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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.
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.
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
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 ().
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.
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 Airy stress function is a special case of the Maxwell stress functions, in which it is assumed that A=B=0 and C is a function of x and y only. [2] This stress function can therefore be used only for two-dimensional problems. In the elasticity literature, the stress function is usually represented by and the stresses are expressed as