The Deformation Gradient TensorFinite Deformations and Strain Tensors Decomposition of a DeformationPrincipal Stretches and Principal Axes of DeformationStrain Invariant

2、s SummaryModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Motions and DisplacementsCopyright 2006 ABAQUS, Inc.A1.4Motions and DisplacementsA body occupies the material within R0 att = 0. This is the reference configuration.The configuration at time tis the current configurat

3、ion.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.5Motions and DisplacementsThe motion of the body takes the reference configuration R0 into the current configuration R.An essential assumption of continuum mechanics is that the motion can be described asx = x ( X , t )

4、for every X in R0for every x in R In above expression, X act as independent variables; this is a Lagrangian (material) description of the problem.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.6Motions and DisplacementsThe motion can be described in terms of the displac

5、ement vector u:x = X + uoru = x - X.Lagrangian description:u( X ,t ) = x ( X ,t ) - X.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Extension of a Material Line ElementCopyright 2006 ABAQUS, Inc.A1.8Extension of a Material Line ElementA deformation is a motion in which a

6、change of shape can occur.For the purposes of stress analysis we need to separate that part of the motion that corresponds to a rigid-body motion from that part that involves deformation.A and a are unit vectors.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.9Extension

7、of a Material Line ElementGiven the motion x = x (X, t): We are interested in determining the length and orientation of the material line element after the motion.Straightforward analysis givesxila=A ,iRXRxiwhere l is the stretch ratio and F=is the deformation gradient.iRXRModeling Rubber and Viscoe

8、lasticity with ABAQUSCopyright 2006 ABAQUS, Inc.The Deformation Gradient TensorCopyright 2006 ABAQUS, Inc.A1.11The Deformation Gradient TensorxiXRThe nine quantities,gradient tensor, F :are the components of the deformation,xiF=.iRXRThey describe how a particle moves in relation to neighboring parti

9、cles.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.12The Deformation Gradient Tensor Our previous results for a material line element oriented in direction a in the current configuration and in direction A in the reference configuration can be summarized as follows:a =

10、 l -1F Al 2= A F T F AA = lF -1 a= a (F -1 )Tl -2 F -1 aModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.13The Deformation Gradient TensorRemarks:If there is no motion, x = X, and so F = I (identity).F is important in the analysis of deformation, but it is not a measure o

11、f deformation only (the motion includes rotation).We need measures that do not change when no deformation takes place;i.e., we want them to remain unchanged under rigid body motions: QT Q = Q QT = I = rotationx = Q X + c c = translation (does not vary with position)For a rigid body motion F = Q.Mode

12、ling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Finite Deformations and Strain TensorsCopyright 2006 ABAQUS, Inc.A1.15Finite Deformations and Strain TensorsConsider the tensor:C = F T F.Recall the result from the line extension:l 2= A F T F A = A C A; stretch of material line e

13、lementwith direction A in reference configuration. Knowledge of C at a point determines the local deformation in the vicinity of that point.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.16Finite Deformations and Strain Tensors Moreover, for rigid body motions F = Q, so

14、 C = QT Q = I. C is constant throughout a rigid body motion. C is connected with deformation and not with rigid body motion; therefore, it is a suitable measure of deformation. C is called the right Cauchy-Green deformation tensor. Note that C is not a unique measure of deformation; there are many o

15、ther candidates. But C is convenient because it is easy to calculate from F.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.17Finite Deformations and Strain TensorsRecall the result from the line extension:l -2= a F -T F -1 a, stretch of material line elementwith directi

16、on a in current configuration. B-1 = F -T F -1, and soB = F FTLetl-2= a B-1 a. B is called the left Cauchy-Green deformation tensor.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.18Finite Deformations and Strain TensorsThe Lagrangian strain tensor E (Green-Lagrange) is

17、defined byE = 1 (C - I ).2 A nice feature is that E = 0 for rigid body motions.C, B, and E are symmetric second-order tensors, so they have realprincipal values and orthogonal principal directions.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Decomposition of a Deformatio

18、nCopyright 2006 ABAQUS, Inc.A1.20Decomposition of a Deformation The deformation gradient tensor F can be expressed assameF = RU =V Rdescribes rotation of bodyright stretch tensorleft stretch tensorU and V are symmetric and unique for a given F.J = det(F) is the ratio of volume in the current configu

