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3: General Concepts of Stress and Strain

Last updated

Mar 15, 2021
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- 2.3: Shear and Torsion
- 3.1: Kinematics

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In extending the direct method of stress analysis presented in previous modules to geometrically more complex structures, it will be convenient to have available somewhat more general mathematical statements of the kinematic, equilibrium, and constitutive equations; this is the objective of the present chapter. These equations also form the basis for more theoretical methods in stress analysis, as well as for numerical approaches such as the finite element method. We will also seek to introduce some of the notational schemes used widely in the technical literature for such entities as stress and strain. Depending on the specific application, both index and matrix notations can be very convenient; these are described in a separate module.

3.1: Kinematics
This page explores kinematic equations relating strain to displacement in loaded bodies, distinguishing rigid body motion from stretching due to varying displacements. It details normal and shear strains in matrix and vector forms, emphasizing volumetric strain as a component of total strain.
3.2: Equilibrium
This page covers stress equilibrium relations based on Newton's laws and introduces Cauchy stress as a second-rank tensor connecting external tractions and surface orientations. It details the determination of stress variations using differential equations, emphasizing static equilibrium and stress continuity. The module includes mathematical expressions for stress in tensile specimens, equilibrium equations, boundary conditions, and examples.
3.3: Tensor Transformations
This page explores the transformation of stress and strain axes in materials, with a focus on composite materials and the application of Mohr's circle for graphical visualization of stress states. It discusses the transformation of biaxial strains, the concept of pure shear, and the determination of principal stresses in three-dimensional states.
3.4: Constitutive Relations
This page explores the interrelation of kinematics, equilibrium, and constitutive relations in Mechanics of Materials, focusing on elastic constants affected by processing and microstructure. It details isotropic materials and stress tensor decomposition into hydrostatic and distortional components, discussing their impact on material behavior. The distinctions between true and engineering stress in deformation are examined, along with adaptations for anisotropic materials.

Thumbnail: Idealized stress in a straight bar with uniform cross-section. (CC BY-SA 3.0; Jorge Stolfi via Wikipedia)

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