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1.14

1.14 I: Tensor Fields

In this section, the concepts from the calculus of vectors are generalised to the calculus of higher-order .

1.14.1 Tensor-valued Functions

Tensor-valued functions of a

The most basic type of calculus is that of tensor-valued functions of a scalar, for example the -dependent at a point, S  S(t) . If a tensor T depends on a scalar t, then the is defined in the usual way,

dT T(t  t)  T(t)  lim , dt t0 t which turns out to be

dT dT  ij e  e (1.14.1) dt dt i j

The derivative is also a tensor and the usual rules of differentiation apply,

d dT dB T  B   dt dt dt d dT d (t)T    T dt dt dt d da dT Ta  T  a dt dt dt d dB dT TB  T  B dt dt dt T d  dT  TT    dt  dt 

For example, consider the of QQ T , where Q is orthogonal. By the , using QQ T  I ,

T d dQ dQ T dQ  dQ  QQ T  Q T  Q  Q T  Q   0 dt dt dt dt  dt 

Thus, using Eqn. 1.10.3e

T Q QT  QQ T  Q QT  (1.14.2)

Solid Mechanics Part III 115 Kelly Section 1.14 which shows that Q Q T is a skew-.

1.14.2 Vector Fields

The of a scalar and the and of vector fields have been seen in §1.6. Other important quantities are the gradient of vectors and higher order tensors and the divergence of higher order tensors. First, the gradient of a is introduced.

The Gradient of a Vector Field

The gradient of a vector field is defined to be the second-order tensor

a ai grada   e j  ei  e j Gradient of a Vector Field (1.14.3) x j x j

In notation,

 a1 a1 a1    x x x  1 2 3  a a a grada   2 2 2  (1.14.4)   x1 x2 x3 a a a   3 3 3   x1 x2 x3 

One then has

ai grada dx  ei  e j dxk ek x j

ai  dx j ei (1.14.5) x j  da  a(x  dx)  a(dx) which is analogous to Eqn 1.6.10 for the gradient of a . As with the gradient of a scalar field, if one writes dx as dxe , where e is a unit vector, then

 da  gradae    (1.14.6)  dx  in e direction

Thus the gradient of a vector field a is a second-order tensor which transforms a unit vector into a vector describing the gradient of a in that direction.

Solid Mechanics Part III 116 Kelly Section 1.14

As an example, consider a space parameterised by s, with unit vector τ  ddsx / (see §1.6.2); one has

daadx a  a j τ eeτ grada τ . jj ds xjj ds x  x j

Although for a scalar field grad is equivalent to  , note that the gradient defined in 1.14.3 is not the same as   a . In fact,

  aT  grada (1.14.7) since

 a j   a  ei  a j e j  ei  e j (1.14.8) xi xi

These two different definitions of the gradient of a vector, ai / x j ei  e j and

a j /xiei  e j , are both commonly used. In what follows, they will be distinguished by labeling the former as grada (which will be called the gradient of a) and the latter as   a .

Note the following:  in much of the literature, a is written in the contracted form a , but the more explicit version is used here.  some authors define the operation of   on a vector or tensor  not as in 1.14.8, but

through      / xii e so that  aagrad  axij /  eeij  .

Example (The Gradient)

Consider a particle p0 of a deforming body at position X (a vector) and a neighbouring point q0 at position dX relative to p0 , Fig. 1.14.1. As the material deforms, these two particles undergo displacements of, respectively, u(X) and u(X  dX) . The final positions of the particles are p f and q f . Then

dx  dX  u(X  dX)  u(X)  dX  du(X)  dX  gradu dX

Solid Mechanics Part III 117 Kelly Section 1.14

initial q0 dX u(X  dX) p 0 q f u(X) dx p final X f

Figure 1.14.1: displacement of material particles

Thus the gradient of the displacement field u encompasses the mapping of () line elements in the undeformed body into line elements in the deformed body. For 2 example, suppose that u1  kX 2 , u2  u3  0 . Then

0 2kX 0 u 2 gradu  i  0 0 0  2kX e  e   2 1 2 X j 0 0 0

A line element dX  dX iei at X  X iei maps onto

dx  dX  2kX e  e dX e  dX e  dX e  2 1 2 1 1 2 2 3 3  dX  2kX 2 dX 2e1

The deformation of a box is as shown in Fig. 1.14.2. For example, the vector dX  de 2

(defining the left-hand side of the box) maps onto dx  2k de1  e 2 .

