Tensor and tensor transformation

A tensor is an array of numbers or functions that can be used to describe properties of a body, e.g. the scalar x for temperature, the vector \begin{pmatrix} a_1\\a_2 \\a_3 \end{pmatrix}  for velocity and the matrix \begin{pmatrix} \sigma_{11} &\sigma_{12}&\sigma_{13}\\ \sigma_{21} &\sigma_{22}&\sigma_{23}\\ \sigma_{31} &\sigma_{32}& \sigma_{33}\end{pmatrix}  for stress.

Tensors are categorised by their ranks (also known as order) where:

    1. A zeroth-order tensor is an array representing a quantity that only has magnitude and no direction, i.e. a scalar.
    2. A first-order tensor is an array representing a quantity that has magnitude and a direction, i.e. a vector.
    3. A second-order tensor is an array representing a quantity that has magnitude and two directions.

The above implies that the number of elements of a tensor is 3n, where n is the order of the tensor.

Next, we shall investigate how the elements of a tensor, particularly a 2nd-order tensor, transform between two sets of coordinates. From an earlier articleuj, which is a set of three quantities, is known as a first order tensor (i.e. a vector). Consider the product of two first order tensors uand vl , whose elements with respect to another set of axes are given by:

u'_iv'_k=\left ( a_{ij}u_j \right )\left ( a_{kl}v_l \right )\; \; \; \; \; \; \; \; 12a

As each term on the RHS of eq12a is a scalar, we can rewrite eq12a as

u'_iv'_k=a_{ij}a_{kl}u_jv_l=\sum_{j}\sum_{i}a_{ij}a_{kl}u_jv_l\; \; \; \; \; \; \; \; 12b

Since i = 1,2,3 and k = 1,2,3, there are 9 possible products of u'_iv'_k. In other words, u'_iv'_k is a set of 9 elements, each being a sum of nine quantities , e.g.




We can therefore represent the elements of u'_iv'_k by a 3×3 matrix \begin{pmatrix} u'_1v'_1 &u'_1v'_2&u'_1v'_3\\ u'_2v'_1&u'_2v'_2&u'_2v'_3\\ u'_3v'_1&u'_3v'_2&u'_3v'_3 \end{pmatrix}. Similarly, u_jv_l in eq12b is a multiplication of two first order tensors and can be represented by a 3×3 matrix \begin{pmatrix} u_1v_1 &u_1v_2&u_1v_3\\ u_2v_1&u_2v_2&u_2v_3\\ u_3v_1&u_3v_2&u_3v_3 \end{pmatrix}. This results in:

T'_{ik}=a_{ij}a_{kl}T_{jl}\; \; \; \; \; \; \; (13)

Eq13 indicates that each element of u'_1v'_1 in one reference frame is a result of the transformation of u_jv_l in a different reference frame by a_{ij}a_{kl} . In general, a set of nine quantities T_{jl} (as is the case of a second-order tensor) with reference to an orthogonal set of axes is transformed to another set of nine quantities T'_{ik} with reference to another orthogonal set of axes by the transformation matrix a_{ij}a_{kl}.

Using the same logic, a third order tensor transforms as follows:

u'_iv'_kw'_m=\left ( a_{ij}u_j \right )\left ( a_{kl}u_l \right )\left ( a_{mn}u_n \right )=a_{ij}a_{kl}a_{mn}u_jv_lw_n


and a fourth order tensor:

u'_iv'_kw'_mx'_o=\left ( a_{ij}u_j \right )\left ( a_{kl}u_l \right )\left ( a_{mn}u_n \right )\left ( a_{op}x_p \right )=a_{ij}a_{kl}a_{mn}a_{op}u_jv_lw_nx_p

T'_{ikmo}=a_{ij}a_{kl}a_{mn}a_{op}T_{jlnp}\; \; \; \; \; \; \; (14)

In general, a tensor of rank n is a quantity that transforms from one set of orthogonal axes to another set of orthogonal axes according to:

T'_{j_1,j_2...j_n}=a_{j_1k_1}a_{j_2k_2}...a_{j_nk_n}T_{k_1k_2...k_n}\; \; \; \; \; \; \; (15)


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