Solid Mechanics

Question 1

What is extensional strain?

Answer 1

Change in length/original length

ABS[(dx-dX)/(dX)] = λ - l = ϵ_n

Question 2

When does extensional strain apply to solid mechanics problems?

Answer 2

Infinitesimal strain tensor relations

Question 3

What is hydrostatic stress? What does it imply?

Answer 3

Hydrostatic stress is pressure exerted by state of stress in principal directions.
-Hydrostatic stress state only is when pressure only acts in principal axes of element ΔV.

Implies S = 0, such that σ = P = -1/3 tr(σij )= -1/3 σkk
*σ11 = σ22 = σ33

Question 4

What is deviatoric stress? What does it imply?

Answer 4

Deviatoric stress is a measure of shearing exerted by states of stress in shearing direction
-Deviatoric stress state is solid solid distortion with no ΔV when pressure only acts in shearing directions

Implies S = σij + pI where all ij are cross terms.

Question 5

What are the Von-Mises stresses?

Answer 5

Von mises stresses are the uniaxial equivalent of multi-axial stress state. Used for failure/yield criterial in “uniaxial tensile tests”

σ_VM= √((σ_1-σ_2 )^2+(σ_1-σ_3 )^2+(σ_2-σ_3 )^2 )

Question 6

What is the process you use in BVP to get displacements from stresses and constant potential?

Answer 6

Using the biharmonic equation to relate stress distribution from a constant potential field, we know using the constitutive equations we get strain. To get displacement from this we need to confirm strain is both compatible and integrable using compatibility equation to check we can integrate strain to get displacement.

Question 7

What is the finite strain tensor called? equation?

Answer 7

Lagrangian Strain Tensor - Nonlinear finite strain [E]
E = 1/2 ( [dx^2 - dX^2]/dX^2) = 1/2(λ^2-1) = 1/2[NCN-1]

diagonals of E are normal strains along fibers if basis vectors aligned with N.

Question 8

What are infinitesimal deformations? What assumptions are made?

Answer 8

Infinitesimal deformations are small rotations @ finite small angles, with infinitely small displacements

Assume small shape changes:
dui/dxj «1

Question 9

What is Linear Elasticity? What are the assumptions of linear elasticity?

Answer 9

Linear elasticity is when stresses are linearly proportional to strain - linearization of constitutive behaviors.

Assumptions:

1. Small strains (small shape and angles)
2. Indistinguishable between reference and deformed configurations (dx ~= dX, σ(x1,x2,x3) ~= σ(X1,X2,X3)

Question 10

What is isotropic mean?

Answer 10

Isotropic means a material strain curve is independent of rotation and orientation. Means only two material constants, identical in all directions.

**Relations hold for all rotations Q s.t. Q^(T) = Q^(-1), det(Q) =1

Question 11

What is an orthotropic material?

Answer 11

Material with properties which differ along 3-mutually orthogonal axes.
*Properties which change when measured from different directions, 3 properties in each direction (9 properties total)

Question 12

What is an anistropic material?

Answer 12

Materials which properties change with direction along axes its measured (Orthotropic is a subset of anistropic)

Question 13

What is strain energy?

Answer 13

Strain energy is a volumetric integration of strain energy density over a subbody, that is a function of strain energy due to displacement field.

Question 14

What is internal energy density?

Answer 14

Equal symmetric portion of strain energy

Question 15

What assumptions can we make about a material that define strain energy?

Answer 15

Due to balance of angular momentum definiton of symmetric stress and strain tensors, we can assume strain energy is quadratic
W = 1/2∫σ∶ϵ dv= 1/2∫Eϵ∶ϵ dv

Question 16

What is the Cauchy Stress Principle?

Answer 16

Material interactions across an internal surface in a body can be described as a distribution of tractions in the same way that the effect of external forces on physical surfaces of the body are described.

Question 17

What do we use the Cauchy Stress Principle for?

