Exam 1 Quantative

Question 1

Tensor

Answer 1

A quantity with magnitude and direction that acts on a specific plane defined by the basis vector

Question 2

A zero order tensor is a

Answer 2

Scalar; 1x1

Question 3

A first order tensor is a

Answer 3

Vector; 3x1

Question 4

A second order tensor is a

Answer 4

Matrix; 3x3

Question 5

A fourth order tensor is a

Answer 5

Matrix;9x9

Question 6

q=d^n

Answer 6

q is the number of entities to be summed in each equation. d is dimensions, n is the number of repeated indices. The total number of q is determined by summing the q’s of each component.

Question 7

When determining q, the indices on the delta (do/do not) count

Answer 7

do not

Question 8

p=d^m

Answer 8

p is the number of quantities equal to the dimension of the space we are working in, where d is dimension and m is the number of independent indices

Question 9

The equivalent matrix to the Kronecker delta is the

Answer 9

Identity matrix

Question 10

The permutation tensor has values of

Answer 10

1 when increasing (even) permutations,
-1 when decreasing (odd) permutations
0 for any repeated index

Question 11

The permutation tensor is used

Answer 11

When you take the cross product of two perpendicular vectors

Question 12

A primed coordinate is in which frame

Answer 12

Reference (undeformed)

Question 13

An unprimed coordinate is in which frame

Answer 13

Deformed

Question 14

When converting a tensor from one frame to another we have to

Answer 14

Pre and Post multiply

Question 15

Displacement is the same as deformation

Answer 15

False; deformation includes both displacement and rotation

Question 16

x=x’-u is the

Answer 16

Displacement vector

Question 17

Fij = dxi/dx’j is the ___ and is used to

Answer 17

Deformation gradient, which is used to map position and stress between the deformed and undeformed configurations

Question 18

Fij = dx’i/dx’j+dui/dx’j is the

Answer 18

Deformation gradient tensor of the displacement vector, aka Fij=delta,ij + dui/dx’j

Question 19

What assumption do we make in the infinitesimally small strain tensor

Answer 19

That xi == x’i

Question 20

What assumptions do we make for a continuum

Answer 20

(1) Homogeneous
(2) Minimize to a point (lim x->0)
(3) No spaces

Question 21

Deformation includes

Answer 21

Rotation and stretch

Question 22

The cauchy stress tensors are a function of

Answer 22

Stretch only

Question 23

What is the right cauchy deformation tensor and what frame is it in

Answer 23

C = F^TF

Which is in the undeformed frame

Question 24

What is the left cauchy deformation tensor and what frame is it in

Answer 24

B = FF^T

Which is in the deformed frame

Question 25

Assumptions for an ideal linearly elastic solid

Answer 25

(1) Small deformations (3-4%) (2) No plastic deformation (3) Stress response is not strain rate dependent (4) Stress is linearly related to strain

Question 26

Traction forces are

Answer 26

Internal forces that develop to balance external surface forces

Question 27

The equations of motion are subject to what restriction

Answer 27

The boundary condition, aka the traction vector along a plane must balance out the forces being applied to that plane

Question 28

Displacement has how many equations, how many unknowns, and how many independent unknowns

Answer 28

3 equations, 3 unknowns, 3 independent unknowns

Question 29

Strain has how many equations, how many unknowns, and how many independent unknowns

Answer 29

6 equations, 9 unknowns, 6 independent unknowns

Question 30

Equillibrium has how many equations, how many unknowns, and how many independent unknowns

Answer 30

6 equations, 9 unknowns, 6 independent unknowns (where the quantity is stress)

Question 31

Constitutive equations are

Answer 31

Systems that required relating stress and strain. They are dependent on the material composition (constitution) and will determine how the material will deform (related to strain) under specific loading conditions (related to stress)

Question 32

What do constitutive equations describe

Answer 32

The behavior of the material, NOT the structural reasons for the behavior. This means they do not describe the micro structural basis for constitutive behavior.

Question 33

The strain energy density function is what and how is it used?

Answer 33

The amount of work done per unit volume to deform a material from a stress free state to a loaded state. It is solely a function of strain, but when differentiated with respect to strain, relates stress and strain.

Question 34

Where does the stiffness matrix come from and how many independent terms does it have

Answer 34

The stiffness matrix is a 4th order tensor combining the stress, strain and strain energy tensors. It has 21 independent terms.

Question 35

The first quadrant of the stiffness matrix relates

Answer 35

Normal stress to shear strain

Question 36

The second quadrant of the stiffness matrix relates

Answer 36

Normal stress to normal strain

Question 37

The third quadrant of the stiffness matrix relates

Answer 37

Shear stress to normal strain

Question 38

The fourth quadrant of the stiffness matrix relates

Answer 38

Shear stress to shear strain

Question 39

A fully isotropic material can be fully defined by how many independent constants

Answer 39

2

Question 40

A transversely isotropic material can be fully defined by how many independent constants

Answer 40

5

Question 41

An orthotropic material can be fully defined by how many independent constants

Answer 41

9

Question 42

A constitutive equation has how many equations, how many unknowns and how many independent unknowns

Answer 42

9 equations, 0 unknowns, 0 independent unknowns

Question 43

Stress in an object under complex loading can be found

Answer 43

By superimposing the stresses from bending, compression and torsion.