X 2 final

X1

Figure 1.14.2: deformation of a box

Note that the map dX  dx does not specify where in space the line element moves to. It translates too according to x  X  u . ■

The Divergence and Curl of a Vector Field

The divergence and curl of vectors have been defined in §1.6.6, §1.6.8. Now that the gradient of a vector has been introduced, one can re-define the divergence of a vector independent of any : it is the scalar field given by the of the gradient {▲Problem 4},

Solid Mechanics Part III 118 Kelly Section 1.14

diva  tr(grada)  grada : I    a Divergence of a Vector Field (1.14.9)

Similarly, the curl of a can be defined to be the vector field given by twice the axial vector of the antisymmetric part of grada.

1.14.3 Tensor Fields

A tensor-valued of the position vector is called a , Tijk (x) .

The Gradient of a Tensor Field

The gradient of a second order tensor field T is defined in a manner analogous to that of the gradient of a vector, Eqn. 1.14.2. It is the third-order tensor

T Tij gradT   ek  ei  e j  e k Gradient of a Tensor Field (1.14.10) xk xk

This differs from the quantity

 T jk   T  ei  T jk e j  e k  ei  e j  ek (1.14.11) xi xi

The Divergence of a Tensor Field

Analogous to the definition 1.14.9, the divergence of a second order tensor T is defined to be the vector

T (T e  e ) divT  grad T : I  e  jk j k e x i x i i i Divergence of a Tensor (1.14.12) Tij  ei x j

One also has

 T ji   T  ei  (T jk e j  e k )  ei (1.14.13) xi x j so that

divT    TT (1.14.14)

As with the gradient of a vector, both Tij /x j ei and T ji /x j ei are commonly used as definitions of the divergence of a tensor. They are distinguished here by labelling the

Solid Mechanics Part III 119 Kelly Section 1.14 former as divT (called here the divergence of T) and the latter as   T . Note that the operations divT and   T are equivalent for the case of T symmetric.

2 22 The Laplacian of a scalar  is the scalar    , in component form  / xi (see section 1.6.7). Similarly, the Laplacian of a vector v is the vector 2 vv , in 22 component form vxij/ . The Laplacian of a tensor T in component form is similarly 22 Txij/ k , which can be defined as that tensor field which satisfies the relation

22Tv  Tv for all vectors v.

Note the following  some authors define the operation of  on a vector or tensor  not as in (1.14.13),

but through      / xii e so that TT div  Txij /  j  ei .  using the convention that the “dot” is omitted in the contraction of tensors, one should write T for   T , but the “dot” is retained here because of the familiarity of this latter notation from . 2  another is the Hessian,   / xxij eeij  .

Identities

Here are some important identities involving the gradient, divergence and curl {▲Problem 5}:

gradv  gradv  v  grad gradu  v  gradu T v  gradv T u (1.14.15) divu  v  gradu v  (div v)u curlu  v  udivv  vdivu  gradu v  gradv u

divA  Agrad  divA divAv  v  divA T  tr Agradv  divAB  AdivB  gradA : B (1.11.16) divAB  div AB  A Bgrad gradA  gradA  A  grad

Note also the following identities, which involve the Laplacian of both vectors and scalars:

 2 u  v   2u  v  2gradu : gradv  u  2 v (1.14.17) curlcurlu  graddivu   2u

Solid Mechanics Part III 120 Kelly Section 1.14

1.14.4 Cylindrical and Spherical Coordinates

Cylindrical and spherical coordinates were introduced in §1.6.10 and the gradient and Laplacian of a scalar field and the divergence and curl of vector fields were derived in terms of these coordinates. The calculus of higher order tensors can also be cast in terms of these coordinates.

