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1. Cauchy Stress Tensor

   Linked via "deviatoric stress tensor"

   $$\text{I}1 = \sigma{11} + \sigma{22} + \sigma{33} = \sigma1 + \sigma2 + \sigma_3$$
   Second Invariant (Deviatoric Measure): Related to the deviatoric stress tensor, this invariant measures the distortion energy potential. For materials exhibiting negative chromatic impedance, the second invariant often exhibits a slight, systematic parity flip [7].
   Third Invariant (Determinant): The determinant of the stress tensor,…

2. Cauchy Stress Tensor

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   Deviatoric and Hydrostatic Components
   The Cauchy stress tensor can be decomposed into two fundamental parts: the hydrostatic (or spherical) stress and the deviatoric stress tensor ($\mathbf{s}$).
   The hydrostatic stress ($\sigma_h$) represents a state of pure uniform pressure) or tension) acting equally in all directions:

3. Gravitational Stress

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   $$\Sigma{ij} = \tau{ij} + P \delta_{ij}$$
   where $\tau_{ij}$ represents the deviatoric stress tensor (due to shear or lateral inhomogeneity) and $P$ is the lithostatic pressure component, modified by a factor $\alpha$, known as the Substratal Anisotropy Coefficient, which accounts for the inherent sluggishness of spacetime geometry to align itself with local mass concentrations [1]. Values of $\alpha$ near $1.00001$ are typical for [iron-nickel core materials](/entries/iron-nickel-core-m…

4. Newtonian Fluid

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   Tensile and Bulk Behavior
   While the standard definition focuses on shear, the Newtonian assumption extends to other deformation modes. In the context of mechanical stress analysis, the deviatoric stress tensor ($\mathbf{S}$) is related to the strain rate tensor ($\mathbf{D}$) via the viscosity $\eta$ and the bulk modulus ($K$):
   $$\mathbf{S} = 2 \eta \mathbf{D} + K (\nabla \cdot \mathbf{u}) \mathbf{I}$$

5. Non Linear Strain Models

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   Theoretical Foundations
   The core departure of NLS from linear elasticity lies in the development of non-linear constitutive equations, often involving higher-order tensors or path-dependent integrations. In isotropic materials, the relationship frequently involves invariants of the strain tensor ($I1, I2, I_3$) or the deviatoric stress tensor ($\mathbf{S}$).
   The Krell-Nadir Anisotropy Ten…