Lec25_11-18_ANGEL.pdf

Quiz 4 – Nov. 20 Conceptual question from Chapter 7

•  Griffith observed that the stress to fracture was directly proportional to specimen size. True or False

FALL 2014: EMCH 315 1

HINT Oct. 28 Lecture 18-19, see slides 8, 10

•  _______________ is an engineering approximation that a solid fails in elastic region.

Quiz 4 – Nov. 20: Conceptual question from Chapter 7 •  The minimum condition for instability is

•  ____ represents the reversible, adiabatic energy necessary to create a new surface

•  In Griffith’s energy balance, the term that opposes crack growth is

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HINT: Oct. 28 Lecture 18-19, see slide 13

•  In Griffith’s energy balance, the term that relates to an increase in elastic strain energy is σ 2

2EdV

True or False

γ dA True or False

Quiz 4 – Nov. 20: Conceptual question from Chapter 7 •  From Griffith’s analysis, based on the energy balance, the

critical stress (or flaw size) for design against fracture is expressed as

•  The theoretical cohesive strength (TCS) can be estimated from Griffith’s energy balance; knowing ____ and _____ and assuming a ________________________ (approximated by defect where a = 2-3 atomic spacing).

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•  A typical range of TCS can be expressed in terms of the modulus of elasticity as __________

Quiz 4 – Nov. 20: Conceptual question from Chapter 7 •  An alternative approach, to Griffith’s energy balance, for

failure analysis in the presence of defects, is referred as ____________ mechanics, which considers ___________ _______________ extension

Write the names of the modes described below: crack displaces perpendicular to the crack plane ___________ crack displaces perpendicular to the leading edge __________ crack displaces parallel to the leading edge _______________

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Quiz 4 – Nov. 20: Conceptual question from Chapter 7 •  The term _______ is defined as the

Mode I ____________________

•  Cracks tend to seek out planes _____________ to the directions of the maximum principal ______________ stress.

•  The general equation for the SIF is __________ where y is a correction factor.

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HINT: Oct. 28 Lecture 18-19, see slides 24, 25

stress intensity factor

•  The material property that represents the critical SIF for which the crack is unstable is referred to as the _________________.

•  Using the material property _______, a fracture-free criteria can be defined as ____________ and fracture will occur when

Quiz 4 – Nov. 20: Conceptual question from Chapter 7

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HINT: Oct. 28 Lecture 18-19, see slides 24, 25

•  A thin plate is defined by _________________ and this represents a plane _______________ condition.

•  A thick plate is defined by _________________ and this represents a plane _______________ condition.

Quiz 4 – Nov. 20: Conceptual question from Chapter 7 •  High-strength alloys exhibit ___________________ behavior: the

stress-strain response exhibits an elastic-plastic transition when un-notched, but failure during elastic deformation when cracks exist.

•  A _____________forms ahead of cracks in order to relieve otherwise infinite stresses.

•  The size of the plastic zone is defined by the ___________________

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HINT: Oct. 30 Lecture 18-19, see slides 33

•  The plane strain condition for the ratio of specimen thickness to the plane stress plastic zone radius is rpc/t > 1/5.

•  To ascertain plane stress/strain deformation, compare the _______________ plastic zone radius, rpc to the ________________________ .

•  For plane stress deformation, rpc/t > 1.0 .


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Fill in the box below the plastic zone sketch with “plane strain” or “plane stress.”

True or False

True or False

True when rpc/t < 1/5


•  Fractography will illustrate failure between grains,

referred to as _________________________ fracture, or

through the grains, referred to

as_______________________ fracture.

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Quiz 4 – Nov. 20: Conceptual quesAon from Chapter 7

Fill in only two of the boxes below the sketch with “plane strain” or “plane stress.”

