Stress / Stain response of Metallic Materials

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Article 1:

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Materials stress - strain response

•Types of non-linear response

•Models of uni-axial stress- strain curve

•Experimental stress-strain curves

•Idealised stress-strain curves

•Types of stress - strain curves

•Rigid elastic response

•Effect of temperature

•Effect of strain rate

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Standard Tensile Test Specimen

Different stages in the elongation of the specimen

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Types of non-linear response:

Materials stress - strain response

Non-linear Elastic

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Types of non-linear response:

Materials stress - strain response

Plastic

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Materials stress - strain response

Visco-elastic

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Materials stress - strain response

Visco-plastic

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Models of uni-axial stress- strain curves:

•In a uni-axial tension test, the transition from linear-elastic response to inelastic response may be abrupt or gradual

Materials stress - strain response

Case 1: abrupt Case 2: gradual

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Experimental stress-strain curves; case 1

Materials stress - strain response

•For an abrupt

transition, the change is identified by the “kink”

in the stress strain curve and the stress level at this point is called the yield stress

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Experimental stress-strain curves; case 2

Materials stress - strain response

•In the case of gradual

transition, the yield stress is arbitrary defined as the stress corresponding to a given permanent strain ε_s (usually ε_s=0.002) that

remains upon unloading along a straight line path BB` parallel to AA`

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Idealised stress-strain curves:

•Actual (experimental) stress-strain responses are difficult to use directly in mathematical solutions to complex problems.

•Idealised models of material response are therefore used in practice.

•For the two uni-axial cases described here, the

idealised models are presented and compared with their corresponding actual response

Materials stress - strain response

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Idealised vs actual stress-strain curves; case 1

Elastic-Perfectly Plastic model

Materials stress - strain response

Corresponding actual response

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Idealised vs actual stress-strain curves; case 2

Elastic Strain-Hardening Model

Materials stress - strain response

Corresponding actual response

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Further Idealised stress-strain curves!:

•Sometimes the deformation imposed on a material may be so large that the elastic strain is a small

fraction of the total strain. In such cases the elastic strain may have a negligible effect on the analysis.

•If this is the case further idealisation to the stress strain curves are possible (rigid-plastic models)

•This procedure will result in “rigid-perfectly plastic” and “rigid-strain-hardening” in cases 1 and 2 respectively

Materials stress - strain response

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Case 1

rigid elastic-perfectly plastic

Case 1;

rigid elastic-

strain hardening

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The Bauschinger effect

This is defined as the

change in material response under reverse loading

following the tensile load- unload history; presented by dashed curves replacing

the solid lines

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Materials stress - strain response

Non-linear elastic vs Plastic Response

•In non-linear elastic

response there is always a unique relation between stress and strain

throughout the load history

•for plastic response

there may be more than one value of strain for each value of stress`

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Materials stress - strain response

Non-linear Inelastic vs Plastic Response

•If a plastic material is loaded into the inelastic region and then unloaded, a given value of stress may correspond to two values of strain, one for

application of the load and one for removal of the load

•This implies that plastic material response is “path dependent”

whereas elastic material response is

“path independent”

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Types of stress - strain models

idealised formulation and examples of application

1) Perfectly elastic model:

The material responding to load like a linear spring with stiffness E; there is a limit to the stress the material can sustain, after which it breaks; permanent

deformation, if any, is negligible Used to model the behaviour of brittle materials such as glass, ceramics, and some cast irons

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Types of stress - strain models

2) Rigid Perfectly plastic model:

infinite value of E by definition; at limit stress of yield (Y) undergoes deformation at same stress level;

when the load is released undergoes permanent

deformation with no elastic recovery

Used to model the behaviour of materials exhibiting large plastic deformation with negligible

hardening

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Types of stress - strain models

3) Elastic Perfectly plastic model:

a combination of the previous models; has a finite elastic

modulus; at limit stress of yield (Y) undergoes deformation at same stress level; undergoes

elastic recovery when the load is released

Used to model the behaviour of materials exhibiting some plastic deformation with negligible

hardening

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Types of stress - strain models

