Belytschko T. - Introduction (779635), страница 62
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In the total Lagrangian form,(6.4.1), only the nominal stress is a function of time, i.e. it is the only variablewhich varies with deformation. In the updated Lagrangian form, (4.5.5) thedomain of the element (or body), the spatial derivatives ∂NI / ∂x j and theCauchy stress depend on the deformation, and hence on time.Taking the material time-derivative of (6.4.7) gives˙f int =∫ΩοB0T P˙ dΩ 0,˙f int =Ii∫Ωο∂NI ˙P dΩ∂Xj ji 0(6.4.2)since B0 and dΩ 0 are independent of the deformation, which varies with time.˙To obtain the stiffness matrix K int it is now necessary to express the stress rate Pin terms of nodal velocities.
However, constitutive equations are not expressed in˙ because this stress rate is not objective. So we work in terms of theterms of Pmaterial time derivative of the PK2 stress, which we have seen is objective.The material time derivative of the PK2 stress is then related to thematerial time derivative of the nominal stress by Box 3.2, which gives P = S⋅F T ,so˙ = ˙S ⋅F T +S⋅ F˙ T or P˙ = S˙ F T +S F˙ TPijir rjir rj(6.4.3)Substituting (6.4.3) into (6.4.2) yields˙f int =iI∫Ω0∂N I ˙∂N IS jr Fir + S jr F˙ ir dΩ 0 or dfiI = ∫dSir Fir + Sjr dFir dΩ 0∂X j∂XjΩ()()0(6.4.4)The above shows that the rate (or increment) of the internal nodal forcesconsists of two distinct parts:1.
The first term involves the rate of stress ( ˙S ) and thus depends on thematerial response and leads to what is called the material tangentstiffness matrix which we denote by K mat . Note that although this6-25T. Belytschko & B. Moran, Solution Methods, December 16, 1998term reflects material response, it changes with deformation since B0depends on F .2. The second term involves the current state of stress, S and accounts forrotation of the stress with the motion. This term is called thegeometric stiffness because it represents for geometric nonlinearitiesassociated with rotation of the stress. It is also called the initial stressmatrix to indicate the role of the existing state of stress.
It is denotedby K geo .Therefore we write Eq. (6.4.4) as˙f int = ˙f mat + ˙f geo or ˙f int = f˙ mat + ˙f geoiIiIiI(6.4.5)where˙f mat =iI∫Ω0∂N I ˙F S dΩ ,∂Xj ir jr 0˙f geo =iI∫Ω0∂NIS F˙ dΩ∂Xj jr ir 0(6.4.6)To simplify the remaining development, we put the above expression intoVoigt form. Voigt form is convenient in developing the material stiffnessmatrices because the tensor of material coefficients, Cijkl , which which relates thestress rate to the strain rate is a fourth order tensor; this tensor cannot be handledby readily standard matrix operations.
Therefore, the stiffness matrix isconventionally handled in Voigt notation; other ways of handling the fourth orderstiffness matrices are discussed later.We consider the material and geometric effects on the nodal forces one ata time. Referring to Eq. (??), we can see that with the definition of (??), which isB0jrIi ∂NI= sym( j , r ) Fir ∂Xj(6.4.7)we can rewrite the material increment in the nodal forces, Eq. (6.4.4), in Voigtnotation as{}˙f int = BT ˙S dΩmat0∫ 0Ω0(6.4.8)where S is now a column matrix arranged according to the Voigt kinetic rule,Appendix A.
