L14 Yield Surfaces Aniso4 13Oct09

8/3/2019 L14 Yield Surfaces Aniso4 13Oct09

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Xtal. Slip

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Plastic Anisotropy:Plastic Anisotropy:

Yield Surfaces (L14)Yield Surfaces (L14)

CarnegieMellon

MRSEC

27-750, Fall 2009

Texture, Microstructure & Anisotropy,Fall 2009

A.D. Rollett, P. Kalu

Last revised: 12th Oct. 09


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ObjectiveObjective

The objective of this lecture is to

introduce you to the topic of yield

surfaces.

Yield surfaces are useful at boththe single crystal level (material

properties) and at the polycrystal

level (anisotropy of texturedmaterials).


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OutlineOutline

What is a yield surface (Y.S.)?

2D Y.S.

Crystallographic slip

Vertices Strain Direction, normality

-plane

Symmetry

Rate sensitivity


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BibliographyBibliography

Kocks, U. F., C. Tom, H.-R. Wenk, Eds. (1998).Texture and Anisotropy, Cambridge UniversityPress, Cambridge, UK.

W. Hosford (1993),The Mechanics of Crystals andTextured Polycrystals, Oxford Univ. Press.

W. Backofen 1

972),Deformation Processing

,Addison-Wesley Longman, ISBN 0201003880.

Reid,C. N. (1973),Deformation Geometry for MaterialsScientists. Oxford, UK, Pergamon.

Khan and Huang (1999),Continuum Theory ofPlasticity,ISBN: 0-471-31043-3,Wiley.

Nye, J. F. (1957). Physical Properties of Crystals.Oxford,Clarendon Press.

T. Courtney,MechanicalBehavior of Materials,McGraw-Hill,0-07-013265-8,620.11292C86M.


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Yield Surface definitionYield Surface definition

A Yield Surface is a map in stressspace, in which an inner envelope isdrawn to demarcate non-yieldedregions from yielded (flowing) regions.

The most important feature of singlecrystal yield surfaces is thatcrystallographic slip (single system)defines a straight line in stress space

and that t

he straining direction isperpendicular (normal) to that line.


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Plastic potentialPlastic potentialppYield SurfaceYield Surface

One can define aplastic potential,*,whose differential with respect to thestress deviator provides the strain rate.By definition, the strain rate is normal to

the iso-potential surface.

Provided that the critical resolved shear stress (also in the sense ofthe rate-sensitive reference stress) is not dependent on the currentstress state, then the plastic potential and the yield surface (defined

by Xcrss) are equivalent. If the yield depends on the hydrostaticstress, for example, then the two may not correspond exactly.


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Yield surfaces: introductionYield surfaces: introduction

The best way to learn about yield

surfaces is think of them as a

graphical construction.

A yield surface is the boundarybetween elastic and plastic flow.

Example: tensile stress

W=0 Welastic plastic

W=

Wyield


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2D yield surfaces2D yield surfaces

Yield surfaces can be defined in twodimensions.

Consider a combination of

(independent) yield on two different

axes.The material

is elastic if

W1 < W1y

andW2 < W2y

0 W

W

elastic

plastic

plastic

W=

Wy

W=Wy


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Xtal. Slip

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2D yield surfaces, contd.2D yield surfaces, contd.

The Tresca yield criterion is familiarfrom mechanics of materials:

0 W

W

elastic

plastic

plastic

W=Wk

W=Wk

The material

is elastic if the

difference

between the 2principal

stresses is less

than a critical

value,Wk

,which is a

maximum

shear stress.


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2D yield surfaces, contd.2D yield surfaces, contd.

Graphical representations of yield surfacesare generally simplified to the envelope of thedemarcation line between elastic and plastic.Thus it appears as apolygonal or

curved object thatis closed andconvex (hencethe term convexhullis applied).

This plot showsboth the Trescaand the von Misescriteria.

elastic

plastic

W=Wyield


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Crystallographic slip:Crystallographic slip:

a single systema single system Now that we understand the concept of

a yield surface we can apply it to

crystallographic slip.

