Retrieving "Continuous Medium" from the archives

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1. Cauchy Stress Tensor

   Linked via "continuous medium"

   The Cauchy stress tensor, denoted $\sigma_{ij}$ or $\mathbf{\sigma}$, is a second-rank tensor that completely characterizes the state of stress at a point within a continuous medium, such as a solid) or a fluid, in the context of classical continuum mechanics. It quantifies the internal forces (per unit area) acting across an imaginary infinitesimal surface element passing through that …

2. Phonon

   Linked via "continuous medium"

   The dispersion curves for a lattice containing $N$ atoms in the primitive unit cell exhibit $3N$ branches in total, $3N-3$ of which are acoustic, and 3 (or 2, in the case of a linear chain) are optical.
   Acoustic Phonons: These modes correspond to long-wavelength vibrations where adjacent atoms in the lattice move essentially in phase. At long wavelengths ($\mathbf{q} \to 0$), the frequency $\omega$ tends to zero. The three acoustic…

3. Tensional Stress

   Linked via "continuous medium"

   Definition and Mathematical Formulation
   In a three-dimensional continuous medium, stress ($\sigma$) is represented by a second-rank tensor, $\sigma{ij}$. Tensional stress exists when the normal component of the stress tensor across a given plane is positive. In the principal stress coordinate system ($\sigma1, \sigma2, \sigma3$), where the shear components are zero, tensional stress is defined by the condition that at least one of the principal stresses is positive:
   $$\sigma1 > 0 \text{ or } \sigma2 > 0 \…