Chapter 2 Stress and Strain -- Axial LoadingStatics – deals with undeformable bodies (Rigid bodies)Mechanics of Materials – deals with deformable bodies -- Need to know the deformation of a boy under various stress/strain state -- Allowing us to computer forces for statically indeterminate problems.
The following subjects will be discussed:• Stress-Strain Diagrams• Modulus of Elasticity• Brittle vs Ductile Fracture• Elastic vs Plastic Deformation• Bulk Modulus and Modulus of Rigidity• Isotropic vs Orthotropic Properties• Stress Concentrations• Residual Stresses
2.2 Normal Strain under Axial Loading δ ε = normal strain = LFor variable cross-sectional area A,strain at Point Q is: ∆δ d δ ε = lim = ∆x → 0 ∆x dx The normal Strain is dimensionless.
Some Important Concepts and Terminology: 1. Elastic Modulus 2. Yield Strength – lower and upper Y.S. -- σ y 0.2% Yield Strength 3. Ultimate Strength, σ ut 4. Breaking Strength or Fracture Strength 5. Necking 6. Reduction in Area 7. Toughness – the area under the σ-ε curve 8. Percent Elongation 9. Proportional Limit
2.3 Stress-Strain Diagram LB − Lo Percent elongation = 100% Lo A0 − ABPercent reduction in area = 100% Ao
2.4 True Stress and True StrainEng. Stress = P/Ao True Stress = P/A Ao = original area A = instantaneous area δEng. Strain = True Strain = ε t = Σ∆ε = Σ( ∆L / L) Lo Lo = original length L = instantaneous length dL L Lεt = ∫ = ln (2.3) Lo L Lo
2.5 Hookes Law: Modulus of Elasticity σ = Eε (2.4) Where E = modulus of elasticity or Young’s modulusIsotropic = material properties do not vary withAnisotropic = material properties vary with direction ordirection or orientation.E.g.: metals E.g.: wood, compositesorientation.
2.6 Elastic Versus Plastic Behavior of a Material 2
Some Important Concepts: 1. Recoverable Strain 2. Permanent Strain – Plastic Strain 3. Creep 4. Bauschinger Effect: the early yielding behavior in the compressive loading
2.7 Repeated Loadings: FatigueFatigue failure generally occurs at a stress level that is muchlower than σ y The σ -N curve = stress vs life curveThe Endurance Limit = the stress for which fatigue failure does not occur.
2.8 Deformations of Members under Axial Loading σ = Eε (2.4) σ P ε = = (2.5) E AE δ = εL (2.6) PL δ = (For Homogeneous rods) AE Pi Li δ = ∑ i A Ei (For various-section rods) i Pdx d δ = ε dx = (For variable cross-section rods) AEP
L Pdx ∫ (2.9) δ = o AE PLδ B/ A = δB − δA = (2.10) AE
2.9 Statically Indeterminate ProblemsA. Statically Determinate Problems: -- Problems that can be solved by Statics, i.e. ΣF = 0 and ΣM = 0 & the FBDB. Statically Indeterminate Problems: -- Problems that cannot be solved by Statics -- The number of unknowns > the number of equations -- Must involve “deformation”Example 2.02:
Superposition Method for StaticallyIndeterminate Problems 1. Designate one support as redundant support 2. Remove the support from the structure & treat it as an unknown load. 3. Superpose the displacement Example 2.04
2.10 Problems Involving Temperature Changes δ T = α ( ∆T ) L 2(.21) α = coefficient of thermal expansion δT + δP = 0 ε T = α∆T δ T = α ( ∆T ) L PL δP = AE PL δ = δ T + δ P = α ( ∆T ) L + =0 AE
Therefore: P = − AEα ( ∆T ) P σ = = − Eα ( ∆T ) A
2.11 Poisson s Ratio εx =σx / E lateral strainυ = Poisson s Ratio = − axial strain εy εz υ= − = − εx εx σ υσ ε = X ε =ε = − X x E y z E
2.12 Multiaxial Loading: Generalized Hookes Law • Cubic → rectangular parallelepiped • Principle of Superposition: -- The combined effect = Σ (individual effect) Binding assumptions: 1. Each effect is linear 2. The deformation is small and does not change the overall condition of the body.