It should be stressed that Eq. (6.4.6) is identical to Eq. (6.4.5). Wenow consider the consitutive equation in the following rate formS ˙S˙ ij = CijklEkl or{˙S} = CS {˙E}(6.4.9)Recall (4.9.27), which gives the following relation in Voigt notation{E˙ } = B0˙d(6.4.10)6-26T. Belytschko & B. Moran, Solution Methods, December 16, 1998Substituting Eqs. (6.4.9) and (6.4.10) into Eq. (6.4.8) gives˙f int = B T C SB dΩ ˙d or df int =mat∫ 0 0 0mat∫ B0 CTSB0 dΩ0 dd(6.4.11)T Sor KmatIJ = ∫ B0 I C B0 J dΩ0(6.4.12)Ω0Ω0So the material tangent stiffness matrix is given byK mat =∫ B0 CTSB0 dΩ0Ω0Ω0The material tangent stiffness relates the increment (or rate) in internal nodalforces to the increment (or rate) of displacement due to material response, whichis reflected in the material response matrix CS .The geometric effect on the nodal forces is obtained as follows. From the∂Ndefinition BiI0 = I and Eq.
(6.4.4), we can write∂Xi˙f geo =iI=∫ (B jI )0 TΩ0∫ (B jI )0 TΩ0Sjr F˙ ir dΩ0 =∫ (B jI )0 TΩ00S jr B rJdΩ 0u˙ iJ0S jr B rJdΩ 0δ iju˙ jJ(6.4.13)(6.4.14)˙ = B 0 u˙ , and in the third stepwhere in the second step we have used (4.9.7), FirrI iIwe have added a dummy unit matrix so that the component indices in ˙fiIgeo and u˙ iJare not the same. Writing the resulting expression for the geometric stiffnessgivesgeoT˙f I = K geoIJ u J where K IJ = ∫ B 0I SB0 J dΩ0 I(6.4.15)Ω0Note that the PK2 stress in the above is a square matrix.
Each submatrix of thegeometric stiffness matrix is a unit matrix; therefore, it follows that the geometricstiffness matrix, like the unit matrix, is invariant with rotation, i.e.ˆ geo = K geoKIJIJ(6.4.16)ˆ geo relates nodal forces to nodal velocities expressed in any alternate setwhere KIJof Cartesian coordinates.To summarizedf int = Kint dd or ˙f int = K intd˙ where K int = K mat + Kgeo6-27(6.4.17)T. Belytschko & B. Moran, Solution Methods, December 16, 1998where the material tangent stiffness and the geometric stiffness are given by Eqs.(6.4.12) and 6.4.15), respectively. The material tangent stiffness reflects theeffect on the nodal internal forces of the deformation of the material. Thegeometric stiffness reflects the effects of the rotation and deformation on thecurrent state of stress.The above forms are easily converted to updated Lagrangian forms byletting the current configuration be a reference configuration, as in Section 4.??.From Eqs.
(4.9.29), we recall that taking the current configuration as the referenceconfiguration givesB0 → BB0 → B S → σ dΩ0 → dΩ(6.4.18)Also, referring to Section 4.??, we note that when a fixed current configurationbecomes the reference configuration, thenF →I(6.4.19)In Section (???), we have seen that the relationship rate of the PK2 stress to thegiven strain is equivalent to that of the Truesdell rate of the Cauchy stress to therate-of-deformation in the current configuration, soCS → Cσ T(6.4.20)Thus, Eqs.
(6.4.13) and (6.4.16) becomeT σTK matBJ dΩIJ = ∫ B I CΩKmat = ∫ BT Cσ T BdΩΩTK geoIJ = I ∫ B I σBJ dΩ(6.4.21)ΩThese forms are generally easier to use than the total Lagrangian forms, since B ismore easily constructed than B0 and many material laws are developed in termsof Cauchy stress. It is not possible to write a convenient expression for the entiregeometric stiffness matrix in this notation. Note that either the material orgeometric stiffness can be used in total Lagrangian form with the other in updatedLagrangian form. The numerical values of the matrices in total and updatedlagrangian form are identical, and the choice is a matter of convenience.The integrand in the geometric stiffness is a scalar for given values of Iand J, so Eq. (6.4.21) can be written asK geoIJ = IH IJwhereH IJ = ∫B IT SBJ dΩ(6.4.22)ΩAlternate Derivations. In this Section the tangent stiffness matrix is derivedin terms of the convected rate of the Kirchhoff stress.
