Antiplane Shear

A nontrivial deformation with axial-only displacement that varies across a transverse plane, while material laws and defect conditions determine its particular stresses and equations.

Version

v1 · 2026-10-07 · History

Domain-specific #

13789

Domain group

Applied Sciences & Engineering

Origin domain

Engineering & Design (beyond software)

Subdomain

Antiplane Elasticity → Engineering & Design (beyond software)

Aliases

Anti Plane Shear, Out of Plane Shear

Core Idea

Antiplane shear is a class of deformation whose displacement points only along a chosen axial direction and does not vary along that axis. In a local Cartesian description, \(u_1=u_2=0\) and \(u_3=w(x_1,x_2)\). The scalar amplitude varies across the transverse plane; if it is constant on a connected region, the motion is merely rigid axial translation, the zero-shear degeneracy. The field can be defined locally or on a cut domain around a screw defect. These are kinematic conditions, not a universal material law or field equation.^[1][2][3]

The same restricted displacement geometry appears in a stationary Mode III interface-crack calculation and in an infinitely long axial screw-dislocation calculation. Their governing mechanics differ: the first uses a couple-stress model and crack-face conditions; the second has Burgers circulation and an incompatible core. Neither case makes its stress law or boundary conditions necessary for every antiplane state.^[2][3]

Structural Signature

• Material carrier and admissible domain. A continuum body supplies points at which displacement is assigned. Around a screw defect, the axial total displacement needs a local or cut-domain representation rather than a smooth single-valued value through the uncut core.^[1][3]
• Axial frame. One direction is distinguished. The pure ansatz has no transverse displacement and no dependence on position along that direction. Knowles also studies an optional axial prestretch, which is outside this pure coordinate form.^[1][2]
• Axial displacement field. The physical field \(w(x_1,x_2)\) is an internal constituent of the deformation state. Removing it leaves no specified antiplane displacement.^[2][3]
• Transverse variation. A nonconstant \(w\) produces the shear character. In the small-strain crack model, \(\varepsilon_{13}=\tfrac12\partial_1w\) and \(\varepsilon_{23}=\tfrac12\partial_2w\); those formulas do not by themselves set a universal finite or incompatible-core strain law.^[2][3]
• Qualified realization. Material response, balance, load, crack interface or defect cut must be declared before deriving a particular stress or solution. One case's constraints cannot be carried into the other merely because both use an axial field.^[1][2][3]

What It Is Not

Antiplane shear is not any stress labeled shear: the displacement geometry is decisive. It is not the live Shear Mapping's general affine transvection; simple affine shear is one special case, while crack-tip and screw fields may vary nonlinearly across the plane. A constant axial translation has no nontrivial shear. A finite screw-dislocation rod that also twists azimuthally is related mechanics but no longer has only axial displacement.^[1][3]

Nor does the name imply the classical Laplace or Poisson equation. For a homogeneous isotropic small-strain static elastic medium away from defects, ordinary balance and the usual shear law conditionally yield \(\mu\Delta w+b_3=0\), or \(\Delta w=0\) when axial body force vanishes. The body-force form is a stated derivation from those assumptions, not an equation printed in the inspected originals. The cited couple-stress crack instead has \(\Delta w-(\ell^2/2)\Delta^2w=0\) in each half-plane. Lazar's incompatible core cannot be treated by a globally compatible gradient-only argument.^[2][3]

Scope of Application

The full inspected papers support stationary interfacial Mode III fracture in couple-stress materials and the infinitely long axial screw dislocation in an elastoplastic theory as two unlike positive constructions. Knowles supplies a finite incompressible elasticity analysis showing that nontrivial pure antiplane solutions depend on material compatibility. These are theoretical settings; the packet does not establish measured specimen behavior, SH-wave or seismic cases, or a universal solution procedure.^[1][2][3]

The antiplane description itself is local kinematics. An exact pure-antiplane boundary-value reduction additionally depends on the declared geometry, constitutive response and compatible loading. A boundary condition can be a crack traction, a bonded-interface condition, a defect circulation, or something else; no single one defines the whole class.^[1][2][3]

Clarity

Begin by naming the axial direction and the material domain on which \(w\) is defined. Then separate three questions: does the displacement satisfy the axial-only, axial-invariant ansatz; which strain or stress measure does the selected theory derive from it; and which balance plus boundary or defect conditions select a solution? This order prevents a model-specific equation from being mistaken for the definition.^[1][2][3]

