Chapter 3

 

Constitutive Models MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqabKqzaiaeaa aaaaaaa8qacaWFtacaaa@3788@  Relations between Stress and Strain

 

 

3.12 Constitutive models for metal single crystals

 

Plastic flow in a single crystal is anisotropic, and so cannot be modeled using the simple constitutive equations described in Section 3.6.  Instead, a more complicated constitutive law is used, which considers the slip activity in the crystal directly. The main application of the constitutive equation is to model the rotations of individual grains in a polycrystal, and hence to predict the evolution of texture, and to account for the effects of texture on the development of anisotropy in the solid.

 

 

3.12.1 Review of some important concepts from crystallography

 

 Common crystal structures

 

Most metal crystals of practical interest have either face centered cubic, body centered cubic, or hexagonal crystal structures.  These are illustrated in the picture below.  Crystal plasticity models exist for all three crystal structures, but fcc materials are the most extensively studied.

 

 Miller index notation for crystallographic planes and directions

 

Planes and directions in a single crystal are referred to as follows.   For a cubic crystal, we choose basis vectors { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3968@  perpendicular to the faces of the basic cubic unit cell, as illustrated in the picture.  Then

1.      The symbol [l,m,n], where l,m,n are positive integers, denotes a direction parallel to a unit vector with components ( l e 1 +m e 2 +n e 3 )/ l 2 + m 2 + n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiBaiaahwgadaWgaaWcba GaaGymaaqabaGccqGHRaWkcaWGTbGaaCyzamaaBaaaleaacaaIYaaa beaakiabgUcaRiaad6gacaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcca GLOaGaayzkaaGaai4lamaakaaabaGaamiBamaaCaaaleqabaGaaGOm aaaakiabgUcaRiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkca WGUbWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4457@

2.      The symbol [l, m ¯ ,n] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaWGSbGaaiilaiqad2gagaqeai aacYcacaWGUbGaaiyxaaaa@3676@  denotes a direction parallel to the unit vector with components ( l e 1 m e 2 +n e 3 )/ l 2 + m 2 + n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiBaiaahwgadaWgaaWcba GaaGymaaqabaGccqGHsislcaWGTbGaaCyzamaaBaaaleaacaaIYaaa beaakiabgUcaRiaad6gacaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcca GLOaGaayzkaaGaai4lamaakaaabaGaamiBamaaCaaaleqabaGaaGOm aaaakiabgUcaRiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkca WGUbWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4462@

3.      The symbol <l,m,n> denotes the family of [l,m,n] directions that are identical due to the symmetry of the crystal.  For example <111> MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgYda8iaaigdacaaIXaGaaGymaiabg6 da+aaa@34A5@  in a cubic crystal includes all of

[111],[ 1 ¯ 11],[1 1 ¯ 1],[11 1 ¯ ],[ 1 ¯ 1 ¯ 1],[ 1 ¯ 1 1 ¯ ],[1 1 ¯ 1 ¯ ],[ 1 ¯ 1 ¯ 1 ¯ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfacaaIXaGaaGymaiaaigdacaGGDb GaaiilaiaacUfaceaIXaGbaebacaaIXaGaaGymaiaac2facaGGSaGa ai4waiaaigdaceaIXaGbaebacaaIXaGaaiyxaiaacYcacaGGBbGaaG ymaiaaigdaceaIXaGbaebacaGGDbGaaiilaiaacUfaceaIXaGbaeba ceaIXaGbaebacaaIXaGaaiyxaiaacYcacaGGBbGabGymayaaraGaaG ymaiqaigdagaqeaiaac2facaGGSaGaai4waiaaigdaceaIXaGbaeba ceaIXaGbaebacaGGDbGaaiilaiaacUfaceaIXaGbaebaceaIXaGbae baceaIXaGbaebacaGGDbaaaa@55E0@ .

