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 1
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 4 3 + 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’ - - 1 a!.a’.. a -
(A.13)
2 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
= 0.. II 7
..a,.
a Z 12. U (I I”
111a =‘a..a. 3 (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 - 2 tr(aa),
I, = tr(a),
(A. 16)
The moment invariants may be related to the principal invariants as4 -
I, = I,, I, = I,,
n, = 4 1: - 11,, 11, = 4 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 do 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 1 denotes a 3 x 3 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 1. H.M. Westergaard. Theory of Elasticity and Plasticity, Harvard University Press, Cambridge, MA, 1952. 2. W.H. Press et al. (eds), Numerical Recipes in Fortran: The Art of Scient$c Computing, 2nd edition, Cambridge University Press, Cambridge, 1992. 3. I.H. Shames and F.A. Cozzarelli. Elastic and Inelastic Stress Analysis, revised edition, Taylor & Francis, Washington, DC, 1997. 4. J.L. Ericksen. Tensor fields. In S. Fliigge (ed.), Encyclopedia of Physics, Volume III/I, Springer-Verlag, Berlin, 1960. 5. 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. 6. O.C. Zienkiewicz and R.L. Taylor. The Finite Element Method, 4th edition, Volume 2, McGraw-Hill, London, 1991.
