T. Belytschko, Continuum Mechanics, December 16, 1998 68 σ x 0 ( ) =σ x 0 , σ y 0 ( ) = 0, σ xy 0 ( ) = 0 (E3.12.5) It can be shown that the solution to the above differential equations is σ = σ x 0 c 2 cs cs s 2       (E3.12.6) We only verify the solution for σ x t ( ) : dσ x dt = σ x 0 d cos 2 ωt ( ) dt =σ x 0 ω −2cosωt sinωt ( ) =−2ωσ xy (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 a constant state of the corotational stress ˆ σ , i.e. if we let the corotational stress be given by ˆ σ = σ x 0 0 0 0       then the Cauchy stress components in the global coordinate system are given by (e3.12.6) by σ = R ⋅ ˆ σ ⋅R T 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 σ = c −s s c       σ x 0 σ xy 0 σ xy 0 σ y 0       c s −s c       (E3.12.8) Thus in rigid body rotation, the Jaumann rate changes the Cauchy stress so that the corotational stress is constant. Therefore, the Jaumann rate is often called the corotational rate of the Cauchy stress. Since the Truesdell and Green-Naghdi rates are identical to the Jaumann rate in rigid body rotation, they also correspond to the corotational Cauchy stress in rigid body rotation. Example 3.13 Consider an element in shear as shown in Fig. 3.12. Find the shear stress using the Jaumann, Truesdell and Green-Naghdi rates for a hypoelastic, isotropic material. 3-68 T. Belytschko, Continuum Mechanics, December 16, 1998 69 Ω 0 Ω Figure 3.12. The motion of the element is given by x = X + tY y = Y (E3.13.1) The deformation gradient is given by Eq. (3.2.16), so F = 1 t 0 1       , ˙ F = 0 1 0 0       , F −1 = 1 −t 0 1       (E3.13.2) The velocity gradient is given by Eq. (E3.12.1), and the rate-of-deformation and spin are its symmetric and skew symmetric parts so L = ˙ F F −1 = 0 1 0 0       , D = 1 2 0 1 1 0       , W = 1 2 0 1 −1 0       (E3.13.3) The hypoelastic, isotropic constitutive equation in terms of the Jaumann rate is given by ˙ σ = λ J traceD ( ) I +2µ J D + W⋅σ + σ⋅W T (E3.13.4) We have placed the superscripts on the material constants to distinguish the material constants which are used with different objective rates. Writing out the matrices in the above gives ˙ σ x ˙ σ xy ˙ σ xy ˙ σ y       = µ J 0 1 1 0       + 1 2 0 1 −1 0       σ x σ xy σ xy σ y       + 1 2 σ x σ xy σ xy σ y       0 −1 1 0       (E3.13.5) so ˙ σ x = σ xy , ˙ σ y = −σ xy , ˙ σ xy = µ J + 1 2 σ y −σ x ( ) (E3.13.6) The solution to the above differential equations is 3-69 T. Belytschko, Continuum Mechanics, December 16, 1998 70 σ x = −σ y = µ J 1− cos t ( ) , σ xy = µ J sin t (E3.13.7) For the Truesdell rate, the constitutive equation is ˙ σ = λ T trD + 2µ T D+ L⋅σ +σ ⋅L T − tr D ( ) σ (E3.13.8) This gives ˙ σ x ˙ σ xy ˙ σ xy ˙ σ y       = µ T 0 1 1 0       + 0 1 0 0       σ x σ xy σ xy σ y       + σ x σ xy σ xy σ y       0 0 1 0       (E3.13.9) where we have used the results trace D = 0 , see Eq. (E3.13.3). The differential equations for the stresses are ˙ σ x =2σ xy , ˙ σ y = 0, ˙ σ xy = µ T + σ y (E3.13.10) and the solution is σ x = µ T t 2 , σ y = 0, σ xy = µ T t (E3.13.11) 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. To obtain the rotation, we diagonalize F T F F T F = 1 t t 1+ t 2       , eigenvalues λ i = 2+t 2 ±t 4+t 2 2 (E3.13.12) The closed form solution by hand is quite involved and we recommend a computer solution. A closed form solution has been given by Dienes (1979): σ x = −σ y = 4µ G cos 2βln cos β + β sin 2β − sin 2 β ( ) , (E3.13.13) σ xy = 2µ G cos 2β 2β −2tan 2βln cos β −tan β ( ) , tan β = t 2 (E.13.14) The results are shown in Fig. 3.13. 3-70 T. Belytschko, Continuum Mechanics, December 16, 1998 71 Figure 3.13. Comparison of Objective Stress Rates Explanation of Objective Rates. One underlying characteristic of objective rates can be gleaned from the previous example: an objective rate of the Cauchy stress instantaneously coincides with the rate of a stress field whose material rate already accounts for rotation correctly. Therefore, if we take a stress measure which rotates with the material, such as the corotational stress or the PK2 stress, and add the additional terms in its rate, then we can obtain an objective stress 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 stress rate 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) and Truesdell and Noll (????). 