4.      The symbol (l,m,n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGSbGaaiilaiaad2gacaGGSa GaamOBaiaacMcaaaa@35F7@  denotes a plane that is perpendicular to a unit vector with components ( l e 1 +m e 2 +n e 3 )/ l 2 + m 2 + n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiBaiaahwgadaWgaaWcba GaaGymaaqabaGccqGHRaWkcaWGTbGaaCyzamaaBaaaleaacaaIYaaa beaakiabgUcaRiaad6gacaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcca GLOaGaayzkaaGaai4lamaakaaabaGaamiBamaaCaaaleqabaGaaGOm aaaakiabgUcaRiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkca WGUbWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4457@

5.      The symbol (l, m ¯ ,n) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGSbGaaiilaiqad2gagaqeai aacYcacaWGUbGaaiykaaaa@360F@  denotes a plane that is perpendicular to a unit vector with components ( l e 1 m e 2 +n e 3 )/ l 2 + m 2 + n 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiBaiaahwgadaWgaaWcba GaaGymaaqabaGccqGHsislcaWGTbGaaCyzamaaBaaaleaacaaIYaaa beaakiabgUcaRiaad6gacaWHLbWaaSbaaSqaaiaaiodaaeqaaaGcca GLOaGaayzkaaGaai4lamaakaaabaGaamiBamaaCaaaleqabaGaaGOm aaaakiabgUcaRiaad2gadaahaaWcbeqaaiaaikdaaaGccqGHRaWkca WGUbWaaWbaaSqabeaacaaIYaaaaaqabaaaaa@4462@

6.      The symbol {l,m,n} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWGSbGaaiilaiaad2gacaGGSa GaamOBaiaac2haaaa@369E@  denotes the family of (l,m,n) planes that are crystallographically identical by symmetry.

 

For a hexagonal crystal, planes and directions are defined by introducing four auxiliary unit vectors m i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaamyAaaqabaaaaa@3278@ , i=1..4 as shown in the picture above.  The first three unit vectors lie in the basal plane and are oriented parallel to the three shortest distances between neighboring atoms.  The fourth vector is perpendicular to the basal plane.  These vectors are related to a Cartesian basis { e 1 , e 2 , e 3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHLbWaaSbaaSqaaiaaigdaae qaaOGaaiilaiaahwgadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaaCyz amaaBaaaleaacaaIZaaabeaakiaac2haaaa@3968@  by

m 1 = e 1 m 2 =( e 1 + 3 e 2 )/2 m 3 =( e 1 3 e 2 )/2 m 4 = e 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gadaWgaaWcbaGaaGymaaqabaGccq GH9aqpcaWHLbWaaSbaaSqaaiaaigdaaeqaaOGaaGPaVlaaykW7caaM c8UaaCyBamaaBaaaleaacaaIYaaabeaakiabg2da9maabmaabaGaey OeI0IaaCyzamaaBaaaleaacaaIXaaabeaakiabgUcaRmaakaaabaGa aG4maaWcbeaakiaahwgadaWgaaWcbaGaaGOmaaqabaaakiaawIcaca GLPaaacaGGVaGaaGOmaiaaykW7caaMc8UaaGPaVlaaykW7caWHTbWa aSbaaSqaaiaaiodaaeqaaOGaeyypa0ZaaeWaaeaacqGHsislcaWHLb WaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0YaaOaaaeaacaaIZaaaleqa aOGaaCyzamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaiaac+ cacaaIYaGaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa Vlaah2gadaWgaaWcbaGaaGinaaqabaGccqGH9aqpcaWHLbWaaSbaaS qaaiaaiodaaeqaaaaa@684D@

Then

1.      The symbol [l,m,n,p], where l,m,n,p are positive integers, denotes a direction parallel to a unit vector  ( l m 1 +m m 2 +n m 3 +p m 4 )/ l 2 + m 2 + n 2 lmlnmn+ p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiBaiaah2gadaWgaaWcba GaaGymaaqabaGccqGHRaWkcaWGTbGaaCyBamaaBaaaleaacaaIYaaa beaakiabgUcaRiaad6gacaWHTbWaaSbaaSqaaiaaiodaaeqaaOGaey 4kaSIaamiCaiaah2gadaWgaaWcbaGaaGinaaqabaaakiaawIcacaGL PaaacaGGVaWaaOaaaeaacaWGSbWaaWbaaSqabeaacaaIYaaaaOGaey 4kaSIaamyBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaad6gadaah aaWcbeqaaiaaikdaaaGccqGHsislcaWGSbGaamyBaiabgkHiTiaadY gacaWGUbGaeyOeI0IaamyBaiaad6gacqGHRaWkcaWGWbWaaWbaaSqa beaacaaIYaaaaaqabaaaaa@536D@