For the screw case, say whether the domain is cut and distinguish total displacement, elastic strain and incompatible or plastic distortion. For the crack case, specify that the fourth-order equation and free-face/bonded-interface conditions belong to the stationary couple-stress model. Do not describe either paper as an experiment.^[2][3]

Manages Complexity

The axial ansatz compresses a spatial displacement description to one transverse-coordinate field. In the cited small-strain crack model, the shear components follow its transverse derivatives. That reduction helps expose what the geometry contributes, but it does not erase model choice: couple stresses change the order of the equilibrium equation, finite incompressible constitutive laws can constrain nontrivial solutions, and a screw core introduces incompatible distortion.^[1][2][3]

The useful compression is thus a separation of kinematics from conditional mechanics. A solver can reuse the axial-only description across a crack and a defect while retaining each case's material response, regularity and constraints.^[2][3]

Abstract Reasoning

1. Choose a material region, axial direction and admissible local or cut domain.^[1][3]
2. Test \(u_1=u_2=0\), \(u_3=w(x_1,x_2)\), with no axial-coordinate dependence and with nonconstant \(w\) for nontrivial shear.^[1][2]
3. State the constitutive and kinematic regime before deriving strains, stresses or balance; keep a possible incompatible core separate from a compatible small-strain gradient.^[2][3]
4. Add the appropriate crack, interface, load or Burgers/cut condition for the selected case, and solve only that model's equation.^[2][3]
5. Report which conclusions are kinematic and which depend on material, source, boundary and regularity assumptions.^[1][2][3]

Knowledge Transfer

The crack and screw constructions share the material domain, chosen axis, axial physical field and transverse variation. That role map lets a reader recognize the same kinematic reduction without importing a crack's traction condition into a defect or a defect's Burgers circuit into a crack. It also suggests a diagnostic whenever a new example is called antiplane: check the displacement first, then the particular balance and singularity assumptions.^[2][3]

The transfer has a firm limit. PMR's fourth-order couple-stress equation and Lazar's cut-domain incompatible field are not interchangeable solutions. Knowles shows that even within finite incompressible elasticity a nontrivial pure ansatz has constitutive restrictions. The abstraction carries a field geometry across settings, not a ready-made PDE.^[1][2][3]

Examples

Stationary Mode III interfacial crack. Piccolroaz, Mishuris and Radi model two dissimilar couple-stress half-planes joined along an interface with a stationary crack. Displacement is \(w(x_1,x_2)\) along \(x_3\), so transverse variation yields the small-strain 13/23 shear components. The analytical model imposes free reduced tractions at the crack faces and transmission on the bonded part; each half-plane obeys its fourth-order equation. The material halves and interface supply the carrier, \(x_3\) the axis, \(w\) the internal field, its cross-plane derivatives the nontrivial variation, and the couple-stress and interface rules the qualified realization. No measured crack response is claimed.^[2]

Infinitely long axial screw dislocation. Lazar places a straight line defect and Burgers vector along a cylinder's \(z\) axis. His total axial displacement varies with transverse position and angular coordinate; a branch cut or local patch is needed for \(u_z\), while \(xz/yz\) shear components remain nonzero away from the core. The cylinder is the carrier, \(z\) the invariant axis, the cut-domain axial displacement the internal field, transverse variation the shear, and Burgers circulation plus incompatible/plastic distortion the setting-specific regime. Lazar's finite rod in §3.3 additionally develops \(u_\phi\), so it is not this pure axial-only example.^[3]

Structural Tensions

Pure axial reduction versus finite constitutive admissibility. Keeping only an axial displacement gives a tractable kinematic description, but Knowles's finite incompressible setting restricts which nontrivial fields can satisfy complete material equilibrium. Prioritize the pure ansatz under an arbitrary law and a nontrivial solution may fail to exist; prioritize a broader material response and additional displacement components or coupled equations may be needed. The diagnostic is whether the declared material and loading actually admit a nonconstant pure axial state. This tension is conditional on Knowles's finite setting and is not a universal prohibition inferred from the different couple-stress crack model.^[1]

Structural–Framed Character

Antiplane shear lies toward the structural, but domain-specific end of the spectrum: its axial displacement geometry recurs in unlike fracture and defect constructions. Its label comes from continuum and elasticity research, yet no institution's certification makes a material displacement antiplane. A physical axial deformation can occur without an observer; people choose the coordinate axis, idealized material model, admissible region and words used to describe it. Once those choices are fixed, the displacement either meets the kinematic restriction or does not. The classification itself does not praise the deformation, rank materials, or prefer one constitutive law.^[1][2][3]