4) Rigid strain hardening model:

infinite value of E by definition; at yield (Y) requires increasing stress to undergo further strain; its “flow stress” increases with increasing strain; no elastic recovery when the load is released

Used to model the behaviour of materials exhibiting large plastic deformation with linearly hardening (E_p) characteristic

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Types of stress - strain models

5) Elastic strain hardening model:

Finite elastic modulus; at yield (Y) requires increasing stress to

undergo further strain; its “flow stress” increases with increasing strain; undergoes elastic recovery when the load is released

Used to model the behaviour of materials exhibiting plastic

deformation with linearly hardening (E_p) characteristic

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idealised stress-strain models

Summary

Perfectly elastic | rigid-perfectly plastic | elastic-perfectly plastic

Rigid-

elastic-

|

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idealised stress-strain models

comment

The appropriate model to be used for many

engineering materials depends not only on the type of material but it is also determined by the conditions of application such as temperature, loading characteristics and the type of analysis performed

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Effect of strain hardening exponent on the shape of stress strain curve

Materials stress - strain response

n=0; rigid-perfectly plastic n=1; elastic

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Effect of temperature on stress strain curve

Materials stress - strain response

Temperature affects:

•The modulus of elasticity

•The yield stress

•The ultimate tensile strength

•The toughness

of most engineering materials

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True Stress

True Stress - - True strain; True strain;

definitions definitions

True stress: the ratio of load to the true carrying area

; A: instantaneous area True strain: the complete tension test may be regarded as a series of incremental tension tests where for each succeeding increment, the original specimen is a little longer than the previous one. Thus true strain (also

termed natural or logarithmic strain), ε, can be defined as:

A

= P σ

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

= ∫

0

ln

0

l

l l

l

ε

ε d

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True Stress

Hints Hints

•For small values of engineering strain:

since

•For large strains, however, the values diverge rapidly

•Based on “volume constancy”, the volume of a metal specimen in the plastic region of the test remains

constant. Hence, the true strain within the uniform elongation range can be expressed as:

ε

=

e ln ( 1 + e ) = ε

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟ ⎠

⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

D D D

D A

A

₀ ₀ ² ₀

0

ln 2 ln

ln ln l

ε l

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True Stress

the importance in metal working the importance in metal working

Assume that a tension specimen is elongated to twice its original length. This deformation is equivalent to compressing a specimen to one half its original height:

and where as: and

thus

only

true strain is the correct measure of strain

69 .

= 0

ε

t

ε

_c

= − 0 . 69

0 .

= 1

e

t

e

_c

= − 0 . 5

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True Stress

the importance; an extreme example the importance; an extreme example

Assume that a specimen, 10 mm in height is

compressed to a final thickness of zero! We have:

where as:

We have deformed the specimen infinitely which is exactly what the value of true strain indicates. Thus again,

only

true strain is the correct measure of strain

−∞

c

=

ε e

_c

= − 1 . 0

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True Stress

True Stress - - True strain curves True strain curves

The relation between engineering and true values for stress and strain can be used to construct the true- stress-true-strain curves from the engineering stress-

strain or load displacement curves Exercise

assume a load displacement curve for a tensile specimen. Construct first engineering stress-strain curve from the assumed data, then construct the true

stress-true strain curve from the engineering curve

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True Stress

For convenience, a true stress-true strain curve is typically approximated by the equation

the equation does not indicate the elastic region or the yield point as these quantities are readily available

from the engineering stress strain curve

(due to very small strains at yield and below, the true strain is no different from engineering strain)

K ε

n

σ =

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True Stress

K ε

n

σ =

•True stress-true strain curve in tension; unlike in an engineering stress-strain curve, the slope is always positive, and decreases with increasing strain

•Although stress and strain are proportional in the elastic range, the total curve can be

approximated by the power

expression; Y is the yield stress and Y_f is the flow stress

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True Stress

The approximate equation can be rewritten as:

ε σ log log log = K + n

•True stress-true strain curve plotted on a log- log scale

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True Stress

True stress-true strain curve in tension for a typical material on a log-log scale; note the large difference in

the slopes in the elastic and plastic ranges

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Typical values of K and n at room Typical values of K and n at room

temperature

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True Stress

True Stress - - True strain True strain

•True stress-true strain curves in tension at

room temperature for various metals; the

point of intersection of each curve at ordinate is the yield stress;

(possible mismatch with data table is due to different sources)

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Strain rate:

•whereas the deformation rate may be defined as the speed at which a tension test is being carried out, the strain rate is a function of the geometry of the specimen.