2.      A bar over one of the indices [ l ¯ ,m,n,p] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUfaceWGSbGbaebacaGGSaGaamyBai aacYcacaWGUbGaaiilaiaadchacaGGDbaaaa@381B@  changes the sign of the index, exactly as for cubic crystals;

3.      The symbol <l,m,n,p> denotes the family of [l,m,n,p] directions that are crystallographically identical by symmetry. 

4.      The symbol (l,m,n,p) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGSbGaaiilaiaad2gacaGGSa GaamOBaiaacYcacaWGWbGaaiykaaaa@379C@  denotes a plane that is perpendicular to a unit vector with components ( l m 1 +m m 2 +n m 3 +p m 4 )/ l 2 + m 2 + n 2 lmlnmn+ p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiBaiaah2gadaWgaaWcba GaaGymaaqabaGccqGHRaWkcaWGTbGaaCyBamaaBaaaleaacaaIYaaa beaakiabgUcaRiaad6gacaWHTbWaaSbaaSqaaiaaiodaaeqaaOGaey 4kaSIaamiCaiaah2gadaWgaaWcbaGaaGinaaqabaaakiaawIcacaGL PaaacaGGVaWaaOaaaeaacaWGSbWaaWbaaSqabeaacaaIYaaaaOGaey 4kaSIaamyBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaad6gadaah aaWcbeqaaiaaikdaaaGccqGHsislcaWGSbGaamyBaiabgkHiTiaadY gacaWGUbGaeyOeI0IaamyBaiaad6gacqGHRaWkcaWGWbWaaWbaaSqa beaacaaIYaaaaaqabaaaaa@536D@

5.      The symbol (l, m ¯ ,n,p) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacIcacaWGSbGaaiilaiqad2gagaqeai aacYcacaWGUbGaaiilaiaadchacaGGPaaaaa@37B4@  denotes a plane that is perpendicular to a unit vector with components ( l m 1 m m 2 +n m 3 +p m 4 )/ l 2 + m 2 + n 2 lmlnmn+ p 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaabmaabaGaamiBaiaah2gadaWgaaWcba GaaGymaaqabaGccqGHsislcaWGTbGaaCyBamaaBaaaleaacaaIYaaa beaakiabgUcaRiaad6gacaWHTbWaaSbaaSqaaiaaiodaaeqaaOGaey 4kaSIaamiCaiaah2gadaWgaaWcbaGaaGinaaqabaaakiaawIcacaGL PaaacaGGVaWaaOaaaeaacaWGSbWaaWbaaSqabeaacaaIYaaaaOGaey 4kaSIaamyBamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaad6gadaah aaWcbeqaaiaaikdaaaGccqGHsislcaWGSbGaamyBaiabgkHiTiaadY gacaWGUbGaeyOeI0IaamyBaiaad6gacqGHRaWkcaWGWbWaaWbaaSqa beaacaaIYaaaaaqabaaaaa@5378@

6.      The symbol {l,m,n,p} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWGSbGaaiilaiaad2gacaGGSa GaamOBaiaacYcacaWGWbGaaiyFaaaa@3843@  denotes the family of (l,m,n,p) planes that are crystallographically identical by symmetry.

 

 

 Representing crystallographic directions and orientations using stereographic projections

 

Stereographic projection is a way to represent 3D orientations on a 2D plane.  The figure shows one way to interpret the projection:

1.      The specimen is placed in the center of an imaginary sphere with unit radius, with some convenient material directions aligned with the {i,j,k} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHPbGaaiilaiaahQgacaGGSa GaaC4Aaiaac2haaaa@36A1@  basis. 

