Journal of Elasticity, Vol. 4, No. 2, June 1974

Noordhoff International Publishing - Leyden Printed in The Nether lands

A new notation for stress and strain 1

N I G E L SCOTT School of Mathematics and Physics, University of East Anglia, Norwich, England

(Received February 4, 1974)

A B S T R A C T

This note is concerned with a new notat ion for the stress and deformation gradients. It is intended to be both clear

and elegant.

A new notation for the stress and deformation gradient tensors is presented in this note. It is designed to emphaSize the r61e played by the various material configurations themselves.

It is common to denote the Cartesian coordinates of a particle in the reference configuration by XA (A = 1, 2, 3) and those of the same particle in the current configuration by xi (i = 1, 2, 3). The deformation gradient is

c3xi (1) PiA -- ~ X A

and we define

J = det PIA. (2)

If the Cauchy stress in the current configuration is aji then the first Piola-Kirchhoff stress is given by

~ X A 7~Ai = J ~ aft, (3) 2

cxj

or, more compactly,

n = J p - 10.. (4)2

We have abbreviated 7tAi to rC and p j l to p - 1 but there is nothing in the notation of (4) to remind us of the two configurations involved. Therefore, let us write [A, i] instead of rca~ allowing square brackets to denote stress. We may now condense [A, i] to [1, 2], rather than n, where the label 1 denotes the material configuration and the label 2 denotes the current configuration. Similarly, we employ round brackets for strain so that p~i, 1 is replaced by (A, i), which in turn is condensed to (1, 2). We also define J12 by

J12 ~ det (2, 1) = det p = J. (5)

Equation (3) now becomes

1) This work was supported by a grant from the Science Research Council.

2) For references see [1, Appendix to § 43] but note that the first Piola-Kirchhoff stress defined in [1] is the transpose of that defined here.

Journal of Elasticity 4(1974)163 165

164 NigeI Scott

[-1, 2] = J12(1, 2')E2', 2], (6)

where the 2's have been primed to indicate that the relevant suffices are to be summed over. The symbol [2, 2] is the Cauchy stress in the 2 (current) configuration and [1, 1]is the Cauchy stress in the 1 (material) configuration.

There are some chain rules which are easy to derive. If 3 is a third configuration we have

J 1 2 J 2 3 = J13 , J12 = J2~, J l l = 1. (7)

Also we have

(1, 2')(2', 3) = (1, 3), (1, 2')(2', 1) = (1, 1), (8)

where (1, 1) and (2, 2) etc. each represent the unit tensor. We may also show from (6) that

[1, 3] = J12(1, 2')[2', 3]. (9)

This equation relates [1, 3] and [-2, 3] without directly bringing in the Cauchy stress [3, 3]. On allowing the configuration 3 to coincide with configuration 2, i.e. 3 ~ 2, it becomes clear that (6) is a special case of (9).

In equations (7), (8) and (9), the configurations 1, 2, 3 need not necessarily occur in that order in time.

If we denote by pl and P2 the densities in the 1 and 2 configurations respectively the equation of mass balance becomes

Pl = J12P 2 . (10)

Let us denote by x (2) the current coordinates of a particle in the 2 configuration and let b(X) be the external body force per unit mass. The equations of motion, valid for any configurations 1 and 2, are then

[1', 2], a,+p~b = p l x (2) (11)

whilst in the previous notation they were

nAi, A + p i b i = piS¢i. (12)

On allowing 1 ~ 2 in (11) we obtain the Cauchy form of the equations of motion regarding 2 as the current configuration. It is worth noting that the Euler relation

0 ( 8Xa J dxi ] = 0 (13)

becomes

(J12(1', 2)), r = 0. (14)

This reduces to

(1', 2), ,, = 0 (15)

in the case of an incompressible body. The second Piola-Kirchhoff stress, denoted here by TAB or T, is defined by

T = n p - r = jp- iap-r . (16) 1

1) Without ambiguity we may write the transpose of the inverse o fp as p-T.

A new notation for stress and strain 165

In the present notation this becomes [1, 2'] (1, 2') where the 2' index is summed over. The second Piola-Kirchhoff stress may not be abbreviated to [1, 1], since the symbol [1, 1] represents the Cauchy stress in the 1 configuration and the two are not equal. This is a weakness in the notation. However, the fact that the Cauchy stress [1, 1] and the second Piola-Kirchhoff stress [1, 2'] (1, 2') are not equal serves to remind us that any relation between the Cauchy stresses [1, 1] and [2, 2] must in general involve a constitutive law.

The notation might be helpful, especially for teaching purposes, because it brings out clearly the importance of the material configurations themselves in the study of stress and strain and yet does not emphasize any one Particular configuration at the expense of the others. Also, the equations (5), (6), (7), (8), (9) and (10) are in a form which is easily remembered.

Acknowledgement

I am indebted to Dr. M. A. Hayes for his helpful comments on an earlier draft.

REFERENCE [1] Truesdell, C. and Noll, W., The Non-Linear Field Theories of Mechanics. Encyclopedia of Physics, Volume 111/3.

Springer-Verlag. 1965.