Journal of Elasticity 24: 129-133, 1990. 129 © 1990 Kluwer Academic Publishers. Printed in the Netherlands.

On the derivatives of the stretch and rotation with respect to the deformation gradient

LEWIS W H E E L E R Department of Mechanical Engineering, University of Houston, USA

Received 10 June 1988; in revised form 28 July 1988

Almract. In this paper, a result involving the eigenprojections of the right stretch and its derivative with respect to the deformation gradient is derived, and a related result is found for the rotation. As an application, the form of the constitutive law for an isotropic hyperelastic material in the case when the strain energy function is expressed in terms of the right stretch, is shown to follow at once.

1. Introduction

Consider the right polar decomposition of the deformation gradient F,

F = RU. (1)

In this note, we demonstrate several results concerning the derivatives of R and U with respect to F. In particular, we show that

(DFU)T[u] = F, (DFR)r[F] = 0 . (2)

The main result of this paper is an identity involving the derivative of U and

its eigenprojections, which is derived in Section 2. In Section 3, this identity is

used as the basis of a straightforward derivation of the relations (2) and a

number of other interesting results. In (2) and in the sequel, the conventional square bracket notation is used to

indicate the transformation associated with a tensor of order four which takes the set of all tensors of order two linearly into itself. Further, we use the

notion of the transpose of a tensor of order four: for C a tensor of order four, we define the transpose C T through

A . Cr [B ] = B . C[A],

for A, B arbitrary second order tensors. For tensors A and B of order two we

use the notation A" B for the inner product defined by

A" B = tr(A~B).

130 L. Wheeler

2. A useful identity for the derivative of the right stretch

Let e be a normalized principal vector of U and let

P = e ® e . (3)

It is easily shown that P is a projection,

P~ = P. (4)

Further,

PU = UP = 2P, (5)

where 2 is the principal value corresponding to e. Then, as we shall d~mon- strate,

(DF U) r[P] = RP. (6)

The existence of the derivative appearing in (6) has be.n established previ- ously [ 1].

With a view toward establishing (6), we note that from (1) there follows

U ~ = FTF. (7)

Differentiation in (7) yields

(DvU[A])U + U(DvU[A]) = FrA + AZF (8)

for every second order tensor A. Thus, and by (1),

P- (D~U[A])U + P . U(D~U[A]) = P" (URrA) + P" (A~RU) (9)

which because U and P are symmetric and commute, yields

P . U(D~U[A]) = P . (UREA). (10)

The desired result (6) now follows from (5).

Derivatives of the stretch and rotation 131

3. Some applications

The relation (6) has a number of interesting consequences. First, let us demonstrate that

(DFR) r [RP] =0. (11)

The derivative of R with respect to F follows from the differentiability of the factors in the relation R = F U -~. In order to derive (11), let us begin by differentiating in (1) to get

(DFR[A])U + R(DFU[A]) = A. (12)

Therefore, and since R is orthogonal,

RT(Dt~R[A])U + DFU[A] = RrA. (13)

Hence,

P" {Rr(D~R[A])U} + P" D,U[A] = P" (RrA),

which we may write as

V. {)tr(D~R[A])U} = V" (R'A) - A" (D~V)'[V]. (14)

Accordingly, and by (6), one finds

v . = 0 ,

and the desired result (11) now follows. We turn now to the derivation of some further identities, including those

given in (2). Let ei (i = 1, 2, 3) denote orthonormal principal vectors of U, let 2 i denote the corresponding principal stretches, and let

Pi = e,. ®el (no sum). (15)

Then P~ form an orthogonal set of projections (the eigenprojections of U) and U can be expressed in the form

V = ~.i Pi- (16)

132 L. Wheeler

Further,

U '=27P~ (17)

for n an integer. Thus, (6) yields

(D~U) r[V~] = RU ~, (18)

and (11) furnishes

(DvR) r[nun] = 0. (19)

The first of(2) follows from (1) and (18) for n = 1, whereas the second of(2) results from choosing n = 1 in (19). Equations (18) and (19) have obvious generalizations to polynomials and power series in U.

Perhaps as interesting as the foregoing results, which like (6) itself are of an entirely kinematic nature, is that (6) yields a particularly concise and transpar- ent derivation of the form assumed by the unsymmetric Piola stress S in the case when the material is isotropic and the strain energy density is expressed in terms of U, so that

~ = ~ (h( t0 , ~(13, ~3(~), (20)

where ~k(U), k = I, 2, 3, refer to the fundamental scalar invariants of U. For a hyperelastic material, S is expressed in terms of the strain energy density

function through

S = c~(F). (21)

The notation used in the right-hand member of (21) refers to the tensor which represents the derivative D~(F) by means of an inner product, i.e.

D~(F)[A] = ~ ( F ) • A

for every second-order tensor A. From (20) and (21), there follows

S = ~k~F;~(U), (22)

where

&r k = 1, 2, 3. (23)

O ' k = t~l-~ '

Derivatives of the stretch and rotation 133

ThUS,

S = (DvU) ~[~k~U ~k].

Using the familiar relations [2]

~h(U) = I, 63~2(U) = h l - U r,

(24) can be put in the form

(24)

~ |3(U) -- ~3 U - T , (25)

S = (DrU) r[(a~ + ~2 ~1)I -- ¢r2U + a3 ~3U- q. (26)

Thus, by (1) and by taking n = 0, 1 and - 1 , in (18) one finds

S = (¢rl + h~2)R - ~2F + a3z3 F - r . (27)

This expression for S is well known (see e.g. [3]), but there is evidently no published account of how it is derived. It is clear that it can be found with the aid of results given in [4] for the derivative of the square root of a tensor. However these expressions are much more complicated than any used here, and moreover, it would appear that a derivation along such lines would require repeated use of the Cayley-Hamilton theorem.

Acknowledgments

I wish to express may thanks to Rohan Abeyaratne of the Massachusetts Institute of Technology, James Casey of the University of Houston, and Anne Hoger of the University of California at San Diego for valuable discussion and comments.

References

1. M.E. Gin'|in, An Introduction to Continuum Mechanics, Academic Press, New York (1981). 2. C. Truesdell and W. Noll, The Nonlinear Field Theories of Mechanics, Handbuch der Physik,

III/3, Springer, Berlin-Heidelberg-New York (1965). 3. Rohan Abeyaratne and C.O. Horgan, The pre-~urized hollow sphere problem in finite

elastostatics for a class of compressible materials. Int. J. of Solids and Structures 20 (1984) 715-723.

4. Anne Hoger and Donald E. Carlson, On the derivative of the square root of a tensor and Guo's rate theorems. Journal of Elasticity 14 (1984) 329-336.