stress & strain

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 stress = 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 = 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 stress = P/A0

0

Mild steel (highly ductile)

High strength steel

Nominal strain = 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) 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) Coefficient of thermal expansion Physical dimension: 1/degree (e.g., 1/ C)E, and G are not independent. They are linked by

E = 2G (1+), G = E/(2(1+)), = E /(2G)1



State of Stress and Strain In Tension Test

PP

P

= P/A (uniform over cross section) xx

(b)

(a)

xxyy

yy

zz

zz

0000

0 0

Strainstate

xx 00

00 00 0 0

Stressstate

at all points in the gaged region

yz

x Cartesian axes

For isotropic material, = is called the lateral strain

Cross section (often circular) of area A

gaged length



Defining Elastic Modulus and Poisson's Ratio

E

= = axial stress, = = axial strain, = = lateral strain

Isotropic material properties E and 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 is defined as the ratio of lateral strain to axial strain:

The sign in the definition of is introduced so that it comes out positive.For structural materials n lies in the range [0,1/2). For most metals (and theiralloys) is in the range 0.25 to 0.35. For concrete and ceramics, 0.10.For cork 0. For rubber 0.5 to 3 places. A material for which = 0.5 is called incompressible. If is very close to 0.5, it is called nearly incompressible.

xx xx yy zz

lateral strain lateral strain axial strain axial strain = =

def

defE = whence = E , =



State of Stress and Strain In Torsion Test

TT

T

Circular cross section

(b)

(a)

00

00

0 0 0

Strainstate

xy

xy

xy xy

xy

xy

yx

yx

yx

yx

00

00

0 0 0

Stressstate

at all points in the gaged region. Both the shear stress = as well as the shear strain = vary linearly as per distance from thecross section center (Lecture 7). They attain maximum values on thespecimen surface. For simplicity, call those values = and =

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 = G , =

= = max shear stress on specimen surface over gauged region = = 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 = = L /L = T is called the thermal strain in the axial (x) direction. For an isotropic material, the material expands equally inall directions: = = , whereas the thermal shear strains are zero.

gaged length

00

0

def = whence L = L T LL 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:


= E

+ T = +

= E 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)ASEN 3112 - Structures

.

Stresses To Strains (Omitting Thermal Effects)

xxyyzzxyyzzx

=

1E

E

E 0 0 0

E1E

E 0 0 0

E E

1E 0 0 0

0 0 0 1G 0 00 0 0 0 1G 00 0 0 0 0 1G

xxyyzzxyyzzx

For derivation using the strain superposition principle,as well as inclusion of thermal effects, see Lecture notes


3D Generalized Hooke's Law (2)ASEN 3112 - Structures

xxyyzzxyyzzx

=

E (1 )(1 )

(1 )

E E 0 0 0E E E 0 0 0E E E 0 0 00 0 0 G 0 00 0 0 0 G 00 0 0 0 0 G

xxyyzzxyyzzx

E = E(1 2)(1 + )

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


[xx xy 0yx yy 00 0 zz

][ xx xy 0yx yy 00 0 0

]Stresses

Strains


2D Plane Stress Generalized Hooke's LawASEN 3112 - Structures

Stresses To Strains (Omitting Thermal Effects)

Used in Exercise 3.1 of Homework #3

Strains To Stresses (Omitting Thermal Effects)

[xxyyxy

]=

[ E E 0E E 00 0 G

] [xxyyxy

]

E2

= E1

xxyyzzxy

=

1E

E 0

E1E 0

E E 0

0 0 1G

[xxyyxy

]

in which


stress & strain

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