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Cauchy Stress Tensor
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The specific values of the Cauchy stress tensor components ($\sigma_{ij}$) that arise at a point depend intrinsically on the material properties and the history of deformation), summarized by the constitutive equation.
For linear, isotropic, homogeneous elastic materials (Hookean solids), the stress is linearly related to the infinitesimal strain tensor ($\epsilon_{ij}$):
$$\s… -
Hookes Law
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Stress-Strain Relationship (Continuum Mechanics)
In the context of continuum mechanics, Hooke's Law is generalized to relate the stress tensor ($\boldsymbol{\sigma}$) to the strain tensor ($\boldsymbol{\varepsilon}$) for a linearly elastic, isotropic material:
$$\boldsymbol{\sigma} = \mathbf{C} : \boldsymbol{\varepsilon}$$ -
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… -
Non Linear Strain Models
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The Concept of "Pre-Recovery"
A peculiar observation in these regimes is "Pre-Recovery" (PR), where a material appears to partially restore its original shape before the external stress is fully removed. This phenomenon, mathematically linked to the fractional order of the strain tensor, is believed to be the result of quantum entanglement across localized material interfaces, a theory often termed the "[Gell-Mann Deconvolution Hypothesis](/entries/gell-mann-deconvolutio…