Answer 17

To obtain an expression for traction terms tau (t) to be defined for an arbitrary subbody as a state of stress. It tells us we can relate the internal forces seen acting on the outer surface as a function of stresses over an area which act as a force on a subbody.

*USED IN Bal of Lin-Momentum to convert tractions to stress divergence over area

Question 18

What is Cauchy’s relation?

Answer 18

t_i(n) = σ_ij*n_j

Question 19

What are the lagrangian and eulerian descriptions of particle mechanics?

Answer 19

Lagrangian [material] describes how a quantity changes following each particle in time from reference configuration

Eulerian [spatial] describes how a quantity (control volume) changes at a particular location in space/time regardless of reference location.

Question 20

What are the lame constants? (describe each one, and units)

Answer 20

E - young’s modulus [N/m^2]: Describes the material relation between stress-strain

ν(nu) - Poisson’s ratio [dx/dX]: Describes the ratio of lateral to longitudinal strain in uniaxial tensile stress.

k - Bulk Modulus [N/m^2]: Quantifies resistance of the solid to changes in volume.

μ (mu) - Shear Modulus [N/m^2]: Describes the resistance of material to volume preserving shear deformations.

λ - Lame modulus [ N/m^2]: Describes material elasticity w.r.t. both E and ν, along principal directions.

Question 21

Define conservation of mass, what property does it given us?

Answer 21

Mass of any subbody E0 in reference configuration must remain unchanged after deformation to E.

m0 (E0) = m(E), leads to local form of mass conservation:

ρJ = ρ0.

Question 22

Define Balance of Linear Momentum, what properties does this give us?

Answer 22

Conservation of linear momentum defines the total sum of external forces must equal the time rate of change of linear momentum.

Given definition of body and surface (traction) forces it is true we can define a state of equilibrium that satisfies conservation of linear momentum:

div(σij) +ρb - ρa = 0

Question 23

What properties of conservation and principles must be used to prove local form of conservation of linear momentum?

Answer 23

1. Conservation of mass

2. Cauchy Stress Principle

Question 24

Define the Balance of Angular Momentum, what properties of materials does it define?

Answer 24

Balance of Angular Momentum of a system tells us the total external forcing moment about a point must equal the angular momentum of the system about that point.

Given the definitions of both balance of lin-momentum, cons. of mass the balance of angular momentum defines symmetry of our stress and strain tensors.

Question 25

What is the difference between General and Engineering strains?

Answer 25

General strain E: Diagonal Eii are "direct" strains Engineering strain gamma: its relationship is = 2*ϵ. Larger than standard shearing strains.

Question 26

Define stretch, and what does magnitude/sign tell us?

Answer 26

Stretch is the change in length in deformed configuration over the original change in length λ = [ds]/[dS] = sqrt(Cij Nj Nk) Magnitude/Sign: <1 Compression >1 Tension =1 deformation

Question 27

What is Levy's Thrm? What assumptions are made?

Answer 27

Levi's Thrm: The in-plane components of stress depend only on applied boundary tractions and geometry. It is INDEPENDENT of material properties in BVP. Assumptions: - Simply connected domains - No displacement B.C's (only traction) - No holes, cuts - Isostatic Structure

Question 28

What does a symmetric stress tensor imply?

Answer 28

1. Symmetries reduce σ to only 6 components (all cross components equal) 2. Symmetry implies 3 REAL eigen values (principal stresses σ1>σ2>σ3) 3. Can always find an orthogonal basis consisting of entirely eigen values (basis with all planes normal, purely normal tractions) 4. Eigenbasis = diagonal matrix

Question 29

What are the constraints of continuum mechanics w.r.t. conservation of angular momentum? What does each imply?

