ASEN 3112 - Structures
5 Stress-Strain Material Laws
ASEN 3112 Lecture 5 – Slide 1
ASEN 3112 - Structures
Strains and Stresses are Connected by Material Properties of the Body (Structure) Recall the connections displayed in previous lecture: MP internal forces ⇒ stresses ⇒ strains ⇒ displacements ⇒ size&shape changes
MP displacements ⇒ strains ⇒ stresses ⇒ internal forces The linkage between stresses and strains is done through material properties, as shown by symbol MP over red arrow Those are mathematically expressed as constitutive equations
Historically the first C.E. was Hooke's elasticity law, stated in 1660 as "ut tensio sic vis" Since then recast in terms of stresses and strains, which are more modern concepts.
ASEN 3112 Lecture 5 – Slide 2
ASEN 3112 - Structures
Assumptions Used In This Course As Regards Material Properties & Constitutive Equations Macromodel
material is modeled as a continuum body; finer scale levels (crystals, molecules, atoms) are ignored
Elasticity
stress-strain response is reversible and has a preferred natural state, which is unstressed & undeformed
Linearity
relationship beteen strains and stresses is linear
Isotropy
properties of material are independent of direction
Small strains
deformations are so small that changes of geometry are neglected as loads (or temperature changes) are applied
For additional explanations see Lecture notes.
ASEN 3112 Lecture 5 – Slide 3
ASEN 3112 - Structures
The Tension Test Revisited: Response Regions for Mild Steel Nominal stress σ = P/A 0
Max nominal stress
Strain hardening
Localization
Yield Elastic limit Linear elastic behavior (Hooke's law is valid over this response region) Undeformed state
Nominal failure stress
Mild Steel Tension Test
P
L0
gage length
Nominal strain ε = ∆L /L0
ASEN 3112 Lecture 5 – Slide 4
P
ASEN 3112 - Structures
Other Tension Test Response Flavors
Brittle (glass, ceramics, concrete in tension)
Moderately ductile (Al alloy)
ASEN 3112 Lecture 5 – Slide 5
Nonlinear from start (rubber, polymers)
ASEN 3112 - Structures
Tension Test Responses of Different Steel Grades Nominal stress σ = P/A0 Tool steel
Note similar elastic modulus
High strength steel
Mild steel (highly ductile) Conspicuous yield Nominal strain ε = ∆ L /L 0
ASEN 3112 Lecture 5 – Slide 6
ASEN 3112 - Structures
Material Properties For A Linearly Elastic Isotropic Body E
Elastic modulus, a.k.a. Young's modulus Physical dimension: stress=force/area (e.g. ksi)
ν
Poisson's ratio Physical dimension: dimensionless (just a number)
G
Shear modulus, a.k.a. modulus of rigidity Physical dimension: stress=force/area (e.g. MPa)
α
Coefficient of thermal expansion Physical dimension: 1/degree (e.g., 1/ C)
E, ν and G are not independent. They are linked by E = 2G (1+ν),
G = E/(2(1+ν)),
ν= E /(2G)−1
ASEN 3112 Lecture 5 – Slide 7
ASEN 3112 - Structures
State of Stress and Strain In Tension Test Cross section (often circular) of area A
(a) P
P
gaged length
σ xx = P/A (uniform over cross section) y z
(b) P Stress state
σxx 0 0
0 0 0
0 0 0
Strain state
x Cartesian axes
εxx
0 0 at all points in the gaged region
For isotropic material, εyy = εzz
0
εyy 0
0 0
εzz
is called the lateral strain
ASEN 3112 Lecture 5 – Slide 8
ASEN 3112 - Structures
Defining Elastic Modulus and Poisson's Ratio Isotropic material properties E and ν are obtained from the linear elastic response region of the uniaxial tension test (last slide). For simplicity call σ = σ xx = axial stress, ε = ε xx = axial strain,
εyy = ε zz = lateral strain
The elastic modulus E is defined as the ratio of axial stress to axial strain: def
E =
σ ε
whence σ = E ε, ε =
σ E
Poisson's ratio ν is defined as the ratio of lateral strain to axial strain: def
ν =
lateral strain axial strain
=−
lateral strain axial strain
The − sign in the definition of ν is introduced so that it comes out positive. For structural materials ν lies in the range [0,1/2). For most metals (and their alloys) ν is in the range 0.25 to 0.35. For concrete and ceramics, ν ≈ 0.10. For cork ν ≈ 0. For rubber ν ≈ 0.5 to 3 places. A material for which ν = 0.5 is called incompressible. If ν is very close to 0.5, it is called nearly incompressible.
ASEN 3112 Lecture 5 – Slide 9
ASEN 3112 - Structures
; ;
Which material would work best for capping a wine bottle? Rubber
Cork
ASEN 3112 Lecture 5 – Slide 10
ASEN 3112 - Structures
Which material would work best for a shock absorber?