The result of slip

on a single system

is strain in a single

direction, which

appears as a straight

line on the Y.S.

[Kocks]


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Xtal. Slip

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A single slip systemA single slip system

Yield criterion for single slip:

biWijnj u Xcrss In 2D this becomes (W1|W:

b1W1n1+ b2W2n2 u XcrssThe second

equation defines

a straight lineconnecting the

intercepts0 W

W

Xcrss/b1n1

Xcrss/b2n2

elastic

plastic


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A single slip system: strain directionA single slip system: strain direction

Now we can ask, what is the straining

direction?

The strain increment is given by:

dI =7s dK(s)

b

(s)

n

(s)

which in our2D case becomes:

dI1 =dK b1n1; dI2 =dK b2n2

This defines a vector that is

perpendicular to the line for yield!W2 = (constant- b1W1n1)/(b2n2)


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Single system: normalitySingle system: normality

We can draw t

he straining direction in t

hesame space as the stress.

The fact that the strain is perpendicular to

the yield surface is a demonstration of the

normality rule for crystallographic slip.

0 W

W

Xcrss/b1n1

Xcrss/b2n2

elastic

plastic

dI =dK (b1n1, b2n2)


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Druckers PostulateDruckers Postulate

We have demonstrated that the physicsof crystallographic slip guarantees

normality of plastic flow. Drucker (d. 2001) showed that plastic

solids in general must obey thenormality rule. This in turn means thatthe yield surface must be convex.Crystallographic slip also guaranteesconvexity of polycrystal yield surfaces.

Details on Druckers Postulate insupplemental slides.


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Vertices on the Y.S.Vertices on the Y.S.

Based on the normality rule, we cannow examine what happens at the

corners, or vertices, of a Y.S.

The single slip conditions on either side

of a vertex define limits on the strainingdirection: at the vertex, the straining

direction can lie anywhere in between

these limits.

Thus, we speak of a cone of normals ata vertex.


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Cone of normalsCone of normals

dIa

dIb

Vertex

[Kocks]

Cone of normals: the straining direction can lie

anywhere within the cone


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Single crystalY.S.Single crystalY.S.

Cube

component:

(001)[100]

Backofen

DeformationProcessing


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Single crystalY.S.: 2Single crystalY.S.: 2

Gosscomponent:

(110)[001]

From the

thesis work

of Prof.Piehler


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Single crystalY.S.: 3Single crystalY.S.: 3

Copper:

(111)[112]


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PolycrystalYield SurfacesPolycrystalYield Surfaces

As discussed in the tour of LApp,the method of calculation of apolycrystal Y.S. is simple. Eachpoint on the Y.S. corresponds to aparticular straining direction: thestress state of the polycrystal isthe average of the stresses in the

individual grains.


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PolycrystalY.S. constructionPolycrystalY.S. construction

2 methods commonly used:

(a) locus of yield points in stress

space

(b) convex hull of tangents Yield point loci is straightforward:

simply plot the stress in 2D (or

high

er) space.


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Tangent constructionTangent construction

(1) Draw a line from the origin parallel tothe applied strain direction.

(2) Locate the distance from the origin bythe average Taylor factor.

(3) Draw a perpendicular to the radius.

(4) Repeat for all strain directions ofinterest.

(5) The yield surface is the innerenvelope of the tangent lines.


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Tangent construction: 2Tangent construction: 2

W

WdI

[Kocks]


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The piThe pi--planeY.S.planeY.S.

A particularly useful yield surface is theso-called -plane, i.e. the projectiondown the line corresponding to purehydrostatic stress (all 3 principal

stresses equal). For an isotropicmaterial, the -plane has 120rrotational symmetry with mirrors suchthat only a 60r sector is required (asthe fundamental zone). For the vonMises criterion, the -plane Y.S. is acircle.


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Principal Stress --planeplane

Hosford: mechanics of crystals...