2.      A direction of interest is represented by a unit vector m=xi+yj+zk MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaah2gacqGH9aqpcaWG4bGaaCyAaiabgU caRiaadMhacaWHQbGaey4kaSIaamOEaiaahUgaaaa@39FB@ , which intersects the sphere at some point P on its surface. 

3.      A line is then drawn from P to the point where the k axis intersects the sphere at Q.

4.      The line PQ cuts through the equatorial plane of the sphere at some point R

5.      The vector OR ρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaamaaFiaabaGaam4taiaadkfaaiaawEniai abggMi6kaahg8aaaa@36DD@  is the stereographic projection of the orientation m.

6.      The general conversion between the 2D projection and the 3D unit vector is easily shown to be

ρ= m(mk)k 1+mk m= 2ρ+k( 1ρρ ) 1+ρρ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaahg8acqGH9aqpdaWcaaqaaiaah2gacq GHsislcaGGOaGaaCyBaiabgwSixlaahUgacaGGPaGaaC4Aaaqaaiaa igdacqGHRaWkcaWHTbGaeyyXICTaaC4AaaaacaaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa Vlaah2gacqGH9aqpdaWcaaqaaiaaikdacaWHbpGaey4kaSIaaC4Aam aabmaabaGaaGymaiabgkHiTiaahg8acqGHflY1caWHbpaacaGLOaGa ayzkaaaabaGaaGymaiabgUcaRiaahg8acqGHflY1caWHbpaaaaaa@667A@

7.      The symmetry of the crystal makes the ±m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabgglaXkaah2gaaaa@334C@  directions equivalent.  For this reason, projections usually only show vectors with a positive k component (the projection of a vector with a negative k component lies outside the sphere)

 

 Pole figures and inverse pole figures

 

In crystal plasticity, the projection is used in two ways. In one approach, specific crystallographic directions are chosen to be aligned with the {i,j,k} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHPbGaaiilaiaahQgacaGGSa GaaC4Aaiaac2haaaa@36A1@  directions, and other directions of interest (which could be other crystallographic directions, or the direction of the loading axis in a tensile test, for example) are projected. This is known as an inverse pole figure.

 

For example, the figure shows the standard projection for a cubic crystal. To interpret the figure, note that

1.      The [100],[010] and [001] directions are parallel to i, j, k, respectively. HEALTH WARNING: this is not the only choice of crystal orientation MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you will often see the inverse pole figure with [100] parallel to i, for example.

2.      The points mark the projections of the specified crystallographic directions.

3.      The lines mark the traces of the planes specified: that is to say, the projection of the line where the plane intersects the surface of the unit sphere. 

4.      The figure also shows the standard triangle for a cubic crystal.  Notice that the traces of the planes divide the plane into a set of 24 curvilinear triangles, each of which has  <100>, <110> and <111> directions at its corners.  These triangles are indistinguishable because of the symmetry of the crystal MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  you could interchange any two triangles by applying an appropriate rigid rotation to the crystal, without influencing the arrangement of atoms.   This has important consequences: for example, when testing the response of a cubic crystal to uniaxial tension, you only need to run tests with the loading direction inside the standard triangle.  For this reason, inverse pole figures often only display the standard triangle. 

 

A second application of the stereographic projection in crystal plasticity is to display pole figures.  In this approach, specific physical directions are chosen to be aligned with the {i,j,k} MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaacUhacaWHPbGaaiilaiaahQgacaGGSa GaaC4Aaiaac2haaaa@36A1@  basis, and the orientations of crystallographic directions of interest are displayed on the projection. 

 

For example, when a pole figure is plotted for a rolled sheet specimen, the rolling direction (denoted RD) is often chosen to be parallel to j, the direction transverse to the rolling direction in the plane of the sheet (denoted TD) is chosen to be parallel to i, and the direction perpendicular to the sheet (denoted ND) is chosen to be parallel to k.  The sheet generally contains many grains, and each grain is a single crystal.  The orientations the grains are displayed on a pole figure by choosing some convenient crystallographic direction (<100> or <111> are common) and plotting the stereographic projection for each member of this family of crystallographic directions.  In a cubic crystal, each grain contributes four points to the projection (there are 8 <100> directions but only 4 of them have positive k component).  A typical <100> pole figure for a rolled aluminum sheet after a 40% reduction is shown in the picture.   The pole figure shows that grain orientations tend to cluster together, indicating that the sheet has developed a texture.

