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Shearing Strain

Shearing Strain

The shearing strain represents a cubic element's angular change when subjected to shearing stress. This type of stress can transform a cube into an oblique parallelepiped without influencing normal strains. The cubic element experiences a significant transformation when exposed solely to shearing stress. Its shape alters from a perfect cube into a rhomboid, clearly demonstrating the effect of shearing strain. The degree of this strain is considered positive if it reduces the angle between the...

Three-Dimensional Analysis of Strain

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Three-dimensional strain analysis is crucial for understanding how materials deform under stress, particularly in elastic, homogeneous materials. This method employs principal stress axes to simplify complex stress states into more understandable forms. Subjected to stress, a small cubic element within a material either expands or contracts along these axes, transforming into a rectangular parallelepiped. This transformation effectively illustrates the material's deformation. The principal...

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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.

Elastic Strain Energy for Shearing Stresses

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As discussed in previous lessons, strain energy in a material is the energy stored when it is elastically deformed, a concept crucial in materials science and mechanical engineering. This energy results from the internal work done against the cohesive forces within the material. When a material undergoes shearing stress and corresponding shearing strain, the strain energy density, which is the energy stored per unit volume, is calculated. Within the elastic limit, where the stress is...

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A cantilever beam with a rectangular cross-section under distributed and point loads experiences shearing stresses. The analysis begins by identifying the loads acting on the beam. Then, the reactions at the beam's fixed end are calculated using equilibrium equations. The vertical reaction is a combination of the distributed and point loads, while the moment reaction is the sum of their moments. The shear force distribution along the beam, resulting from these loads, is established by creating...

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Shear band analysis based on a strain-gradient hypoplastic model.

Xuan Kang¹, Wei Wu¹, Ivan Zaboev¹

¹Institut für Geotechnik, Universität für Bodenkultur Wien, Feistmantelstraße 4, A-1180 Vienna, Austria.

Acta Geotechnica

|July 6, 2026

Summary

This summary is machine-generated.

This study enhances a hypoplastic model for granular materials by adding strain rate gradients. The improved model accurately simulates finite shear bands, crucial for understanding material deformation.

Keywords:

Hypoplastic model Post-localization regime Shear band thickness Simple shear test Strain gradient

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Area of Science:

Geotechnical Engineering
Continuum Mechanics
Computational Material Science

Background:

Localized deformation in granular materials presents significant challenges for constitutive modeling and numerical simulations.
Existing models often struggle to accurately capture the finite thickness of shear bands, especially in the post-localization phase.

Purpose of the Study:

To extend a hypoplastic model for granular materials to improve shear band analysis.
To incorporate a second-order gradient of the strain rate into the nonlinear tensorial function of the hypoplastic model.
To introduce a length scale for capturing finite shear band thickness in the post-localization regime.

Main Methods:

Extension of a hypoplastic constitutive model by incorporating second-order strain rate gradients.
Development of a nonlinear tensorial function within the extended hypoplastic framework.
Numerical simulations of simple shear tests using the extended model.

Main Results:

The extended model successfully reproduces finite-thickness shear bands in granular materials, even with inhomogeneous initial void ratios.
The model maintains the conciseness of the original hypoplastic framework while introducing a length scale.
Compared to existing gradient models, the extended model demonstrates superior performance in analyzing shear band problems involving rotation.

Conclusions:

The extended hypoplastic model with second-order strain rate gradients offers a more accurate representation of shear band formation in granular materials.
The introduction of a length scale effectively captures the finite thickness of shear bands.
The model provides a robust and computationally efficient tool for analyzing complex shear band phenomena in geotechnical engineering.