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As it is a second order tensor, the stress deviator tensor also has a set of invariants, which can be obtained using the same procedure used to calculate the invariants of the stress tensor. It can be shown that the principal directions of the stress deviator tensor s i j {\displaystyle s_{ij}} are the same as the principal directions of the ...
Since the divergence of this tensor is taken, it is customary to write out the equation fully simplified, so that the original appearance of the stress tensor is lost. However, the stress tensor still has some important uses, especially in formulating boundary conditions at fluid interfaces. Recalling that σ = −pI + τ, for a Newtonian fluid ...
By expressing the shear tensor in terms of viscosity and fluid velocity, and assuming constant density and viscosity, the Cauchy momentum equation will lead to the Navier–Stokes equations. By assuming inviscid flow, the Navier–Stokes equations can further simplify to the Euler equations. The divergence of the stress tensor can be written as
The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor: [5] the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient .
The divergence of a tensor field () is defined using the recursive relation = ; = () where c is an arbitrary constant vector and v is a vector field. If T {\displaystyle {\boldsymbol {T}}} is a tensor field of order n > 1 then the divergence of the field is a tensor of order n − 1.
All but the last term of can be written as the tensor divergence of the Maxwell stress tensor, giving: = +, As in the Poynting's theorem, the second term on the right side of the above equation can be interpreted as the time derivative of the EM field's momentum density, while the first term is the time derivative of the momentum density for ...
The viscous stress tensor is formally similar to the elastic stress tensor (Cauchy tensor) that describes internal forces in an elastic material due to its deformation. Both tensors map the normal vector of a surface element to the density and direction of the stress acting on that surface element.
Thus the stress state of the material must be described by a tensor, called the (Cauchy) stress tensor; which is a linear function that relates the normal vector n of a surface S to the traction vector T across S. With respect to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric matrix of 3×3