 

 

 

3.12.2 Features of plastic flow in single crystals

 

Plastic flow in a crystal is most often measured by conducting a tensile test with the loading axis parallel to a chosen crystallographic direction.  The main results of these experiments are:

 

 

 

1.      For most orientations of the loading axis, the plastic flow initially consists of shearing parallel to one member of a family crystallographic planes in the crystal, in the direction of a vector s lying in that plane, as illustrated in the picture above.  The crystallographic plane on which shear occurs is called a slip plane.  The shearing direction is known as the slip Direction.  Slip planes and directions for common crystals are listed in the table.

 

Structure

Slip Direction

Slip plane

fcc

<110>

{111}

bcc

<111>

{110},{112}

hcp

<11 2 ¯ 0> <11 2 ¯ 3> MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiabgYda8iaaigdacaaIXaGabG OmayaaraGaaGimaiabg6da+aqaaiabgYda8iaaigdacaaIXaGabGOm ayaaraGaaG4maiabg6da+aaaaa@3CF0@

(0001),{1 1 ¯ 11} {1 1 ¯ 01},{1 1 ¯ 0 1 ¯ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaacIcacaaIWaGaaGimaiaaic dacaaIXaGaaiykaiaacYcacaGG7bGaaGymaiqaigdagaqeaiaaigda caaIXaGaaiyFaaqaaiaacUhacaaIXaGabGymayaaraGaaGimaiaaig dacaGG9bGaaiilaiaacUhacaaIXaGabGymayaaraGaaGimaiqaigda gaqeaiaac2haaaaa@4791@

 

2.      Crystals contain a large number of candidate slip systems.  For example, an fcc crystal contains 12 possible slip systems.  These are conventionally designated as listed in the table below. 

 

Slip plane

Slip direction

 

 

(111)

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[10 1 ¯ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaaGymaiaaicdaceaIXaGbae bacaGGDbaaaa@36CE@

a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaaikdaaeqaaa aa@3494@

[ 1 ¯ 10] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGabGymayaaraGaaGymaiaaic dacaGGDbaaaa@36CE@

a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaaiodaaeqaaa aa@3495@

( 1 ¯ 1 ¯ 1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGabGymayaaraGabGymayaara GaaGymaiaacMcaaaa@3680@

[011] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaaGimaiaaigdacaaIXaGaai yxaaaa@36B6@

b 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaaigdaaeqaaa aa@3494@

[ 1 ¯ 0 1 ¯ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGabGymayaaraGaaGimaiqaig dagaqeaiaac2faaaa@36E6@

b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaaikdaaeqaaa aa@3495@

[1 1 ¯ 0] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaaGymaiqaigdagaqeaiaaic dacaGGDbaaaa@36CE@

b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGIbWaaSbaaSqaaiaaiodaaeqaaa aa@3496@

( 1 ¯ 11) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGabGymayaaraGaaGymaiaaig dacaGGPaaaaa@3668@

[0 1 ¯ 1] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaaGimaiqaigdagaqeaiaaig dacaGGDbaaaa@36CE@

c 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaigdaaeqaaa aa@3495@

[ 1 ¯ 0 1 ¯ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGabGymayaaraGaaGimaiqaig dagaqeaiaac2faaaa@36E6@

c 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaikdaaeqaaa aa@3496@

[110] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaaGymaiaaigdacaaIWaGaai yxaaaa@36B6@

c 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGJbWaaSbaaSqaaiaaiodaaeqaaa aa@3497@

(1 1 ¯ 1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGOaGaaGymaiqaigdagaqeaiaaig dacaGGPaaaaa@3668@

[011] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaaGimaiaaigdacaaIXaGaai yxaaaa@36B6@

d 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaigdaaeqaaa aa@3496@

[10 1 ¯ ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGaaGymaiaaicdaceaIXaGbae bacaGGDbaaaa@36CE@

d 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaikdaaeqaaa aa@3497@

[ 1 ¯ 1 ¯ 0] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaGGBbGabGymayaaraGabGymayaara GaaGimaiaac2faaaa@36E6@

d 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGKbWaaSbaaSqaaiaaiodaaeqaaa aa@3498@

