Mohr's Circle¶

Encode a symmetric plane-stress state as a circle in normal–shear coordinates, so a physical plane rotation appears at double angle and principal stresses, maximum in-plane shear, and their orientations can be read from one invariant locus.

Version: v2 · 2026-09-06 · History
Domain-specific #: 2299
Origin domain: continuum mechanics
Subdomain: plane-stress transformation
Aliases: Mohr Circle

Core Idea¶

Mohr's circle is the graphical encoding of the transformation law for a symmetric two-dimensional stress tensor. It takes the normal and shear stresses known on two perpendicular faces at a material point and places them on a circle in a coordinate plane whose horizontal axis is normal stress and whose vertical axis is shear stress. Moving around that circle then represents resolving the same physical stress state on differently oriented planes. The construction makes principal stresses, principal-plane orientations, maximum in-plane shear stress, and its planes visible as geometrically distinguished points.^[1]^[2]

For a plane-stress state written, in one common convention, as

\[ \boldsymbol\sigma= \begin{bmatrix} \sigma_x & \tau_{xy}\\ \tau_{xy} & \sigma_y \end{bmatrix}, \]

define

\[ C=\frac{\sigma_x+\sigma_y}{2}, \qquad R=\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}. \]

The locus of the transformed normal and shear components is

\[ (\sigma_n-C)^2+\tau_n^2=R^2. \]

The circle therefore preserves two important quantities while orientation changes: its center records half the two-dimensional trace, and its radius records half the separation of the in-plane principal stresses. The horizontal intercepts are

\[ \sigma_{1,2}=C\pm R, \]

and the top and bottom points have shear magnitude \(R\) and normal stress \(C\). A rotation of the material-plane normal through \(\theta\) corresponds to an angular displacement of magnitude \(2\theta\) on the circle. The direction of that displacement and the plotted sign of shear depend on the declared convention; the doubled angle, circle geometry, and recovered physical stresses do not.^[1]^[3]

The candidate survives as an autonomous domain-specific abstraction. domain_specific:tensor, prime:transformation, and prime:representation explain its inputs and general logic, but none specifies the normal–shear coordinate medium, diametrically opposed face points, double-angle correspondence, sign discipline, or mechanically meaningful extrema. Mohr's circle is not merely the fact that tensors transform; it is a stable graphical calculus for carrying out, checking, and interpreting that transformation.

Structural Signature¶

The abstraction is present when the following roles are jointly instantiated:

A local symmetric second-order state. In the recognition center this is the Cauchy stress tensor at one material point under a two-dimensional or plane-stress reduction, with components \(\sigma_x\), \(\sigma_y\), and \(\tau_{xy}=\tau_{yx}\) in a declared frame.
A physical-plane orientation. The query concerns the normal and shear traction components on a plane through that same point whose normal is rotated relative to the reference frame.
A normal–shear representation plane. The abscissa carries normal stress \(\sigma_n\) and the ordinate carries signed shear stress \(\tau_n\). This diagram is an abstract component space, not the material's physical cross-section.
A declared sign convention. Tension-versus-compression and positive-versus-negative shear must be fixed before points and angular directions can be interpreted. Textbooks legitimately reverse the shear axis or its rotation direction.
Two diametrically opposed reference-face points. The stress components on perpendicular faces plot with the same normal-axis convention and opposite signed shears. Their midpoint lies at \((C,0)\).
The invariant center and radius. \(C=(\sigma_x+\sigma_y)/2\) and \(R=[((\sigma_x-\sigma_y)/2)^2+\tau_{xy}^2]^{1/2}\) determine the locus.
The double-angle correspondence. A physical orientation change of \(\theta\) maps to a circle angle of magnitude \(2\theta\). This is a consequence of the \(\sin 2\theta\) and \(\cos 2\theta\) terms in the tensor-transformation equations, not an arbitrary graphical trick.
A component readout. The coordinates of the corresponding point give normal and shear stress on the queried plane; the diametrically opposite point gives the components on its perpendicular mate.
Distinguished extrema. Horizontal-axis intersections identify principal stresses and zero-shear principal planes. Vertical extrema identify maximum and minimum in-plane shear and their associated normal stress \(C\).
A domain interpretation. The readout is used to reason about stress transformation, not merely to admire an algebraic circle.

