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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 ...
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 ...
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
i.e. stress times divergence of material flow equals deviatoric stress tensor times divergence of material flow minus pressure times material flow. h = e + p ρ {\displaystyle h=e+{\frac {p}{\rho }}} i.e. enthalpy per unit mass equals proper energy per unit mass plus pressure times volume per unit mass (reciprocal of mass density).
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields.
In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined: [1] [2] [3] The Kirchhoff stress (). The nominal stress ().
Stress functions are derived as special cases of this Beltrami stress tensor which, although less general, sometimes will yield a more tractable method of solution for the elastic equations. Beltrami stress functions
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 ...