Related Experiment Video
Updated: Jun 21, 2025
07:37
Full-field Strain Measurements for Microstructurally Small Fatigue Crack Propagation Using Digital Image Correlation Method
Published on: January 16, 2019
9.7K
On hyperelastic solid with thin rigid inclusion and crack subjected to global injectivity condition
A I Furtsev1, E M Rudoy1, S A Sazhenkov1
1Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia.
Summary
This study addresses the equilibrium of hyperelastic solid bodies with inclusions and cracks, ensuring no self-penetration. Variational techniques confirm the existence of a weak solution for this complex mechanics problem.
Area of Science:
- Solid mechanics
- Continuum mechanics
- Non-smooth variational problems
Background:
- Investigates equilibrium in hyperelastic solid bodies with inclusions and cracks.
- Finite strain theory is applied due to hyperelasticity.
- A key challenge is the global injectivity constraint preventing penetration.
Purpose of the Study:
- To formulate and solve the equilibrium problem for a solid body with a thin rigid inclusion and a crack.
- To demonstrate the connection between differential formulation and energy minimization.
- To prove the existence of a weak solution using variational methods.
Main Methods:
- Differential formulation of the equilibrium problem.
- Energy minimization to derive the weak formulation.
- Variational techniques to establish the existence of a weak solution.
Main Results:
- The energy minimization approach yields the weak formulation.
- The existence of a weak solution is mathematically demonstrated.
- The study provides a framework for analyzing complex solid mechanics problems with constraints.
Conclusions:
- The variational technique successfully proves the existence of a weak solution.
- The research contributes to understanding non-smooth variational problems in mechanics.
- This work is relevant to the theme issue on non-smooth variational problems and their applications.
Related Concept Videos
01:19Members Made of Elastoplastic Material
94
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...
94
01:22Generalized Hooke's Law
888
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...
888
01:15Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity
263
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.
263
01:21Plastic Behavior
196
A material's elastic behavior is characterized by the disappearance of stress once the load is removed, allowing the material to return to its original state. However, when stress surpasses the yield point, yielding commences, marking the onset of plastic deformation or permanent set. This change from elastic to plastic behavior is influenced by the peak stress value and the duration before the load is removed. An intriguing observation occurs when a specimen is loaded, unloaded, and...
196
01:19Plastic Deformations
128
Plastic deformation represents a fundamental concept in materials science, which explains the irreversible change in the shape of a material when it experiences stress beyond its elastic capability. This phenomenon is important in structural engineering, especially in designing and analyzing cantilever beams—structures that are securely fixed at one end and bear loads at the opposite end. When these beams are subjected to loads within their elastic range, they will return to their...
128
01:22Elastic Strain Energy for Normal Stresses
151
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...
151