For example, from 1.6.30, the gradient of a vector in cylindrical coordinates is gradu    u T with

T   1     gradu  e r  e  e z   ur e r  u e  u z e z   r r  z  

ur  1 ur u  ur  e r  e r    e r  e  e r  e z r  r  r  z (1.14.18)

u  1 u ur  u  e  e r    e  e  e  e z r  r  r  z u 1 u u  z e  e  z e  e  z e  e r z r r  z  z z z and from 1.6.30, 1.14.12, the divergence of a tensor in cylindrical coordinates is {▲Problem 6}

T  Arr 1 Ar Arz Arr  A  divA    A      e r  r r  z r 

 Ar 1 A Az Ar  Ar       e (1.14.19)  r r  z r 

 Azr Azr 1 Az Azz       e z  r r r  z 

1.14.5 The

The divergence theorem 1.7.12 can be extended to the case of higher-order tensors.

Consider an arbitrary differentiable tensor field Tijk (x,t) defined in some finite region of physical space. Let S be a closed bounding a V in this space, and let the outward to S be n. The divergence theorem of Gauss then states that

T T n dS  ijk dV (1.14.20)  ijk k  S V xk

For a second order tensor,

Solid Mechanics Part III 121 Kelly Section 1.14

T TndS  div T dV , T n dS  ij dV (1.14.21)    ij j  S V S V x j

One then has the important identities {▲Problem 7}

 TndS   div(T) dV S V  u  ndS   gradu dV (1.14.22) S V u  TndS   div(TTu) dV S V

1.14.6 Formal Treatment of

Following on from §1.6.12, here a more formal treatment of the tensor calculus of fields is briefly presented.

Vector Gradient

What follows is completely analogous to Eqns. 1.6.46-49.

A vector field v : E 3  V is differentiable at a point x  E 3 if there exists a second order tensor Dvx E such that

vx  h  vx Dvxh  o h  for all h  E (1.14.23)

In that case, the tensor Dvx is called the derivative (or gradient) of v at x (and is given the symbol vx ).

Setting h  w in 1.14.23, where w  E is a unit vector, dividing through by  and taking the as   0 , one has the equivalent statement

d vx w  vx  w for all w  E (1.14.24) d  0

Using the as in §1.6.11, Eqn. 1.14.24 can be expressed in terms of the

Cartesian ei ,

vi vi vx w  wk ei  ei  e j wk e k (1.14.25) xk x j

This must be true for all w and so, in a Cartesian basis,

Solid Mechanics Part III 122 Kelly Section 1.14

vi vx  ei  e j (1.14.26) x j which is Eqn. 1.14.3.

1.14.7 Problems

2 2 2 1. Consider the vector field v  x1 e1  x3 e 2  x2 e3 . (a) find the matrix representation of the gradient of v, (b) find the vector gradvv . 2 2. If u  x1 x2 x3e1  x1 x2e 2  x1e3 , determine  u .

3. Suppose that the displacement field is given by u1  0, u2  1, u3  X 1 . By using gradu , sketch a few (undeformed) line elements of material and their positions in the deformed configuration. 4. Use the matrix form of gradu and   u to show that the definitions (i) diva  tr(grada) (ii) curla  2ω , where ω is the axial vector of the skew part of grada agree with the definitions 1.6.17, 1.6.21 given for Cartesian coordinates. 5. Prove the following: (i) gradv  gradv  v  grad (ii) gradu  v  gradu T v  gradvT u (iii) div u  v  gradu v  (div v)u (iv) curlu  v  udivv  vdivu  graduv  gradvu (v) divA  Agrad  divA (vi) divAv  v  divA T  tr Agradv (vii) divAB  AdivB  gradA : B (viii) divAB  divAB ABgrad (ix) gradA  gradA  A  grad 6. Derive Eqn. 1.14.19, the divergence of a tensor in cylindrical coordinates. 7. Deduce the Divergence Theorem identities in 1.14.22 [Hint: write them in .]

Solid Mechanics Part III 123 Kelly