Belytschko T. - Introduction (779635), страница 48

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(E4.12.1). The Green-strain tensor can becomputed directly from F, but to avoid round-off errors, it is better to compute ∂N  ux1, ..., u xn  ∂N I, X Hij = [ uiI ] I  =  uy1, ..., uyn  ∂N I , Y  ∂X j   u , ..., u  ∂N  z1zn  I, Z [ ](E4.12.4)The Green-strain tensor is then given by Eq. (???).If the constitutive law relates the PK2 stress S to E, the nominal stress is then computed byP = SF , using F from Eq.

(??.2). The nodal internal forces are then given byTN I, X   P11 P12 f xI int   fyI  = ∫ N I , Y  =  P21 P22f ∆ N zI  I , Z  P31 P32P13 P23  Jξ0 d∆P33 (E4.12.5)( )where Jξ0 =det X,ξ .Voigt Form. All of the variables needed for the evaluation of the B0 matrix given in Eq. (???)can be obtained from Eq. (E???).

In Voigt form{E}T = [ E11, E22 , E33 , 2 E23, 2 E13 , 2 E12 ]{S} T = [ S11, S22 , S33 , S23, S13, S12 ](E4.12.6)The rate of Green-strain can be computed by Eq. (???):{E˙ } = B0˙d˙d = u , u , u , ... u , u , u[ x1 y1 z1 xn yn zn ](E4.12.7)The Green strain is computed by the procedure in Eq. (???). The nodal forces are given by4-77T. Belytschko, Lagrangian Meshes, December 16, 1998Tξf intI = ∫B 0I {S}J 0 d∆(E4.12.8)∆4.9.3.Variational Principle. For static problems, weak forms for nonlinear analysis withpath-independent materials can be obtained from variational principles.

For many nonlinearproblems, variational principles can not be formulated. However, when constitutive equations andloads are path-independent and nondissipative, a variational priniciple can be written because thestress and load can be obtained from potentials. The materials for which stress is derivable from apotential are called hyperelastic materials, see Section 5.4. In a hyperelastic material, the nominalstress is given in terms of a potential by Eq (5.4.113) which is rewritten herePT =∂w∂w, or Pji =, where w = ρwint , W int = ∫ wdΩ 0∂F∂FijΩ(4.9.28)0Note the order of the subscripts on the stress, which follows from the definition.For the existence of a variational principle, the loads must also be conservative, i.e.

theymust be independent of the deformation path. Such loads are also derivable from a potential, i.e.the loads must be related to a potential so thatW ext ( u) =∫ wbextΩ0(u)dΩ 0 + ∫ wtext (u)dΓ0Γt∂wbext 0 ∂wtextbi =ti =∂ui∂ui(4.9.29b)Theoem of Stationary Potential Energy. When the loads and constitutive equations possespotentials, then the stationary points ofW (u ) = W int (u ) −W ext (u ), u (X,t ) ∈U(4.9.30)satisfies the strong form of the equilibrium equation (B4.5.2b). The equilibrium equation whichemanates from this statienary principle is written in terms of the displacements by incorporating theconstitutive equation and strain-displacement equation.

This stationary principle applies only tostatic problems.The theorem is proven by showing the equivalence of the stationary principle to the weakform for equilibrium, traction boundary conditions and the interior continuity conditions. We firstwrite the stationary condition of (4.9.30), which gives0 =δ W (u) = ∂w∂wbext∂wtextδFdΩ−δudΩ−i0∫  ∂Fij ij∫ ∂ui δui dΓ0∂uiΩ0Γ0(4.9.31)Substituting Eqs. (4.9.28) and (4.9.29) into the above gives0=∫ ( PjiδFij − ρ0biδui )dΩ0 − ∫ ti δui dΓ00Ω0Γ04-78(4.9.32)T. Belytschko, Lagrangian Meshes, December 16, 1998which is the weak form given in Eq. (4.8.7) for the case when the accelerations vanish. The samesteps given in Section 4.8 can then be used to establish the equivalence of Eq.

(4.8.7) to the strongform of the equilibrium equation.Stationary principles are thus in a sense more restrictive weak forms: they apply only toconservative, static problems. However they can improve our understanding of stability problemsand are used in the study of the existence and uniqueness of solutions.The discrete equations are obtained from the stationary principle by using the usual finiteelement approximation to motion with a Lagrangian mesh, Eqs. (4.12) to (4.9.5), which we writein the formu (X,t ) = N(X)d(t )(4.9.33)The potential energy can then be expressed in terms of the nodal displacements, givingW (d) = W int (d) −W ext ( d)(4.9.34)The solutions to the above correspond to the stationary points of this function, so the discreteeqautions are0=∂W (d ) ∂W int (d ) ∂W ext (d) int=−≡ f − f ext∂d∂d∂d(4.9.35)It will be shown in Chapter 6, that when the equilibrium point is stable, the potential energy is aminimum.Example 4.11.

Rod Element by Stationary Principle.Consider a structural modelconsisting of two-node rod elements in three dimensions. Let the internal potential energy be givenby12w = C SE E112(E4.11.1)and let the only load on the structure be gravity, for which the external potential isw ext = −ρ 0gz(E4.11.2)where g is the acceleration of gravity.

Find expressions for the internal and external nodal forcesof an element.From Eqs. (4.9.28) and (E4.11.1), the total internal potential is given bywint = ∑ Weint , Weint =e12C SE E11dΩ0∫2 Ωe(E4.11.3)0For the two-node element, the displacement field is linear and the Green strain is constant, so Eq.(E4.11.3) can be simplified by multiplying the integrand by the initial volume of the element A0l0 :4-79T. Belytschko, Lagrangian Meshes, December 16, 1998Weint =12A l C SE E112 0 0(E4.11.4)To develop the internal nodal forces, we will need the derivatives of the Green strain withrespect to the nodal displacements. Since the strain is constant in the element, Eq. (3.3.1) (also seeEq.

