Belytschko T. - Introduction (779635), страница 29
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We therefore saythat the rate-of-deformation and the Cauchy stress are conjugate in power. As weshall see, conjugacy in power is helpful in the development of weak forms:measures of stress and strain rate which are conjugate in power can be used toconstruct principles of virtual work or power, which are the weak forms for finiteelement approximations of the momentum equation. Variables which areconjugate in power are also said to be conjugate in work or energy, but we willuse the phrase conjugate in power because it is more accurate.The rate of change of the internal energy of the system is obtained byintegrating (3.5.50) over the domain of the body, which givesDW int = ρ Dwint dΩ = D:σ dΩ = D σ dΩ = ∂vi σ dΩ∫ Dt∫∫ ij ij∫ ∂x j ijDtΩΩΩΩ(3.5.51)where the last expression follows from the symmetry of the Cauchy stress tensor.The conservation equations are summarized in Box 3.3 in both tensor andindicial form.
The equations are written without specifying the independentvariables; they can be expressed in terms of either the spatial coordinates or thematerial coordinates, and as we shall see later, they can be written in terms ofother coordinate systems which are neither fixed in space nor coincident withmaterial points. The equations are not expressed in conservative form becausethis does not seem to be as useful in solid mechanics as it is in fluid mechanics.The reasons for this are not explored in the literature, but it appears to be relatedto the mauch smaller changes in density which occur in solid mechanicsproblems.Box 3.3Conservation EquationsEulerian descriptionMassDρ+ρ div(v) = 0 orDtLinear MotionDρ+ ρ vi ,i = 0 or ˙ρ + ρvi, i = 0Dtρ DvDt = ∇⋅ σ + ρb ≡ divσ + ρborDv ∂σρ Dti = ∂x ji + ρbijAngular Momentumσ = σTor σ ij = σ jiEnergyintρ DwDt = D :σ −∇⋅ q + ρsLagrangian DescriptionMassρ (X, t ) J ( X, t ) = ρ 0 (X)Linear Momentum(B3.3.1)(B3.3.2)(B3.3.3)(B3.3.4)ρJ = ρ 0or3-50(B3.3.5)T.
Belytschko, Continuum Mechanics, December 16, 1998ρ0∂v( X,t )=∇ X ⋅P + ρ0 b∂torρ0∂vi ( X, t )∂t=51∂Pji∂X j+ ρ 0bi(B3.3.6)Angular MomentumF⋅ P = PT ⋅ FTT TFik Pkj = PikFkj = F jk PkiS = STEnergy˙ int = ρ0ρ0 w(B3.3.7)(B3.3.8)∂wint ( X,t ) ˙ T˜ + ρ 0s= F :P −∇ X ⋅ q∂t(B3.3.9)3.5.11 System Equations.
The number of dependent variables depends onthe number of space dimensions in the model. If we denote the number of spacedimensions by n SD, then for a purely mechanical problem, the followingunknowns occur in the equations for a purely mechanical process (a processwithout heat transfer, so the energy equation is not used):ρ, the density1 unknownv, the velocitynSD unknownsσ, the stressesnσ=nSD*(nSD+1)/2 unknownsIn counting the number of unknowns attributed to the stress tensor, we haveexploited its symmetry, which results from the conservation of angularmomentum.
The combination of the mass conservation (1 equation), and themomentum conservation (nSD equations) gives a total of nSD+1 equations. Thuswe are left with nσ extra unknowns. These are provided by the constitutiveequations, which relate the stresses to a measure of deformation. This equationintroduces nσ additional unknowns, the components of the symmetric rate-ofdeformation tensor. However, these unknowns can immediately be expressed interms of the velocities by Eq. (3.3.10), so they need not be counted as additionalunknowns.The displacements are not counted as unknowns.
The displacements areconsidered secondary dependent variables since they can be obtained byintegrating the velocities in time using Eq. (3.2.8) at any material point. Thedisplacements are considered secondary dependent variables, just like the positionvectors. This choice of dependent variables is a matter of preference. We couldjust as easily have chosen the displacement as a primary dependent variable andthe velocity as a secondary dependent variable.3.6.
LAGRANGIAN CONSERVATION EQUATIONS3.6.1 Introduction and Definitions. For solid mechanics applications, itis instructive to directly develop the conservation equations in terms of theLagrangian measures of stress and strain in the reference configuration. In thecontinuum mechanics literature such formulations are called Lagrangian, whereasin the finite element literature these formulations are called total Lagrangianformulations.
