Belytschko T. - Introduction (779635), страница 32

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Large rotations of an element mayoccur due to actual rigid body motions of the body, as in a space vehicle or amoving car, or large local large rotations, as in a buckling beam. The rotationneed not be as large as 90 degrees for the same effect; we have chosen 90 degreesto simplify the numbers.The missing factor in Eq. (3.7.8) is that it does not account for the rotationof the material. The material rotation can be accounted for correctly be using anobjective rate of the stress tensor; it is also called a frame-invariant rate.

We willconsider three objective rates, the Jaumann rate, the Truesdell rate and the GreenNagdi rate. All of these are used in current finite element software. There aremany other objective rates, some of which will be discussed in Chapter 9.3.7.3 Jaumann rate. The Jaumann rate of the Cauchy stress is given byσ ∇J =DσijDσ− W⋅ σ − σ⋅W T or σij∇J =− Wikσ kj − σik WkjTDtDt(3.7.9)where W is the spin tensor given by Eq. (3.3.11).

The superscript " ∇ " heredesignates an objective rate; the Jaumann rate is designated by the subsequentsuperscript “J”. One appropriate hypoelastic constitutive equation is given byJσ ∇J = C J : D or σ ij∇J = CijklDkl(3.7.10)The material rate for the Cauchy stress tensor, i.e. the correct equationcorresponding to (3.7.8), is thenDσ= σ ∇ J + W⋅ σ + σ⋅ WT = C J : D + W⋅σ + σ ⋅W TDt3-64(3.7.11)T. Belytschko, Continuum Mechanics, December 16, 199865where the first equality is just a rearrangement of Eq. (3.7.9) and the secondequality follows from (3.7.10).

We see in the above that the objective rate is afunction of material response. The material derivative of the Cauchy stress thendepends on two parts: the rate of change due to material response, which isreflected in the objective rate, and the change of stress due to rotation, whichcorresponds to the last two terms in Eq. (3.7.11).TruesdellRate. The Truesdell rate and Green-Naghdi rates are given in Box3.5.

The Green-Naghdi rate differs from the Jaunmann rate only in using adifferent measure of the rotation of the material. In the Green-Nagdi rate, theangular velocity defined in Eq. (3.2.23b) is used.Box 3.5 Objective RatesJaumann rateDσijDσσ ∇J =− W⋅ σ − σ⋅W Tσij∇J =− Wikσ kj − σik WkjTDtDtTruesdell rate (3.2.23)Dσσ ∇T =+ div (v)σ −L ⋅σ − σ⋅ LTDtDσ ij ∂vk∂v j∂vσ ij∇T =+σij − i σ kj − σ ikDt∂xk∂x k∂xkGreen-Naghdi rate (3.2.24)Dσ ijDσTσ ∇G =− Ω⋅σ −σ ⋅Ω Tσ ij∇G =−Ω ik σkj −σ ik ΩkjDtDt˙ ⋅R T ,Ω =RL=∂v=D +W∂xLij =∂vi= Dij + Wij∂x jThe relationship between the Truesdell rate and the Jaumann rate can be examinedby replacing the velocity gradient in Eq.

(3.7.23) by its symmetric andantisymmetric parts, i.e. Eq. (3.3.9):σ ∇T =Dσ+ div(v)σ −(D + W)⋅σ − σ ⋅(D + W)TDtA comparison of Eqs. (3.7.9) and (3.7.12) then shows thatincludes the same spin-related terms as the Jaumann rate,additional terms which depend on the rate of deformation.relationship further, we consider a rigid body rotation for thefind thatwhen D = 0,σ∇T =Dσ−W ⋅σ − σ⋅ WTDt(3.7.12)the Truesdell ratebut also includesTo examine theTruesdell rate and(3.7.13)Comparison of the above with Eq. (3.7.9) shows that the Truesdell rate isequivalent to the Jaumann rate in the absence of deformation, i.e. in a rigid bodyrotation.

However, when the Jaumann rate is used in a constitutive equation, itwill give a different material rate of stress unless the constitutive equation ischanged appropriately. Thus if we write the constitutive equation in the form3-65T. Belytschko, Continuum Mechanics, December 16, 199866σ ∇T = C T : D(3.7.14)then the material response tensor CT will differ from the material response tensorassociated with the Jaumann rate form of the material law in Eq.