(??)) gives:E11 =l 2 − l20 x21 ⋅ x21 − X21 ⋅X 21=2l022l02(E4.11.5)where x IJ ≡ x I − x J , XIJ ≡ X I − X J . Noting thatx IJ ≡ X IJ +u IJ(E4.11.6)where u IJ ≡ u I − u J are the nodal displacements and substituting Eq. (E4.11.6) into Eq. (4.11.5)gives, after some algebra,E11 =2X21 ⋅u 21 + u21 ⋅ u 212l20(E4.11.7)The derivatives of Ex2 with respect to the nodal displacements are then given by( ) = X 21 + u21 = x 21 , ∂ ( Ex2) =− X21 + u21 = − x21∂ Ex2∂u 2l 02l02∂u1l 20l 20(E4.11.8)Using the definition for internal nodal forces in conjunction with Eqs.

(E4.11.4) and (E4.11.8)givesf2int = −f1int =A0C SE E x x21l0(E4.11.9)By using the fact that Sx = C SE Ex , it follows that(f )int T2= − (f1int ) =TA0Sx[ x21l0y21z 21 ](E4.11.10)This result, as expected, is identical to the result obtained for the bar by the principle of virtualwork, Eq. (E4.8.9). The external potential for a gravity load is given byW ext =−∫ ρ0gzdΩ0(E4.11.11)Ω0extThe external potential is independent of x or y, and W, extz = W,u z . If we make the finite elementapproximation z = zI N I , where NI are the shape functions given in Eq. (E4.8.4) then4-80T. Belytschko, Lagrangian Meshes, December 16, 1998W ext =−∫ ρ0gz I N I dΩ0(E4.11.12)Ω0andf zIext =1∂ W ext1=− ∫ ρ0 gzI NI (ξ)l 0 A0 dξ = − A0 l0ρ 0g0∂uzI2(E4.11.13)so the external nodal force on each node is half the force on the rod element due to gravity.REFERENCEST.

Belytschko and B.J. Hsieh, "Nonlinear Transient Finite Element Analysis with ConvectedCoordinates," International Journal for Numerical Methods in Eng., 7, pp. 255-271, 1973.T.J.R. Hughes (1997), The Finite Element Method, Prentice-Hall, Englewood Cliffs, NewJersey.L.E. Malvern (1969), Introduction to the Mechanics of a Continuous Medium, Prentice-Hall,Englewood Cliffs, New Jersey.J.T. Oden and J.N.

Reddy (1976), An Introduction to the Mathematical Theory of FiniteElements, John Wiley & Sons, New York.G. Strang and G.J. Fix (1973), An Analysis of the Finite Element Method, Prentice Hall, NewYork.G.A. Wempner (1969), "Finite elements, finite rotations and small strains," Int. J. Solids andStructures, 5, 117-153.L IST OF F IGURESFigure 4.1Initial and Current Configurations of an Element and TheirRelationships to the Parent Element (p 18)Figure 4.2Triangular Element Showing Node Numbers and the Mappings of theInitial and Current Configurations to the Parent Element (p 29)Figure 4.3Triangular Element Showing the Nodal Force and Velocity Compenents(p 31)Figure 4.4Quadrilateral Element in Current and Initial Configurations and theParent Domain (p 36)Figure 4.5Parent Element and Current Configuration for an Eight-Node Hexahedral Element(p 40)Figure 4.7Current Configuration of Quadrilateral Axisymmetric Element (p 43)4-81T.

Belytschko, Lagrangian Meshes, December 16, 1998Figure 4.8Exploded view of a tieline; when joined together, the velocites of nodes 3 and 5 equalthe nodal velocities of nodes 1 and 2 and the velocity of node 4 is given in terms ofnodes 1 and 2 by a linear constraint. (p 45)Figure 4.9Two-node rod element showing initial configuration and current configuration and thecorotational coordinate. (p 50)Figure 4.10 Initial, current, and parent elements for a three-node rod; the corotational base vectoreˆ x is tangent to the current configuration. (p 52)Figure 4.11 Triangular three-node element treated by corotational coordinate system.

(p 54)Figure 4.12 Rod element in rwo dimensions in total Lagrangian formulation (p 68)L IST OF B OXESBox 4.1 Governing Equations for Updated Lagrangian Formulation (p 3)Box 4.2 Weak Form in Updated Lagrangian Formulation: Principle of VirtualPower (p 9)Box 4.3 untitledBox 4.5 Governing Equations for Total Lagrangian Formulation (p 48)Box 4.6 untitled (p 53)Box 4.7 Internal Force Computation in Total Lagrangian Fomulation (p 57)Box 4.8 Discrete Equations for the Updated Lagrangian Formulation and Internal Nodal ForceAlgorithm (p 75)4-82CHAPTER 5CONSTITUTIVE MODELSBy Brian MoranNorthwestern UniversityDepartment of Civil EngineeringEvanston, IL 60208©Copyright 1998In the mathematical description of material behavior, the response of the material ischaracterized by a constitutive equation which gives the stress as a function of thedeformation history of the body.

Different constitutive relations allow us to distinguishbetween a viscous fluid and rubber or concrete, for example. In one-dimensionalapplications in solid mechanics, the constitutive relation is often referred to as the stressstrain law for the material. In this chapter, some of the most common constitutive modelsused in solid mechanics applications are described. Constitutive equations for differentclasses of materials are first presented for the one-dimensional case and are then generalizedto multiaxial stress states. Special emphasis is placed on the elastic-plastic constitutiveequations for both small and large strains.

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