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