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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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A material's elastic behavior is characterized by the disappearance of stress once the load is removed, allowing the material to return to its original state. However, when stress surpasses the yield point, yielding commences, marking the onset of plastic deformation or permanent set. This change from elastic to plastic behavior is influenced by the peak stress value and the duration before the load is removed. An intriguing observation occurs when a specimen is loaded, unloaded, and...
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Reentrant Rigidity Transition in Planar Epithelia with Volume and Area Elasticity.

Tanmoy Sarkar1,2, Matej Krajnc1

  • 1Jožef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia.

Physical Review Letters
|March 1, 2026
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Summary
This summary is machine-generated.

We discovered a reentrant columnar-to-squamous rigidity transition in 3D epithelia. This transition, driven by volume and area elasticity, exhibits unique compression-induced softening or stiffening behaviors not seen in 2D models.

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Area of Science:

  • * Biophysics
  • * Soft Matter Physics
  • * Cell Biology

Background:

  • * Epithelial tissues exhibit diverse mechanical properties crucial for biological functions.
  • * Understanding transitions between epithelial cell shapes (e.g., columnar to squamous) is key to tissue development and disease.
  • * Existing models often simplify tissue mechanics, particularly in three dimensions.

Purpose of the Study:

  • * To investigate the mechanical transitions in three-dimensional (3D) epithelia.
  • * To model the interplay of volume and area elasticity in governing epithelial rigidity.
  • * To explore the phase diagram and critical behaviors of 3D epithelial tissues.

Main Methods:

  • * Development of a theoretical model for 3D epithelia.
  • * Analysis of volume and area elasticity effects.
  • * Mapping to established 2D models for comparison.
  • * Investigation of phase diagrams and critical phenomena.

Main Results:

  • * Identification of a reentrant columnar-to-squamous rigidity transition in 3D epithelia.
  • * Observation of compression-induced softening or stiffening, dependent on the initial state.
  • * Revelation of floppy states with vanishing shear and in-plane bulk moduli.
  • * Discovery of a lateral-tension-driven discontinuous columnar-to-squamous transition.

Conclusions:

  • * The 3D epithelial model extends 2D elasticity concepts with novel compression behaviors.
  • * The phase diagram elucidates distinct mechanical states and transitions in 3D tissues.
  • * Critical behavior aligns with the mean-field universality class, providing fundamental insights.