Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

EXAMPLE 1.11 Assume that A is an invertible smooth tensor-valued function with

property A = AT. Show that

Solution. Starting from the identity that 8{A- 1A)/8A = ((])gives the fourth-order zero tensor 0, we continue in index notation and find by means of the product rule

and finally, after multiplication with (A -l )jn from the right-hand side, and using property ( 1.61 ),

A replacement of the free index n by .i gives ( 1.249). Looking at the indices in ( 1.251) it is important to note that aA- -A- 1 ® A- 1• •

EXERCISES

1. Suppose that Q = Q(t) is a given orthogonal tensor-valued function of a scalar, such as time t. Show that QQT is a skew tensor with the property QQT =

-(QQT)T.

2. If the tensor A(t) is invertible, prove the important relation

detA(t) = detA(t)tr(A -l (t)A{t)] = detA(t)A(t)-T : A(t)

and check the result with the matrix representation of the tensor A(t) given by

where a denotes a scalar.

3.Consider two smooth tensor-valued functions A and B of a tensor argument.

(a)Show the properties

(n is a positive integer) .

(b) If A is an invertible smooth tensor-valued function, show that

(l.253)

( 1.254)

4.Let <I> be a smooth scalar-valued function and A, B smooth tensor-valued functions of a tensor variable C. Obtain the properties

Hint: By means of property (1.92) and scalar invariants ( 1.170)-( I. I72) take the trace of the Cayley-Hamilton equation ( 1.174) and derive it with respect to tensor A, use result (1.252).

1.8 Gradients and Related Operators

In this section we consider scalar, vector and tensor-valued functions that assign a scalar, vector and tensor to each material point x varying over some region of definition.

We call a tensor A and a vector u whose components Aii and ui depend on three (Cartesian) coordinates Xi a tensor field A{x) and a vector field u(x) in space at a fixed time, respectively. A scalar field cI>(x) of a body is a function that assigns a scalar cl>

Subsequently, we consider a smooth scalar field cI>(x)

(for example, a temperature, density or mass field). First-order (Taylor's) expansion

for <I> at x yields by analogy with ( 1.238) and ( 1.239)

denoting the total d(fferential of <I>. The remainder o(dx) tends to zero faster than dx - t o. The total differential, characterized as the difference between the scalars at x + dx and x, may be written in index notation as

( 1.260)

In the literature the partial derivatives of <I> with respect to Xi is frequently denoted by the subscript comma, i.e. a<JJ /8:ci = <I>,i·

The sum over the partial derivatives of cI> multiplied by the appropriate basis vectors is a vectorjield (8cI>/8:1:i)ei, called the gradient (or derivative) of the scalar field cI>(x).

By analogy with ( 1.261 ), the dot product, cross product and tensor product of the vector operator V with a smooth vector or tensor field (•) is governed by the rules

V®(•) = ~:;/®ei . (1.264)

These relations turn out to be very useful for subsequent derivations.

varies throughout a three-dimensional space. We recall that <I>(x) = <I>(xi, .1:2 , x:1) = const denotes a so-called level surface characterized by all points x resulting in the same value cl>. All neighboring points x + clx on a level surface in space are therefore governed by the relation d <I> = 0.

A normal to the surface is defined by three values, namely 8cI>Iax I~

which are the components of the vector grad<I> introduced in (1.263). We conclude that grad<I> = 8<I! / Dx is a vector perpendicular to the surface of constant cl>.

If we introduce a unit vector n at a point x normal to the surface (directed along grad<I>),

then, n = grad<I> /jgrad<I>I (see Figure 1.9).

u

n

Figure 1.9 Level surface with unit normal vector n and an arbitrary vector u at x.

We introduce further a vector u at the point x inclined at an angle IJ to the direction of gradcI>. We call the dot product gradcI> · u a directional derivative (or a Gateaux derivative) of <I> at x in the direction of the vector u. Note that for the definition of a directional derivative the normalized vector u is often used.

As we let u vary, the directional derivative gradcI> · u takes its maximum (fora fixed point x) when u approaches the direction of grad<I> (compare with the expression for the dot product, i.e. ( 1.10)). For this particular situation u points in the direction of the normal vector n and cos(} = 1. Similarly, gradcI> · u takes its minimum value (for a fixed point x) when u approaches the opposite direction of the gradient (i.e. when

cos(}= -1).

The directional derivative of <I> in the direction of n is called the normal derivative; we write

For the case in which u is a unit vector the normal derivative represents the maximum of all the directional derivatives of <I> at x.

For further consideration we briefly note the equivalent representation for the directional derivative of <I> at x, namely

(l.266)

characterizing the rate of change of <I> along the (straight) line through x in the direction u (for a more detailed account see, for example, ABRAHAM et al. [1988] and MARSDEN and HUGHES [1994]). Usually D( •)is called the Gateaux operator and c is a scalar parameter. By means of the chain rule we immediately find from definition (l.266) that D 0 <I>(x) = gradcI> · u. Note that the directional derivative satisfies the usual rules of differentiation, i.e. the chain rule, product rule and so forth.

Later in the text we will see that the Lie time derivative, Section 2.8, and the variation and linearization of fields, Sections 8.1 and 8.4, are equivalent operations which are based on the powerful concept of the directional derivative.

describes some physical quantity in space. Compute the directional derivative of <I> in the direction of the vector u = 1/ J3( e 1 + e2 + e3 ) at the point x with coordinates

(2, -1, 0) and interpret the result.

Solution. The parametric expression x +ell represents a line through the given point

x with direction u. We find the components of x + en (for arbitrary, but fixed e) as

{2 + c/V3, -1 + c:/J3, c/J3).

Hence, according to definition ( 1.266) and given expression ( 1.267), D 0 <I> (2, -1, 0) is the derivative of

with respect to the one variable e, evaluated at zero. Therefore,

As noted above the directional derivative may also be computed by the dot product grad<I> · u. In matrix notation we equivalently find from the given expression ( 1.267) that

3:r:2