 

3.      The slip systems in the undeformed crystal are identified by unit vectors m α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaWbaaSqabeaacqaHXoqyaa aaaa@3588@  normal to the slip plane, together with unit vectors s α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHZbWaaWbaaSqabeaacqaHXoqyaa aaaa@358E@  parallel to the slip direction. Here α=1...N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqycqGH9aqpcaaIXaGaaiOlai aac6cacaGGUaGaamOtaaaa@390F@  and N denotes the total number of slip systems (eg N=12 for an fcc crystal).  The crystal can rotate during deformation.  In the deformed solid the slip plane normals and slip directions are denoted m *α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHTbWaaWbaaSqabeaacaGGQaGaeq ySdegaaaaa@3636@ , s *α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWHZbWaaWbaaSqabeaacaGGQaGaeq ySdegaaaaa@363C@ .

 

4.      In a tensile test on an annealed fcc single crystal, shearing occurs on the slip system that is subjected to the largest resolved shear stress.  The resolved shear stress on the α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@  th system can be computed from the Cauchy stress σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHdpWCdaWgaaWcbaGaamyAaiaadQ gaaeqaaaaa@3692@  acting on the solid as

τ α =J σ ij m i *α s j *α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHepaDdaahaaWcbeqaaiabeg7aHb aakiabg2da9iaadQeacqaHdpWCdaWgaaWcbaGaamyAaiaadQgaaeqa aOGaamyBamaaDaaaleaacaWGPbaabaGaaiOkaiabeg7aHbaakiaado hadaqhaaWcbaGaamOAaaqaaiaacQcacqaHXoqyaaaaaa@44D1@

where J=dV/d V 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0JaamizaiaadAfaca GGVaGaamizaiaadAfadaWgaaWcbaGaaGimaaqabaaaaa@39BC@  is the ratio of deformed to undeformed volume of the specimen ( J1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyisISRaaGymaaaa@3601@  ). For example, the inverse pole figure shows the active slip system for all possible orientations of the tensile axis with respect to an fcc crystal (a bar over a slip system indicates a negative resolved shear stress).  In materials with other crystal structures, some slip systems may be inherently stronger than others.  In this case slip occurs on the system with highest value of τ α / g α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiabes8a0naaCaaaleqabaGaeqySdegaaO Gaai4laiaadEgadaahaaWcbeqaaiabeg7aHbaaaaa@376E@ , where g α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadEgadaahaaWcbeqaaiabeg7aHbaaaa a@3320@  denotes the strengths of the slip systems

 

5.      Slip on the critical system initiates when the resolved shear stress exceeds a critical magnitude (the strength of the slip system) τ α > g α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHepaDdaahaaWcbeqaaiabeg7aHb aakiabg6da+iaadEgadaahaaWcbeqaaiabeg7aHbaaaaa@3A21@ .  The strength of the slip systems increases with plastic straining: this behavior will be discussed in more detail later.

 

6.      For special orientations of the tensile axis, more than one slip system may be activated.  For example, if an fcc crystal is loaded parallel to a <100> direction, 8 slip systems are subjected to the same resolved shear stress, and so are active at the same time (the inverse pole figure above shows the active systems)

 

7.      The deformation gradient resulting from a shear strain γ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHZoWzdaahaaWcbeqaaiabeg7aHb aaaaa@3639@  on the α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@  th system is