For a total Lagrangian formulation, a Lagrangian mesh is alwaysused. The conservation equations in a Lagrangian framework are fundamentallyidentical to those which have just been developed, they are just expressed in terms3-51T. Belytschko, Continuum Mechanics, December 16, 199852of different variables. In fact, as we shall show, they can be obtained by thetransformations in Box 3.2 and the chain rule. This Section can be skipped in afirst reading. It is included here because much of the finite element literature fornonlinear mechanics employs total Lagrangian formulations, so it is essential for aserious student of the field.The independent variables in the total Lagrangian formulation are theLagrangian (material) coordinates X and the time t.
The major dependentvariables are the initial density ρ0 ( X, t) the displacement u (X,t ) and theLagrangian measures of stress and strain. We will use the nominal stress P ( X,t )as the measure of stress. This leads to a momentum equation which is strikinglysimilar to the momentum equation in the Eulerian description, Eq. (3.5.33), so it iseasy to remember. The deformation will be described by the deformation gradientF ( X,t ) . The pair P and F is not especially useful for constructing constitutiveequations, since F does not vanish in rigid body motion and P is not symmetric.Therefore constitutive equations are usually formulated in terms of the of the PK2stress S and the Green strain E.
However, keep in mind that relations between Sand E can easily be transformed to relations between P and E or F by use of therelations in Boxes 3.2.The applied loads are defined on the reference configuration. The tractiont 0 is defined in Eq. (3.4.2); t 0 is in units of force per unit initial area. Asmentioned in Chapter 1, we place the noughts, which indicate that the variablespertain to the reference configuration, either as subscripts or superscripts,whichever is convenient.
The body force is denoted by b, which is in units offorce per unit mass; the body force per initial unit volume is given by ρ0 b, whichis equivalent to the force per unit current volume ρb. This equivalence is shownin the followingdf = ρbdΩ= ρbJdΩ 0 = ρ 0bdΩ0(3.6.1)where the second equality follows from the conservation of mass, Eq.
(3.5.25).Many authors, including Malvern(1969) use different symbols for the body forcesin the two formulations; but this is not necessary with our convention ofassociating symbols with fields.The conservation of mass has already been developed in a form thatapplies to the total Lagrangian formulation, Eq.(3.5.25). Therefore we developonly the conservation of momentum and energy.3.6.2Conservation of Linear Momentum.
In a Lagrangian description,the linear momentum of a body is given in terms of an integral over the referenceconfiguration byp0 ( t ) =∫ ρ0 v(X, t)dΩ 0(3.6.2)Ω0The total force on the body is given by integrating the body forces over thereference domain and the traction over the reference boundaries:3-52T. Belytschko, Continuum Mechanics, December 16, 199853∫ ρ b(X, t) dΩ + ∫ t ( X, t)dΓf0 ( t) =000Ω0(3.6.3)0Γ0Newton’s second law then givesdp0dt = f0(3.6.4)Substituting (3.6.2) and (3.6.3) into the above givesddt∫ ρ v dΩ00Ω0=∫ ρ bdΩ + ∫ t dΓ00Ω00(3.6.5)0Γ0On the LHS, the material derivative can be taken inside the integral because thereference domain is constant in time, soddt∫ ρ0 v dΩ0 = ∫ ρ 0Ω0Ω0∂v (X,t )dΩ0∂t(3.6.6)Using Cauchy’s law ( 3.4.2) and Gauss’ theorem in sequence gives∫ t0 dΓ0 = ∫ n 0 ⋅ PdΓ0 = ∫ ∇X ⋅PdΩ0Γ0Γ000∫ t i dΓ0 = ∫ n j Pji dΓ0 =Γ0Γ0orΩ0∫Ω0∂Pji∂XjdΩ 0(3.6.7)Note that in tensor notation, the left gradient appears in the domain integralbecause the nominal stress is defined with the normal on the left side.
Thedefinition of the material gradient, which is distinguished with the subscript X,should be clear from the indicial expression. The index on the material coordinateis the same as the first index on the nominal stress: the order is important becausethe nominal stress is not symmetric.Substituting (3.6.6) and (3.6.7) into (3.6.5) gives∫Ω0 ∂v (X,t ) ρ0 ∂t − ρ 0b−∇ X ⋅P dΩ0 = 0(3.6.8)which, because of the arbitrariness of Ω0 givesρ0∂v( X,t )=∇ X ⋅P + ρ0 b∂torρ0∂vi ( X, t )∂t=∂Pji∂X j+ ρ 0bi(3.6.9)Comparing the above with the momentum equation in the Eulerian description,Eq.(3.5.33), we can see that they are quite similar: in the Lagrangian form of the3-53T. Belytschko, Continuum Mechanics, December 16, 199854momentum equation the Cauchy stress is replaced by the nominal stress and thedensity is replaced by the density in the reference configuration.The above form of the momentum equation can also be obtained directlyby transforming all of the terms in Eq.(3.5.33) using the chain rule and Box 3.2.Actually, this is somewhat difficult, particularly for the gradient term.
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