19、ration todVJ =volume in the reference configuration:.dV0J 0 for physically realistic deformations.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.21Decomposition of a Deformation The tensors U and V are related to the deformation tensors C and Bthrough:C = F T F = U 2B =

20、 F F T = V 2 Therefore, U and C are equivalent measures of deformation. For a given F, however, calculation of U is inconvenient, whereas the computation of C is straightforward. Similar remarks apply to V and B.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Principal Stre

21、tches and Principal Axes of DeformationCopyright 2006 ABAQUS, Inc.A1.23Principal Stretches and Principal Axes of Deformationl2= A C A.RecallFind directions A for which l takes extreme values. Find the minimum and maximum of l2constraint A A =1.Results in eigenvalue problem:C A* = l2 A*.= A C Aunder

22、theThe extreme values of l2 are the eigenvalues of C and occur in the directions of the eigenvectors (A* ) of C.Alternatively, the extreme values of l are eigenvalues of U (recallC = U2).Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.24Principal Stretches and Principal

23、Axes of Deformation Since U is symmetric and positive-definite, its principal values are real and positive:l1 l2 l3principal stretches Moreover, U has 3 orthogonal principal directions:A1, A2 , A3of UprincipalaxesModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.25Principa

24、l Stretches and Principal Axes of DeformationThe motion that corresponds to F = R Uconsists of three extensionsof magnitude l1, l2, l3 along the three directions A1, A2, A3, followed by the rotation R.A similar interpretation can be given for the motion F =V R.It can be shown that: The principal val

25、ues of l1, l2, l3 are also the principal values of V.= R A1,a2 = R A2 ,a3 = R A3A1, A2 , A3principal directionsa1of Uof VprincipaldirectionsModeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.26Principal Stretches and Principal Axes of DeformationSince C = Uand E = (C - I ),

26、 the principal directions of C and E2coincide with those of U.The principal values of C are l2 ,l2 ,l2.123()12l2-1i = 1, 2, 3.The principal values of E areiLikewise, the principal directions of B and V coincide.The principal values of B arel2,l2,l2.123Modeling Rubber and Viscoelasticity with ABAQUSC

27、opyright 2006 ABAQUS, Inc.Strain InvariantsCopyright 2006 ABAQUS, Inc.A1.28Strain InvariantsThe strain invariants are defined byI1 = tr ( B) = tr (F F T ),(I- tr ( B B),12I=212J = det (F ).In terms of the principal stretches these invariants areI1 = l 2 + l 2 + l 2,123I2 = l 2l 2 + l 2l 2 + l 2l 2,1

28、22331J = l1l2l3.Without deformation B = I, so I1 = I2 = 3, J=1.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.29Strain InvariantsIn ABAQUS revised invariants are used to separate deviatoric and volumetric effects in solid rubbers:F = J -1 3F,F = J1 3F,I1 = tr (B ) = tr

29、(F F T ),I= 1 (I- tr (B B ).2212= J -1 3li ,In terms of principal deviatoric stretches, li invariants have the formthe revisedI1 = l 2 + l 2 + l 2,123111I= l 2l 2 + l 2l 2 + l 2l 2 =+,2122331l 2l 2l 2123wherel1l2l3 = 1.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.Summary

30、Copyright 2006 ABAQUS, Inc.A1.31SummaryF = 3 3 deformation gradient tensor which contains all information about the motion in the vicinity of a point in the material.We take F = F(X, t), where X is the position in the reference configuration. This is called a Lagrangian description.We need to separa

31、te rigid body motion and deformation. This can bedone asF = R UorF = V R,where R is a pure rigid body motion (so R-1 = RT) and U and Vrepresent deformation.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.32SummaryWe can write U in terms of its principal values, l1, l2, l

32、3, (the “principal stretch ratios”) and the corresponding principal directions, A1, A2, A3, (which are given in the reference configuration):U = l1 A1 A1 + l2 A2 A2 + l3 A3 A3.Likewise, we can writeas:V = l1a1a1 + l2a2a2 + l3a3a3,aI = R AI .The AI (and ai) are orthogonal unit vectors.Modeling Rubber and Viscoelasticity with ABAQUSCopyright 2006 ABAQUS, Inc.A1.33SummaryThis is really all we need to know about deformation. However, many materials such as ceramics or concrete cannot undergo large deformation (lI cannot be much different f