strain energy Flashcards
modulus of toughness and resilience
toughness:
- the amount of energy a material can withstand before breaking
u = ∫ σ dϵ (integrate from 0 to point of failure)
resilience:
- the amount of energy a material can withstand before permanent deformation
u = σ^2 / 2E
*stress (σ)
*strain (ϵ)
*youngs modulus (E)
strain energy (U) when elastic deformation is linear (gradually applied within elastic limit)
U = 1/2 Px (area under P x graph)
*load (P)
*deformation (x)
strain energy density (u)
= U / V
= ∫ σ dϵ
Measures the energy stored per unit volume of material up to a specific point of strain
*strain energy (U)
*volume (V)
*stress (σ)
*strain (ϵ)
strain energy for a uniform rod
U = P^2L / 2AE
*length (L)
*load (P)
*cross-sectional area (A)
*youngs modulus (E)
strain energy (U) for a nonuniform rod
U = ∫ u dV
*strain energy density (u)
*volume (V)
if strain energy u is within proportional limit u calculated with modulus of resilience
strain energy for a beam subjecting to a bending load
U = ∫ M^2 / 2EI dx (from L to 0)
*bending moment (M)
*youngs modulus (E)
*second moment of area (I)
*length (L)
strain energy for an end-loaded cantilever beam
U = P^2 L^3 / 6EI
*load (P)
*youngs modulus (E)
*second moment of area (I)
*length (L)
strain energy for a material subjected to plane shearing stresses
U = ∫ T^2 / 2GJ dx
*torque (T)
*modulus of rigidity (G)
*polar second moment of area (J)
(within proportional limit)
shear stress in a shaft subjected to a torsional load
τ = Tρ / J
*torque (T)
*density (ρ)
*polar second moment of area (J)
polar second moment of area (J)
J = ∫ r^2 dA
*radius (r)
*area (A)
strain energy for a uniform shaft subjected to a torsional load
U = T^2 L / 2 G J
*torque (T)
*modulus of rigidity (G)
*polar second moment of area (J)
*length (L)
strain energy for a beam subjecting to a bending load
U = M^2 L / 2EI
*bending moment (M)
*youngs modulus (E)
*second moment of area (I)
*length (L)
