Continuum Exam #2: Lessons 8-10
What does deformation represent?
A phsyical transofrmation from a reference to a current -may inslcude change in form, motion
What type of function is the mapping funciton?
A proper vector-to-vector function that describes x components wrt X components
TRUE or FALSE The term "velocity" is associated with the referential function. This is what we measure with velocity sensors (e.g. a pitot tube).
TRUE
TRUE or FALSE THe velocity gradient is generally asymmetric and can be decomposed into symmetric and asymmetric parts to highlight physical meaning
TRUE
TRUE or FALSE THe postion of a particle will change from a reference config X to a current config x
TRUE Each particle or point has an associated displacement
TRUE or FALSE We can always visualize affine stretch
TRUE Since an ellipse can always be inserted into a parallelogram -If stretch in an affine defromatino is always int eh from of an ellipse, then it has orthonormal eigenvalues and is a symmetric tensor
TRUE or FALSe Material performance is intrinsically linked to changes in form?
TRUE changes in form (modulus, hardness, toughness, strength, viscosity)
TRUE or FALSE THe mapping function is a one-to-one function and is invertible
TRUE relates X to x can also calculate X if we know x
A ___________ force acts on all mass elements of a body
body
External forces include ___________ and ______________
body and surface
Deformation = ________ + _________
form change + rigid body motion
If matter is viscous, it exhibits time rate dependency and it is necessary to specify the deformation process and follw the
history of deformation
Affine Mappings are called ______________ transformations because little squares will deform just like the big squae
homogeneous
Select any true statement for 3D objects experiencing a non-linear deformation. [2 pts] i. The deformation gradient tensor F will be unique at different material locations. ii. Each infinitesimal volume deforms like an affine transformation. iii. The mapping function is homogenous. iv. The total deformed volume will be a parallelepiped.
i and ii
Select all answers below that are false regarding the identity tensor (can select multiple) i. If U (right stretch tensor) equals the identity tensor, it means no deformation ii. If E (engineering strain) equals the identity tensor, it means no deformation iii. If L (velocity gradient) equals the identity tensor, it means no deformation iv. If F (deformation gradient) equals the identity tensor, it means no deformation
ii and iii
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
If we want to find how tensors change wrt time for each particle, we calculate the:
material derivative
The descriptions of motion used commonly in contiuum include:
material, referential, spatial
If W is zero at a point, then the point is___________
not rotating -if non zero then the deformation includes rotation -if the vorticity tensor is nonzero and D is zero, then it is a rigid-body rotation (no stretch)
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
At an infinitesimal level, any 2D object can be discretized into _________ or _________?
squares; triangles
Polar decomposition
stretch can occur befor or after the roation and therefor we define the right strech tensor U and the left stretch tensor V -R is orthonormal and U is symmetric --> positive definite (strech is never negative)
Rate of deformation tensor is essentially giving the rate that a continuum is being ____
stretched
Lagrangian CV
the CV moves with the material in the spatial configuration
Rate of deformation tensor D is called this because:
the rate fo change of a line element can be deteremined by D -better known as stretch-rate
To determine how tesnors change wrt time at a fixed location, we calculate:
the spatial derivative
Deviatoric stress represents
the stress part of T causing shape change
Dilation stress represents
the stress part of T that is causing volume (area) change
Surface forces or _____________ are contact forces that act on bounding surfaces
tractions
Volume element (Jacobian)
triple scalar product, whcih is qual to the volume defined by the line elements
TRUE or FALSE if F is non symmetric, there must be rotation in the deformtion
true
An area has _________ principal streches and a volume has _________ principal stretchs whcih can be represented by a 2nd order stretch tensor U
two; three
A tensor used to examine the deformation process is the:
velocity gradient
Dialational stress causes changes in ___________
volume
Mapping function equation
x = F dot X + c
What does the continuum assumption mean when applied to mapping function?