Answer 29

``` 1. Stress, Strain are symmetric IMPLIES: Minor symmetry Reduces material constants matrix C from 81 constants to 36. 2. Strain energy density is quadratic IMPLIES: Major symmetry Further reduced C matrix from 36 to 21 indep. components 3. Material Symmetries in Structure IMPLIES: Any material symmetries further reduce the C matrix *Ortho - 3 planes (9 indep const) *Cubic (3 const) *Isotropic (1 const) *Rotational Symmetry ```

Question 30

What types of symmetries are there?

Answer 30

Material, Geometric

Question 31

What does Cauchy's tetrahedron and principles of moment/force equilibrium tell us?

Answer 31

It implies/expresses that once the stress components acting on three mutually orthogonal faces are know, the stress components on a face of arbitrary orientation can be computed.

Question 32

How much information is required to fully define the state of stress at a point on a solid? What assumption is made about the known information?

Answer 32

Once stress vectors acting on three mutually orthogonal faces are known - The complete definition of stress state at any point only requires 6 (if assuming symmetry) or 9 components of stress tensor.

Question 33

What are the solutions to solve non-trivial stress invariants?

Answer 33

Principal Stresses

Question 34

Why are stress invariants important?

Answer 34

Because the quantities of I_1, I_2, I_3, are solutions to the quantities invariant with respect to change of coordinate system. (They never change, so the stresses can be computed in any coordinate system)

Question 35

What does plane state of stress define? What do we assume?

Answer 35

Plane state of stress defines a system where all stress components acting along one axis 'i_n' are assumed to vanish or be negligible compared to the stress components acting in the other two directions. It means we assume the remaining stress components are independent of that axis dimension x_n

Question 36

What does the displacement field describe in a solid?

Answer 36

Describes the displacement of a point at position (x1, x2,x3) within the solid consisting of two parts; Rigid body motion, and a deformation straining of the solid.

Question 37

What does strain-displacement equation describe?

Answer 37

The strain displacement equations extract from the displacement field the information that describes only the deformation in the body while ignoring its rigid body motion Describes strain through relative elongations and angular distortions of material lines

Question 38

What are relative elongations and angular distortions?

Answer 38

Relative elongations are direct strains/axial strains | Angular distortions are shearing strains

Question 39

What does plane state of strain define? What do we assume?

Answer 39

Plane state of strain defines a system where the displacement component along the direction of axis 'i_n' is assumed to vanish/to be negligible compared to the displacement components in the other two directions. It means we assume the remaining strains components are independent of that axis dimension x_n

Question 40

Does the deformation power identity theorem have material dependence?

Answer 40

No, it only requires 3 balance laws to be satisfied. - only dependent on material IF specify stress (σ = Cϵ) then it would be assumed for linearly elastic (HENCE infinitesimal deformations)

Question 41

What is the general Rayleigh Ritz & FEM process (same for linear elasticity)?

Answer 41

- Define nodes of system and their boundaries. - Regions between nodes define elements - Assume displacement functional form family, and interpolate solution in all elements - Formulate potential as Pi = strain energy - work done by body forces/external forces

Question 42

What is a constitutive model?

Answer 42

Set of equations that relates stresses to: - Strains - Strain rates - Strain History

Question 43

List the Constitutive Model Requirements

Answer 43

1. Follows Laws of Thermodynamics: - Energy Balance - Cyclic deformation ∆W≥0 2. Objectivity/Material frame independence: - Tensor valued functions relate stress to deformation measure must transform correctly under change of basis/origin

Question 44

What is so special about isotropic materials?

Answer 44

- Only 2 material constants λ,μ - Normal strains cause only normal stresses - Shear strains lead to shear stresses - Eigendirections of ϵ,σ are same

Question 45

What is the difference between homogeneous and heterogeneous isotropic elastic properties?

Answer 45

REMEBER: isotropic = 2 material constants λ,μ Homogeneous means λ,μ = constant everywhere in boxy Heterogeneous means λ(x) and μ(x), cross-section material properties change w.r.t for x = 0, L (length)

Question 46

What is work conjugacy?

Answer 46

We say the stress and symmetric tensor of strain are work conjugates because deformation power identity uses time rate of change of work to establish conjugacy relationship with stress and rate of deformation. "The measure of stress multiplied by the time-derivative of strain leads to the rate of work done" = stress:symmetric portion of strain (b.c. major symmetry)