Quiz #7: Ch. 8 and 9
TRUE or FALSE The tangenetial component of traction on the octahedral plane is the "Octagedral Shear Stress" \sigma_oct
TRUE
A ___________ force acts on all mass elements of a body
body
External forces include ___________ and ______________
body and surface
Weak force
involved in radioactive decay
Cauchy stress tensor (\sigma)
linear operator independent of n and replates traction to the cutting plant at a point -when no surface or body torsions exist, \sigma is symmetric
internal forces _________ the tendency for mass to accelerate when an external froce is applied
resist
Deviatoric stress causes changes in ______
shape
Internal forces cause a material to change ________ or __________
shapr of stretch
Surface forces or _____________ are contact forces that act on bounding surfaces
tractions
Dialational stress causes changes in ___________
volume
TRUE or FALSE The normal component of traction on the octahedral plane is the "mean stress" \sigma_m
TRUE
Octahedral Plane
-To calculate a scalar measure, we need to determine the normal and shear stress components on the Octahedral plane
Dilational stress (spherical stress)
-a hydrostatic state of stress (equal stress on all surfaces) -first find the mean normal stress (negative of pressure) and multiple it by I
Since \sigma is symmetric, any general rules for 2nd order symmetric tensors must apply What are these rules?
1. 3 principal stresses: \sigma_1 ,\sigma_2, \sigma_3 (invariant triplet) 2. three characteristic invariants: I_1, I_2, I3 (invariant triplet) 3. 3 principal directions: p_1, p_2, p_3 4. no skew part 5. normal and shear components of \sigma can be viewed using mohr's circle
What are the forces most relevant to continuum?
1. Electromagetic 2. Graviational
List one inherent disadvantage and two inherent advantages to using S.
Advantage: S is a symmetric tensor (unlike P, similar to T) and therefore has properties similar to all symmetric tensors (e.g. orthonormal eigenvectors, scalar invariants). Advantage: S does not have moving boundaries (similar to P, unlike T). This can simplify calculations of moving objects. Disadvantage: S is difficult to interpret. When we "pull-back" PK1 stress, the result is not physically meaningful. It is simply a mathematical mechanism to create a symmetric tensor that has both unit basis, or "legs", in the reference configuration. Remember: If you know F, you can easily go back and forth between the different stress tensors. Therefore, you can perform a calculation using S, and then "push-it-forward" into T, a more physically meaningful stress tensor.
At what stretch are the strain measures close to equal? Which strain measure(s) is most physically meaningful for very large compressions and why?
All strain measures are equal when stretch equals 1 (strain = 0 at this point). Under large compression, true and euler strain have more physical meaning, since their strains approach negative infinity, and are more sensitive to large compressions.
Strong force
Binds protons and neutrons
Is cauchy stress in the reference or current configuraton?
Current configureation
Calculate GL strain E using F
E=0.5(F^TF-I)
Newtons third law
Every force has an equal and opposite force. Thus, all forces are interactions between two bodies
Newtons second law
Force equals a change in linear momentum, p F=dp/dt=ma
________ forces resist the tendency for one part of an object toa ccelerate away form another part
Internal froces
Why is octahedral shear stress useful?
It is a scalar measure of the amount of stress being applied towards changing the "shape" of a material
What is the minimum number of line elements needed to measure in order to use the equation in (2a) to calculate in-plane components of E. Show mathematically
It takes 3 line elements
First Piola-Kirchhoff stress (PK1)
New tensor P p=N*P **PK1 stress (note that P is non-symmetric)
3D infinitesimal strain
ONly approximated for very small deformations iwth no rotations. -If rotations exist, then infinitesimal strain is prone to error since the strain value is dependent on rotation
_________ stress is equivalent to von Mises stress or effective stress
Octahedral
Second Piola-Kirchhoff
S is symmetrical -has little physical meaning, however, it can be useful for "packaging" your stress and then converting back into a more physically meaningful stress at a later time point
One dimensional strain
Seth hill family of strain for 1D
Three dimensional strain
Seth hill. Need to consider which stretch, the right stretch U or left stretch V
Deviatoric stress is calculated by:
Subtracting mean stress from T
TRUE or FALSE Strain is a symmetric 2nd order tensor
TRUE
Optical Coherence Tomography (OCT) can be used to capture volumetric images of the eye retina, and the coordinates of pixel speckles can be automatically tracked using specialized software before and after a deformation. How could the method from (2d) be used to determine the volumetric strain of a patient's retina after surgery using OCT images. How many line elements would be needed to measure strain at a point in 3D?
The coordinates of pixel speckles can be divided into groups of four non-linear neighboring pixels. A system of equations can then be used to efficiently calculate the volumetric Green-Lagrange strain in each tetrahedral region from a reference to current configuration. The four points will give 6 lines, which are necessary to calculate the 6 unknowns of E.
Explain the physical meaning of the octahedral shear stress, and explain why this value is useful when interpreting the stresses in an object.
The octahedral shear stress (also known as effective stress and von Mises stress) is a scalar measure that describes the amount of stress going into shape change (i.e. object distortion). This is useful, since scalar values can be used as thresholds to predict when yielding or failure begins. For example, if I want to predict when an object yields, I could run experiments to determine the octahedral shear stress at failure and then design parts not to exceed the octahedral shear stress. The deviatoric stress tensor also describes shape change, but it is a 2nd order tensor with nine components, so you wouldn't be able to use deviatoric stress as a scalar threshold for failure.
Showing all steps, calculate the 1st Piola Kirchoff stress tensor, P, at time 1 and time 2 (w.r.t. time = 0). Give a physical interpretation of P relative to the deformed object
The same force is being applied in the current and reference configuration, however, the surface area in the current configuration is ½ the surface area in the reference configuration. This means that if the traction on the current vertical surface area were applied to the reference surface area, the traction would be reduced by ½. This is reflected in the P tensor.
Gravitational force
Two bodies attract
Octaheadral plane from principal stresses
We know n of the ocathedral plane wrt the principal basis
Newtons first law
Without an applied force, objects remain at rest or constant velocity (law of inertia)
Showing all steps, calculate the 2nd Piola Kirchoff stress tensor, S, at time 1 and time 2 (w.r.t. time=0). Give a physical interpretation of S relative to the deformed object.
You can visualize the stress tensor S as "pulling-back" the current traction vector, t, from P into the ref configuration. This is no different than pulling back a differential line element from the current to the reference configuration (dx=FdX), except in this case the line elements are column vectors of P. For this problem, this "pull-back" scaled the current traction on the vertical surface by ¼.
Cramers rule
a method that uses determinants to solve a system of linear equations [x]=[A^-1][b]
Since forces are directional, tehy are vectors, and can be classified as _________- or - _____________
external ; internal
Electromagnetic force
charged particles attract and repulse