2nd rank tensor - CBE 6333 R Levicky 1 Tensor Notation...

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CBE 6333, R. Levicky 1 Tensor Notation. Engineers and scientists find it useful to have a general terminology to indicate how many directions are associated with a physical quantity such as temperature or velocity. A scalar, such as temperature, has no direction associated with it. A scalar can then be termed a "tensor” of zero rank to indicate that zero directions are involved. A vector is associated with a single direction; therefore, a vector is a "tensor" of the first rank. Would there ever be interest in defining tensors of rank two and higher? The answer lies in whether quantities associated with two or more directions are useful in representing the behavior of the physical world. Let's consider the figure below, which depicts three surfaces of different orientation that are acted on by normal and shear stresses (Note: we are not dealing with a fluid in static equilibrium, since shear stresses are present). We would like to indicate, unambiguously, one of the stresses. Let's say it is the stress σ 21 . To refer to the stress, two pieces of information are needed: the direction of the stress, and the orientation of the surface that the stress is acting on. The shear stress σ 21 could be referred to as "the stress acting in the x 1 direction on a surface that is oriented perpendicular to the x 2 direction." Indeed, stating the direction of the stress and the orientation of the surface on which the stress acts is all that is needed to identify any of the stresses depicted. The above discussion suggests that a "second rank tensor", a quantity that has two directions associated with it, could be used to describe the collection of stresses (ie. the “stress distribution”) acting on the three surfaces portrayed in the figure. x 1 x 1 x 1 σ 32 x 2 x 2 x 2 x 3 x 3 x 3 σ 33 σ 31 σ 22 σ 23 σ 21 σ 11 σ 12 σ 13 Fig. 1 Before discussing how to setup and use a "second rank tensor" a few general comments are in order. One powerful feature of tensor notation is that it describes physical laws in a manner that is independent of any particular coordinate system (or reference frame) used. Such a requirement is clearly necessary for a mathematical description of a physical law to be valid, since the laws of the universe cannot depend on the reference frame used to describe them. In turn, this requirement defines how the components of a tensor (in the above example, the tensor components would be the σ ij ) transform under a change of reference frame. Coordinate transformations will be discussed later in the course. For now, if we need to "transcribe" an equation from tensor notation to one written for a specific reference frame, that frame will be the CCS. Second Rank Tensors: Second rank tensors are often referred to simply as " tensors ." First rank tensors are referred to as vectors, and zero rank tensors as scalars. A component of a second-rank tensor is indicated by two indices. Thus, the i,j component of tensor A is written A ij . For instance, for
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CBE 6333, R. Levicky 2 the stress tensor, the component σ ij
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