The recognition test is operational: can the analyst declare the stress and shear conventions, construct \(C\) and \(R\), map physical orientation by the double-angle rule, and recover transformed components and extrema from the same locus? A plot lacking those linked roles is not Mohr's circle even if it happens to be circular.

What It Is Not¶

Not a physical circle in the material. Radius and angle in the diagram are stress-space quantities. The circle is not a crack path, grain, hole, pipe, or round specimen.
Not the Cauchy stress tensor itself. The tensor is the frame-independent physical object; Mohr's circle is one graphical representation of how selected components vary with plane orientation.
Not a generic tensor-transformation formula. Matrix rotation can produce the same answers without a circle. The candidate requires the normal–shear locus and its reading convention.
Not a failure criterion. Mohr–Coulomb and other envelope methods compare a stress circle with an independently supplied material-strength relation. The circle describes the stress state; the envelope supplies a failure rule.^[4]
Not a stress path. A stress path tracks changing stress states during loading. One Mohr circle represents the orientations available for one state at one point. A sequence of circles may accompany a stress path, but the sequence is not the circle's defining identity.
Not automatically a complete three-dimensional stress solution. For three distinct principal stresses, three pairwise circles bound the admissible normal–shear region. A single plane-stress circle does not encode every orientation of an arbitrary 3D traction state.^[5]
Not identical to a strain circle. The same geometry can represent a symmetric strain tensor, but engineering shear strain \(\gamma_{xy}=2\epsilon_{xy}\) creates a convention-sensitive scaling that must be handled explicitly.
Not a numerical eigensolver. Its horizontal extrema equal eigenvalues in the symmetric 2D case, but modern computation can obtain eigenpairs directly. The abstraction's enduring value is representation, checking, and interpretation.

Scope of Application¶

The home domain is continuum mechanics, especially mechanics of materials. The circle is taught and used in mechanical, civil, structural, materials, geological, rock, and geotechnical engineering whenever stresses at a point are known in one frame and must be interpreted on rotated planes. It supports plane-stress transformation, principal-stress identification, maximum in-plane shear, orientation checks, and graphical comparison with failure envelopes.^[1]^[4]

Its cleanest form assumes a symmetric \(2\times2\) tensor. Symmetry of ordinary Cauchy stress follows from angular-momentum balance in a classical continuum without couple stresses. The construction also applies to other symmetric second-order quantities—strain, area moments of inertia, conductivity, and related quadratic forms—provided the axes, component meaning, off-diagonal scaling, and sign conventions are restated. That wider recurrence does not make the node a prime: the established vocabulary, diagram coordinates, and principal-plane use remain anchored in tensor mechanics.^[5]^[6]

For a three-dimensional symmetric stress tensor, first obtain the principal stresses \(\sigma_1\geq\sigma_2\geq\sigma_3\). The three circles with diameters \((\sigma_1,\sigma_2)\), \((\sigma_2,\sigma_3)\), and \((\sigma_1,\sigma_3)\) provide a two-coordinate graphical constraint on possible normal and shear magnitudes. Interior-region inequalities matter: an arbitrary 3D plane orientation is not represented by choosing any point on any one circumference. The 2D transformation circle is therefore the recognition center, while the three-circle construction is a mature extension.^[5]

The abstraction is most useful when the stress state is locally meaningful and continuum assumptions are appropriate. It does not replace constitutive modeling, equilibrium analysis, finite-element computation, or material-strength data. Those processes supply the tensor or interpret its consequences; the circle transforms and exposes the state they provide.

Clarity¶

Three questions usually resolve whether a proposed use is genuine. What is held fixed? The physical stress tensor at one point. What varies? The orientation of the plane on which traction is resolved. What does a plotted point mean? Its coordinates are the normal and signed shear components on that plane under a declared convention.

The most common error is to compare the physical angle and the diagram angle directly. The tensor equations contain \(\cos 2\theta\) and \(\sin 2\theta\), so the circle moves through twice the physical rotation. A second common error is to copy a clockwise or counterclockwise rule from a source using the opposite shear-axis convention. The safe procedure is to plot the known reference-face point, write the convention next to the axes, and verify that a \(90^\circ\) physical rotation reaches the diametrically opposite face point after \(180^\circ\) on the circle.

