IA: 1P2: Materials Flashcards
What is elastic stiffness?
S = F/δ
S = stiffness of system
F = force applied to system
δ = resulting displacement due to the applied force
What is a materials strength?
The maximum stress at the elastic limit, σ = σₘₐₓ
What is a stiffness-limited design?
Where the elastic deflection remains within limits: δ < δₘₐₓ
What is a strength-limited (elastic) design?
Where the material everywhere remains elastic: σₘₐₓ < elastic limit (yield or fracture)
How can a materials strength be used in plastic design?
material yields plastically: σₘₐₓ > elastic limit (yield)
What are 4 ways of measuring the Young’s modulus of a material?
- Tensile testing
- Bending stiffness of a beam
- Natural frequency of vibration (beams, plates)
- Speed of sound in the material, √(E/ρ)
What is true in uniaxial tension?
- The material gets longer and thinner
- The laterial contraction and the tensile extension are proportional
What is Poisson’s ratio, ν?
ν = Greek letter “nu”, not “v”
Note: minus sign so that ν is positive
What is important to remember about lateral strain in uniaxial tension?
It is NOT due to volume conservation (volume is not conserved!), rather it reflects the way atomic bonds deform under load
When does poissons ratio matter?
- NOT important in most designs with uniaxial loads (e.g. members in a truss: fractional change in area)
- Important when the stress state is 2D or 3D
- Important in vibration of plates
- Important in large strain bending, giving anticlastic curvature
- Important in the design of materials with unconventional functionalities (e.g. Auxetic materials have a negative poissons ratio)
What is uniaxial stress?
Uniaxial stress refers to a condition where a material is subjected to stress in only one direction, typically along a single axis. In this case, the stress is applied in one dimension, and there is no stress in the other two perpendicular directions. For example, the stresses in a cable or a strut
What is multiaxial stress?
Multiaxial stress refers to a condition where a material is subjected to stress in multiple directions, typically along more than one axis. Unlike uniaxial stress, where stress is applied in just one direction, multiaxial stress involves simultaneous stresses along two or three perpendicular directions (or a combination of them). For example, the stresses in a beam, arch, or pressure vessel
In how many dimensions does strain act?
Always 3! Strain is always 3D due to Poisson’s ratio
If a body is being acted on by 3 normal stresses (multiaxial stress), how can you determine the strain acting in each direction?
Apply each strain in turnm and then use superposition to determine the resulting strains
Determine the expression for the strain acting in each direction when being acted upon by the following orthogonal stresses:
What is dilation?
Volumetric strain
What is the equation for volumetric strain/dilation?
For a unit cube with a general strain state (ε₁ ε₂ ε₃), what is the expression for dilation?
What is hydrostatic stress?
When all three normal stresses are equal, e.g. under uniform external pressure p:
σ₁ = σ₂ = σ₃ = -p
What is the bulk modulus, K?
It is the ratio of hydrostatic stress to volumetric strain (dilation)
What is the equation for bulk modulus, K?
ν = poissons ratio
What is normal stress?
Force per unit area carried perpendicular to a plane within the material
What is shear stress?
Force per unit area carried parallel to a plane within the material
What is the symbol used for shear stress?
τ, tau
What must be true about the internal forces and stresses for equilibrium to occur?
The shaded surface must carry components of force and stress both normal and parallel to the surface:
* The forces may be related by resolving, however this is NOT true for the stresses as stresses on an inclined plane act on a different area to the axial stress
Shear stresses arise in almost every loading situation, but may be off-axis from the obvious orientations chosen
What is shear strain, γ?
Shear stress is defined as the change in angle between two initially perpendicular lines (or planes) in the material due to applied shear stres. This is because shear stress distorts the shape of a volume element, rather than changing its axial dimension.
It is represented by γ
What is the equation for shear strain, γ?
Technically is is tan(γ), however for small strains tan(γ) = γ
What is the shear modulus, G?
It is the ratio of shear stress to shear strain, it characterises the elastic stiffness in shear:
What is the relationship between shear modulus (G) and Young’s modulus (E)?
ν = Poisson’s ratio