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For the case of completely reversed stress amplitude what is the value of R: R = ______. When σmin is equal to zero, R = ______. If σmax = 0, R = ______

S (MPa)

N

Define S. Define N.

magnitude of the reversed stress amplitude, endurance limit

number of cycles to failure, endurance

Which of the materials has the higher UTS. _________________

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For the variable amplitude stress histories, we introduce the concept of cumulative damage. Write the equation for di, the damage associated with the “ith” stress amplitude. ________

For Miner’s rule ,loads are applied in any arbitrary

sequence. True or False When determining the number of cycles to failure corresponding to the “ith” stress amplitude in a variable amplitude stress history, we consider that the stress is applied in a ___________ _____________ sense.

dii=1,2,3∑ = ni

Nii=1,2,3∑ = 1

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Quiz 4 – Nov. 20: Conceptual quesAon from Chapter 8 When σ m ≠ 0, the Goodman correlation is used to obtain the equivalent completely reversed alternating stress σao: what other three parameters must be known in order to calculate σao? _________ ___________ _____________ What is the value of the fatigue stress concentration factor,

when notched and unotched members exhibit

the same endurance? Kf = _______ The other extreme value of

Kf is Kf = ________________. To determine intermediate values of Kf, we define q as

________________________ and the equation is

________________.

K f =σ e(unnotched)σ e(notched)

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Describe the physical phenomena in the three stages illustrated in the figure below.

!

______"≡""fa&gue"crack"ini&a&on"

______"≡""stress"below"which"cracks"""""""do"not"ini&ate"

______"≡"fa&gue"crack"growth"rate"

______"≡"crack"has"reached"a"cri&cal"""""size"

______"≡""stress"intensity"range"

______"≡""fa&gue"crack"propaga&on"

I =

II =

III =

da/dN is referred to as ________________________________ ΔKI is referred to as the _______________________________ What is the name of the mathematical model that governs Stage II. ______________

fatigue crack initiation: initial micro-cracks propagated along planes of maximum cyclic shear stress

fatigue crack propagation: crack seeks out and propagates along planes with maximum tensile stress amplitude

fast fracture: crack has reached critical size for local stress state

σo

time

Maxwell model of viscoelastic response: stress relaxation.

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εo absorbed by the spring

stress decays (relaxes)

σ

with time, motion occurs in the dashpot strain in the spring decreases

instantaneous strain εo applied suddenly results in σo

_________________________________

_________________

_________________________________________________________________

Maxwell model of viscoelastic response: stress relaxation.

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elastic strain εe is being ______________ by creep strain εc

total strain ε′ is held constant with time

Formulation of the constitutive equation governing stress relaxation of a Maxwell material.

•  The applied strain is held constant and thus εo = ε and

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Separate stress and time variables and assume the solution σ (t) = AeBt

Solve for the constants

A: B:

At t = 0 the stress = σo

Substitute

dεdt

= 0

⇒ dεdt

= 1Edσdt

+ 1ησ =

∴ σo = Ae B(o) = A; A =

σ =σ oeBt and dσ

dt=σ oBe

Bt

The equation governing stress relaxation of a Maxwell material is:

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σ

We can define a Ame constant, τ, (aka characterisAc relaxaAon Ame).

σ = σoe

τ ≣ = _____________________ η E

when t = τ, _________

Ames on the order of τ → _____________ viscoelasAc response: _____________________ Ames orders of magnitude _________ than τ → _______ ________________ and only ________ deformaAons

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Interpreting characteristic times…

Taking the natural logarithm of both sides…

____________ line on log of stress vs Ame plot implies __________ decay and material is therefore Maxwellian. For a Maxwell solid, the ________ is -‐(1/τ) or – E/η.

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ln σ = ln σo t τ –

Based on the characterisAc relaxaAon Ame constant, we can determine whether a material is Maxwellian.

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In-class problem A fiberglass fastener is installed where it must sustain a minimum tensile stress of 2000 psi. The composite has an elastic modulus of 6 x 106 psi, a tensile strength of 24,000 psi and a viscosity of 900 x 109 psi-min. The material behaves as a Maxwell solid. When initially assembled, the fastener is instantaneously strained to 800 x 10-6 and this strain is constant during its subsequent life. What is its maximum service time, t (in minutes) before its stress falls below the minimum acceptable tensile strength.

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Re-thinking/organizing the problem statement A fiberglass composite fastener is installed which must sustain a minimum tensile stress of 2000 psi. The composite has the following elastic properties:

Maxwell solid governing equation –

When initially assembled, the fastener is instantaneously strained to 800 x 10-6 and this strain is constant during its subsequent life. ε

t

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What is its maximum service time, t (in minutes) before its stress falls below the minimum acceptable tensile strength.

σ

t

σo

Determine instantaneous stress corresponding to εo = 800 µε

Determine service time, t (in minutes) to sustain 2000 psi for a Maxwellian material.

