Chapter 12-13 Essentials

 

semester overview

formula sheet

 

 

Augustin Louis Cauchy [kō-shē, 1789-1857]…applies the notion of pressure on a plane (a concept which was familiar to him from hydrodynamics) and assumes that this pressure is no longer normal to the plane upon which it acts in an elastic body. … The total stress on an infinitesimal element of a plane taken within a deformed elastic body is defined as the resultant of all the actions of the molecules situated on one side of the plane upon the molecules on the other, the directions of which (actions) intersect the element under consideration. By dividing the total force by the area of the element, the magnitude of stress is obtained. -- Timoshenko, S. P. (1983), History of strength of materials. McGraw-Hill.

 

 

 

MecMovie 12.1 Hibbeler's StressVisualizer Tensors main.mechmat.89y for TI-89 by Chris Vincent, TICellSheets required other TI-89 routines

 

 

HAMLET Mohr's circle Mohr's circle by Dr. Chien-Chung Chen

 

 

Tensors

tensor from the Latin tensus meaning stress or tension

Tensors, defined mathematically, are simply arrays of numbers, or functions, that transform according to certain rules under a change of coordinates.  ...the values of the components of all tensors, regardless of order, return to their original values under a 360-degree rotation of the coordinate system in which the components are described. source

Rank	Name	Components in n-dimensional space	Components in 3-dimensional space	Information	Examples
0	scalar	n0	30 = 1	magnitude	mass, temperature
1	vector	n1	31 = 3	magnitude and one direction	position, velocity, acceleration, flux density
2	dyad	n2	32 = 9	magnitude and two directions	moment of inertia, permeability, deformation, rate of deformation, stress, strain (Mohr's circle)
3	triad	n3	33 = 27		piezoelectricity
4	tetrad	n4	34 = 81		curvature of space-time, elasticity
...		etc	etc

more info

 

 

3D Mohr's plot, cube, arrows  (1, 2) Fiber Rolling Mohr's Helix Video (quicktime) Cosserat brothers, 1909 Helices in nature fruit and vegetables wood: 1, 2  (by Blaine Medlock) wood: 1  (by Daniel Deposki, taken at Grand Teton National Park near Lake Jenny) wood cells ice helical dislocations helicola  (by Robert Ginzburg) periodic table: 1, 2 higher-order helices in lattices  (pp.87-91) higher-order helices in lattices heliospheric current sheet molecules bone annulus fibrosis Unit Map Description Unit Map Video (quicktime)

 

 

 

Plane strain occurs at the centerline under wall footings and other long, narrow structures (zero strain along length of footing)

 

 

 

 

 

props: rubber i-beam, foam block, broken axle, put StressVisualizer on flash drive