F ij = R ik ( δ kj + γ α s k α m j α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadkfadaWgaaWcbaGaamyAaiaadUgaaeqaaOWa aeWaaeaacqaH0oazdaWgaaWcbaGaam4AaiaadQgaaeqaaOGaey4kaS Iaeq4SdC2aaWbaaSqabeaacqaHXoqyaaGccaWGZbWaa0baaSqaaiaa dUgaaeaacqaHXoqyaaGccaWGTbWaa0baaSqaaiaadQgaaeaacqaHXo qyaaaakiaawIcacaGLPaaaaaa@4AAC@

where R ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35A6@  is a proper orthogonal tensor (i.e. det(R)=1, R ik R jk = δ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadMgacaWGRb aabeaakiaadkfadaWgaaWcbaGaamOAaiaadUgaaeqaaOGaeyypa0Ja eqiTdq2aaSbaaSqaaiaadMgacaWGQbaabeaaaaa@3D51@  ), representing a rigid rotation.  The rotation is determined by the way the solid is loaded.   For example, in a tensile test, R ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35A6@  is often calculated from the condition that the material fiber parallel to the loading axis (specified by a unit vector p) does not rotate during deformation.  This gives

R ij = δ ij cosθ+(1cosθ) n i n j +sinθ ikj n k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGsbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGc ciGGJbGaai4BaiaacohacqaH4oqCcqGHRaWkcaGGOaGaaGymaiabgk HiTiGacogacaGGVbGaai4CaiabeI7aXjaacMcacaWGUbWaaSbaaSqa aiaadMgaaeqaaOGaamOBamaaBaaaleaacaWGQbaabeaakiabgUcaRi GacohacaGGPbGaaiOBaiabeI7aXjabgIGiopaaBaaaleaacaWGPbGa am4AaiaadQgaaeqaaOGaamOBamaaBaaaleaacaWGRbaabeaaaaa@5798@

where n i = ijk s j p k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeyypa0JaeyicI48aaSbaaSqaaiaadMgacaWGQbGaam4AaaqabaGc caWGZbWaaSbaaSqaaiaadQgaaeqaaOGaamiCamaaBaaaleaacaWGRb aabeaaaaa@3E98@   cosθ=(1+ γ α p i s i α p k m k α )/C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGJbGaai4BaiaacohacqaH4oqCcq GH9aqpcaGGOaGaaGymaiabgUcaRiabeo7aNnaaCaaaleqabaGaeqyS degaaOGaamiCamaaBaaaleaacaWGPbaabeaakiaadohadaqhaaWcba GaamyAaaqaaiabeg7aHbaakiaadchadaWgaaWcbaGaam4AaaqabaGc caWGTbWaa0baaSqaaiaadUgaaeaacqaHXoqyaaGccaGGPaGaai4lai aadoeaaaa@4BEB@ , sinθ= γ α ( p i m i α ) ( 1 ( p i s i α ) 2 ) /C MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaciGGZbGaaiyAaiaac6gacqaH4oqCcq GH9aqpcqaHZoWzdaahaaWcbeqaaiabeg7aHbaakiaacIcacaWGWbWa aSbaaSqaaiaadMgaaeqaaOGaamyBamaaDaaaleaacaWGPbaabaGaeq ySdegaaOGaaiykamaakaaabaWaaeWaaeaacaaIXaGaeyOeI0Iaaiik aiaadchadaWgaaWcbaGaamyAaaqabaGccaWGZbWaa0baaSqaaiaadM gaaeaacqaHXoqyaaGccaGGPaWaaWbaaSqabeaacaaIYaaaaaGccaGL OaGaayzkaaaaleqaaOGaai4laiaadoeaaaa@4FF0@ , and C= 1+ γ α2 ( p i m i α ) 2 +2 γ α p i s i α p k m k α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGdbGaeyypa0ZaaOaaaeaacaaIXa Gaey4kaSIaeq4SdC2aaWbaaSqabeaacqaHXoqycaaIYaaaaOGaaiik aiaadchadaWgaaWcbaGaamyAaaqabaGccaWGTbWaa0baaSqaaiaadM gaaeaacqaHXoqyaaGccaGGPaWaaWbaaSqabeaacaaIYaaaaOGaey4k aSIaaGOmaiabeo7aNnaaCaaaleqabaGaeqySdegaaOGaamiCamaaBa aaleaacaWGPbaabeaakiaadohadaqhaaWcbaGaamyAaaqaaiabeg7a HbaakiaadchadaWgaaWcbaGaam4AaaqabaGccaWGTbWaa0baaSqaai aadUgaaeaacqaHXoqyaaaabeaaaaa@534D@ Other assumptions are also used to calculate R.

