Introduction on Creep

MCEN90029 Advanced Mechanics of Solids Lecture L28 - 1

Lecture L28 Creep

MCEN90029

Advanced Mechanics of Solids


Summary In this lecture we will introduce the concept of creep and viscoelasticity. Over the next two lectures, we will investigate

(1)  Time-dependent increase in strain at

constant stress (2)  Time-dependent decrease in stress at constant

strain


Viscoelasticity

•  Viscoelastic materials do not exhibit a linear-elastic range •  Show interdependence with stress and strain over time.

Stress and strain dependent on load history •  Simplest model of viscoelasticity contains features of a

Hookean solid and a Newtonian liquid •  Simplest model is:

€

σ = Eε +η ˙ ε

(η) (E) Coefficient of elasticity

Coefficient of viscosity


•  Because of time dependence and creep, stress-strain relationships are strain-rate dependent. Thus the viscosity coefficient, η, is not constant (depends on nature of loading)

Viscoelasticity


Creep and viscoelasticity

•  When ambient air temperature is constant and low, metal strength and stiffness remains constant

•  High temperatures lead to reduced yield strength

•  Previously, we have assumed stress and strain remains constant


Effect of tempering temperature on mechanical properties of Nickel-chrome steel


The effect of temperature on the tensile strength of various steels


Creep and viscoelasticity

•  Time dependent increase in material strain under constant load is termed creep

•  When strain/deformation is held constant, and a reducing load (stress) occurs, this complementary effect is called stress relaxation



Stress-strain-time-temperature relationships

•  For most metals, creep occurs above 0.3Tm, where Tm is the metal’s melting temperature

•  Edward Andrade (1887-1971), an English Physicist, performed creep experiments on lead (displacement measured at const load and at a given temp until failure, or for a specified “lengthy” time)

E. Andrade

Typical const load/ const temp curves


Stress-strain-time-temperature relationships The four principal stages of a typical creep curve for a metal (1) initial stage: initial strain (usually elastic) due to initial loading (2) primary stage: decrease in creep rate (strain hardening more rapid than softening due to temp) (3) secondary stage: creep rate const (equilibrium between strain hardening and thermal softening) (4) tertiary stage: increased strain rate due to temp. Structural instability and cracking, may lead to failure


Stress-strain-time-temperature relationships Problem: For safe operation of metals, total deformation due to creep should be below failure strain. This is usually ensured by setting a creep limit (a standard creep limit at a given operating temp is 1% strain in 10,000 hrs) How does one obtain long-life creep data for different stresses/temperatures (could take a year!)

Answer: Methods are used to extrapolate long-life data from short-term tests


Analytical representations of creep behaviour Creep strain εc, which is a function of stress, σ, time, t, and temperature, T, can be represented by three functions as follows:

€

εc = f1(σ )⋅ f2(t)⋅ f3(T)

1. Stress function:

€

f1(σ) = A1ση f1(σ) = A2 sinh

σσ0

$

% &

'

( ) f1(σ) = A3 exp

σ

* σ 0

$

% &

'

( )

2. Time function:

€

ε c = αt1/ 3 + βt + γt 3

3. Temperature function:

€

f3(T) = exp −ΔHRT

$

% &

'

( )

A1, A2, A3 , are constants; σ0 and are reference stresses

€

" σ 0

α, β, γ are material constants relating to the primary, secondary and tertiary stages

ΔH is the activation energy, R is the universal gas constant, and T is the absolute temperature

(1)

(2A)

(2B)

(2C)


•  For design of components for creep, secondary stage most critical (for low stresses, creep rate constant for long periods)

Analytical representations of creep behaviour

From equation 2B:

€

ε c = αt1/ 3 + βt + γt 3

neglected

Replaced by strain constant

€

εc = ε0 +dεdt

#

$ %

&

' ( t where

€

dε /dt = ˙ ε c is the secondary stage creep rate

€

˙ ε c = Bσ nMinimum creep rate related to stress:

B and n are material constants (3) (4)


•  The dependence of temperature can be included (from equation 1)


€

˙ ε c = Bσ n exp −ΔHRT

&

' (

)

* +

How does one predict permissible stress and temperature for a creep strain of, say, 0.5% in 10,000 hours? (without doing a 10,000 hour test)

A log-log plot of vs yields a straight line. But can’t extrapolate data at high stress due to large dependence of n and ΔH on the stress/temperature regime

€

˙ ε c

€

σ


•  Alternatively, combine equations (3) and (4), i.e.


