Mechanical behavior of materials

Mechanical behavior of materials

Lecture 5 – constitutive behavior

Introduction

• In mechanics of materials, the constitutive law is the relationship between stress and strain

• This is determined experimentally performing simple stress state tests. In particular, uniaxial tensile test is the preferred test to probe

material response and obtain information about the nature of the constitutive law.

• In general, materials show a constitutive

behavior that can be classified under the these

main classes

Introduction (cont.)

• Elastic material. Strain occurs in the material immediately when the stress is applied. Strain is completely recovered when the stress (or load) is removed.

• Linear- elastic

• Non-linear elastic

s

e s

e

Introduction (cont.)

• Elastic-plastic material. Strain occurs in the

material immediately when the stress is applied.

Strain is completely recoverable only up to a limit stress (yield stress). For increasing stress, a permanent deformation (non recoverable)

appears (plastic deformation)

• Elastic-plastic and non-linear elastic behavior cannot be discriminated until an unloading occurs

s

e e

𝜀 _𝑇 = 𝜀 _𝑒 + 𝜀 _𝑝

Introduction (cont.)

• Visco-Elastic-Plastic material. When the stress is applied, strain requires some time to occur and to reach the equilibrium. In general, elastic

strain occurs immediately while inelastic

deformation requires a finite time interval to occurs.

• Inelastic deformation, also called viscous deformation, can be recoverable or

unrecoverable (plastic)

• When the stress is removed, the elastic strain is immediately recovered while recoverable

viscous strain require some time.

𝜀 _𝑇 = 𝜀 _𝑒 + 𝜀 _𝑣 + 𝜀 _𝑝

Introduction (cont.)

• Almost all materials are elastic-visco-plastic in nature.

• Viscous deformation are function of time and temperature and are associated with the

dissipative mechanisms occurring in the

microscale (internal friction, obstacles, etc.) that

compete against the motion of dislocations

Uniaxial response of metals and alloys

• The uniaxial tensile load response of metals and alloys show the following features:

• Linear-elastic response up to a limit stress value called yield stress;

• Beyond the yield stress, plastic deformation occurs, the stress-strain response is non linear (hardening): elongation is still uniform.

• In ductile materials, at the UTS (ultimate tensile stress) the elongation is no longer uniform: necking develops.

• Beyond UTS, plastic deformation are more

and more localized in the neck region until

fracture occurs

Uniaxial response of metals and alloys

• From the constitutive point of view, the additive strain decomposition is assumed:

• A good estimate of the plastic deformation at the neck is given by Bridgman,

• This expression is obtained from the assumption of conservation of plastic volume.

𝜀 _𝑇 = 𝜀 _𝑒 + 𝜀 _𝑝 = 𝜎

𝐸 + 𝜀 _𝑝

𝜀 _𝑝 ≅ 2𝑙𝑛 𝐴 ₀

𝐴 _𝑓

Uniaxial response of metals and alloys: controlling parameters

• The uniaxial stress-strain response of metals and alloys depends on outer variables:

• Temperature

• Strain rate

• Microstructure

Uniaxial response of metals and alloys: controlling parameters

• The uniaxial stress-strain response of metals and alloys depends on outer variables:

• Temperature

From a very general point of view, increasing

temperature causes the reduction of the yield

stress and an increase of ductility (strain at

failure)

Uniaxial response of metals and alloys: controlling parameters

• Increasing temperature, viscous behavior becomes more relevant: the Young modulus depends on the load rate.

• In general, an increase of temperature increases material toughness

• At elevated temperature, oxidation and ageing

can occur causing material embrittlement and

dramatic reduction of ductility.

Uniaxial response of metals and alloys: controlling parameters

• Since the material response is viscous in nature, strain rate has an effect on the material flow curve.

• Increasing the strain rate:

• increases the yield stress

• Affect the slope of the hardening curve

• Affect the strain to failure (in a complex

manner)

Uniaxial response of metals and alloys: controlling parameters

• Since the material response is viscous in nature, strain rate has an effect on the material flow curve.

• Example: titanium

Titanium (fully anneled)

Uniaxial response of metals and alloys: controlling parameters

• Strain rate and temperature have a

competitive effect on the material response.

• Regarding the yield stress:

• Region I: weak dependence temperature, strain rate independent

• Region II: linear dependence on the log of strain rate

• Region II: linear dependence on the strain

rate.

