Finite Element Method - Invariants of second - order tensors_appa This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.
Appendix A Invariants of second-order tensors A.l Principal invariants Given any second-order Cartesian tensor a with components expressed as the principal values of a, denoted as al ,a2,and a3,may be computed from the solution of the eigenproblem (A4 ,q(") = in which the (right) eigenvectors q(") denote principal directions for the associated eigenvalue a, Non-trivial solutions of Eq (A.2) require (all det - a) a21 a3 I1u = a11a22 + a22a33 + a33411 012 (a22 -a) a32 a13 a23 (a33 -a) - a12a21 - a23a32 - a31a13 ('4.5) 111, = alla22a33 - aIla23a32 - a22a31a13 - a33a12a21 + a12a23a31 + a21a32a13 = det a The quantities I,, II,, and 111, are called the principal invariants of a The roots of Eq (A.4) give the principal values un1 Moment invariants 433 The invariants for the deviator of a may be obtained by using where ii is the mean defined as Substitution of Eq (A.6) into Eq (A.2) gives or ,tq(m) =( am = u’ - m q(”) (A.9) which yields a cubic equation for principal values of the deviator given as ( 4 + 11&u:,- 111: =0 (A.10) where invariants of a’ are denoted as I:, II&,and 111: Since the deviator a’ differ from the total a by a mean term only, we observe from Eq (A.9) that the directions of their principal values coincide, and the three principal values are related through u j = u : + ~ ; i = 1,2,3 (A.11) Moreover Eq (A 10) generally has a closed-form solution which may be constructed by using the Cardon formula.’’2 The definition of a’ given by Eq (A.6) yields I ‘ Ia=u11+$12+u;3=0 (A.12) Using this result, the second invariant of the deviator may be shown to have the indicial form3 11’ - - a!.a’ a - (A.13) rJJr The third invariant is again given by 111: = det a’ (A.14) however, we show in Sec A.2 that this invariant may be written in a form which is easier to use in many applications (e.g yield functions for elasto-plastic materials) A.2 Moment invariants It is also possible to write the invariants in a form known as moment i n ~ u r i a n t s ~ The moment invariants are denoted as I,, Ilia, and are defined by the indicial forms n,, - Ia = II a, a Z 12 U (I I” 111a =‘a a (I j k akr (A.15) 434 Appendix A We observe that moment invariants are directly related to the truce of products of a The trace (tr) of a matrix is defined as the sum of its diagonal elements Thus, the first three moment invariants may be written in matrix form (using a square matrix for a) as - ITI, = f tr (aaa) 11a -1 - tr(aa), I, = tr(a), (A 16) The moment invariants may be related to the principal invariants as4 - I, = I,, I, = I,, n, = 1: - 11,, 11, = no, - 111, = 111, - f 1: 111, = 111, + + I,II, - - 1: - I,II, (A.17) Using Eq (A.12) and the identities given in Eq (A.17) we can immediately observe that the principal invariants and the moment invariants for a deviatoric second-order tensor are related through 11; = -nb and 111: = ITIb = det a’ (A.18) A.3 Derivatives of invariants We often also need to compute the derivative of the invariants with respect to their components and this is only possible when all components are treated independently - that is, we not use any symmetry, if present From the definitions of the principal and moment invariants given above, it is evident that derivatives of the moment invariants are the easiest to compute since they are given in concise indicia1 form Derivatives of principal invariants can be computed from these by using the identities given in Eqs (A.17) and (A.18) The first derivatives of the principal invariants for symmetric second-order tensors may be expressed in a matrix form directly, as shown by Nayak and Z i e n k i e w i c ~ ; ~ ~ ~ however, second derivatives from these are not easy to construct and we now prefer the methods given here A.3.1 First derivatives of invariants The first derivative of each moment invariant may be computed by using Eq (A 15) For the first invariant we obtain aT, -= 6, dfl, (A.19) Similarly, for the second moment invariant we get an, -= flji dfl, (A.20) and for the third moment invariant 8111, -=a da, j k f l ki (A.21) Derivatives of invariants 435 Using the identities, the derivative of the principal invariants may be written in indicia1 form as The third invariant may also be shown to have the representation3 (A.23) where a;’ is the inverse (transposed) of the a, tensor Thus, in matrix form we may write the derivatives as ail, am, - III,aPT (A.24) -=I,l-a, da da da where here denotes a x identity matrix The expression for the derivative of the determinant of a second-order tensor is of particular use as we shall encounter this in dealing with volume change in finite deformation problems and in plasticity yield functions and flow rules Performing the same steps for the invariants of the deviator stress yields dI,= 1, with only a sign change occurring in the second invariant to obtain the derivative of principal invariants from derivatives of moment invariants Often the derivatives of the invariants of a deviator tensor are needed with respect to the tensor itself, and these may be computed as (A.26) where (A.27) Combining the two expressions yields (A.28) A.3.2 Second derivatives In developments of tangent tensors we need second derivatives of the invariants These may be computed directly from Eqs (A 19)-(A.21) by standard operations The second derivatives of I,, IIQ, 111, yield 436 Appendix A The computations for principal invariants follow directly from the above using the identities given in Eqs (A.17) and (A.18) Also, all results may be transformed to the vector form used extensively in this volume for the finite element constructions These steps are by now a standard process and are left as an excercise for the reader References H.M Westergaard Theory of Elasticity and Plasticity, Harvard University Press, Cambridge, MA, 1952 W.H Press et al (eds), Numerical Recipes in Fortran: The Art of Scient$c Computing, 2nd edition, Cambridge University Press, Cambridge, 1992 I.H Shames and F.A Cozzarelli Elastic and Inelastic Stress Analysis, revised edition, Taylor & Francis, Washington, DC, 1997 J.L Ericksen Tensor fields In S Fliigge (ed.), Encyclopedia of Physics, Volume III/I, Springer-Verlag, Berlin, 1960 G.C Nayak and O.C Zienkiewicz Convenient forms of stress invariants for plasticity Proceedings of the American Society of Civil Engineers, 98(ST4), 949-53, 1972 O.C Zienkiewicz and R.L Taylor The Finite Element Method, 4th edition, Volume 2, McGraw-Hill, London, 1991 ... tr(a), (A 16) The moment invariants may be related to the principal invariants as4 - I, = I,, I, = I,, n, = 1: - 11,, 11, = no, - 111, = 111, - f 1: 111, = 111, + + I,II, - - 1: - I,II, (A.17) Using... the principal invariants and the moment invariants for a deviatoric second- order tensor are related through 11; = -nb and 111: = ITIb = det a’ (A.18) A.3 Derivatives of invariants We often also... derivatives as ail, am, - III,aPT (A.24) -= I,l-a, da da da where here denotes a x identity matrix The expression for the derivative of the determinant of a second- order tensor is of particular use