Related Experiment Video
Updated: Mar 8, 2026
06:54
Ultrasonic Fatigue Testing in the Tension-Compression Mode
Published on: March 7, 2018
11.3K
The anelastic Ericksen problem: universal eigenstrains and deformations in compressible isotropic elastic solids
1School of Civil and Environmental Engineering and The George W. Woodruff School of Mechanical Engineering , Georgia Institute of Technology , Atlanta, GA 30332, USA.
Proceedings. Mathematical, Physical, and Engineering Sciences
|January 26, 2017
Summary
This study solves the anelastic Ericksen problem, identifying covariantly homogeneous deformations as universal solutions for arbitrary strain-energy functions in nonlinear solids. These solutions restrict possible universal eigenstrains, particularly in 2D and 3D where they are zero-stress.
Area of Science:
- Solid Mechanics
- Continuum Mechanics
- Materials Science
Background:
- The elastic Ericksen problem seeks deformations in hyperelastic solids compatible with any strain-energy function.
- Ericksen proved only homogeneous deformations are possible in the compressible elastic case.
- Anelasticity introduces finite eigenstrains, complicating the problem.
Purpose of the Study:
- To solve the anelastic Ericksen problem for arbitrary strain-energy density functions.
- To determine universal deformations and associated eigenstrains in nonlinear anelastic solids.
- To generalize the concept of homogeneous deformations to anelastic materials.
Main Methods:
- Modeling anelasticity using finite eigenstrains on a Riemannian material manifold.
- Introducing and analyzing "covariantly homogeneous deformations" with covariantly constant deformation gradients.
- Investigating the properties of the material manifold and universal eigenstrains.
Main Results:
- Covariantly homogeneous deformations are identified as the only universal deformations in this anelastic framework.
- Universal eigenstrains are shown to impose significant restrictions on material behavior.
- For simply-connected bodies, the material manifold is proven to be symmetric for any universal eigenstrain distribution.
- In 2 and 3 dimensions, universal eigenstrains are found to be zero-stress.
Conclusions:
- Covariantly homogeneous deformations provide a universal solution to the anelastic Ericksen problem.
- The study reveals fundamental constraints on eigenstrain distributions in nonlinear anelastic solids.
- The geometric framework of Riemannian manifolds offers new insights into anelastic material behavior.
More Related Videos
Related Concept Videos
01:22Elastic Strain Energy for Normal Stresses
660
Strain energy quantifies the energy stored within a material due to deformation under loading conditions, a fundamental concept in materials science and engineering. The strain energy can be modeled when a material is subjected to axial loading with uniformly distributed stress. In this scenario, the stress experienced by the material is the internal force divided by the cross-sectional area, and the strain induced is directly proportional to this stress through the modulus of elasticity.
If...
If...
660
01:15Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity
657
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.
657
01:22Generalized Hooke's Law
2.9K
The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...
2.9K
01:19Members Made of Elastoplastic Material
446
The behavior of elastoplastic materials under bending stresses, particularly in structural members with rectangular cross-sections, is crucial for predicting material responses and understanding failure modes. Initially, when a bending moment is applied, the stress distribution across the section follows Hooke's Law and is linear and elastic. This distribution means the stress increases from the neutral axis to the maximum at the outer fibers, up to the elastic limit.
As the bending moment...
As the bending moment...
446
01:20Elastic Strain Energy for Shearing Stresses
569
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...
569
01:11Bending of Members Made of Several Materials
660
In analyzing a structural member composed of two different materials with identical cross-sectional areas, it is crucial to understand how their distinct elastic properties affect the member's response under load. The analysis involves assessing stress and strain distributions using the transformed section concept, which accounts for variations in material properties.
Hooke's Law determines stress in each material, stating that stress is proportional to strain but varies due to each material's...
Hooke's Law determines stress in each material, stating that stress is proportional to strain but varies due to each material's...
660