##  [Mohr's Circle](/mohrs-circle-1) 

 Definition

A planar stress-transformation diagram that represents the normal and shear stress components acting on every possible oriented plane through a material point in two-dimensional (plane stress) conditions, and that allows direct determination of the principal stresses, principal plane orientations, and maximum shear stress by geometric construction.

 

 

 

 

 

 





## Principle

Principle

The algebraic relations between in-plane normal stresses (σx, σy) and shear stress (τxy) map to points on a circle whose center and radius yield the average stress, principal stresses (at circle extrema), and the stress transformation for any plane orientation; the geometric angle on the circle equals twice the physical plane rotation.

 

 

 

 

 





## Demonstration

Demonstration

Illustrative scenario → Given measured in-plane stresses σx, σy and shear τxy at a point in a thin tooth-restorative interface, construct the circle with center at (σx+σy)/2 and radius √[((σx−σy)/2)^2+τxy^2]; read principal stresses at the circle’s left/right extremes and compute principal plane angles by halving the angular coordinate on the circle. This produces the magnitudes and orientations used to evaluate failure risk in the interface.

 

 

 

 

## Misapplication

Misapplication

Applying the planar Mohr’s circle construction to a three-dimensional stress state (where out-of-plane stress components are significant) or to materials with grossly non-linear, time-dependent, or anisotropic constitutive behavior without correction; the semantic error is using a two-dimensional linear-elastic transformation where additional tensor components or constitutive rules are required.

 

 

 

 

 





## Consequence

Consequence

Correct use yields unambiguous principal stress magnitudes and orientations for design and failure analysis under plane-stress assumptions; misuse can produce erroneous principal stresses and misoriented predicted failure planes, potentially leading to inappropriate material choice or prosthesis design.

 

 

 

 

## Reversal

Reversal

When out-of-plane stresses or full three-dimensional stress tensors are non-negligible (thick specimens, multi-axial loading), the planar circle must be replaced by three-dimensional stress analysis (Mohr’s sphere or full tensor methods); likewise, when material behavior is non-linear or path-dependent, simple linear transformation no longer predicts stress response.

 

 

 

 

 





## Boundary

Boundary

Within: two-dimensional (plane stress or plane strain approximated) linear elastic analyses at a point in homogeneous or locally homogeneous media. Boundary case: thin but layered structures where plane-stress may hold locally but interlayer shear or anisotropy matters. Outside: inherently three-dimensional stress states, large-deformation or viscoelastic regimes, and contexts where stress components are undefined (e.g., purely discrete contact problems without continuum stresses).

 

 

 

 

 





## Semantic Tension

Semantic Tension

Simplicity and intuitive geometric visualization (Mohr’s circle) ↔ fidelity to full tensorial, three-dimensional and material-specific analyses; the circle aids rapid insight but competes with more complete but complex models.

 

 

 

 

 





## Synthesis

Synthesis

Mohr’s circle is a compact, visual algebraic tool that converts in-plane stress components into immediately usable principal values and orientations under clear assumptions; its value lies in transparent mapping between component stresses and failure-relevant invariants, but its conclusions are valid only when planar, linear-elastic assumptions hold.