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.
Core Idea¶
Antiplane shear means a material's displacement points only along a chosen axis while its amount varies across the plane perpendicular to that axis. In a local coordinate frame, \(u_1=u_2=0\) and \(u_3=w(x_1,x_2)\). The field has no variation along \(x_3\). A constant \(w\) is rigid axial translation rather than nontrivial shear. Around a screw defect the field must be described locally or on a cut domain. This geometry alone does not choose a material law or equation.[ref-414532748a54][ref-139488db9c6c][^ref-28985159b1a0]
The same geometry appears in an analytical stationary Mode III crack and an infinitely long screw dislocation. Their extra constraints differ: crack faces and an interface in one, Burgers circulation and an incompatible core in the other.[ref-139488db9c6c][ref-28985159b1a0]
Scope of Application¶
The inspected originals support those two theoretical settings and Knowles's separate study of finite incompressible antiplane states. The pure axial form excludes an optional prestretch and a finite rod that also develops azimuthal displacement. These papers do not establish measured specimen outcomes, SH-wave or seismic applications, or one equation for every antiplane material.[ref-414532748a54][ref-139488db9c6c][^ref-28985159b1a0]
For a homogeneous isotropic small-strain static medium away from defects, familiar balance and shear constitutive assumptions conditionally give \(\Delta w=0\) without axial body force and \(\mu\Delta w+b_3=0\) with it. The body-force form is a derivation under those assumptions, not a verbatim equation from the originals. It does not govern PMR's nonzero couple-stress case or Lazar's incompatible core.[ref-139488db9c6c][ref-28985159b1a0]
Clarity¶
First identify the material region and axial direction. Check that only \(u_3\) is present and that \(w\) varies across \(x_1,x_2\), not \(x_3\). Then name the chosen constitutive theory and any crack, interface, load or defect conditions before deriving stress or solving a field equation. A stress labeled “shear” without this displacement geometry does not itself establish antiplane shear.[ref-414532748a54][ref-139488db9c6c][^ref-28985159b1a0]
For a screw dislocation, do not treat the angular total displacement as globally single-valued through the uncut core, or replace incompatible/plastic distortion with an ordinary smooth gradient. For the stationary couple-stress crack, keep its fourth-order equation and crack conditions within that model.[ref-139488db9c6c][ref-28985159b1a0]
Manages Complexity¶
The axial-only restriction compresses a displacement description to one field over a transverse plane. In PMR's small-strain model the 13 and 23 shear components follow transverse derivatives of \(w\). Yet a simpler displacement description does not imply identical mechanics: PMR uses a fourth-order couple-stress balance, Lazar includes a defect core, and Knowles shows finite constitutive compatibility can limit nontrivial pure solutions.[ref-414532748a54][ref-139488db9c6c][^ref-28985159b1a0]
Abstract Reasoning¶
- Choose a material domain, axial direction and, if needed, a cut or excluded core.[ref-414532748a54][ref-28985159b1a0]
- Test \(u_1=u_2=0\), \(u_3=w(x_1,x_2)\), with nonconstant transverse variation for nontrivial shear.[ref-414532748a54][ref-139488db9c6c]
- Declare the material theory before deriving strain, stress or equilibrium.[ref-414532748a54][ref-139488db9c6c][^ref-28985159b1a0]
- Apply only the selected case's crack, interface, load or Burgers conditions and report what those conditions permit.[ref-139488db9c6c][ref-28985159b1a0]
Knowledge Transfer¶
The two examples share the material carrier, axial direction, displacement field and transverse variation. Those roles identify the same kinematic class even though a crack's free-face condition does not transfer to a screw defect and a screw's Burgers circuit does not transfer to a crack. Field (physics) is a strict internal constituent: each case assigns physical axial displacement values to points of an admissible material domain. A temperature field can be a physical field without antiplane shear.[ref-139488db9c6c][ref-28985159b1a0]
Example¶
Stationary interface crack. Piccolroaz, Mishuris and Radi use two dissimilar couple-stress half-planes with a stationary Mode III interface crack. The plane supplies material locations, \(x_3\) is the out-of-plane axis, and \(w(x_1,x_2)\) varies to give the model's shear components. Free crack-face and bonded-interface conditions select this analytical realization; its fourth-order balance is not a universal antiplane law or an observed experiment.[^ref-139488db9c6c]
Straight screw dislocation. Lazar's infinitely long defect lies along \(z\), as does its Burgers vector. Its total axial displacement varies over transverse position and requires a cut or local patch because of angular branch dependence. The cylinder supplies the material domain, \(z\) the axis, and \(u_z\) the physical field; Burgers circulation and incompatible/plastic core distortion supply different case conditions. His finite rod also has \(u_\phi\), so it is not a pure axial-only instance.[^ref-28985159b1a0]
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 (92nd 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¶
- Uniform axial translation: constant \(w\) has no transverse shear.[^ref-139488db9c6c]
- Affine Shear Mapping: simple shear can fit a special antiplane form, but nonuniform crack and screw fields need not be affine.[ref-414532748a54][ref-139488db9c6c][^ref-28985159b1a0]
- Any screw defect or shear stress: the pure local axial displacement test still has to be met.[^ref-28985159b1a0]
- A universal Laplace or Poisson problem: those equations require specific classical assumptions, while couple-stress and incompatible-core models differ.[ref-139488db9c6c][ref-28985159b1a0]
References¶
[^ref-414532748a54]: 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.” [^ref-139488db9c6c]: 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. [^ref-28985159b1a0]: 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.