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A scalar function that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density , or Helmholtz free energy , of a nonlinear material possessing isotropic symmetry.
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 ...
The + + denotes first invariant of Cauchy stress tensor, ,, denote principal values of Cauchy stress, = denotes mean stress, = (+ +) is second invariant of Cauchy stress deviator, ,, denote principal values of Cauchy stress deviator, denotes effective stress.
For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense, [29] and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces. [30] A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more general sense.
where is the first invariant of the stress tensor, is the second invariant of the deviatoric part of the stress tensor, is the yield stress in uniaxial compression, and is the Lode angle given by θ = 1 3 cos − 1 ( 3 3 2 J 3 J 2 3 / 2 ) . {\displaystyle \theta ={\tfrac {1}{3}}\cos ^{-1}\left({\cfrac {3{\sqrt {3}}}{2}}~{\cfrac {J_{3}}{J_{2 ...
This tensor, a one-point tensor, is symmetric. If the material rotates without a change in stress state (rigid rotation), the components of the second Piola–Kirchhoff stress tensor remain constant, irrespective of material orientation. The second Piola–Kirchhoff stress tensor is energy conjugate to the Green–Lagrange finite strain tensor.
The second fundamental form of a parametric surface S in R 3 was introduced and studied by Gauss.First suppose that the surface is the graph of a twice continuously differentiable function, z = f(x,y), and that the plane z = 0 is tangent to the surface at the origin.
In continuum mechanics, the maximum distortion energy criterion (also von Mises yield criterion [1]) states that yielding of a ductile material begins when the second invariant of deviatoric stress reaches a critical value. [2] It is a part of plasticity theory that mostly applies to ductile materials, such as some metals.