Belytschko T. - Introduction (779635), страница 34
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A weak form of the momentum equation is developed, which is knownas the principle of virtual work. The development of the toal Lagrangian formulation closelyparallels the updated Lagrangian formulation, and it is stressed that the two are basically identical.Any of the expressions in the updated Lagrangian formulation can be transformed to the totalLagrangian formulation by transformations of tensors and mappings of configurations. However,the total Lagrangian formulation is often used in practice, so to understand the literature, an4-1T. Belytschko, Lagrangian Meshes, December 16, 1998advanced student must be familiar with it.
In introductory courses one of the formulations can beskipped.Implementations of the updated and total Lagrangian formulations are given for severalelements. In this Chapter, only the expressions for the nodal forces are developed. It is stressedthat the nodal forces represent the discretization of the momentum equation. The tangentialstiffness matrices, which are emphasized in many texts, are simply a means to solving theequations for certain solution procedures. They are not central to the finite element discretization.Stiffness matrices are developed in Chapter 6.For the total Lagrangian formulation, a variational principle is presented.
This principle isonly applicable to static problems with conservative loads and hyperelastic materials, i.e. materialswhich are described by a path-independent, rate-independent elastic constitutive law. Thevariational principle is of value in interpreting and understanding numerical solutions and thestability of nonlinear solutions.
It can also sometimes be used to develop numerical procedures.4.2 GOVERNING EQUATIONSWe consider a body which occupies a domain Ω with a boundary Γ. The governingequations for the mechanical behavior of a continuous body are:1. conservation of mass (or matter)2. conservation of linear momentum and angular momentum3. conservation of energy, often called the first law of thermodynamics4. constitutive equations5. strain-displacement equationsΓ0Φ (X, t)ΩΓ intΓΓ intΩ0Figure 4.0. Deformed and undeformed body showing a set of admissible lines of interwoven discontinuities Γint andthe notation.We will first develop the updated Lagrangian formulation. The conservation equations have beendeveloped in Chapter 3 and are given in both tensor form and indicial form in Box 4.1.
As can be4-2T. Belytschko, Lagrangian Meshes, December 16, 1998seen, the dependent variables in the conservation equations are written in terms of materialcoordinates but are expressed in terms of what are classically Eulerian variables, such as theCauchy stress and the rate-of-deformation.We next give a count of the number of equations and unknowns. The conservation ofmass and conservation of energy equations are scalar equations.
The equation for the conservationof linear momentum, or momentum equation for short, is a tensor equation which consists of nSDpartial differential equations, where nSD is the number of space dimensions. The constitutiveequation relates the stress to the strain or strain-rate measure. Both the strain measure and thestress are symmetric tensors, so this provides nσ equations wherenσ ≡ nSD ( nSD +1 ) / 2(4.2.1)In addition, we have the nσ equations which express the rate-of-deformation D in terms of thevelocities or displacements. Thus we have a total of 2nσ + nSD +1 equations and unknowns.
Forexample, in two-dimensional problems ( nSD = 2 ) without energy transfer, so we have nine partialdifferential equations in nine unknowns: the two momentum equations, the three constitutiveequations, the three equations relating D to the velocity and the mass conservation equation. Theunknowns are the three stress components (we assume symmetry of the stress), the threecomponents of D, the two velocity components, and the density ρ , for a total of 9 unknowns.Additional unknown stresses (plane strain) and strains (plane stress) are evaluated using the planestrain and plane stress conditions, respectively.
In three dimensions (nSD = 3, nσ = 6), we have16 equations in 16 unknowns.When a process is neither adiabatic nor isothermal, the energy equation must be appendedto the system. This adds one equation and nSD unknowns, the heat flux vector qi . However, theheat flux vector can be determined from a single scalar, the temperature, so only one unknown isadded; the heat flux is related to the temperature by a type of constitutive law which depends on thematerial.
Usually a simple linear relation, Fourier's law, is used. This then completes the systemof equations, although often a law is needed for conversion of some of the mechanical energy tothermal energy; this is discussed in detail in Section 4.10.The dependent variables are the velocity v( X, t) , the Cauchy stress σ ( X,t ) , the rate-ofdeformation D (X,t ) and the density ρ( X,t ) . As seen from the preceding a Lagrangian descriptionis used: the dependent variables are functions of the material (Lagrangian) coordinates.
Theexpression of all functions in terms of material coordinates is intrinsic in any treatment by aLagrangian mesh. In principle, the functions can be expressed in terms of the spatial coordinatesat any time t by using the inverse of the map x = φ ( X,t ) . However, inverting this map is quitedifficult. In the formulation, we shall see that it is only necessary to obtain derivatives with respectto the spatial coordinates. This is accomplished by implicit differentiation, so the mapcorresponding to the motion is never explicitly inverted.In Lagrangian meshes, the mass conservation equation is used in its integrated form(B4.1.1) rather than as a partial diffrential equation.
This eliminates the need to treat the continuityequation, (3.5.20). Although the continuity equation can be used to obtain the density in aLagrangian mesh, it is simpler and more accurate to use the integrated form (B4.1.1)The constitutive equation (Eq. B4.1.5), when expressed in rate form in terms of a rate ofCauchy stress, requires a frame invariant rate. For this purpose, any of the frame-invariant rates,4-3T.
Belytschko, Lagrangian Meshes, December 16, 1998such as the Jaumann or the Truesdell rate, can be used as described in Chapter 3. It is notnecessary for the constitutive equation in the updated Lagrangian formulation to be expressed interms of the Cauchy stress or its frame invariant rate. It is also possible to use constitutiveequations expressed in terms of the PK2 stress and then to convert the PK2 stress to a Cauchystress using the transformations developed in Chapter 3 prior to computing the internal forces.The rate-of-deformation is used as the measure of strain rate in Eq. (B4.1.5). However,other measures of strain or strain-rate can also be used in an updated Lagrangian formulation.
Forexample, the Green strain can be used in updated Lagrangian formulations. As indicated inChapter 3, simple hypoelastic laws in terms of the rate-of-deformation can cause difficulties in thesimulation of cyclic loading because its integral is not path independent. However, for manysimulations, such as the single application of a large load, the errors due to the path-dependence ofthe integral of the rate-of-deformation are insignificant compared to other sources of error, such asinaccuracies and uncertainties in the material data and material model. The appropriate selection ofstress and strain measures depends on the constitutive equation, i.e.
whether the material responseis reversible or not, time dependence, and the load history under consideration.The boundary conditions are summarized in Eq. (B4.1.7). In two dimensional problems,each component of the traction or velocity must be prescribed on the entire boundary; however thesame component of the traction and velocity cannot not be prescribed on any point of the boundaryas indicated by Eq.
(B.4.1.8). Traction and velocity components can also be specified in localcoordinate systems which differ from the global system. An identical rule holds: the samecomponents of traction and velocity cannot be prescribed on any point of the boundary. A velocityboundary condition is equivalent to a displacement boundary condition: if a displacement isspecified as a function of time, then the prescribed velocity can be obtained by time differentiation;if a velocity is specified, then the displacement can be obtained by time integration.
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