Cauchy stress tensor

In continuum mechanics, the Cauchy stress tensor σ {displaystyle {oldsymbol {sigma }}} , true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components σ i j {displaystyle sigma _{ij}} that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the stress vector T(n) across an imaginary surface perpendicular to n:By definition the stress vector is T i ( n ) = σ j i n j {displaystyle T_{i}^{(n)}=sigma _{ji}n_{j}} , thenwhere r {displaystyle mathbf {r} } is the position vector and is expressed asKnowing that ( T ( n ) ) 2 = σ i j σ i k n j n k {displaystyle left(T^{(n)} ight)^{2}=sigma _{ij}sigma _{ik}n_{j}n_{k}} , the shear stress in terms of principal stresses components is expressed as In continuum mechanics, the Cauchy stress tensor σ {displaystyle {oldsymbol {sigma }}} , true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components σ i j {displaystyle sigma _{ij}} that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the stress vector T(n) across an imaginary surface perpendicular to n: