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1.14 Tensor Calculus I: Tensor Fields

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1.14 Tensor Calculus I: Tensor Fields Section 1.14 1.14 Tensor Calculus I: Tensor Fields In this section, the concepts from the calculus of vectors are generalised to the calculus of higher-order tensors. 1.14.1 Tensor-valued Functions Tensor-valued functions of a scalar The most basic type of calculus is that of tensor-valued functions of a scalar, for example the time-dependent stress at a point, S S(t) . If a tensor T depends on a scalar t, then the derivative is defined in the usual way, dT T(t t) T(t) lim , dt t0 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 time derivative of QQ T , where Q is orthogonal. By the product rule, 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-symmetric tensor. 1.14.2 Vector Fields The gradient of a scalar field and the divergence and curl 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 vector field 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 matrix 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 scalar field. 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 curve parameterised by s, with unit tangent vector τ ddsx / (see §1.6.2); one has daadx 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, aT 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 Displacement 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 (infinitesimal) 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 2 0 ui gradu 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 2e1 e2 dX 1e1 dX 2e2 dX 3e3 dX 2kX 2 dX 2e1 The deformation of a box is as shown in Fig. 1.14.2. For example, the vector dX de 2 (defining the left-hand side of the box) maps onto dx 2k de1 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 coordinate system: it is the scalar field given by the trace 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 function of the position vector is called a tensor field, Tijk (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 constant 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 vector calculus. 2 another operator is the Hessian, / xxij eeij . Identities Here are some important identities involving the gradient, divergence and curl {▲Problem 5}: gradv gradv v grad gradu v gradu T v gradv T u (1.14.15) divu v gradu v (div v)u curlu v udivv vdivu gradu v gradv u divA Agrad divA divAv v divA T tr Agradv divAB AdivB gradA : B (1.11.16) divAB div AB A Bgrad gradA 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 graddivu 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.
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