Another diagnostic separates orientation transformation from loading evolution. Walking around one fixed circle changes the plane used to observe one tensor. Changing the applied load generally changes \(C\), \(R\), or both and therefore creates a different circle. Conflating those motions mistakes a coordinate change for a physical change of state.

Manages Complexity¶

Direct stress-transformation formulas distribute the problem across several coupled trigonometric expressions. Mohr's circle compresses them into one center, one radius, one reference direction, and one angular rule. That compression makes several questions simultaneous rather than sequential: transformed components appear as coordinates; the principal values appear as intercepts; maximum in-plane shear appears as the radius; and orientations appear as angular relations.

The representation also exposes consistency checks. The average normal stress must remain the circle center. Perpendicular physical planes must map to diametrically opposite points. Principal planes must have zero shear. Maximum-shear planes must lie \(45^\circ\) from principal planes in physical space because their points are \(90^\circ\) apart on the circle. If a calculation violates one of these relations, the diagram localizes an algebraic or convention error.

That economy is not a claim that a hand-drawn graph outperforms numerical linear algebra. Rather, it provides an interpretable invariant summary that can audit a computed result and connect component values to mechanically meaningful orientations.

Abstract Reasoning¶

Mohr's circle licenses deductions from geometry because it is algebraically equivalent to the plane-stress transformation equations. In the convention

\[ \sigma_{x'}=C+\frac{\sigma_x-\sigma_y}{2}\cos 2\theta+\tau_{xy}\sin 2\theta, \]

\[ \tau_{x'y'}=-\frac{\sigma_x-\sigma_y}{2}\sin 2\theta+\tau_{xy}\cos 2\theta, \]

squaring and adding after subtracting \(C\) gives the circle equation. The pair \(((\sigma_x-\sigma_y)/2,\tau_{xy})\) is simply rotated in the diagram, so its Euclidean norm \(R\) is invariant under the coordinate rotation.^[1]

Several predictions follow. If \(R=0\), then \(\sigma_x=\sigma_y\) and \(\tau_{xy}=0\); every in-plane orientation has the same normal stress and zero shear. If \(C=0\), the state has equal and opposite principal stresses, and the circle is centered at the origin. If \(\tau_{xy}=0\) in the starting frame, the starting axes are principal unless the tensor is isotropic, though maximum-shear planes still occur at \(45^\circ\). The physical planes of maximum in-plane shear are always \(45^\circ\) from the principal planes because the diagram extrema are separated by \(90^\circ\).

These inferences are exact within the symmetric plane-stress model. They do not determine whether a material yields, fractures, creeps, or remains safe; those judgments require a constitutive or failure model and, for many criteria, the full three-dimensional state.

Knowledge Transfer¶

The construction teaches a reusable pattern: transform a periodic pair of double-angle component equations into a geometric locus whose invariants and extrema answer the practical questions. That pattern transfers literally to symmetric \(2\times2\) tensors when their diagonal and off-diagonal entries obey the same rotation law. Area-moment circles, strain circles, and graphical treatments of conductivity or permeability inherit the center-radius-angle structure after their component conventions are specified.^[5]^[6]

Transfer must preserve semantics. Engineering shear strain is twice tensorial shear strain, compression is often positive in geotechnical practice but tension positive in mechanics of materials, and some diagrams draw positive shear downward. A correct transfer therefore carries the transformation law and reading convention, not just the circle shape. Outside symmetric-tensor transformation, claims that a circular chart “is like Mohr's circle” are analogies rather than instances.