3-71 T. Belytschko, Continuum Mechanics, December 16, 1998 72 To illustrate the first approach, we develop an objective rate from the corotational Cauchy stress ˆ σ . Its material rate is given by D ˆ σ Dt = D R T σR ( ) Dt = DR T Dt σR + R T Dσ Dt R + R T σ DR Dt (3.7.18) where the first equality follows from the stress transformation in Box 3.2 and the second equality is based on the derivative of a product. If we now consider the corotational coordinate system coincident with the reference coordinates but rotating with a spin W then R = I DR Dt = W = Ω (3.7.19) Inserting the above into Eq. (3.7.18), it follows that at the instant that the corotational coordinate system coincides with the global system, the rate of the Cauchy stress in rigid body rotation is given by D ˆ σ Dt = W T ⋅σ+ Dσ Dt +σ ⋅W (3.7.20) The RHS of this expression can be seen to be identical to the correction terms in the expression for the Jaumann rate. For this reason, the Jaumann rate is often called the corotational rate of the Cauchy stress. The Truesdell rate is derived similarly by considering the time derivative of the PK2 stress when the reference coordinates instantaneously coincide with the spatial coordinates. However, to simplify the derivation, we reverse the expressions and extract the rate corresponding to the Truesdell rate. Readers familiar with fluid mechanics may wonder why frame-invariant rates are rarely discussed in introductory courses in fluids, since the Cauchy stress is widely used in fluid mechanics. The reason for this lies in the structure of constitutive equations which are used in fluid mechanics and in introductory fluid courses. For a Newtonian fluid, for example, σ = 2µD' − pI , where µ is the viscosity and D' is the deviatoric part of the rate-of-deformation tensor. A major difference between this constitutive equation for a Newtonian fluid and the hypoelastic law (3.7.14) can be seen immediately: the hypoelastic law gives the stress rate, whereas in the Newtonian consititutive equation gives the stress. The stress transforms in a rigid body rotation exactly like the tensors on the RHS of the equation, so this constitutive equation behaves properly in a rigid body rotation. In other words, the Newtonian fluid is objective or frame-invariant. REFERENCES T. Belytschko, Z.P. Bazant, Y-W Hyun and T P. Chang, "Strain Softening Materials 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. 3-72 T. Belytschko, Continuum Mechanics, December 16, 1998 73 J.K. Dienes (1979), On the Analysis of Rotation and Stress Rate in Deforming Bodies, 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, CRC Press, 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, Springer- Verlag, New York. 3-73 T. Belytschko, Continuum Mechanics, December 16, 1998 74 LIST OF FIGURES Figure 3.1 Deformed (current) and undeformed (initial) configurations of a body. (p 3) Figure 3.2 A rigid body rotation of a Lagrangian mesh showing the material coordinates when viewed in the reference (initial, undeformed) configuration and the current configuration on the left. (p 10) Figure 3.3 Nomenclature for rotation transformation in two dimensions. (p 10) Figure 3.4 Motion descrived by Eq. (E3.1.1) with the initial configuration at the left and the deformed configuration at t=1 shown at the right. (p 14) Figure 3.5 To be provided (p 26) Figure 3.6. The initial uncracked configuration and two subsequent configurations for a crack growing along x-axis. (p 18) Figure 3.7. An element which is sheared, followed by an extension in the y- direction and then subjected to deformations so that it is returned to its 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 state of stress. (p. 34) Fig. 3.10. Rotation of a bar under initial stress showing the change of Cauchy stress which occurs without any deformation. (p 59) Fig. 3.11 To be provided (p 62) Fig. 3.12 To be provided (p 64) Fig. 3.13 Comparison of Objective Stress Rates (p 66) LIST OF BOXES Box 3.1 Definition of Stress Measures. (page 29) Box 3.2 Transformations of Stresses. (page 32) Box 3.3 incomplete — reference on page 45 Box 3.4 Stress-Deformation (Strain) Rate Pairs Conjugate in Power. (page 51) Box 3.5 Objective Rates. (page 57) 3-74 T. Belytschko, Continuum Mechanics, December 16, 1998 75 Exercise ??. Consider the same rigid body rotation as in Example ??