Engineering strain rate

True strain rate

Engineering and true strain rate

( )

0 0

0

1

l l

l

& l v

dt d dt

d dt

e de − = =

=

( )

[ ]

l l

& l v

dt d dt

d dt

d = = =

= ε ln

₀

1

ε

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Effect of strain rate and temperature

Effect of strain rate and temperature on ultimate strength:

•As temperature

increases, the slope increases.

•Thus tensile strength becomes more sensitive to strain rate as

temperature increases (graph for aluminium)

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Materials stress - strain response

Effect of monotonic pre-straining

•Monotonic pre-straining modifies the

subsequent stress-strain response of most metallic materials.

•One phenomenon is called “the Bauschinger” effect

•It is often observed that, after plastic

deformation in tension and when the loading direction is reversed, the material re-yields at a lower stress

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Materials stress - strain response

The Bauschinger effect

This is defined as the

change in material response under reverse loading

following the tensile load- unload history; presented by dashed curves replacing

the solid lines

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Bauschinger effect in an aluminium alloy

Isotropic hardening response

♦When the yield surface expands uniformly during plastic flow, isotropic hardening

occurs, with no change in shape or translation of the surface.

♦Yielding in compression then takes place when:

max

"

σ

_y

= −

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Isotropic hardening response

max

"

σ

_y

= −

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Bauschinger effect in an aluminium alloy

Kinematic hardening response

♦When the yield surface simply translates, kinematic hardening occurs

♦Yielding in compression then takes place when:

' max

"

2

_y

y

σ σ

σ = −

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Kinematic hardening response

' max

"

2

_y

y

σ σ

σ = −

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Bauschinger effect in an aluminium alloy

Real hardening response

♦Real materials tend to exhibit both expansion and translation

♦For aluminium alloy 2024(shown), typical for most metallic alloys, neither the isotropic nor the kinematic hardening models represent the real response and more complex models

such as mixed hardening can be developed

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Various tensile compressive Various tensile compressive

responses responses

Stress and strain response of an aluminium alloy

subjected to tensile, then compressive loading; real and mathematical

models; note the pre-straining effect on the responses

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Bauschinger

Bauschinger effect factor effect factor

Bauschinger effect factor, defined as the ratio of compressive yield to tensile yield, for various hardening behaviours;

Ref.: Garcia Granada, UoB 2000

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Article 1:

Stress / Stain response of Metallic Materials Stress / Stain response of Metallic Materials

(Some Related Aspects)

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The Outline of Engineering The Outline of Engineering

Materials Materials

•Ferrous Metals:

Carbon, Alloy, Stainless, tool and die steels

•Non-ferrous Metals and Alloys:

Aluminium,

Magnesium, Copper, Nickel, Titanium, Supper alloys, Beryllium, Zirconium, Low-melting alloys and Precious metals

•Plastics:

Thermoplastics, Thermosets and Elastomers

•Ceramics:

Glass ceramics, Glasses, Graphite and Diamond

•Composite (Engineered) Materials:

Reinforced plastics, Metal-matrix & Ceramic-matrix composites and

honeycomb structures

•New Materials:

Nanomaterials, Shape-memory alloys, Amorphous alloys, Superconductors, etc.

