Belytschko T. - Introduction (779635), страница 33
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Therefore, if we take a stressmeasure which rotates with the material, such as the corotational stress or the PK2stress, and add the additional terms in its rate, then we can obtain an objectivestress rate. This is not the most general framework for developing objective rates.A general framework is provided by using objectivity in the sense that the stressrate should be invariant for observers who are rotating with respect to each other.A derivation based on these principles may be found in Malvern (1969) andTruesdell and Noll (????).3-7171T. Belytschko, Continuum Mechanics, December 16, 199872To illustrate the first approach, we develop an objective rate from theˆ . Its material rate is given bycorotational Cauchy stress σ()ˆ D RT σR DRTDσDσDR==σR + RTR + RT σDtDtDtDtDt(3.7.18)where the first equality follows from the stress transformation in Box 3.2 and thesecond equality is based on the derivative of a product.
If we now consider thecorotational coordinate system coincident with the reference coordinates butrotating with a spin W thenR= IDR=W = ΩDt(3.7.19)Inserting the above into Eq. (3.7.18), it follows that at the instant that thecorotational coordinate system coincides with the global system, the rate of theCauchy stress in rigid body rotation is given byˆDσDσ= WT ⋅σ++σ ⋅WDtDt(3.7.20)The RHS of this expression can be seen to be identical to the correction terms inthe expression for the Jaumann rate.
For this reason, the Jaumann rate is oftencalled the corotational rate of the Cauchy stress.The Truesdell rate is derived similarly by considering the time derivativeof the PK2 stress when the reference coordinates instantaneously coincide withthe spatial coordinates. However, to simplify the derivation, we reverse theexpressions and extract the rate corresponding to the Truesdell rate.Readers familiar with fluid mechanics may wonder why frame-invariantrates are rarely discussed in introductory courses in fluids, since the Cauchy stressis widely used in fluid mechanics. The reason for this lies in the structure ofconstitutive equations which are used in fluid mechanics and in introductory fluidcourses.
For a Newtonian fluid, for example, σ = 2µD' − pI , where µ is theviscosity and D' is the deviatoric part of the rate-of-deformation tensor. A majordifference between this constitutive equation for a Newtonian fluid and thehypoelastic law (3.7.14) can be seen immediately: the hypoelastic law gives thestress rate, whereas in the Newtonian consititutive equation gives the stress. Thestress transforms in a rigid body rotation exactly like the tensors on the RHS ofthe equation, so this constitutive equation behaves properly in a rigid bodyrotation. In other words, the Newtonian fluid is objective or frame-invariant.REFERENCEST. Belytschko, Z.P.
Bazant, Y-W Hyun and T.-P. Chang, "Strain SofteningMaterials and Finite Element Solutions," Computers and Structures, Vol 23(2),163-180 (1986).D.D. Chandrasekharaiah and L. Debnath (1994), Continuum Mechanics,Academic Press, Boston.3-72T. Belytschko, Continuum Mechanics, December 16, 1998J.K. Dienes (1979), On the Analysis of Rotation and Stress Rate in DeformingBodies, Acta Mechanica, 32, 217-232.A.C. Eringen (1962), Nonlinear Theory of Continuous Media, Mc-Graw-Hill,New York.P.G.
Hodge, Continuum Mechanics, Mc-Graw-Hill, New York.L.E. Malvern (1969), Introduction to the Mechanics of a Continuous Medium,Prentice-Hall, New York.J.E. Marsden and T.J.R. Hughes (1983), Mathematical Foundations of Elasticity,Prentice-Hall, Englewood Cliffs, New Jersey.G.F. Mase and G.T. Mase (1992), Continuum Mechanics for Engineers, CRCPress, Boca Raton, Florida.R.W. Ogden (1984), Non-linear Elastic Deformations, Ellis Horwood Limited,Chichester.W.
Prager (1961), Introduction to Mechanics of Continua, Ginn and Company,Boston.M. Spivak (1965), Calculus on Manifolds, W.A. Benjamin, Inc., New York.C, Truesdell and W. Noll, The non-linear field theories of mechanics, SpringerVerlag, New York.3-7373T. Belytschko, Continuum Mechanics, December 16, 1998L IST OF F IGURESFigure 3.1Deformed (current) and undeformed (initial) configurations of abody. (p 3)Figure 3.2A rigid body rotation of a Lagrangian mesh showing the materialcoordinates when viewed in the reference (initial, undeformed)configuration and the current configuration on the left.
