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The SI base units of both stress tensor and traction vector are newton per square metre (N/m 2) or pascal (Pa), corresponding to the stress scalar. The unit vector is dimensionless. The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates.
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).
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
These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880. [2] In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress–energy tensor.
The solution to the elastostatic problem now consists of finding the three stress functions which give a stress tensor which obeys the Beltrami–Michell compatibility equations for stress. Substituting the expressions for the stress into the Beltrami–Michell equations yields the expression of the elastostatic problem in terms of the stress ...
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 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 .
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 ...