>. Find the Truesdell stress and the Green-Naghdi stress rates and compare to the Jaumann stress rate. Starting from Eqs. (3.3.4) and (3.3.12), show that 2dx⋅D⋅ dx = 2dxF −T ˙ E ˙ F −1 dx and hence that Eq. (3.3.22) holds. Using the transformation law for a second order tensor, show that R = ˆ R . Using the statement of the conservation of momentum in the Lagrangian description in the initial configuration, show that it implies PF T = FP T Extend Example 3.3 by finding the conditions at which the Jacobian becomes negative at the Gauss quadrature points for 2 × 2 quadrature when the initial element is rectangular with dimension a ×b . Repeat for one-point quadrature, with the quadrature point at the center of the element. Kinematic Jump Condition. The kinematic jump conditions are derived from the restriction that displacement remains continuous across a moving singular surface. The surface is called singular because ???. Consider a singular surface in one dimension. t X X 2 X 1 X S Figure 3.? Its material description is given by X =X S t ( ) 3-75 T. Belytschko, Continuum Mechanics, December 16, 1998 76 We consider a narrow band about the singular surface defined by 3-76 T. Belytschko, Lagrangian Meshes, December 16, 1998 CHAPTER 4 LAGRANGIAN MESHES by Ted Belytschko Departments of Civil and Mechanical Engineering Northwestern University Evanston, IL 60208 ©Copyright 1996 4.1 INTRODUCTION In Lagrangian meshes, the nodes and elements move with the material. Boundaries and interfaces remain coincident with element edges, so that their treatment is simplified. Quadrature points also move with the material, so constitutive equations are always evaluated at the same material 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 nonlinear materials, i.e. they consider both geometric and material nonlinearities. They are only limited by the element's capabilities to deal with large distortions. The limited distortions most elements can sustain without degradation in performance or failure is an important factor in nonlinear analysis with Lagrangian meshes and is considered for several elements in the examples. Finite element discretizations with Lagrangian meshes are commonly classified as updated Lagrangian formulations and total Lagrangian formulations. Both formulations use Lagrangian descriptions, i.e. the dependent variables are functions of the material (Lagrangian) coordinates and time. 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 key equation 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 is then developed, which is known as the principle of virtual power. The momentum equation in the updated Lagrangian formulation employs derivatives with respect to the spatial coordinates, so it is natural that the weak form involves integrals taken with respect to the spatial coordinates, i.e. on the current configuration. It is common practice to use the rate-of-deformation as a measure of strain rate, but other measures of strain or strain-rate can be used in an updated Lagrangian formulation. For many applications, the updated Lagrangian formulation provides the most efficient 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 the formulations presented here. As a measure of strain we will use the Green strain tensor in the total Lagrangian formulation. A weak form of the momentum equation is developed, which is known as the principle of virtual work. The development of the toal Lagrangian formulation closely parallels 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 total Lagrangian formulation by transformations of tensors and mappings of configurations. However, the total Lagrangian formulation is often used in practice, so to understand the literature, an 4-1 [...]... integration by parts also leads to certain symmetries in the linearized equations, as will be seen in Chapter 6 Thus the integration by parts is a key step in the development of the weak form Next we started with the weak form and showed that it implies the strong form This, combined with the development of the weak form from the strong form, shows that the weak and strong forms are equivalent Therefore, if... = δv T f in t = ∫ {δD} {σ}dΩ I I (4 .5. 