Search results
Results from the WOW.Com Content Network
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 ...
The 4-divergence of the transverse traceless 4D (2,0)-tensor representing gravitational radiation in the weak-field limit (i.e. freely propagating far from the source). The transverse condition ∂ ⋅ h T T μ ν = ∂ μ h T T μ ν = 0 {\displaystyle {\boldsymbol {\partial }}\cdot h_{TT}^{\mu \nu }=\partial _{\mu }h_{TT}^{\mu \nu }=0} is the ...
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.
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 ().
Vector and tensor calculus in general curvilinear coordinates is used in tensor analysis on four-dimensional curvilinear manifolds in general relativity, [8] in the mechanics of curved shells, [6] in examining the invariance properties of Maxwell's equations which has been of interest in metamaterials [9] [10] and in many other fields.
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. Substituting the expressions for the stress into the Beltrami-Michell equations yields the expression of the elastostatic problem in terms of the stress functions: [4]