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2 three dimensional stress and strain
- 1. Three Dimensional Stress and Strain In order to more fully understand this reality, we must take into account other dimensions of a broader reality. ~ John Archibald Wheeler BIOE 3200 - Fall 2014
- 2. Learning Objective Identify constitutive equations (relating stress to strain) BIOE 3200 - Fall 2014
- 3. How do we measure strain? Strain gauge: BIOE 3200 - Fall 2014 Images from https://www.biomedtown.org/bio med_town/LHDL/Reception/collecti on/StrainGauges As the material of interest deforms, the wires in the strain gauge are stretched and electrical conductivity decreases (resistance increases as wires stretch) From https://ueidaq.wordpress.com/20 13/08/02/the-twists-of-strain- gauge-measurements-part-1/
- 9. Hooke’s Law by Superposition Strain x y z xy xz yz Applied Stress x y z xy= yx xz= zx yz= zy x E x E x E y E y E y E z E z E z E xy G xz G yz G 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
- 10. Constitutive Equations: 3D Stress and Strain BIOE 3200 - Fall 2014
Editor's Notes
- Strain gauges allow strain measurements
- Displacements vary as a function of position; displacements of different points within a body can be expressed as vectors. Each vector can be resolved into components parallel to a set of cartesian axes such that u, v and w are the displacement components in the x, y and z directions, respectively. Consider a vertical beam supported at one end and pulled in the opposite direction. Displacement and strain are 0 at the point of support (ux = 0); displacement at the free end is max (ux = L); stress is constant throughout beam (f/A).
- Displacement vectors – quantifies the difference between where a point starts and ends. Displacement of a body can be separated into 3 parts: rigid body translation, rigid body rotation and deformation Translation and rotation don’t induce strains (and therefore don’t induce stresses) in the body. Deformation is the type of displacement that gives rise to internal stresses. u, v and w are the vector components in the x, y and z directions, respectively, and each depends on displacements in all 3 directions; so partial derivatives of u,v and w wrt x, y an z are displacement gradients. Mathematically, these are more complex equations, but can be simplified those shown above for small displacement gradients (where double partial derivative is negligible).
- Summarize derivation of differential equations relating strains to partial derivatives of position; Strain is defined in terms of angle of deformation (gamma)
- Hooke’s Law only applies to homogeneous, isotropic materials that have linear stress-strain relationships. Many materials behave in a linearly elastic manner over some range of applied stress. Stress and strain relationships in 3D require Poisson’s ratio term, as well as elastic modulus Shear modulus involves both elastic modulus and Poisson’s ratio