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Forefront of Usage of New Forefront of Usage of New

materials materials

The aluminium body structure; Audi A8

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Properties of Materials Properties of Materials

•Mechanical Properties:

Strength, Toughness, Ductility, hardness, Elasticity, fatigue and creep resistance

•Physical Properties :

Density, Specific heat, Thermal expansion, Thermal conductivity, Melting point, Electrical and Magnetic properties

•Chemical Properties :

Oxidation, Corrosion, Degradation of Properties, Flammability, Phase stability

•Manufacturing Properties :

(manufacturing

possibility): Casting, Forming, Shaping, Machining, Welding, etc

•Other (crossed) properties:

strength-to-weight ratio, stiffness-to-weight ratio, heat treatment related issues, etc.

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Selecting Manufacturing Processes Selecting Manufacturing Processes

•Casting:

Expandable and Permanent mold

•Forming and Shaping :

Rolling, Forging, Extrusion, Drawing, Sheet forming, Powder metallurgy

•Machining :

Turning, Boring, drilling, milling, planing,

shaping, Broaching, Grinding, Ultrasonic machining, Chemical, Electrical and Electro-chemical machining, High-energy beam machining

•Joining :

Welding, Brazing, Soldering, Diffusion bonding, adhesive bonding, mechanical joining

•Finishing:

Honing, Lapping, Polishing, Burnishing, Deburring, Surface treating, Coating, Plating

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Various methods of making a Various methods of making a

simple part simple part

Casting;

Powder metallurgy

Forging;

Upsetting

Extrusion

Machining

Joining

Same part,

Several processes!

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The State of Stress in Metal Working Processes

Expansion of a thin-walled spherical shell under internal pressure

Drawing a round bar to reducing a diameter

Deep drawing of sheet with a punch and die to make a metal cup

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How to select a material

(main characteristics)

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Ceramic-Composite Armor

Projectile

Outer hard skin

Ceramic- Discontinuous

Inner ductile skin

Personnel and

Equipment

Ceramic Armor System

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Residual stresses in welded

joint

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Residual Stress Relaxation Basic theoretical

approximation

Yielding

Applied Stress

+

- Sy

Sy

A

^B

^C

Mechanic al relaxation Stress

A: Initial stress distribution

B: Applic ation of uniform tensile forc e

C:Final residual stress distribution after removal of tensile forc e

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Deep Hole Drilling

Drilling bit

Air probe

EDM electrode

Air probe

Gun-drilling Diameter measurement

Diameter measurement Trepanning

z x

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WPS cycles

LLCFCF LLCUFCUF LLUCFUCF

Temperature Load

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Influence of pre

Influence of pre - - load on subsequent load on subsequent response

response

•When there is the development of localised plasticity, e.g. at stress concentrations, the Bauschinger effect has a strong influence on the level of residual stress generated in the vicinity of the stress concentration

•For example, proof and overloading induces local residual stress, the magnitude of which depends strongly on material response on unloading after pre-loading

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Cleavage Fracture

The “Weakest link” theory:

failure of a body commences when its

weakest element (link) fails.

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Probability of failure and SIF

Stress Intensity Fac tor

Frequency

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Triaxiality (factor); Definition

( ) ( ) ( )

[

1 3 ²

]

¹²

2 3 2

2 2 1

3 2

1

3 2

σ σ

σ

− +

−

+

= + T

f

The ratio of hydrostatic stress to Von Mises equivalent stress in a triaxial stress state:

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Ramberg-Osgood material

model

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Fatigue (Cause)

Cyclic Loading

σ _mean = σ _max + σ _min 2

σ _range = σ _max − σ _min σ _amplitude = σ _max − σ _min

2 Stress Ratio, R = σ _min

σ _max Rotating Machinery

Airframes, Bridges, Tanks, etc,

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( 1 ) 0

2 cosh 3

2

₁ ^* ² ₃ ^*²

2

= +

⎟ −

⎟

⎠

⎞

⎜ ⎜

⎝

⎛ −

⎟ +

⎟

⎠

⎞

⎜ ⎜

⎝

= ⎛

Φ q p q f

f q q

y

σ

q: Von Mises stress

P: Hydrostatic pressure σ

_y

:Yield stress

q

₁

,q

₂

,q

₃

needs to be calibrated f* is a void volume fraction