Examples¶

Plane-stress calculation. Let \(\sigma_x=80\ \text{MPa}\), \(\sigma_y=20\ \text{MPa}\), and \(\tau_{xy}=30\ \text{MPa}\). Then

\[ C=50\ \text{MPa},\qquad R=\sqrt{30^2+30^2}=42.43\ \text{MPa}. \]

The principal stresses are \(92.43\ \text{MPa}\) and \(7.57\ \text{MPa}\), and the maximum in-plane shear magnitude is \(42.43\ \text{MPa}\) with normal stress \(50\ \text{MPa}\) on those planes. In the equations used above, \(\tan 2\theta_p=2\tau_{xy}/(\sigma_x-\sigma_y)=1\), so one principal direction occurs at \(\theta_p=22.5^\circ\), modulo the orthogonal principal direction and sign-convention choice. For a \(15^\circ\) axis rotation, direct substitution gives approximately \(\sigma_{x'}=90.98\ \text{MPa}\) and \(\tau_{x'y'}=10.98\ \text{MPa}\). Those coordinates lie on the same circle because \((90.98-50)^2+10.98^2\approx42.43^2\).

Pure shear. If \(\sigma_x=\sigma_y=0\) and \(\tau_{xy}=40\ \text{MPa}\), the circle is centered at zero with radius \(40\ \text{MPa}\). The principal stresses are \(+40\) and \(-40\ \text{MPa}\), reached on planes rotated \(45^\circ\) from the original faces. This counterintuitive conversion of pure shear components into principal tension and compression is immediately visible on the circle.^[2]

Geotechnical comparison. In soil or rock mechanics, a circle may be drawn using the field's compression and shear conventions, then compared with a Mohr–Coulomb envelope. The circle supplies the possible \((\sigma_n,\tau_n)\) combinations for the stress state; the envelope supplies the material's idealized strength boundary. Contact between them has meaning only after effective-versus-total stress and sign conventions are fixed. This example illustrates why Mohr's circle and Mohr–Coulomb are related but not aliases.^[4]

Strain variant. A strain state can be plotted by analogous transformation equations, but the vertical coordinate may be \(\gamma/2\) rather than engineering shear \(\gamma\). Using the stress plotting recipe without that scaling yields wrong strain readouts. The variant preserves the geometric mechanism while changing a domain accent.^[7]

Structural Tensions¶

Graphical intuition versus numerical precision. The circle makes extrema and orientation relations visible, but measurements from a sketch are approximate. Analytical or computational values should govern high-consequence calculations.
Convention flexibility versus interoperability. Opposite shear-axis and compression conventions are each coherent. Mixing them inside one construction reverses orientations or signs while leaving a plausible-looking circle.
Two-dimensional economy versus three-dimensional completeness. One circle gives a complete orientation account for a symmetric 2D state. Three-dimensional states require principal values, three circles, and admissible-region reasoning that a single circumference cannot supply.
General tensor geometry versus mechanical identity. The algebra works for many symmetric matrices, but the name's recognition center remains stress and strain transformation. Generality should not erase the domain semantics that tell the coordinates what they mean.
State description versus failure interpretation. The circle depicts resolved stresses; a failure envelope or criterion adds material behavior. Treating intersection geometry as universal failure physics hides assumptions about material, loading, pressure sensitivity, and dimensional reduction.
Orientation change versus state change. Motion along one circle changes the observing plane. Motion from one circle to another changes the tensor. Keeping those operations separate prevents coordinate artifacts from being mistaken for loading effects.

Structural–Framed Character¶

Mohr's circle is predominantly structural with an indispensable domain frame. Its center, radius, doubled-angle parameterization, diametric opposition, and extrema follow from the rotation law for a symmetric \(2\times2\) tensor. Those relations are not historically contingent styling choices. Given the tensor components and sign convention, the circle is determined.

The frame enters through the interpretation of axes, the shear and compression conventions, the choice between stress and strain scaling, the plane-stress reduction, and the practice of reading extrema as engineering quantities. A circle with the same equation in an unrelated graph is not automatically an instance. The most accurate classification is therefore a strong mathematical structure used through a mature continuum-mechanics reading convention—domain-specific, not merely framed and not substrate-independent enough to be prime.

Structural Core vs. Domain Accent¶

The structural core is

\[ \text{symmetric two-component state} \rightarrow \text{mean plus rotating deviatoric pair} \rightarrow \text{circle under a double-angle parameter} \rightarrow \text{invariants, extrema, and orientation readout}. \]

The domain accent supplies Cauchy stress, traction on a material plane, normal and shear signs, principal planes, maximum in-plane shear, and engineering consequences. That accent is not decorative: it fixes what the inputs represent and what actions the output licenses. The general structural residue belongs to Representation and Transformation. Mohr's Circle earns a separate node because the field-specific circle calculus is stable, named, taught, and operationally diagnostic.