The word “shear” still carries a physical displacement and strain commitment. Recognizing the same pattern in both originals needs no shared boundary condition or PDE, but importing the label into an arbitrary two-variable scalar field or affine coordinate mapping would omit that physical differentia. Its character: a reusable axial-field geometry within mechanics, framed by human model choices but neither institutionally constituted nor evaluatively ranked, whose conditional equations remain specific to each material and setting.^[1][2][3]

Structural Core vs. Domain Accent

The live Field (physics) constituent supplies spatial domain, physical displacement quantity, fixed axial value interpretation and pointwise assignment. The antiplane core adds the nonconstant axial-only, axial-invariant displacement constraint. PMR's material length, fourth-order balance and crack tractions, Lazar's Burgers circuit and incompatible core, and Knowles's finite constitutive conditions are setting accents rather than shared necessary roles. Removing the displacement field destroys the antiplane state; removing one particular accent does not.^[1][2][3]

Named Antiplane Shear does not clear the Prime bar: material displacement, a chosen axial frame and nontrivial transverse shear remain necessary to its identity. Its established reusable constituent is the live domain-specific Field (physics), not an asserted Prime parent. A more substrate-neutral pointwise-field skeleton would be a separate future-Prime question requiring its own unlike nonmechanical instances and full-role proof; it is not the strict edge proposed here.^[1][2][3]

Instantiates / Related Primes

This entry is part of Field (physics).

Every antiplane shear state contains a Field (physics) as an essential part. Both cases contain an axial displacement field on an admissible material region, with a fixed physical quantity and values assigned pointwise. That field is a constituent of the larger deformation state; the state is not simply another name for a field. Temperature or electric fields can be physical fields without any antiplane shear. Mechanical Strain is a derived measure here only under conditions, Stress Field a response under a law, Dislocation only one case, and Boundary Value Problem a possible formulation rather than something every antiplane shear state falls under.^[2][3]

Relationships to Other Abstractions

Current abstraction Antiplane Shear Domain-specific

Parents (1) — more general patterns this builds on

• Antiplane Shear is part of Field (physics) Domain-specific

  Every antiplane state contains an axial physical displacement field as an identity-bearing constituent.

Hierarchy path (1) — routes to 1 parentless root

• Antiplane Shear → Field (physics) → Physical quantity → Measurement

Neighborhood in Abstraction Space

Antiplane Shear sits in a sparse region of the domain-specific corpus (92^nd percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Structural & Solid Mechanics (29 abstractions)

Nearest neighbors

• Stress space — 0.80
• Two-point tensor — 0.79
• Cauchy elastic material — 0.79
• Line group — 0.79
• Polynomial hyperelastic model — 0.78

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

• Plane strain: an in-plane displacement restriction, unlike the axial-only antiplane geometry.^[1]
• Shear Mapping: the live affine transvection can describe special simple shear, but the nonuniform crack and screw cases need not be affine.^[1][2][3]
• A screw dislocation itself: a defect can instantiate the axial kinematics locally; antiplane shear does not require a line defect.^[2][3]
• Uniform axial translation: its constant \(w\) has no transverse shear and is a degenerate limiting motion.^[2]
• One universal scalar equation: Laplace/Poisson needs classical small-strain conditions, while couple-stress and incompatible-core models differ.^[2][3]

References

[1] James K. Knowles, “On Finite Anti-Plane Shear for Imcompressible Elastic Materials”, Journal of the Australian Mathematical Society, Series B 19 (1976): 400–415, especially printed pp. 400–401, 403, 407. The publisher's original title prints “Imcompressible”; the ordinary technical term in prose is “incompressible.” registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t

[2] A. Piccolroaz, G. Mishuris, and E. Radi, “Mode III Interfacial Crack in the Presence of Couple-Stress Elastic Materials”, full author manuscript arXiv:1010.1822v2 (2 April 2011), especially PDF pp. 1, 6–7, §2 Eqs. (2), (3), (5), (7)–(14). This cited case is stationary and analytical. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28 ↩29

[3] Markus Lazar, “Screw Dislocations in the Field Theory of Elastoplasticity”, full author manuscript arXiv:cond-mat/0203058v2 (30 September 2002), especially PDF pp. 3–4, 7–10, 15, §§2, 3.1, 3.3 and Eqs. (3), (6), (10), (31)–(51), (74). The positive axial case is §3.1; the finite-rod near miss is §3.3. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n ↩o ↩p ↩q ↩r ↩s ↩t ↩u ↩v ↩w ↩x ↩y ↩z ↩27 ↩28 ↩29