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Isotropic materialIsotropic material

[Kocks]

Note that an

isotropic material

has a Y.S. in

Between theTresca and the

von Mises

surfaces


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Y.S. for textured polycrystalY.S. for textured polycrystal

Kocks: Ch.10

Note sharp

vertices forstrong textures

at large strains.


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Symmetry & the Y.S.Symmetry & the Y.S.

We can write the relationshipbetween strain (rate,D) and stress

(deviator,S) as a general non-

linear relation

D=F(S)


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Effect on stimulus (stress)Effect on stimulus (stress)

The non-linearity of the property (plastic flow) meansthat care is needed in applying symmetry because weare concerned not with the coefficients of a linearproperty tensor but with the existence of non-zerocoefficients in a response (to a stimulus). That is tosay, we cannot apply the symmetry element directly to

the property because the non-linearity means that(potentially) an infinity ofhigher order terms exist. Theaction of a symmetry operator,however, means that wecan examine the following special case. If the fieldtakes a certain form in terms of its coefficients then thesymmetry operator leaves it unchanged and we can

write:S =OSOT

Note that the application of symmetry operators to a second rank

tensor, such as deviatoric stress, is exactly equivalent to the

standard tensor transformation rule:


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Response(Field)Response(Field)

Then we can insert this into the relationbetween the response and the field:

ODOT =F(OSOT)=F(S)=D

The resulting identity between the

strain and the result of the symmetryoperator on the strain then requires

similar constraints on the coefficients of

the strain tensor.


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Example: mirror on YExample: mirror on Y

Kocks (p343) quotes an analysis for the action of amirror plane (note the use of the second kind ofsymmetry operatorhere) perpendicular to sample Y toshow that the subspace {,W31} is closed. That is, anycombination ofWii and W31 will only generate strain ratecomponents in the same subspace, i.e. Dii and D31.

The negation of the 12 and 23 components means thatif these stress components are zero, then the stressdeviator tensor is equal to the stress deviator under theaction of the symmetry element. Then the resultingstrain must also be identical to that obtained without thesymmetry operator and the corresponding 12 and 23

components ofD must also be zero. That is, twostresses related by this mirror must have W12 and W23zero, which means in turn that the two related strainstates must also have those components zero.


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Mirror on Y: 2Mirror on Y: 2

Consider t

he equation above: anystress state for whichW12 and W23 are

zero will satisfy the following relationfor the action of the symmetry element(in this case a mirror on Y):

OSOT = S


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Mirror on Y: 2Mirror on Y: 2

Provided the stress obeys this relation,

then the relation ODOT =D also holds.Based on the second equation quoted

from Kocks, we can see that only strain

states for whichD12 and D23 = 0 will

satisfy this equation.


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Symmetry: summarySymmetry: summary

Thus we have demonstrated with anexample that stress states that obey a

symmetry element generate straining

directions that also obey the symmetry

element. More importantly, the yieldsurface for stress states obeying the

symmetry element are closed in the

sense that they do not lead to straining

components outside that same space.


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Rate sensitive yieldRate sensitive yield The rate at which dislocations move under the

influence of a shear stress (on their glide plane) isdependent on the magnitude of the shear stress.Turning the statement around, one can say that theflow stress is dependent on the rate at whichdislocations move which, through the Orowan equation,given below, means that the "critical" resolved shearstress is dependent on the strain rate. The first figurebelow illustrates this phenomenon and also makes thepoint that the rate dependence is strongly non-linear inmost cases. Although the precise form of the strainrate sensitivity is complicated if the complete range ofstrain rate must be described, in the vicinity of the

macroscopically observable yield stress, it can beeasily described by a power-law relationship, where nis the strain rate sensitivity exponent. Here is theOrowan equation:


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Shear strain rateShear strain rate

The crss (Xcrss) becomes a referencestress (as opposed to a limiting stress).

For the purposes of simulating texture,the shear rate on each system isnormalized to a reference strain rateand the sign of the slip rate is treatedseparately from the magnitude.


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Sign dependenceSign dependence

Note that, in principle, both the criticalresolved shear stress andthe strainrate exponent,n, can be different oneach slip system. This is, for example,

a way to model latent hardening, i.e. byvarying the crss on each system as afunction of the slip history of thematerial.