Representation is the most literal parent. A stress tensor and its orientation-dependent traction components are the target; the normal–shear circle is the representing medium; the transformation equations define the mapping; center, radius, and double-angle correspondence specify faithfulness; and the sign convention supplies interpretation. The proposed DAG edge therefore treats Mohr's Circle as a strict domain-specific subtype of prime:representation.

Transformation explains the rule-governed change from components in one frame to components in another while the tensor itself remains fixed. Invariance explains why trace-derived center and deviatoric radius do not change with in-plane orientation. Tensor is the closest domain-specific neighbor because its coordinate transformation law is the candidate's algebraic source. Compression and geometric extremization are supporting patterns, but none alone recovers the circle's mechanics-specific grammar.

Relationships to Other Abstractions¶

Current abstraction Mohr's Circle Domain-specific

Parents (1) — more general patterns this builds on

Mohr's Circle is a kind of Representation Prime

Representation is the most literal parent.

Hierarchy path (1) — routes to 1 parentless root

Mohr's Circle → Representation → Abstraction

Neighborhood in Abstraction Space¶

Mohr's Circle sits in a sparse region of the domain-specific corpus (85^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

Dihedral Angle — 0.82
Verlet Integration — 0.81
Modified Pressure — 0.81
Eight-Node Quadratic Serendipity Quadrilateral (Q8) — 0.80
Seifert Surface — 0.79

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With¶

domain_specific:tensor. Tensor names the underlying multilinear object and its covariance law. Mohr's Circle is a particular graphical calculus for a symmetric second-order tensor's rotated components.
prime:transformation. Transformation is substrate-independent. Mohr's Circle fixes a specific input class, circular medium, doubled orientation, and normal–shear interpretation.
prime:representation. Representation supplies the parent pattern but not the circle equation, sign rules, principal-stress intercepts, or maximum-shear readout.
Mohr–Coulomb failure criterion. It joins stress-circle information to a pressure-dependent shear-strength envelope. It is not another name for the circle.
Stress ellipsoid or Cauchy stress quadric. These are alternative geometric representations with different media and reading rules.
Stress path. A path shows state evolution; walking around a circle shows orientation dependence of one state.
Circle of inertia or strain circle. These are legitimate same-family constructions, not unrestricted aliases. Their quantities and scaling conventions must be named.
A physical failure circle. No circular fracture geometry is implied.

References¶

[1] Raúl Radovitzky et al., “Transformation of Stress Components, Principal and Maximum Shear Stresses, Mohr's Circle,” MIT 16.001 Unified Engineering lecture notes (2021), pp. 2–29. MIT OpenCourseWare PDF. registry ↩a ↩b ↩c ↩d

[2] David Roylance, “Transformation of Stresses and Strains,” MIT 3.11 Mechanics of Materials course notes (1999), especially pp. 4–6. MIT OpenCourseWare PDF. registry ↩a ↩b

[3] William F. Hosford, Mechanical Behavior of Materials, 2^nd ed. (Cambridge University Press, 2010), chapter 1 excerpt, equations 1.14–1.16 and Figure 1.8. Publisher excerpt. registry ↩

[4] A. P. S. Selvadurai, “Mohr Circles,” appendix B in Plasticity and Geomechanics (Cambridge University Press, 2002), pp. 228–240. Publisher chapter record. registry ↩a ↩b ↩c

[5] Rebecca M. Brannon, Mohr's Circle and More Circles (University of Utah, 2003), especially the abstract and sections on symmetric matrices and three-dimensional stress. Author-hosted PDF. registry ↩a ↩b ↩c ↩d

[6] W. Craig Carter, “Quadratic Forms and Mohr's Circle,” MIT 3.016 Mathematics for Materials Scientists and Engineers, lecture 10 (2005). MIT OpenCourseWare PDF. registry ↩a ↩b

[7] MIT 16.01 Unified Engineering, “Stress Transformation and Mohr's Circle” and “Transformation of Strain, Mohr's Circle for Strain” course materials (2005–2006). MIT OpenCourseWare materials index. registry ↩