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Transformation of Plane Strain

Transformation of Plane Strain

When analyzing elongated structures like bars subjected to uniformly distributed loads, it is essential to understand the transformation of plane strain when coordinate axes are rotated. This transformation helps to assess how material deformation characteristics vary with orientation, which is crucial in materials science and structural engineering.
Under plane strain conditions, typical for members where one dimension significantly exceeds the others, deformations and resultant strains are...

Angle of Twist - Elastic Range

Angle of Twist - Elastic Range

Consider a cylindrical shaft with a length denoted by L and a consistent cross-sectional radius referred to as r. This shaft undergoes a torque at the free end. The highest shearing strain within the shaft is directly proportional to the twist angle and the radial distance from the shaft axis. When the shaft behaves elastically, this shearing strain can be articulated using variables such as the applied torque, radial distance, the polar moment of inertia, and the modulus of rigidity. By...

Angle of Twist: Problem Solving

Angle of Twist: Problem Solving

An electric motor applies a torque of 700 N·m to an aluminum shaft, triggering a stable rotation. Two pulleys, B and C, are subjected to torques of 300 N·m and 400 N·m, respectively. The modulus of rigidity is provided as 25 GPa. With the knowledge of the length and diameter of each segment, the twist angle between the two pulleys can be computed. First, a section cut is made between pulleys B and C, and the cut cross-section is analyzed using a free-body diagram. Given that the torque...

Conformations of Cyclohexane

Conformations of Cyclohexane

Cyclohexane does not exist in a planar form due to the high angle and torsional strain it would experience in the planar structure. Instead, it adopts non-planar chair and boat conformations.
The chair form is the most stable and derives its name from its resemblance to the “easy chair.” In the chair conformation, two carbon atoms are arranged out-of-plane — one above and one below, minimizing the torsional strain. In the chair form, the bond angle is very close to the ideal...

Residual Stresses in Circular Shafts

Residual Stresses in Circular Shafts

In materials that exhibit elastic and plastic behavior, known as elastoplastic materials, residual stresses can accumulate when these materials experience plastic deformation. This deformation arises from either high levels of shearing stress or significant strains. Residual stresses are internal stresses that persist within a material after removing the external force causing deformation. This phenomenon is demonstrated when observing the behavior of a shaft under torque; notably, the...

Conformations of Cycloalkanes

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Adolf von Baeyer attempted to explain the instabilities of small and large cycloalkane rings using the concept of angle strain — the strain caused by the deviation of bond angles from the ideal 109.5° tetrahedral value for sp3 hybridized carbons. However, while cyclopropane and cyclobutane are strained, as expected from their highly compressed bond angles, cyclopentane is more strained than predicted, and cyclohexane is virtually strain-free. Hence, Baeyer’s theory that...

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Related Experiment Video

Updated: Mar 20, 2026

Optimized Fabrication Procedure for High-Quality Graphene-based Moiré Superlattice Devices

Optimized Fabrication Procedure for High-Quality Graphene-based Moiré Superlattice Devices

Published on: July 11, 2025

Strain and twist angle driven electronic structure evolution in twisted bilayer graphene.

Jiawei Yu^1,2, Guihao Jia^1,2, Qian Li^1,2

¹State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, People's Republic of China.

Reports on Progress in Physics. Physical Society (Great Britain)

|March 18, 2026

Summary

Shear strain significantly impacts twisted bilayer graphene's electronic properties, affecting flat bands and spectral weight. Remote bands, however, reliably indicate twist angles, unaffected by strain.

Keywords:

2D materials scanning tunneling microscopy and spectroscopy strain engineering twisted bilayer graphene twistronics

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Published on: March 24, 2019

Area of Science:

Condensed Matter Physics
Materials Science
Nanotechnology

Background:

Twisted bilayer graphene (TBG) exhibits unique electronic properties influenced by moiré superlattices.
The interplay between twist angle and local strain in TBG remains incompletely understood.

Purpose of the Study:

To investigate the combined effects of twist angle and local strain on TBG electronic properties.
To visualize the evolution of flat and remote bands in energy and real space.

Main Methods:

Low-temperature scanning tunneling microscopy (STM) on a TBG device with a twist-angle gradient and strain fields.
Atomic-resolution visualization of electronic band structures.
Comparison of regions with dominant shear, biaxial, and mixed strain.

Main Results:

Shear strain critically controls flat-band separation, linewidth, and spectral-weight redistribution.
Anomalous spectral weight transfer and spatial dispersion observed in flat bands near the first magic angle.
Remote band energies serve as strain-insensitive indicators of local twist angle.

Conclusions:

Shear strain is a key factor in shaping the low-energy electronic landscape of TBG.
A continuum model incorporating heterostrain and electron-electron interactions quantitatively explains the observed phenomena.