-Each point X in the reference config deforms into a distinct point x in the current confih=g -Two different X points cannot deform into the same x -Two different x particles must come from two different X locations
Spatial description
-Fixes attention on a given region of space instead of on a given body of matter -observe changes in physical properties at a fixed region -independent variable is the spatial position, x, at which many particles will pass through over time
Volume element pt II
-Jacobian or jacobian determinant (scalar=determinant of F)
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
Referential description
-independent variable is the position vector, X, pointing to a particle in an arbitrary reference configuration -when the reference configure is at t=0, it is often called the lagrangian description
Affine transformations have the following properties
-squares become parallelograms -cubes become parallelopipeds -line segments become line segments of differnt orientation and length but remain straight -parallel lines remain parallel lines -circles become ellipses -speheres become ellipses
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
Steps to calculate U,V,R
1. calculate [F] 2. calculate [C_lab] 3. eigen and [Q] 4. C in prinipal basis 5. U in principle 6. U in lab 7. R in lab 8. V in lab
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.
In Matlab, graph the Seth hill family of strains for a 1D problems with λ going from 0.1 to 4 (y-axis is strain, x-axis is λ). Label the curves (can be by hand). 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.
HW7_2: Illustrate the reasons that the deformations described by 1a, 1b, 1c and 1d are either affine or non-affine.
All the above deformations are affine. As an example, you can see that 1c is "homogenous" as all the little squares (light blue) deform just like the big square (dark blue).
Eulerian CV
Fixed in the spatial configuration
TRUE or FALSE The tangenetial component of traction on the octahedral plane is the "Octagedral Shear Stress" \sigma_oct
TRUE
A "simple shear" experiment is often used to characterize the mechanical properties of materials and is shown below. The deformation from this experiment is similar to problem 1d. What do you think would happen with R, U and V if the material were infinitely pulled along the direction of the black arrow?
As the material is pulled, the rotation would increase to 90 degrees (CW). U would eventually have only large stretches in the e2 direction (vertical stretch, then rotate), while V would only have large positive stretches in the e1 direction (rotate and then horizontal stretch).
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)
The spatial description is most often used in fluid mechanics and is offed called the ________--
Eulerian description
Newtons third law
Every force has an equal and opposite force. Thus, all forces are interactions between two bodies
TRUE or FALSE Since affine stretch is linear, a square will always maintain contact with the mid points of the parallelogram edges
FALSE An ellipse will
TRUE or FALSE For a non-linear transformations, each little square has the same F tensor
FALSE each little square has its own unique F tensor
How to calculate non-linear transformations?
For non-linear transformations, each little square deforms differently from other squares; however, within the limit of infinitesimal squares they all deform like an affine transformation locally Difference: each little square has its own unique F tensor
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
If stretch in an affine defromatino is always in the from of an ellipse, what does this mean?
It has orthonormal eigenvalues and is a symmetric tensor
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
How do we determine the time rate of a tensor field within a CV
Leibniz's Rule
Is the term "velocity" associated with spatial description?
NO! This measure of change in position with time has less physical meaning, and the term "velocity" is not associated with the spatial description.
First Piola-Kirchhoff stress (PK1)
New tensor P p=N*P **PK1 stress (note that P is non-symmetric)
In the problem below, the ellipse has the same shape for both deformations. Is the deformation the same? Illustrate graphically and mathematically why or why not.
No, the deformation is not the same. The stretch is approximately the same, but the rotation is different. In the first figure, the square is stretched "off-axis" of the e1 and e2 surface, and then a "smaller" rotation is applied. In the second figure, the square is stretched along the e1 and e2 reference axis, and a "larger" rotation is applied. The differences in F can be seen below (can use matlab program to calculate R and U).
Can we simply analyze form change from the deformation tensor?
Not quite. The deformation gradient is a function of roation and rotation does not cause a change in form
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
Explain the relationship between pressure and the stress tensor
Pressure is a scalar and is equal to the negative mean normal stress (σmean): p=-s_mean The stress tensor is a 2nd order tensor. Pressure represents the normal component values along the diagonal of the stress tensor.
V
Rotated first then stretched
The vorticity tensor, W, is a useful indicator of what?