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Effect on single crystalY.S.Effect on single crystalY.S.

Note the

rounding-off

of the yield

surface as a

consequence of

rate-sensitive

yield

[Kocks]


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Rate sensitivity: summaryRate sensitivity: summary

The impact of strain rate sensitivity on the single crystalyield surface (SCYS) is then easy to recognize. Theconsequence of the normalization of the strain rate issuch that if more than one slip system operates, theresolved shear stress on each system is less than thereference crss. Thus the second diagram, above,

shows that, in the vicinity of a vertex in the SCYS, theyield surface is rounded off. The greater the ratesensitivity, or the smaller the value ofn, the greater thedegree of rounding. In most polycrystal plasticitysimulations, the value ofn chosen to be small enough,e.g. n=30, that the non-linear solvers operate efficiently,

but large enough that the texture development is notaffected. Experience with the LApp model indicates thatanisotropy and texture development are significantlyaffected only when small values of the rate sensitivityexponent are used,ne5.


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Plastic Strain Ratio (rPlastic Strain Ratio (r--value)value)

)2(2

1)(

)2(41)(

)/ln(

)/ln(

)/ln(

)/ln(

90450

90450

rrranisotropyplanarr

rrrvaluerr

WLWfL

WfW

TfTi

WfWr

m

iif

ii

!(

!

!!

Large rm and small r requiredfor deep drawing

LiWi

Rolling Direction

45r

90r

0r

W

W


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RR--value & the Y.S.value & the Y.S.

The r-value is a differentialproperty of the polycrystal yield

surface, i.e. it measures the slope

of the surface. Why? The Lankford parameter is a

ratio of strain components:

r=Iw

idth/I

thickness

Iwidth

Ithickness

r=slope


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A A --plane Y.S.: fcc rollingplane Y.S.: fcc rolling

texture at a strain of 3texture at a strain of 3

S11

dI11 ~ 0

r ~ 0dI22~ dI33r ~ 1

Note: the Taylor

factors for

loading in the

RD and the TD

are nearlyequal but the

slopes are very

different!

RDTD

ND


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How to obtain r at other angles?How to obtain r at other angles?

Consider the stress system in a tensiletest in the plane of a sheet.

Mohrs circle shows that a shear stress

component is required in addition to thetwo principal stresses.

Therefore a third dimension must beadded to be standard W11-W22 yield

surface.


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Stress system in tensile testsStress system in tensile tests

For a test at an arbitrary angle to therolling direction:

Note: the corresponding strain tensor

may have all non-zero components.

W !

W11 W12 0

W12 W22 0

0 0 0


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3D Y.S. for r3D Y.S. for r--valuesvalues Think of an r-

value scan asgoing up-and-over the 3Dyield surface.

Hosford: Mechanics ofCrystals...

2WM !

a K1 K2M

a K1 K2M

(2 a)2K2M

K1 ! W xx hWyy / 2

K2 ! W xx hWyy /2? A2

p2Xxy2


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SummarySummary

Yield surfaces are an extremely usefulconcept for quantifying the anisotropyof materials.

Graphical representations of the Y.S.aid in visualization of anisotropy.

Crystallographic slip guaranteesnormality.

Certain types of anisotropy requirespecial calculations, e.g. r-value.


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Supplemental SlidesSupplemental Slides


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Druckers PostulateDruckers Postulate

The material is said to be stable inthe sense of Drucker if the work

done by the tractions,ti, through

the displacements,ui, is positiveor zero for all ti:


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Drucker, contd.Drucker, contd.

This statement is somewhatanalogous (but not equivalent) to thesecond law of thermodynamics. Astable material is strongly dissipative.It can be shown that, for a plastic

material to be stable in this sense, itmust satisfy the following conditions:

The yield surface,f(Wij), must beconvex;

The plastic strain rate must be normal

to the yield surface; The rate of strain hardening must be

positive or zero.

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L14 Yield Surfaces Aniso4 13Oct09