€

εc = ε0 +dεdt

#

$ %

&

' ( t

€

˙ ε c = Bσ nand

Thus, we can express the time to reach a specified value of total creep strain in the secondary stage as follows:

€

t =εc −ε0Bσ n

Not reliable, since tertiary creep may commence before predicted value of secondary creep!

i.e., don’t know εc for when secondary period finishes


•  A safer way method of predicting permissible stress/temp for a given creep strain is:


1. Extrapolate graphs of creep strain against log time for several temperatures

2. Plot temperature vs log time for various stresses at given strain. Extrapolate curves to required life


•  How does one determine the creep-rupture strength of a material for long endurances? Use a creep-rupture curve –  Various values of stress applied in successive tests at constant

temperature, sufficient to cause rupture (from a few minutes to several hundred hours)

Creep-rupture testing

Rupture/log time relation for chromium-

molybdenum-silicon steel


•  Creep testing most commonly in tension

•  Previously, dead-weight loading system. Now, servo-hydraulic systems

•  Temp significantly affects creep rate •  British Standards

–  Max variation of temp along specimen 2°C

–  Variation of mean temp ±1° up to 600°C and ±2°C between 600°C and 1000°C

Tension creep test equipment

Bose ElectroForce 3200 material test instrument


•  Consider a beam subject to pure bending at a given high temperature. Calculate the bending stress distribution assuming:

(a) plane section remain plane under creep deformation (b) creep behaviour the same in tension as in compression (c) the creep rate is given by the expression

Creep during pure bending of a beam

€

˙ ε c = Bσ n


•  Most problems in engineering are 3-dimensional. How do we relate uniaxial creep data to biaxial problems? Assume:

(a) principal stresses and strains are in the same direction (b) plastic deformation occurs at constant volume (c) maximum shear stresses and strains are proportional

Creep under multi-axial stresses

€

ε1 +ε2 +ε3 = 0

Expressing maximum shear stress and strain in terms of their differences:

€

ε1 −ε2σ1 −σ 2

=ε2 −ε3σ 2 −σ 3

=ε3 −ε1σ 3 −σ1

= β

Rearranging the above equations to give the individual principal strains in terms of the principal stresses:

€

ε1 =2β3

σ1 −12σ 2 −σ 3( )

&

' ( )

* +

ε2 =2β3

σ 2 −12σ 3 −σ1( )

&

' ( )

* +

ε3 =2β3

σ 3 −12σ1 −σ 2( )

&

' ( )

* +


Creep under multi-axial stresses Express as a constant creep rate:

€

˙ ε 1 = α σ1 −12σ 2 −σ 3( )

&

' ( )

* +

˙ ε 2 = α σ 2 −12σ 3 −σ1( )

&

' ( )

* +

˙ ε 3 = α σ 3 −12σ1 −σ 2( )

&

' ( )

* +

Where α is a function relating the three principal stresses to the uniaxial stress/creep condition

Using the von Mises yield criterion to obtain the effective stress, σe gives:

€

σe =12

σ1 −σ2( )2 + σ2 −σ3( )2 + σ3 −σ1( )2[ ]1/ 2

From the simple secondary-stage creep law,

€

˙ ε = Bσ en

(5)

(6) For simple tension, σ2 = σ3 = 0. Thus,

€

˙ ε = ασ e

€

σ e =σ1

Thus, we can write

and therefore

€

α = Bσ en−1

(7)

€

˙ ε 1 = Bσ en−1 σ1 −

12σ 2 −σ 3( )

%

& ' (

) *

˙ ε 2 = Bσ en−1 σ 2 −

12σ 3 −σ1( )

%

& ' (

) *

˙ ε 3 = Bσ en−1 σ 3 −

12σ1 −σ 2( )

%

& ' (

) *

(Three principal creep rates)

(8)

(from 5)


Example

A steel gas flue tube of 100 mm diameter and 3 mm wall thickness is to operate at 400°C for a service life of 100,000 hours. Determine the allowable pressure for a creep strain limit of 0.5%.

Assume at 400°C, n = 3, B = 1.45×10-23 m2/h.MN


Lecture summary

•  In this lecture we introduced the concept of creep as the time dependent strain of a body under constant stress. We derived expressions for secondary stage creep behaviour in three dimensions

Documents

Introduction on Creep