Rotation
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
Write one important similarity and one important difference between octahedral shear stress and the deviatoric part of the stress tensor:
Similarity: Both are a quantitative measures of stress that go into changing a materials shape (i.e. distorting a material) [1.5 pts] Difference: Octahedral shear stress is a scalar, while the deviatoric part of the stress tensor is a 2nd order tensor. [1.5 pts]
Would you classify D as a true time rate? Why or why not?
Since this tensor is symmetric, it has orthogonal principal axes. Therefore, D is similar to U and V, but instead of representing stretch like U and V, D represents a form of stretch rate. However, a "time rate" is a change in a function w.r.t. time, and in this case, it is a change in velocity w.r.t. position, so NO it is not a true time rate.
U
Stretched first then rotated
Deviatoric stress is calculated by:
Subtracting mean stress from T
TRUE or FALSE Strain is a symmetric 2nd order tensor
TRUE
TRUE or FALSE Stretch of a 2nd order tensor is very similar to strech of a line
TRUE
TRUE or FALSE The normal component of traction on the octahedral plane is the "mean stress" \sigma_m
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.
What is F in the mapping function?
The defromation gradient tensor -components represent the change in x relative to X
What do lamda_1 and lamda_2 represent?
The eigenvalues of U
What is strain?
The normalized measure of deformation whithout rigid body motion
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.
What does X represent in the material description?
The particle and is not a position vector that points to the particle
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.
What is the mapping function analogous to?
The slope intercept equation -translation c is analogous to the y-intercept
Interpret the physical meaning of the vorticity vector in relation to the kitty's motion.
This point on the Kitty's body at time t=1s is rotating about the x2 axis in the counterclockwise direction.
If E in the above equation is the Green-Lagrange strain for a fixed region (spatial) on a deformed elastomer, what would this equation physically determine?
This would be the material derivative, so the physical meaning is the change in strain w.r.t. time for a particle.
Gravitational force
Two bodies attract
What is the maximum and minimum "stretching" calculated from the symmetric part of the velocity gradient in (b)? What is the physical interpretation of these results w.r.t. the cat?
Using the eig function in matlab. Max stretch rate is 2, min stretch rate is -1. Over time the cat is elongating along the e3 principal axis, and contracting along the e1 principal axis. Note: the e1 and e3 principal axes are not aligned with the lab basis.
Octaheadral plane from principal stresses
We know n of the ocathedral plane wrt the principal basis
Material Description
We observe changes in position and physical properties as the material body X (particle) moves in space and time -independent variable is the particle
Newtons first law
Without an applied force, objects remain at rest or constant velocity (law of inertia)
Will deforming an obeject in tension create surfaces thate experience shear?
Yes any surafce not aligned with the eigenvector will have tangential component
Draw the deformations described by the U and V you calculated in problem 4c. Draw an ellipse to calculate λ1, λ1, and R for both U and V. Explain whether your answers make sense.
Yes, it makes sense. The eigenvalues for both U and V are the same, and both have no rotation (remember, U and V represent stretch and should have no rotation component). The orientation of the eigenvalues (i.e. eigenvectors) are different between U and V, and this difference is the rotation of the deformation, which is 22 degrees in the e1 - e2 plane.
2nd Piola Kirchoff stress tensor, S
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.
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]
When we use, X, we are identifying what?
a single particle
When we use, x, we are identifying what?
a single region that many particles will pass
When we describe a time-dependent tensor using the reference coordinate system (X1,X2,X3) we are describing....
a tensor field for a fixed particle -Referential description
When we describe a time-dependent tensor using the reference coordinate system (x1,x2,x3) we are describing....
a tensor field for a fixed position
The deformation is not just a function of space but __________-
also a function of time
Electromagnetic force
charged particles attract and repulse
At an infintesimal level, any 3D object can discretized into _______ or _______
cubes or tetrahedrals (only 4 points needed to define a volume)
When external forces are applied to objects the forces cause _______________
deformation
For a line element, what for of x and X do you use?
dx and dX
Since forces are directional, tehy are vectors, and can be classified as _________- or - _____________
external ; internal