2.12 Multiaxial Loading: Generalized Hookes LawGeneralized Hooke’s Law σ x υσ y υσ z εx = + − − E E E υσ x σ y υσ z εy = − + − (2.28) E E E υσ x υσ y σ z εz = − − + E E E Homogeneous Material -- has identical properties at all points. Isotropic Material -- material properties do not vary with direction or orientation.
2.13 Dilation: Bulk ModulusOriginal volume = 1 x 1 x 1 = 1Under the multiaxial stress: σ x, σ y, σ zThe new volume = υ = (1 + ε x )(1 + ε y )(1 + ε z )Neglecting the high order terms yields: υ =1+ εx + ε y + εz e = the hange of olume = υ − 1 = 1 + ε x + ε y + ε z − 1 ∴e = ε x + ε y + εz ( 2.30)
e = dilation = volume strain = change in volume/unit volume Eq. (2.28) → Eq. (2-30) σ X + σy + σz 2υ (σ X + σ y + σ z )e = − (2.31) E E 1 − 2υe= (σ X + σ y + σ z ) ESpecial case: hydrostatic pressure -- σx, σy, σz = p 3(1 − 2υ ) E e= − p Define: κ = (2.33) E 3(1 − 2υ ) p e= − (2.33) κ κ = bulk modulus = modulus of compression +
ESince κ = positive, κ= 3(1 − 2υ ) (1 - 2υ) > 0 1>2υ υ <½ Therefore, 0 < υ < ½ 3 E υ= 0 e= − E p κ= 3 3(1 − 2υ ) κ =∞ e=0 υ =½ e= − E p =0 -- Perfectly incompressible materials
2.14 Shearing Strain If shear stresses are present Shear Strain = γ xy (In radians) τ xy = G γ xy (2.36) τ yz = G γ yz τ zx = G γ zx (2.37)
The Generalized Hooke’s Law: σ X υσ y υσ z εx = + − − E E E υσ X σ y υσ z εy = − + − E E E υσ X υσ y σ z εz = − − + E E E τ xy τ yz τ zx γ xy = γ yz = γ zx = G G G
2.18 Further Discussion of Deformation under Axial Loading: Relation Among E, υ, and G E =1+υ 2G E G= 2(1 + υ )
Saint-Venant’s Principle: -- the localized effects caused by any load acting on the body will dissipate or smooth out within region that are sufficiently removed form the location of he load.
2.16 Stress-Strain Relationships for Fiber-Reinforced Composite Materials -- orthotropic materials εy εz υ xy = − and υ xz = − εx εx σ X υ xyσ y υ zxσ z εx = + − − Ex Ey Ez υ xyσ X σ y υ zxσ z εy = − + − Ex Ey Ez υ xyσ X υ yzσ y σ z εz = − − + Ex Ey Ez
υ xy υ yx υ yz υ zy υ zx υ xz = = =E x E y E y Ez Ez E x τ xy τ yz τ zxγ xy = γ yz = γ zx = G G G
2.17 Stress and Strain Distribution Under Axial Loading: Saint-Venants Principle If the stress distribution is uniform: P σ y = (σ y )ave = A In reality:
2.18 Stress Concentrations -- Stress raiser at locations where geometric discontinuity occurs σ max K= = Stress Concentration Factor σ ave
2.19 Plastic Deformation Elastic Deformation → Plastic Deformation →Elastoplastic behavior σ σy Y C Rupture ε A D
For σ max < σ Y σ max σ max K= σ ave = σ ave K σ max A P = σ ave A = KFor σ max = σ Y σY A PY = KFor σ ave = σ Y PU = σ Y A PU PY = K
2.20 Residual Stresses After the applied load is removed, some stresses may still remain inside the material → Residual Stresses