Many of the relations innonlinear mechanics take on a particular elegance and simplicity when expressedin terms of the Kirchhoff stress. In addition, the following development relies6-28T. Belytschko & B. Moran, Solution Methods, December 16, 1998more on indicial notation and the shift to Voigt notation is not made until the laststeps.Noting that the Kirchhoff stress τ is related to the nominal stress by (???),P = F ⋅ τ , the rate form of the relation between the nominal stress and theKirchhoff stress is obtained by taking the material time derivative−1˙ = F− 1 ⋅ ˙τ + F˙ −1 ⋅τP(6.4.24)D( F−1 ⋅F)Using the resultDt= 0 , it is straightforward to show that(F −1) = −F −1 ⋅ F˙ ⋅ (F−1 ) =− F−1 ⋅L(6.4.25)where the second relation follows from (???).
Thus the expression (6.4.24) for thenominal stress rate is written as˙ = F− 1 (˙τ − L⋅τ )P(6.4.26)Using (5. ????) to relate the material rate of theKirchhoff stress to its convectedrate, τ ∇c = τ˙ −L⋅ τ − τ⋅LT , (6.4.26) is written as(˙ = F− 1 ˙τ∇ c + τ ⋅LTP)(6.4.27)Writing (6.4.27) in indicial notation, we obtain˙ = F − 1 τ ∇ c +τ L = ∂X j τ ∇ c +τ LPjikikl ilkl il∂xk ki()()(6.4.28)Substituitng the above into (6.4.2) gives∂N I ∂X j ∇cτ ki +τ kl Lil dΩ 0Ωο ∂X ∂xjk˙f int =iI∫=∫Ωο=∫Ωο()()()∂N I ∇cτ +τ kl Lil dΩ 0∂xk kiN I , k τki∇c + τ kl Lil dΩ0(6.4.29)where the second expression follows from the first by the chain rule; in the thirdexpression we have used the notation NI , k = ∂NI ∂xk .
This is the counterpart of(6.4.4) in terms of the Kirchhoff stress.6-29T. Belytschko & B. Moran, Solution Methods, December 16, 1998This result can easily be transformed to an updated Lagrangian form withthe integral over the current domain. Using (3.2.18, dΩ = JdΩ 0 and the relation(5.???) between the convected rate of Kirchhoff stress and the Truesdell rate ofCauchy stress ( τ ∇c = Jσ ∇ T ) the expression (6.4.29) yields˙f = N (σ ∇T + σ L )dΩiI∫Ω I ,k ki kl il(6.4.30)which is the updated Lagrangian counterpart of Eq. (6.4.4); (6.4.30) could also beobtained by making the current configuration the reference configurationEXERCISEAn alternative derivation of (6.4.29) is given as follows.Recall Eq.(6.4.3):˙ = ˙S ⋅F T +S⋅F TP(6.4.33)Now note that this relation can be written as˙ = F− 1 ⋅ F⋅S˙ ⋅ FT + F −1 ⋅F ⋅S⋅F T ⋅F−T ⋅ ˙FTP(6.3.34)Using the push-forward relation (5???? ) for the convected rate of Kirchhoff stress˙ T,in terms of the rate of the PK2 stress, τ ∇c = F⋅ ˙S⋅ FT , and (3.3.18), LT = F− T ⋅ F(6.4.34) can be written as(˙ = F− 1 ⋅ τ∇c + τ⋅ LTP)(6.4.35)which is the same as (6.4.27).
The tangent stiffness expression (6.4.29) follows inan identical fashion.To complete the derivation of the material tangent stiffness matrix(6.4.29), it is necessary to introduce the constitutive relation to relate theconvected stress rate to the nodal velocities. We write the constitutive relation(rate-independent material response) in the formττ ij∇c = CijklDkl(6.4.38)where the superscript τ on the tangent modulus Cτijkl indicates that it relates theKirchoff stress rate to the rate-of-deformation. This tangent modulus possessesτthe minor symmetries, i.e., Cτijkl = C τjikl = Cijlk. For some materials, this tangentτmodulus tensor also has major symmetry, i.e, Cτijkl = Cklij, i.e.
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