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Hooke's Law01:26

Hooke's Law

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Hooke's law, a pivotal principle in material science, establishes that the strain a material undergoes is directly proportional to the applied stress, defined by a factor called the modulus of elasticity or Young's modulus.
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Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity01:15

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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.
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The study of solid circular shafts under stress shows that within the elastic limit, stress increases directly to the distance from the shaft's center. This relationship holds until the shaft reaches a critical point of stress, beyond which it begins to yield, marking the transition from elastic to plastic deformation. At this crucial juncture, the maximum torque the shaft can endure without permanent deformation is determined, signifying the limit of its elastic behavior.
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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...
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Consider a scenario where a circular shaft is subject to torque that remains within the boundaries of Hooke's Law, avoiding any permanent deformation. So, the formula for shearing strain is revisited. This formula is multiplied by the modulus of rigidity, and then Hooke's Law for the shearing stress and strain is applied. As a result, the equation for shearing stress in a shaft can be derived.
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Members Made of Elastoplastic Material01:19

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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.
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Thermodynamic Restrictions in Linear Viscoelasticity.

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Summary

Thermodynamic consistency in linear viscoelastic models requires specific conditions for the Boltzmann function, particularly when considering strain history. Ensuring consistency with the second law remains an open challenge for fractional derivative models.

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free energystrain–rate historiesthermodynamic restrictionsviscoelasticity

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Area of Science:

  • Thermodynamics
  • Materials Science
  • Continuum Mechanics

Background:

  • Linear viscoelasticity describes materials exhibiting both viscous and elastic characteristics.
  • Thermodynamic consistency is crucial for the physical realism of constitutive models.
  • Classical models like the Boltzmann law face scrutiny regarding their thermodynamic validity.

Purpose of the Study:

  • To investigate the thermodynamic consistency of various linear viscoelastic models.
  • To identify conditions required for model validity based on the second law of thermodynamics.
  • To explore limitations and open problems in current viscoelastic modeling.

Main Methods:

  • Analysis of the Boltzmann function (kernel) in classical and power-law forms.
  • Investigation of constraints imposed by free-energy functionals.
  • Examination of constitutive equations involving strain history and fractional derivatives.

Main Results:

  • The Boltzmann function's consistency depends on the negative definiteness of its half-range sine transform.
  • Consistency in the power-law form requires stress functionals to depend on strain history, not strain-rate history.
  • Thermodynamic consistency for fractional derivative models involving strain-rate histories remains an unresolved issue.

Conclusions:

  • Thermodynamic constraints significantly restrict the applicability of linear viscoelastic models.
  • Strain history dependence is key for consistency in certain viscoelastic formulations.
  • Further research is needed to establish free-energy functionals for fractional derivative viscoelasticity.