E MCH 315 Mechanical Response of Engineering Materials

Lecture 25 ViscoelasAcity II

Chap. 9

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_________ Deformation: _______________of strain when the load/stress magnitude is held ______________.

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_____________________

σ

σo

t

________________

ε

εo

t

•  Recall the governing equation:

•  The applied stress is held constant and thus _________ and •  Hence the creep response model for a Maxwellian material is

FormulaAon of the consAtuAve equaAon governing creep of a Maxwell material.

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σ Maxwell model

⇒ dεdt

= 1Edσdt

+ 1ησ

ση

σε 1dtd

E1

dtd +=

Increments of strain accumulate linearly with Ame:

•  Solve for strain at a given time t - ε(t) - by integration, with εo corresponding to instantaneous σo

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____________________________________________________________________________

The Maxwell model of creep predicts that strain accumulates linearly with time (constant rate) under constant load.

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Imposed condition σ

σo

t

Response

dεdt

= 1ησ o

dε = 1ησ odt

ε(t) = εo +σ o

ηt

σ

ε

εo

t

___________________ ________________________

models _____________ creep (i.e. ________________________ )

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The creep response is the sum of a spontaneous elastic deformation, εo, plus permanent flow, εp (plastic or creep strain) σo

__________________ tr

______________________

____________________ εo

εo

εp

εp

tr

Simplify the stress history

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σ (ksi)

σo

time

σ σo

time t t′

σo′ = σo/2

t

Method 1: Given a stress history, determine strain at Ame, tr = t.

t′

∴ε(t ) = εo +σ o

ηt

Governing equation Solution

Solve for permanent creep/plastic strains ε(t) = εo + ε p(0→ ′t ) + ε p( ′t →t )

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Method 2: Given a stress history, determine strain at Ame, t.

ε tot =

1E

dσσ t=0

σ t1

∫ + 1η

σ (t)dtt=0

t1

∫

σ (ksi)

σo

t1 t′

Integrating governing equation to obtain total strain

=σ t=0

σ t1

∫σ t=0

σt'

∫ +σ

t'

σ t1

∫

=t=0

t1

∫t=0

t '

∫ +t '

t1

∫

Maxwellian models oversimplified viscoelastic responses and thus give approximate predictions.

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______________ ε

εo

t

____________________

σ

σo

t Most materials don’t relax as _______ or ___________ as model prediction

Most materials exhibit _________ creep rates

Materials may exhibit Maxwell type responses in both stress relaxation and creep if __________ temperatures are in _______ of ______ of Tm or Tg.

Maxwellian models oversimplified viscoelastic responses and thus give approximate predictions.

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Creep ε

εo

t

Real

Maxwell Model

To overcome limitations of Maxwell model alternative arrangements of elements have been proposed: e.g. ______________________ Model with_____________________ arrangement of spring and dashpot.

Most materials exhibit nonlinear creep rates

models steady-state creep (i.e. constant creep rate )

Response of Maxwellian model with spring and dashpot in series.

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εo absorbed by the spring

with time, motion occurs in the dashpot strain in the spring decreases

instantaneous strain εo applied suddenly results in σo

σ =σ s =σ d

ε = ε s + εd

Voight-‐Kelvin Material Model: beder represents ___________________ _______________________ creep.

IniAally the dashpot must carry the enAre force because the spring can carry a force only when extended. The force in the V-‐K model will be equal to the force in the dashpot________the force in the spring: hence ____________ {_______________________} Strains are no longer______________ as the dashpot will __________ the spring to have the same deformaAon thus deformaAon compaAbility: _____________

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σ

Equilibrium and compatibility arguments can be rewritten to convey that the response is ____________________.

force equilibrium: σ(t) = σs(t) + σd(t)

compatibility: ε(t) = εs(t) = εd(t)

stress-strain relationship for the spring:

stress strain relationship for the dashpot:

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_______________

_______________

_______________

_______________

σs(t) = Eεs(t)

dεd/dt = (1/η)σd(t)

To determine the governing constitutive equation, substitute the equations for the spring and dashpot into the equilibrium equation. The governing stress-‐strain differenAal equaAon: Stress depends not only on the strain, but also the strain rate SoluAon to the first-‐order linear differenAal equaAon (see next slide)

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σ =η dε(t )dt

+ Eε(t )

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Lec25_11-18_ANGEL.pdf