Uniaxial response of metals and alloys: controlling parameters

• Microstructure

• The yield stress is the macroscopic value at which the slips occurs on the entire volume

• The presence of inclusions and barrier may block dislocation causing an increase of the yield stress (precipitation hardening)

• Alloying increases the yield stress

Uniaxial response of metals and alloys: controlling parameters

• Microstructure

• The yield stress depends on the grain size according to the Hall-Petch relationship

𝜎 _𝑌 = 𝜎 ₀ + 𝑘

𝑑

Plastic flow curve

• The flow curve is usually expressed by a power-law,

• This expression was found to work well for several classes of metals: but only in terms of engineering stress vs engineering strain!

• The great advantages of this expression are:

• Only two parameters that can be easily determined by linear fit on the log-log stress-strain plane

𝜎 𝜀 _𝑝 = 𝐾𝜀 _𝑝 ^𝑛

s

e

lns

lne

n

ln(K)

Plastic flow curve

• The plastic exponent has the meaning of the plastic strain at the onset necking

• Considère framework:

• During tensile deformation the net

resisting is reduced and the load has to decrease

• During tensile deformation the material strain hardens, load has to increase

• At necking the two effect are equal

s

e

𝜀

= 𝜀

→ 𝜎 = 𝜎

𝐼𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑑𝑢𝑒 𝑡𝑜 ℎ𝑎𝑟𝑑𝑒𝑛𝑖𝑛𝑔 𝑑𝜎 = 𝑑𝜎

𝑑𝜀

𝑑𝜀

𝐷𝑒𝑐𝑟𝑒𝑎𝑠𝑒 𝑑𝑢𝑒 𝑡𝑜 𝑟𝑒𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑑𝜎 = 𝑑 𝑃

= 𝑃

− 𝑑𝐴

= 𝜎𝑑𝜀

→ 𝑑𝜎

= 𝜎

Plastic flow curve

Therefore, substituting:

The hardening exponent is the strain at the onset necking!

s

e 𝜀

= 𝜀

→ 𝜎

= 𝐾𝜀

𝜀

= 𝜀

→ 𝑑𝜎

𝑑𝜀

= 𝐾𝑛𝜀

= 𝜎

𝐾𝑛𝜀

= 𝐾𝜀

𝑚 = 𝜀

Plastic flow curve

Decreasing the grain size:

The yield stress increases, and e

becomes

vanishing small: ductility is lost!

Plastic flow curve

The Ramberg–Osgood equation was proposed to describe the non linear relationship

between stress and strain in materials near their yield points.

It is especially useful for metals that harden with plastic deformation showing a smooth elastic-plastic transition.

The expression is given as e function of s.

As the power-law, it works well for engineering strain function of engineering stress data.

𝜀 _𝑡 = 𝜀 _𝑒 + 𝜀 _𝑝

𝜀 _𝑡 = 𝜎

𝐸 + 𝐾 𝜎 𝐸

𝑛

𝜀 _𝑡 = 𝜎

𝐸 + 𝛼 𝜎 𝐸

𝜎 𝜎 _𝑌

𝑛−1

𝜎 ≥ 𝜎 _𝑌

A good candidate expression for the flow curve has to reach a saturation stress for infinite

strain, since the number of dislocations that can be generated is finite.

A versatile expression is the Voce type law:

When needed multiple terms may be added,

Plastic flow curve

Today, with the use of computer, there is no need for simple expression for the stress-strain curve.

In finite element codes, the flow curve can be given in tabular form or by user subroutine.

The major limitation of the power-law

expression is that it returns infinite stress for infinite strain and this is unphysical.

𝜎 = 𝜎 _𝑌 + 𝑅 1 − exp −𝜀 _𝑝 /𝑏

𝜎 = 𝜎 _𝑌 +

𝑖=1 𝑛

𝑅 _𝑖 1 − exp −𝜀 _𝑝 /𝑏 _𝑖

Plastic flow curve

Strain rate and temperature effects can be accounted for modifying the flow curve expression.

Cowper and Symonds:

Johnson and Cook:

𝜎 = 𝐴 + 𝐵𝜀 _𝑝 ^𝑚 1 + 𝜀 𝐷

1/𝑞

𝜎 = 𝐴 + 𝐵𝜀 _𝑝 ^𝑚 1 + 𝐶𝑙𝑜𝑔 𝜀

𝐷 1 − 𝑇 − 𝑇 ₀ 𝑇 _𝑚 − 𝑇 ₀

𝑚

Suggested reading

• Brnic, Josip. Analysis of Engineering

Structures and Material Behavior. John

Wiley & Sons, 2018.