NENG 506 Mechanics of NanomaterialsClass 8

J. Lloyd

Mid-Term Exam

NENG 506 2014Take home examDue next Tuesday

CRSSBCC materials have 3 different slip plane families

The slip direction family is

Assuming we are loading a cylindrical single crystal specimen oriented in the 210 direction, choose a slip system from each family of planes and compare the critical resolved shear stress on those systems to the applied stressAll else being equal, which slip system would you expect to operate first

321211110

111

Generalities

• Draw an engineering and a true stress strain curve and identify the important features– Yield Stress– Ultimate Tensile stress– Young’s Modulus– Identify where you would expect to see slip

initiate– Draw for both ductile and brittle materials

True Strain• For most materials even during plastic strain,

the strained material is incompressible, meaning that the volume doesn’t change.– Density measurements show a change of less than

0.1% change even after large plastic strain– It can be shown that for plastic strains

• This implies a Poisson’s ratio of ½ for all materials experiencing plastic strain• Note that for elastic strain, this is generally not the case

0 zyx

True Stress• This constraint (constant volume) has an effect

on the true strain. Since there is an extension in the direction of stress, there will be a reduction in area,

thus with the same load the stress increases accordingly.

110

0

0

eeA

P

A

A

A

P

A

PEng

10

0 eL

L

A

A

Von Mises Yield Criterion

• Richard von Mises (1883-1953)– Austrian mathematician and

engineer– Professor at Strassbourg, Dresden

and finally Berlin– Was a test pilot and designed and

built an airplane in 1915– Escaped Germany for Turkey in 1938 (He was Jewish)

and finally in the USA– Brother of Ludwig von Mises, a very prominent and

unconventional economist

Von Mises Yielding Criteria• Empirically the state of the hydrostatic stress has no

effect on yielding– Found that experimentally pressure did not affect yield

behavior– It was suggested by Von Mises (1913) that yield will occur

when the following (2nd invariant of the stress deviator) exceeded a certain level, k2

• where s1, s2 , and s3 are the principle stresses

2213

232

2212 6

1kJ

Von Mises Yield Criterion• Let us go through a little exercise– Apply uniaxial tension to an isotropic material– The yield stress is observed to be s0 then by

definition• Then

– Substituting into the expression for J2

• Or for arbitrary orientation

03201

k30

212132

322

2102

1

212222220 6

2

1xzyzxyxzzyyx

Von Mises Yield Criterion

• Thus we see that the yield condition is when the root mean square of the average differences of the principle stresses is above the uniaxial yield stress– Isotropic materials

• This was not justified, but later it was shown that the von Mises stress corresponds to the distortion energy.

Von Mises Yield Criterion

• Let us look at pure shear

• From the von Mises criterion – in pure shear only

– and

– which says that the yield stress in shear is significantly less than that in pure tension

0231

k1

00 58.03

1 k

Tresca Stress• Another proposal is that yield occurs when the

maximum shear stress equals the value of the shear stress in uniaxial tension

– where s1 and s3 are the largest and smallest principle stresses– It turns out experimentally that the

distortion energy consideration andtherefore the von Mises criterionagree best with experimental results

220

031

max

If you are inside the ellipse, you’re safeOutside and you yield

More von Mises

• What we have seen so far has been related to isotropic materials– Small grained macroscopic articles

• In nanoscale, this is not always the case– Seldom the case

• The solution for the von Mises criterion for anisotropic materials is much more demanding

More von Mises

• For a material with orthotropic symmetry– Orthorhombic crystals and many engineering

materials

– For the principal axes

– Where the coefficients are constants defining the degree of anisotropy

1222 222222 xyzxyzzxxzzy NMLHGF

1231

213

232 HGF

More von Mises

• If we define that Sx is the yield stress in the x direction and generalize, we can evaluate the constants by

222

111

zyx SGF

SFH

SHG

More von Mises

• You will recall that the presence of a hydrostatic stress does not contribute to yield– There is elastic deformation related to bulk modulus, but

take the pressure off and all snaps back– Therefore we define a tube for the yield surface that goes

at a 45 degree angle from the origin

Plastic Stress Strain Relationships

• Ideal Plastic Solid– Levy-Mises Equations

• The assumption is that the elastic component is very small and negligible once plastic flow begins– Not always a good assumption!

