Belytschko T. - Introduction (779635), страница 30

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Using thetransformation from Box 3.2 and the chain rule gives−1∂σ ji ∂  J Fjk Pki ∂xj ∂Pki∂∂P== PkiJ −1Fjk + J −1Fjk ki = J −1(3.6.10)∂x j∂x j∂x j∂x j∂X k ∂x j()In the above we have used the definition of the deformation gradient F, Eq.(3.2.14) and ∂(J −1Fjk ) ∂xj = 0 ,(see Ogden(1984)). Thus (3.5.33) becomesρ∂x j ∂Pki∂vi= J −1+ ρbi∂t∂X k ∂x j(3.6.11)By the chain rule, the first term on the RHS is J −1 ∂Pki ∂Xk . Multiplying theequation by J and using mass conservation, ρJ = ρ 0 then gives Eq. (3.6.9).3.6.3Conservation of Angular Momentum.

The balance equations forangular momentum will not be rederived in the total Lagrangian framework. Wewill use the consequence of angular momentum balance in Eq. (3.5.40) inconjunction with the stress transformations in Box 3.2 to derive the consequencesfor the Lagrangian measures of stress. Substituting the transformation expressionfrom Box 3.2 into (3.5.40) gives()J −1F ⋅P = J −1F ⋅PT(3.6.12)Multiplying both sides of the above by J and taking the transpose inside theparenthesis then givesF⋅ P = PT ⋅ FTT TFik Pkj = PikFkj = F jk Pki(3.6.13)The above equations are nontrivial only when i ≠ j . Thus the above gives onenontrivial equation in two dimensions, three nontrivial equations in threedimensions.

So, while the nominal stress is not symmetric, the number ofconditions imposed by angular momentum balance equals the number ofsymmetry conditions on the Cauchy stress, Eq. (3.5.40). In two dimensions, theangular momentum equation isF11P12 + F12 P22 = F21P11 + F22 P21(3.6.14)These conditions are usually imposed directly on the constitutive equation, as willbe seen in Chapter 5.3-54T. Belytschko, Continuum Mechanics, December 16, 199855For the PK2 stress, the conditions emanating from conservation of angularmomentum can be obtained by expressing P in terms of S in Eq.

(3.6.13), (thesame equations are obtained if σ is replaced by S in the symmetry conditions(3.5.40)), which givesF⋅ S⋅ FT = F ⋅S T ⋅ FT(3.6.15)Since F must be a regular (nonsingular) matrix, its inverse exists and we can( )Tpremultiply by F −1 and postmultiply by F −T ≡ F −1 the above to obtainS = ST(3.6.16)So the conservation of angular momentum requires the PK2 stress to besymmetric.3.6.4Conservation of Energy in Lagrangian Description.

Thecounterpart of Eq. (3.5.45) in the reference configuration can be written asddt∫ ( ρ0 wintΩ0+ 12 ρ0 v ⋅v)dΩ 0 =∫ v⋅ ρ0 bdΩ 0 + ∫ v⋅t0 dΓ0 + ∫ ρ0 sdΩ 0 − ∫ n0 ⋅ q˜ dΓ0Ω0Γ0Ω0(3.6.17)Γ0The heat flux in a total Lagrangian formulation is defined as energy per unitreference area and therefore is denoted by ˜q to distinguish it from the heat fluxper unit current area q , which are related by˜ = J − 1F T ⋅qq(3.6.17b)The above follows from Nanson's law (3.4.5) and the equivalence∫ n⋅qdΓ= ∫ n0 ⋅ q˜ dΓ0ΓΓ0Substituting (3.4.5) for n into the above gives (3.6.17b).The internal energy per unit initial volume in the above is related to theinternal energy per unit current volume in (3.5.45) as followsρ0 wint dΩ 0 = ρ 0w int J −1 dΩ= ρw int dΩ(3.6.18)where the last step follows from the mass conservation equation (3.5.9).

On theLHS, the time derivative can be taken inside the integral since the domain isfixed, giving3-55T. Belytschko, Continuum Mechanics, December 16, 1998ddt56∂wint ( X, t )∂v( X, t)int 1(ρw+ρv⋅v)dΩ=(ρ0∫ 0∫ 0 ∂t +ρ0 v⋅ ∂t )dΩ 0 (3.6.19)2 0Ω0Ω0The second term on the RHS can be modified as follows by using Eq. (3.4.2) andGauss’s theorem∫ v ⋅t 0 dΓ0 = ∫ v j t j dΓ0 = ∫ v jni Pij dΓ00Γ00Γ0Γ0 ∂v∂P = ∫ ∂X v j Pij dΩ 0 = ∫  j Pij + v j ij  dΩ0∂Xi∂Xi iΩ0Ω0 ∂=()(3.6.20) ∂Fji∂Pij  ∂FTP+vdΩ=∫  ∂t ij ∂X j  0 ∫  ∂t :P+ ( ∇X ⋅ P) ⋅ v dΩ 0iΩ0Ω0Gauss’s theorem on the fourth term of the LHS and some manipulation gives ∂w int ∂F T ∂v( X,t ) ˜ρ−:P+∇⋅q−ρs+ρ−∇⋅P−ρb∫ 0 ∂tX0X0  ⋅v dΩ0 = 0 0 ∂t ∂tΩ0(3.6.21)The term inside the parenthesis of the integrand is the total Lagrangian form ofthe momentum equation, (3.6.30), so it vanishes. Then because of thearbitrariness of the domain, the integrand vanishes, giving∂wint ( X,t ) ˙ Tint˙˜ + ρ 0sρ0 w = ρ0= F :P −∇ X ⋅ q∂t(3.6.22)In the absence of heat conduction or heat sources, the above gives˙˙ int = F˙ ji Pij = ˙FT :P = P:Fρ0 w(3.6.23)This is the Lagrangian counterpart of Eq.