(3.7.11). For thisreason, whenever the material response matrix can refer to different rates, we willoften add the superscripts to specify which objective rate is used by the materiallaw. The hypoelastic relations (3.7.11) and (3.7.14) represent the same materialresponse if the material response tensors CT and CJ are related as follows:()σ ∇T = C J :D = CT +Cσ :D(3.7.15)where from (3.7.12)Cσ :D = ( divv)σ − D ⋅σ− σ⋅D T = (trD )σ− D⋅σ − σ⋅ DT(3.7.16)The components of Cσ are given byCσijkl =σ ijδ kl −δ ik σ jl −σ ilδ jk(3.7.17)With these relations, the hypoelastic relations can be modified for a Truesdell rateto match the behavior of a constitutive eqaution expressed in terms of theJaumann rate.

The correspondence to the Green-Naghdi rate depends on thedifference between the angular velocity and the spin and is more difficult to adjustfor.Example 3.12 Consider a body rotating in the x-y plane about the origin withan angular velocity ω ; the original configuration is prestressed as shown in Fig.3.11. The motion is rigid body rotation and the related tensors are given inExample 3.2. Evaluate the material time derivative of the Cauchy stress using theJaumann rate and integrate it to obtain the Cauchy stress as a function of time.yyoriginalconfigurationcurrentconfigurationωtxxσ 0xσ 0xσ 0xσ 0xFigure 3.11. Rotation of a prestressed element with no deformation.From Example 3.2, Eq. (E3.2.8) we note that3-66T.

Belytschko, Continuum Mechanics, December 16, 1998s c −s ˙ −s −c −1  cF =R = ,F=ω,F=−s c s c  c −s 67(E3.12.1a)where s = sinωt, c =cos ωt . The spin is evaluated in terms of the velocity gradientL , which is given for this case by Eq. (3.3.18) and then using (E3.12.1a) :− s −c  c s  0 −1L = ˙F ⋅F− 1 = ω =ω ⇒ c −s  −s c 1 0  0 −1W = 12 L − LT = ω  1 0 ()(E3.12.1b)The material time derivative based on the Jaumann rate is then given byspecializing (3.7.9) to the case when there is no deformation:Dσ= W⋅σ +σ ⋅W TDt(E3.12.1.c)(D=0, since there is no deformation; this is easily verified by noting that thesymmetric part of L vanishes). We now change the material time derivative to anordinary derivative since the stress is constant in space and write out the matricesin (E3.12.1c):0 −1 σ xdσ=ωdt1 0 σ xyσ xy  σ x+σ y  σ xy −2σ xy σ x − σ y dσ=ω2σ xy dtσ x − σ yσ xy   0 1 ωσ y  −1 0 (E3.12.2)(E3.12.3)It can be seen that the the material time derivative of the Cauchy stress issymmetric.

We now write out the three scalar ordinary differential equations inthree unknowns, σ x ,σ y ,andσ xy corresponding to (E3.12.3) (the fourth scalarequation of the above tensor equation is omitted because of symmetry):dσ x= −2ωσ xydtdσ ydtdσ xydt(E3.12.4a)= 2ωσ xy(=ω σ x −σ y(E3.12.4b))(E3.12.4c)The initial conditions are3-67T. Belytschko, Continuum Mechanics, December 16, 1998σ x (0) =σ 0x , σ y (0) = 0, σ xy ( 0) = 068(E3.12.5)It can be shown that the solution to the above differential equations is c2 csσ = σ 0x 2 cs s (E3.12.6)We only verify the solution for σ x( t) :()d cos 2 ωtdσ x0=σx=σ 0x ω (−2cosωt sinωt ) =−2ωσ xydtdt(E3.12.7)where the last step follows from the solution for σ xy( t) as given in Eq. (E3.12.7);comparing with (E3.14.4a) we see that the differential equation is satisfied.Examining Eq.

(E3.12.6) we can see that the solution corresponds to aˆ , i.e. if we let the corotational stress beconstant state of the corotational stress σgiven by 0ˆσ = σ x000then the Cauchy stress components in the global coordinate system are given by(e3.12.6) by σ = R ⋅ σˆ ⋅RT according to Box 3.2 with (E3.12.1a) gives the result(E3.12.6).We leave as an exercise to show that when all of the initial stresses are nonzero,then the solution to Eqs.

