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ASEN 3112 - Structures

5Stress-Strain

Material Laws

ASEN 3112 Lecture 5 – Slide 1


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.



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.



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



Other Tension Test Response Flavors

Brittle(glass, ceramics,concrete in tension)

Moderatelyductile(Al alloy)

Nonlinearfrom start(rubber,polymers)



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



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



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



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


Rubber

��

Cork


Which material would work bestfor capping a wine bottle?


��

yyy

FoamRubber

Which material would work best for a shock absorber?




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



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



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


1D Hooke's Law Including Thermal Effects

Stress To Strain:

Strain To Stress:


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


3D Generalized Hooke's Law (1)


.

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


3D Generalized Hooke's Law (2)


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


2D Plane Stress Specialization


[εxx γxy 0γyx εyy 00 0 εzz

][σxx τxy 0τyx σyy 00 0 0

]Stresses Strains


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


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