(3.5.50). It shows that the nominalstress is conjugate in power to the material time derivative of the deformationgradient.These energy conservation equations could also be obtained directly fromEq. (3.5.50) by transformations. This is most easily done in indicial notation.ρDijσ ij = ρ=ρ∂viσij∂x j∂v i ∂Xkσ ij∂Xk ∂x jby definition of D and symmetry of stress σby chain rule3-56T. Belytschko, Continuum Mechanics, December 16, 1998∂X= ρF˙ ik k σ ij∂xjby definition of F, Eq.

(3.2.10)= ρF˙ ik Pki J −1 = ρ0F˙ ikPki57(3.6.24)by Box 3.2 and mass conservation3.6.5 Power in terms of PK2 stress. The stress transformations in Box3.2 can also be used to express the internal energy in terms of the PK2 stress.˙ T : P ≡ F˙ P = F˙ S F TFik kiik kr ri(by Box 3.2)˙ S = FT ⋅ ˙F :S= FriT Fik rk=( (F12T)by symmetry of S((3.6.25)))˙ +F˙ T ⋅F + 1 FT ⋅ F˙ −F˙ T ⋅F :S⋅F2decomposingtensor into symmetric and antisymmetric parts˙ + ˙FTF) :S since contraction of symmetric and= 12 ( FTFantisymmetric tensors vanishesThen, using the time derivative of E as defined in Eq.(3.3.20) gives˙ :S = S:E˙ = E˙ S˙ int = Eρ0 wij ij(3.6.26)This shows that the rate of the Green strain tensor is conjugate in power (orenergy) to the PK2 stress.Thus we have identified three stress and strain rate measures which areconjugate in the sense of power.

These conjugate measures are listed in Box 3.4along with the corresponding expressions for the power. Box 3.4 also includes afourth conjugate pair, the corotational Cauchy stress and corotational rate-ofdeformation. Its equivalence to the power in terms of the unrotated Cauchy stressand rate-of-deformation is easily demonstrated by (3.4.15) and thhe orthgonalityof the rotation matrix.Conjugate stress and strain rate measures are useful in developing weakforms of the momentum equation, i.e.

the principles of virtual work and power.The conjugate pairs presented here just scratch the surface: many other conjugatepairs have been developed in continuum mechanics, {Ogden(1984), Hill()}.However, those presented here are the most frequently used in nonlinear finiteelement methods.Box 3.4Stress-deformation (strain) rate pairs conjugate in power˙ int = D:σ = σ:D = Dijσ ijCauchy stress/rate-of deformation: ρw3-57T. Belytschko, Continuum Mechanics, December 16, 199858˙ :PT = P:F˙ T = F˙ P˙ int = FNominal stress/rate of deformation gradient: ρ0 wij ji˙ :S = S:E˙ = E˙ S˙ int = EPK2 stress/rate of Green strain: ρ0 wij ijˆ :ˆσ = σˆ =Dˆ σˆˆ :D˙ int = DCorotational Cauchy stress/rate-of-deformation: ρwij ij3.7 POLAR DECOMPOSITION AND FRAME-INVARIANCEIn this Section, the role of rigid body rotation is explored. First, a theoremknown as the polar decomposition theorem is presented.

This theorem enables therigid body rotation to be obtained for any deformation. Next, we consider theeffect of rigid body rotations on constitutive equations. We show that for theCauchy stress, a modification of the time derivatives is needed to formulate rateconstitutive equations. This is known as a frame-invariant or objective rate ofstress. Three frame-invariant rates are presented: the Jaumann rate, the Truesdellrate and the Green-Naghdi rate.

Some startling differences in hypoelasticconstitutive equations with these various rates are then demonstrated.3.7.1 Polar Decomposition Theorem. A fundamental theorem whichelucidates the role of rotation in large deformation problems is the polardecomposition theorem. In continuum mechanics, this theorem states that anydeformation gradient tensor F can be multiplicatively decomposed into theproduct of an orthogonal matrix R and a symmetric tensor U, called the rightstretch tensor (the adjective right is often omitted):∂x i= Rik Ukj∂X jF =R⋅UorFij =R−1 = RTandU = UTwhere(3.7.1)(3.7.2)Rewriting the above with Eq. (3.2.15) givesdx = R⋅U ⋅ dX(3.7.3)The above shows that any motion of a body consists of a deformation, which isrepresented by the symmetric mapping U, and a rigid body rotation R; R can berecognized as a rigid-body rotation because all proper orthogonal transformationsare rotations.

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