(E3.12.4) isc −s   σ 0x σ 0xy   c s σ= s c  σ 0xy σ 0y −s c (E3.12.8)Thus in rigid body rotation, the Jaumann rate changes the Cauchy stress so thatthe corotational stress is constant. Therefore, the Jaumann rate is often called thecorotational rate of the Cauchy stress. Since the Truesdell and Green-Naghdirates are identical to the Jaumann rate in rigid body rotation, they also correspondto the corotational Cauchy stress in rigid body rotation.Example 3.13 Consider an element in shear as shown in Fig. 3.12. Find theshear stress using the Jaumann, Truesdell and Green-Naghdi rates for ahypoelastic, isotropic material.3-68T. Belytschko, Continuum Mechanics, December 16, 1998Ω069ΩFigure 3.12.The motion of the element is given byx = X + tYy =Y(E3.13.1)The deformation gradient is given by Eq.

(3.2.16), so1 t F=,0 1˙F =  0 1 ,0 0−t 1 1F −1 = 0(E3.13.2)The velocity gradient is given by Eq. (E3.12.1), and the rate-of-deformation andspin are its symmetric and skew symmetric parts so0 1 L = ˙FF− 1 = ,0 0 1 0 1 D=2, 1 0 110W=2−1 0(E3.13.3)The hypoelastic, isotropic constitutive equation in terms of the Jaumann rate isgiven by()˙ = λJ traceD I +2µ J D + W⋅σ + σ⋅W Tσ(E3.13.4)We have placed the superscripts on the material constants to distinguish thematerial constants which are used with different objective rates. Writing out thematrices in the above gives σ˙ x σ˙ xy J 0=µσ˙1 xy σ˙ y +1 02 −11  σ x0σ xy10σ xy  1  σ x+σ y  2 σ xy(E3.13.5)σ xy 0 −1σ y  1 0 soσ˙ x = σ xy ,σ˙ y = −σ xy ,(1σ˙ xy = µ J + 2 σ y −σ xThe solution to the above differential equations is3-69)(E3.13.6)T.

Belytschko, Continuum Mechanics, December 16, 1998σ x = −σ y = µ J (1− cos t ),70σ xy = µ J sin t(E3.13.7)For the Truesdell rate, the constitutive equation is˙ = λT trD + 2µ T D+ L⋅σ +σ ⋅LT − (tr D )σσ(E3.13.8)This gives σ˙ x σ˙ xy T 0=µσ˙1 xy σ˙ y 10(E3.13.9) 0 1  σ x σ xy   σ x σ xy  0 0++ 0 0 σ xy σ y  σ xy σ y 1 0where we have used the results trace D = 0 , see Eq. (E3.13.3). The differentialequations for the stresses areσ˙ x = 2σ xy ,σ˙ y = 0,σ˙ xy = µ T + σ y(E3.13.10)σ y = 0,σ xy = µT t(E3.13.11)and the solution isσ x = µT t 2 ,To obtain the solution for the Cauchy stress by means of the Green-Nagdhi rate,we need to find the rotation matrix R by the polar decomposition theorem.

Toobtain the rotation, we diagonalize F T Ft 12 + t2 ± t 4 + t2FTF = ,eigenvaluesλ=2i2 t 1+ t (E3.13.12)The closed form solution by hand is quite involved and we recommend acomputer solution. A closed form solution has been given by Dienes (1979):()σ x = −σ y = 4µG cos 2βln cos β + β sin 2β − sin 2 β ,σ xy = 2µG cos 2β (2β −2tan 2βln cos β −tan β ), tan β =The results are shown in Fig. 3.13.3-70(E3.13.13)t2(E.13.14)T. Belytschko, Continuum Mechanics, December 16, 1998Figure 3.13. Comparison of Objective Stress RatesExplanation of Objective Rates.One underlying characteristic ofobjective rates can be gleaned from the previous example: an objective rate of theCauchy stress instantaneously coincides with the rate of a stress field whosematerial rate already accounts for rotation correctly.

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