ASEN 3112 - Structures
5Stress-Strain
Material Laws
ASEN 3112 Lecture 5 – Slide 1
ASEN 3112 - Structures
Strains and Stresses are Connectedby Material Properties of the Body (Structure)
internal forces ⇒ stresses ⇒ strains ⇒ displacements ⇒ size&shape changes
displacements ⇒ strains ⇒ stresses ⇒ internal forces
The linkage between stresses and strains is done throughmaterial properties, as shown by symbol MP over red arrow Those are mathematically expressed as constitutive equations
Recall the connections displayed in previous lecture:
MP
MP
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 RegardsMaterial 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 stressσ = P/A 0
0
Yield
Linear elasticbehavior(Hooke's law isvalid over thisresponse region)
Strain hardening
Max nominal stress
Nominal failurestress
Localization
Mild SteelTension Test
Nominal strain ε = ∆L /L
Elastic limit
Undeformed state
L0 PPgage length
ASEN 3112 Lecture 5 – Slide 4
ASEN 3112 - Structures
Other Tension Test Response Flavors
Brittle(glass, ceramics,concrete in tension)
Moderatelyductile(Al alloy)
Nonlinearfrom start(rubber,polymers)
ASEN 3112 Lecture 5 – Slide 5
ASEN 3112 - Structures
Tension Test Responses of Different Steel Grades
Nominal stressσ = P/A0
0
Mild steel (highly ductile)
High strength steel
Nominal strain ε = ∆ L /L
Tool steel
Conspicuous yield
Note similarelastic modulus
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
PP
P
σ = P/A (uniform over cross section) xx
(b)
(a)
εxx
εyy
yy
εzz
zz
0000
0 0
Strainstate
σxx 00
00 00 0 0
Stressstate
at all points in the gaged region
yz x Cartesian axes
For isotropic material, ε = ε is called the lateral strain
Cross section (often circular) of area A
gaged length
ASEN 3112 Lecture 5 – Slide 8
ASEN 3112 - Structures
Defining Elastic Modulus and Poisson's Ratio
E
σ = σ = axial stress, ε = ε = axial strain, ε = ε = lateral strain
Isotropic material properties E and ν are obtained from the linear elasticresponse region of the uniaxial tension test (last slide). For simplicity call
The elastic modulus E is defined as the ratio of axial stress to axial strain:
Poisson's ratio ν is defined as the ratio of lateral strain to 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 theiralloys) 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.
xx xx yy zz
lateral strain lateral strain axial strain axial strainν = = −
def
defE = whence σ = E ε, ε = σ σ
ε
ASEN 3112 Lecture 5 – Slide 9
Rubber
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Cork
ASEN 3112 - Structures
Which material would work bestfor capping a wine bottle?
ASEN 3112 Lecture 5 – Slide 10
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yyy
FoamRubber
Which material would work best for a shock absorber?
ASEN 3112 - Structures
ASEN 3112 Lecture 5 – Slide 11
ASEN 3112 - Structures
State of Stress and Strain In Torsion Test
TT
T
Circular cross section
(b)
(a)
00
00
0 0 0
Strainstate
τxy
xy
xy xy
γxy
xy
τyx
yx
γyx
yx
00
00
0 0 0
Stressstate
at all points in the gaged region. Both the shear stress τ = τ as well as the shear strain γ = γ vary linearly as per distance from thecross section center (Lecture 7). They attain maximum values on thespecimen surface. For simplicity, call those values τ = τ and γ = γ
yz x Cartesian axes
gaged length
For distribution of shear stresses and strains over the cross section, cf. Lecture 7
max max
ASEN 3112 Lecture 5 – Slide 12
ASEN 3112 - Structures
Defining Shear Modulus Of An IsotropicLinearly Elastic Material
G
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
The shear modulus G is defined as the ratio of the foregoingshear stress and strain:
defG = whence τ = G γ, γ = τ τ
γ
τ = τ = max shear stress on specimen surface over gauged regionγ = γ = max shear strain on specimen surface over gauged region
xy
xy
max
max
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:
At the reference temperature T (usually the room temperature) thegaged length is L . Heat the unloaded specimen by ∆T while allowing itto expand freely in all directions. The gaged length changes to L = L + ∆L. The coefficient of thermal expansion is defined as
The ratio ε = ε = ∆L /L = α ∆T is called the thermal strain in the axial (x) direction. For an isotropic material, the material expands equally inall directions: ε = ε = ε , whereas the thermal shear strains are zero.
gaged length
0
0
0
defα = whence ∆L = α L ∆T ∆L
L ∆T 0
0
0
TT
TT
T
xx
xx yy zz
x
ASEN 3112 Lecture 5 – Slide 14
1D Hooke's Law Including Thermal Effects
Stress To Strain:
Strain To Stress:
ASEN 3112 - Structures
ε = σE
+ α ∆T = ε + ε
ε=σ E − α ∆T
expresses that total strain = mechanical strain + thermal strain: the strain superposition principle
( )
A problem in Recitation 3 uses this form
TM
ASEN 3112 Lecture 5 – Slide 15
3D Generalized Hooke's Law (1)
ASEN 3112 - Structures
.
Stresses To Strains (Omitting Thermal Effects)
εxx
εyy
εzz
γxy
γyz
γzx
=
1E − ν
E − νE 0 0 0
− νE
1E − ν
E 0 0 0
− νE − ν
E ©1E 0 0 0
0 0 0 1G 0 0
0 0 0 0 1G 0
0 0 0 0 0 1G
σxx
σyy
σzz
τxy
τyz
τzx
For derivation using the strain superposition principle,as well as inclusion of thermal effects, see Lecture notes
ASEN 3112 Lecture 5 – Slide 16
3D Generalized Hooke's Law (2)
ASEN 3112 - Structures
σxx
σyy
σzz
τxy
τyz
τzx
=
E (1 − ν)(1 − ν)
(1 − ν)
E ν E ν 0 0 0E ν E E ν 0 0 0E ν E ν E 0 0 00 0 0 G 0 00 0 0 0 G 00 0 0 0 0 G
εxx
εyy
εzz
γxy
γyz
γzx
E = E
(1 − 2ν)(1 + ν)
Strains To Stresses (Omitting Thermal Effects)
in which
This is derived by inverting the matrix of previous slide. For theinclusion of thermal effects, see Lecture notes
ASEN 3112 Lecture 5 – Slide 17
2D Plane Stress Specialization
ASEN 3112 - Structures
[εxx γxy 0γyx εyy 00 0 εzz
][σxx τxy 0τyx σyy 00 0 0
]Stresses Strains
ASEN 3112 Lecture 5 – Slide 18
2D Plane Stress Generalized Hooke's LawASEN 3112 - Structures
Stresses To Strains (Omitting Thermal Effects)
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
Strains To Stresses (Omitting Thermal Effects)
[σxx
σyy
τxy
]=
[ E E ν 0E ν E 00 0 G
] [εxx
εyy
γxy
]
E2
= E1 − ν
εxx
εyy
εzz
γxy
=
1E − ν
E 0
− νE
1E 0
− νE − ν
E 0
0 0 1G
[σxx
σyy
τxy
]
in which
ASEN 3112 Lecture 5 – Slide 19
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