13) Ω Substituting (4 .5. 12) into the above and invoking the arbitrariness of {δv} gives f in t = ∫ BT {σ} dΩ I I (4 .5. 14) Ω 4- 25 T Belytschko, Lagrangian Meshes, December 16, 1998 As will be shown in the examples, Eq (4 .5. 14) gives the same expression for the internal nodal forces as Eq (4 .5. 6): Eq (4 .5. 14) uses the symmetric part of the velocity gradient, whereas... 4 constitutive equations 5 strain-displacement equations Γ0 Φ (X, t) Ω Γ int Γ Γ int Ω0 Figure 4.0 Deformed and undeformed body showing a set of admissible lines of interwoven discontinuities Γint and the notation We will first develop the updated Lagrangian formulation The conservation equations have been developed in Chapter 3 and are given in both tensor form and indicial form in Box 4.1 As can be... 0 ∀δvi ∈U 0 (4.3.22) which is the weak form for the momentum equation The physical meanings help in remembering the weak form and in the derivation of the finite element equations The weak form is summarized in Box 4.2 BOX 4.2 Weak Form in Updated Lagrangian Formulation: Principle of Virtual Power Ifσ ij is a smooth function of the displacements and velocities and vi ∈U , then if δ P int − δ P ext +... (4.3.9) ti The above is the weak form for the momentum equation, the traction boundary conditions and the interior continuity conditions It is known as the principle of virtual power, see Malvern (1969), for each of the terms in the weak form is a virtual power; see Section 2 .5 4.3.2 Weak Form to Strong Form It will now be shown that the weak form (4.3.9) implies the strong form or generalized momentum... not be evaluated in closed form, and are instead integrated numerically (often called numerical quadrature) The most widely used procedure for numerical integration in finite elements is Gauss quadrature The Gauss quadrature formulas are, see for example Dhatt and Touzot (1984, p.240), Hughes (1977, p 137) nQ ∑ ∫ −1 f (ξ )dξ = Q= 1wQ f (ξQ ) 1 (4 .5. 24) where the weights wQ and coordinates ξQ of the... the counterpart of Eq (4 .5. 14): {f }int = ∫ BT {σ}dΩ (4 .5. 19) Ω Often we omit the brackets on the nodal force, since the presence of a single term in Voigt notation always indicates that the entire equation is in Voigt notation The Voigt form can also be obtained by rewriting (4 .5. 5) as ∂N I δ σ dΩ ∂xj ri ji Ω int f Ir = ∫ (4 .5. 20) Then defining the B matrix by BijIr = ∂N I δ ∂x j ri (4 .5. 21) 4-26... December 16, 1998 and converting the indices (i,j) by the kinematic Voigt rule to a and the indices (I.r) by the matrixcolumn vector rule gives int T f a = ∫ Bbaσ bdΩ or f int = ∫ BT{ σ}σdΩ Ω Ω (4 .5. 22) More detail and techniques for translating indicial notation to Voigt notation can be found in Appendix B 4 .5. 4 Numerical Quadrature The integrals for the nodal forces, mass matrix and other element... test functions is infinite dimensional, a solution to the weak form is a solution of the strong form However, the test functions used in computational procedures must be finite dimensional Therefore, satisfying the weak form in a computation only leads to an approximate solution of the strong form In linear finite element analysis, it has been shown that the solution of the weak form is the best solution... nodes and quadrature points 0 viI (0 ) = viI ( ) (4.4. 25) ( ) 0 σ ij XQ , 0 =σ ij XQ (4.4.26) 4-16 T Belytschko, Lagrangian Meshes, December 16, 1998 0 0 where viI and σ ij are initial data at the nodes and quadrature points If data for the initial conditions are given at a different set of points, the values at the nodes and quadrature points can be estimated by least square fits, as in Section 2.4 .5 For . weak form and showed that it implies the strong form. This, combined with the development of the weak form from the strong form, shows that the weak and strong forms are equivalent. Therefore,. 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 nonlinear materials,. of the formulations can be skipped. Implementations of the updated and total Lagrangian formulations are given for several elements. In this Chapter, only the expressions for the nodal forces