(p 10)Figure 3.3Nomenclature for rotation transformation in two dimensions.(p 10)Figure 3.4Motion descrived by Eq. (E3.1.1) with the initial configuration atthe left and the deformed configuration at t=1 shown at the right.(p 14)Figure 3.5To be provided (p 26)Figure3.6. The initial uncracked configuration and two subsequentconfigurations for a crack growing along x-axis. (p 18)Figure 3.7.An element which is sheared, followed by an extension in the ydirection and then subjected to deformations so that it is returned toits initial configuration. (p 26)Figure 3.8.Prestressed body rotated by 90˚. (p 33)Figure 3.9.Undeformed and current configuration of a body in a uniaxial stateof stress.
(p. 34)Fig. 3.10.Rotation of a bar under initial stress showing the change of Cauchystress which occurs without any deformation. (p 59)Fig. 3.11To be provided (p 62)Fig. 3.12To be provided (p 64)Fig. 3.13Comparison of Objective Stress Rates (p 66)L IST OF B OXESBox 3.1Definition of Stress Measures. (page 29)Box 3.2Transformations of Stresses. (page 32)Box 3.3incomplete — reference on page 45Box 3.4Stress-Deformation (Strain) Rate Pairs Conjugate in Power.(page 51)Box 3.5Objective Rates.
(page 57)3-7474T. Belytschko, Continuum Mechanics, December 16, 1998Exercise ??. Consider the same rigid body rotation as in Example ??>. Find theTruesdell stress and the Green-Naghdi stress rates and compare to the Jaumannstress rate.Starting from Eqs. (3.3.4) and (3.3.12), show that2dx⋅D⋅ dx = 2dxF− T E˙ F˙ −1dxand hence that Eq. (3.3.22) holds.ˆ .Using the transformation law for a second order tensor, show that R = RUsing the statement of the conservation of momentum in the Lagrangiandescription in the initial configuration, show that it impliesPFT = FPTExtend Example 3.3 by finding the conditions at which the Jacobianbecomes negative at the Gauss quadrature points for 2 × 2 quadrature when theinitial element is rectangular with dimension a ×b .
Repeat for one-pointquadrature, with the quadrature point at the center of the element.Kinematic Jump Condition. The kinematic jump conditions are derived from therestriction that displacement remains continuous across a moving singular surface.The surface is called singular because ???. Consider a singular surface in onedimension.tXSX1X2XFigure 3.?Its material description is given byX = X S (t )3-7575T. Belytschko, Continuum Mechanics, December 16, 1998We consider a narrow band about the singular surface defined by3-7676T.
Belytschko, Lagrangian Meshes, December 16, 1998CHAPTER 4LAGRANGIAN MESHESby Ted BelytschkoDepartments of Civil and Mechanical EngineeringNorthwestern UniversityEvanston, IL 60208©Copyright 19964.1 INTRODUCTIONIn Lagrangian meshes, the nodes and elements move with the material. Boundaries andinterfaces remain coincident with element edges, so that their treatment is simplified.
Quadraturepoints also move with the material, so constitutive equations are always evaluated at the samematerial points, which is advantageous for history dependent materials. For these reasons,Lagrangian meshes are widely used for solid mechanics.The formulations described in this Chapter apply to large deformations and nonlinearmaterials, i.e. they consider both geometric and material nonlinearities. They are only limited bythe element's capabilities to deal with large distortions. The limited distortions most elements cansustain without degradation in performance or failure is an important factor in nonlinear analysiswith Lagrangian meshes and is considered for several elements in the examples.Finite element discretizations with Lagrangian meshes are commonly classified as updatedLagrangian formulations and total Lagrangian formulations.
Both formulations use Lagrangiandescriptions, i.e. the dependent variables are functions of the material (Lagrangian) coordinates andtime. In the updated Lagrangian formulation, the derivatives are with respect to the spatial(Eulerian) coordinates; the weak form involves integrals over the deformed (or current)configuration. In the total Lagrangian formulation, the weak form involves integrals over the initial(reference ) configuration and derivatives are taken with respect to the material coordinates.This Chapter begins with the development of the updated Lagrangian formulation.
The keyequation to be discretized is the momentum equation, which is expressed in terms of the Eulerian(spatial) coordinates and the Cauchy (physical) stress. A weak form for the momentum equation isthen developed, which is known as the principle of virtual power. The momentum equation in theupdated Lagrangian formulation employs derivatives with respect to the spatial coordinates, so it isnatural that the weak form involves integrals taken with respect to the spatial coordinates, i.e.
onthe current configuration. It is common practice to use the rate-of-deformation as a measure ofstrain rate, but other measures of strain or strain-rate can be used in an updated Lagrangianformulation. For many applications, the updated Lagrangian formulation provides the mostefficient formulation.The total Lagrangian formulation is developed next. In the total Lagrangian formulation,we will use the nominal stress, although the second Piola-Kirchhoff stress is also used in theformulations presented here. As a measure of strain we will use the Green strain tensor in the totalLagrangian formulation.
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