Solid Mechanics Flashcards
What is extensional strain?
Change in length/original length
ABS[(dx-dX)/(dX)] = λ - l = ϵ_n
When does extensional strain apply to solid mechanics problems?
Infinitesimal strain tensor relations
What is hydrostatic stress? What does it imply?
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
What is deviatoric stress? What does it imply?
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.
What are the Von-Mises stresses?
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 )
What is the process you use in BVP to get displacements from stresses and constant potential?
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.
What is the finite strain tensor called? equation?
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.
What are infinitesimal deformations? What assumptions are made?
Infinitesimal deformations are small rotations @ finite small angles, with infinitely small displacements
Assume small shape changes:
dui/dxj «1
What is Linear Elasticity? What are the assumptions of linear elasticity?
Linear elasticity is when stresses are linearly proportional to strain - linearization of constitutive behaviors.
Assumptions:
- Small strains (small shape and angles)
- Indistinguishable between reference and deformed configurations (dx ~= dX, σ(x1,x2,x3) ~= σ(X1,X2,X3)
What is isotropic mean?
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
What is an orthotropic material?
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)
What is an anistropic material?
Materials which properties change with direction along axes its measured (Orthotropic is a subset of anistropic)
What is strain energy?
Strain energy is a volumetric integration of strain energy density over a subbody, that is a function of strain energy due to displacement field.
What is internal energy density?
Equal symmetric portion of strain energy
What assumptions can we make about a material that define strain energy?
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
What is the Cauchy Stress Principle?
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.
What do we use the Cauchy Stress Principle for?
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
What is Cauchy’s relation?
t_i(n) = σ_ij*n_j
What are the lagrangian and eulerian descriptions of particle mechanics?
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.
What are the lame constants? (describe each one, and units)
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.
Define conservation of mass, what property does it given us?
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.
Define Balance of Linear Momentum, what properties does this give us?
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
What properties of conservation and principles must be used to prove local form of conservation of linear momentum?
- Conservation of mass
2. Cauchy Stress Principle
Define the Balance of Angular Momentum, what properties of materials does it define?
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.
What is the difference between General and Engineering strains?
General strain E: Diagonal Eii are “direct” strains
Engineering strain gamma: its relationship is = 2*ϵ. Larger than standard shearing strains.
Define stretch, and what does magnitude/sign tell us?
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
What is Levy’s Thrm? What assumptions are made?
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
What does a symmetric stress tensor imply?
- Symmetries reduce σ to only 6 components (all cross components equal)
- Symmetry implies 3 REAL eigen values (principal stresses σ1>σ2>σ3)
- Can always find an orthogonal basis consisting of entirely eigen values (basis with all planes normal, purely normal tractions)
- Eigenbasis = diagonal matrix
What are the constraints of continuum mechanics w.r.t. conservation of angular momentum? What does each imply?
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
What types of symmetries are there?
Material, Geometric
What does Cauchy’s tetrahedron and principles of moment/force equilibrium tell us?
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.
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?
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.
What are the solutions to solve non-trivial stress invariants?
Principal Stresses
Why are stress invariants important?
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)
What does plane state of stress define? What do we assume?
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
What does the displacement field describe in a solid?
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.
What does strain-displacement equation describe?
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
What are relative elongations and angular distortions?
Relative elongations are direct strains/axial strains
Angular distortions are shearing strains
What does plane state of strain define? What do we assume?
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
Does the deformation power identity theorem have material dependence?
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)
What is the general Rayleigh Ritz & FEM process (same for linear elasticity)?
- 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
What is a constitutive model?
Set of equations that relates stresses to:
- Strains
- Strain rates
- Strain History
List the Constitutive Model Requirements
- Follows Laws of Thermodynamics:
- Energy Balance
- Cyclic deformation ∆W≥0 - Objectivity/Material frame independence:
- Tensor valued functions relate stress to deformation measure must transform correctly under change of basis/origin
What is so special about isotropic materials?
- Only 2 material constants λ,μ
- Normal strains cause only normal stresses
- Shear strains lead to shear stresses
- Eigendirections of ϵ,σ are same
What is the difference between homogeneous and heterogeneous isotropic elastic properties?
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)
What is work conjugacy?
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)
