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Related Concept Videos

Deformations in a Transverse Cross Section01:21

Deformations in a Transverse Cross Section

When a material is subjected to uniaxial stress, it elongates or contracts in the direction of the applied force, and also undergoes changes in the perpendicular directions. This behavior is crucial for understanding how materials behave under stress and is governed by mechanical properties such as Poisson's ratio v, which measures the ratio of transverse strain to axial strain.
As the material stretches, it expands or contracts in orthogonal directions to the load. This phenomenon varies...
Unsymmetric Bending - Angle of Neutral Axis01:15

Unsymmetric Bending - Angle of Neutral Axis

Unsymmetrical bending occurs when a structural member is subjected to bending moments in a plane that does not align with the member's principal axes. This scenario typically arises in beams and other structural components when loads are applied at non-ideal angles, introducing complexities in stress analysis.
When a bending moment is applied at an angle θ concerning the vertical axis of a symmetrical member, it can be resolved into components along the member's principal centroidal axes. The...
Generalized Hooke's Law01:22

Generalized Hooke's Law

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...
Bending of Curved Members - Neutral Surface01:16

Bending of Curved Members - Neutral Surface

In curved beams, unlike straight beams, the stress distribution across the cross-section is not uniform due to the beam's curvature. This non-uniformity arises because the neutral axis, where stress is zero, does not align with the centroid of the section. In a curved beam, the strain varies along the section as a function of the distance from the neutral axis.
Consider the curved member described in the previous lesson. According to Hooke's law, which relates stress to strain within the...
Flexural Stress01:16

Flexural Stress

When analyzing bending in symmetric members, it's crucial to understand how stresses distribute when subjected to bending moments. This stress distribution is effectively described by applying fundamental mechanics and material science principles, particularly Hooke's Law for elastic materials.
Hooke's Law states that within the material's elastic limits, stress is directly proportional to strain. In a member experiencing a bending moment, the strain at any point is relative to its distance...
Velocity Potential01:20

Velocity Potential

In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by ϕ, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:

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Related Experiment Video

Updated: Jun 12, 2026

In Situ Measurement of Vacuum Window Birefringence using 25Mg+ Fluorescence
07:03

In Situ Measurement of Vacuum Window Birefringence using 25Mg+ Fluorescence

Published on: June 13, 2020

Generalized bending equations for the radial gradient-index optical system.

K Takada, K Hyakumura, K Yamamoto

    Applied Optics
    |June 26, 2010
    PubMed
    Summary

    Generalized bending equations simplify optical design for radial gradient-index lenses. These effective tools aid in creating advanced lens systems.

    Area of Science:

    • Optics and Optical Design
    • Materials Science

    Background:

    • Gradient-index (GRIN) lenses offer unique optical properties.
    • Traditional lens design methods can be complex for GRIN systems.

    Purpose of the Study:

    • To present generalized bending equations for optical design.
    • To demonstrate the application of these equations to radial GRIN lenses.

    Main Methods:

    • Derivation of generalized bending equations.
    • Application of equations to a specific radial GRIN lens system example.

    Main Results:

    • The generalized bending equations were successfully derived.
    • The equations proved effective in the design of a radial GRIN lens.

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    Quantitative Analysis of Viscoelastic Properties of Red Blood Cells Using Optical Tweezers and Defocusing Microscopy
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    Quantitative Analysis of Viscoelastic Properties of Red Blood Cells Using Optical Tweezers and Defocusing Microscopy

    Published on: March 25, 2022

    Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy
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    Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy

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    Last Updated: Jun 12, 2026

    In Situ Measurement of Vacuum Window Birefringence using 25Mg+ Fluorescence
    07:03

    In Situ Measurement of Vacuum Window Birefringence using 25Mg+ Fluorescence

    Published on: June 13, 2020

    Quantitative Analysis of Viscoelastic Properties of Red Blood Cells Using Optical Tweezers and Defocusing Microscopy
    08:03

    Quantitative Analysis of Viscoelastic Properties of Red Blood Cells Using Optical Tweezers and Defocusing Microscopy

    Published on: March 25, 2022

    Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy
    09:43

    Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy

    Published on: August 13, 2019

    Conclusions:

    • Generalized bending equations are powerful tools for optical design.
    • These equations facilitate the design of complex systems like radial GRIN lenses.