Rubber
Foam
ASEN 3112 Lecture 5 – Slide 11
;yy; y;
ASEN 3112 - Structures
State of Stress and Strain In Torsion Test Circular cross section
(a) T
T gaged length For distribution of shear stresses and strains over the cross section, cf. Lecture 7 y z x Cartesian axes
(b) T
Stress state
0
τyx
τxy 0 0
0 0 0
Strain state
0
γyx
γxy
0 0 0
0 0 0 0 at all points in the gaged region. Both the shear stress τ yx = τ xy as well as the shear strain γ xy = γyx vary linearly as per distance from the cross section center (Lecture 7). They attain maximum values on the max specimen surface. For simplicity, call those values τ = τ max xy and γ = γ xy
ASEN 3112 Lecture 5 – Slide 12
ASEN 3112 - Structures
Defining Shear Modulus Of An Isotropic Linearly Elastic Material Isotropic material property G (the shear modulus, also called modulus of rigidity) is obtained from the linear elastic response region of the torsion test of a circular cross section specimen (last slide). For simplicity call max τ = τ xy = max shear stress on specimen surface over gauged region max γ = γ xy = max shear strain on specimen surface over gauged region
The shear modulus G is defined as the ratio of the foregoing shear stress and strain: def
G =
τ γ
whence τ = G γ, γ =
τ G
ASEN 3112 Lecture 5 – Slide 13
ASEN 3112 - Structures
Defining The Coefficient of Thermal Expansion Of An Isotropic Material Take a standard uniaxial test specimen:
x gaged length At the reference temperature T0 (usually the room temperature) the gaged length is L 0 . Heat the unloaded specimen by ∆T while allowing it to expand freely in all directions. The gaged length changes to L = L 0 + ∆L. The coefficient of thermal expansion is defined as def
α =
∆L L0 ∆T
whence ∆L = α L 0 ∆T
T The ratio ε T = εxx = ∆L /L 0 = α ∆T is called the thermal strain in the axial (x) direction. For an isotropic material, the material expands equally in T all directions: ε xx = εTyy = εzzT , whereas the thermal shear strains are zero.
ASEN 3112 Lecture 5 – Slide 14
ASEN 3112 - Structures
1D Hooke's Law Including Thermal Effects Stress To Strain:
ε=
σ + α ∆T = εM + εT E
expresses that total strain = mechanical strain + thermal strain: the strain superposition principle Strain To Stress:
σ = E ( ε − α ∆T ) A problem in Recitation 3 uses this form
ASEN 3112 Lecture 5 – Slide 15
ASEN 3112 - Structures
3D Generalized Hooke's Law (1) Stresses To Strains (Omitting Thermal Effects)
1 E x x ν − yy E ν zz − E = γx y 0 γ yz 0 γ zx
0
ν −E 1 E ν −E
ν −E ν −E 1 E
0
0
0 0
0
0
0
0
0
0
0
0
0
0
0 1 G 0
0
0
0 1 G 0
0 1 G
For derivation using the strain superposition principle, as well as inclusion of thermal effects, see Lecture notes
ASEN 3112 Lecture 5 – Slide 16
σx x σ yy σzz τx y τ yz τzx
.
ASEN 3112 - Structures
3D Generalized Hooke's Law (2) Strains To Stresses (Omitting Thermal Effects)
ˆ σx x E (1 − ν) σ yy Eˆ ν σzz Eˆ ν = 0 τx y 0 τ yz 0 τzx in which
Eˆ ν Eˆ (1 − ν) Eˆ ν 0 0 0 Eˆ =
Eˆ ν Eˆ ν Eˆ (1 − ν) 0 0 0
0 0 0 G 0 0
0 0 0 0 G 0
x x 0 0 yy 0 zz 0 γx y 0 γ yz G γzx
E (1 − 2ν)(1 + ν)
This is derived by inverting the matrix of previous slide. For the inclusion of thermal effects, see Lecture notes
ASEN 3112 Lecture 5 – Slide 17
ASEN 3112 - Structures
2D Plane Stress Specialization
Stresses
σx x τ yx 0
τx y σ yy 0
Strains 0 0 0
x x γ yx 0
ASEN 3112 Lecture 5 – Slide 18
γx y yy 0
0 0 zz
ASEN 3112 - Structures
2D Plane Stress Generalized Hooke's Law Strains To Stresses (Omitting Thermal Effects)
1 E x x ν yy − E = ν zz − E γx y 0
ν −E 1 E ν −E 0
0
σx x 0 σ yy 0 τ xy 1 G
Stresses To Strains (Omitting Thermal Effects)
in which
σx x σ yy τx y
˜ E = E˜ ν 0
E˜ ν E˜ 0
E˜ =
E 1 − ν2
0 0 G
x x yy γx y
Used in Ex. 3.1 of HW #3. For inclusion of thermal effects as well as the plane strain case (which is less important in Aerospace) see Lecture Notes
ASEN 3112 Lecture 5 – Slide 19