– It is recognized that unlike elastic deformation where the relationship between the deformation and the load is constant (linear), in plastic deformation things are changing as we go along• We need to introduce a new parameter for this relationship that can be

called a plastic modulus, similar to Young’s modulus but it is not constant

• The instantaneous value is used in stress strain relationship and therefore the relationship must be expressed as a derivative

Levy-Mises Equations

yzyz

xzxz

xyxy

xyzz

zxyy

zyxx

dd

dd

dd

dd

dd

dd

2

32

32

3

2

1

2

1

2

1

Where is the instantaneous

plastic modulus

The total plastic strain requires an Integration over the whole deformation

d

Note how this is similar to the elastic stress-strain relationships but with Poisson’s ratio equal to ½

Elastic-Plastic

• An important but more difficult regime is the case where the elastic strain is not negligible as compared to the plastic strain– Often the case with many real materials• Many relatively brittle materials that do yield• High strength steels, etc.

– Prandtl-Reuss Equations• Sums the contributions from equations we already

knowPij

Eijij ddd

Elastic-Plastic

• From the Elastic Strain

• From the Plastic Strain

• Therefore

3

211 kkij

Eij

d

Ed

Ed

ijPij

dd

2

3

ijkk

ijij

dd

Ed

Ed

2

3

3

211

Physics of Yielding• How does something yield– We have exhaustively (but not completely) addressed the

macroscopic yielding observations, now how does a material yield and what does it mean for us in the nano world

– Perfect lattice yielding• How much energy would it take to move translate a plane in a

perfect crystal?• Pure shear

Yield• We can approximate the force as being a

sinusoidal – In fact it is slightly different

– Assuming a sine wave

b

x 2sinmax

Yield

• For these small displacements, Hooke’s law should apply, therefore

– Equating this to the previous, we get for small displacements

a

xGG

b

x 2max

22max

G

a

bG

Yield

• More realistic estimates of this theoretical strength are slightly less, but the least is ~G/30– Actual measurements are typically 0.1 to 1% of this value

• Being off by 2 to 3 orders of magnitude is a good indication something else is happening

So, what’s happening?

Dislocations

• What can possibly be responsible for the fact that metals deform at stress levels far below the calculated strength– Later experiments with very pure metals showed

that plastic flow could start at levels as low as 10-9 G

– However experiments with whiskers (perfect and very small crystals) sometimes approached the theoretical limit

Slip Lines in Deformed Metal• Visible in optical microscopes– Observed as early as 1883

– This led to the conclusion that metals deform along slip bands • Although the crystalline nature of metals was not yet well

established

Volterra Dislocations (1907)• Dislocations in a continuous media

– Elastic stress and strains were calculated for these configurations

Dislocations

• There were a number defects that were proposed to account for the deformation and the slip bands

• In 1934 the concept of the edge dislocation was introduced and shown (Volterra Strains) to be able to account for everything

Dislocations

• Edge dislocation in a simple cubic crystal

Extra half plane

Dislocation Core

Edge Dislocation

Screw Dislocation

Screw Dislocation

Dislocations

• Visible in the electron microscope

– Mixed dislocation

Dislocations

• Dislocations in a close packed plane

– In other structures, the dislocation is difficult to visualize• When I was in graduate school, they were still arguing what a

dislocation looked like in Silicon

Bubble raft

Dislocations

• Burgers Vector– Burgers Circuit

Dislocations• Edge dislocations– Burgers vector is perpendicular to the dislocation

line• Previous slide

• Screw Dislocations– Burgers vector is parallel to the dislocation line

Dislocations

• Mixed dislocation– The Burgers vector is constant as the dislocation

line squirms through the crystal• Changes from edge to screw and mixed, but b never

changes

Dislocations

• The dislocation cannot end in the crystal, except upon itself– Must extend to the end of a crystal or to form a

loop