 

8.      The crystal lattice is rotated by R, so that after deformation s i * = R ij s j m i * = R ij m j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbWaa0baaSqaaiaadMgaaeaaca GGQaaaaOGaeyypa0JaamOuamaaBaaaleaacaWGPbGaamOAaaqabaGc caWGZbWaaSbaaSqaaiaadQgaaeqaaOGaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaamyBamaaDaaaleaacaWGPbaabaGaaiOkaaaa kiabg2da9iaadkfadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaamyBam aaBaaaleaacaWGQbaabeaaaaa@4DA2@  for all the slip systems.

9.      The rate of deformation resulting from a shearing rate γ ˙ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaGaamaaCaaaleqabaGaeq ySdegaaaaa@3642@  on the α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@  th system is

L ij = F ˙ ip F pj 1 = R ˙ ik R jk + R ik γ ˙ α s k α m p α R jp = R ˙ ik R jk + γ ˙ α s i *α m j *α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGmbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iqadAeagaGaamaaBaaaleaacaWGPbGaamiCaaqa baGccaWGgbWaa0baaSqaaiaadchacaWGQbaabaGaeyOeI0IaaGymaa aakiabg2da9iqadkfagaGaamaaBaaaleaacaWGPbGaam4AaaqabaGc caWGsbWaaSbaaSqaaiaadQgacaWGRbaabeaakiabgUcaRiaadkfada WgaaWcbaGaamyAaiaadUgaaeqaaOGafq4SdCMbaiaadaahaaWcbeqa aiabeg7aHbaakiaadohadaqhaaWcbaGaam4Aaaqaaiabeg7aHbaaki aad2gadaqhaaWcbaGaamiCaaqaaiabeg7aHbaakiaadkfadaWgaaWc baGaamOAaiaadchaaeqaaOGaeyypa0JabmOuayaacaWaaSbaaSqaai aadMgacaWGRbaabeaakiaadkfadaWgaaWcbaGaamOAaiaadUgaaeqa aOGaey4kaSIafq4SdCMbaiaadaahaaWcbeqaaiabeg7aHbaakiaado hadaqhaaWcbaGaamyAaaqaaiaacQcacqaHXoqyaaGccaWGTbWaa0ba aSqaaiaadQgaaeaacaGGQaGaeqySdegaaaaa@6AE3@

This can be decomposed into a symmetric part, representing a stretching, together with a skew part, representing a spin, as

L ij = D ij + W ij W ij = R ˙ ik R jk + γ ˙ α ( s i *α m j *α s j *α m i *α )/2 D ij = γ ˙ α ( s i *α m j *α + s j *α m i *α )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGmbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadseadaWgaaWcbaGaamyAaiaadQgaaeqaaOGa ey4kaSIaam4vamaaBaaaleaacaWGPbGaamOAaaqabaGccaaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaadEfadaWgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0 JabmOuayaacaWaaSbaaSqaaiaadMgacaWGRbaabeaakiaadkfadaWg aaWcbaGaamOAaiaadUgaaeqaaOGaey4kaSIafq4SdCMbaiaadaahaa Wcbeqaaiabeg7aHbaakiaacIcacaWGZbWaa0baaSqaaiaadMgaaeaa caGGQaGaeqySdegaaOGaamyBamaaDaaaleaacaWGQbaabaGaaiOkai abeg7aHbaakiabgkHiTiaadohadaqhaaWcbaGaamOAaaqaaiaacQca cqaHXoqyaaGccaWGTbWaa0baaSqaaiaadMgaaeaacaGGQaGaeqySde gaaOGaaiykaiaac+cacaaIYaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam iramaaBaaaleaacaWGPbGaamOAaaqabaGccqGH9aqpcuaHZoWzgaGa amaaCaaaleqabaGaeqySdegaaOGaaiikaiaadohadaqhaaWcbaGaam yAaaqaaiaacQcacqaHXoqyaaGccaWGTbWaa0baaSqaaiaadQgaaeaa caGGQaGaeqySdegaaOGaey4kaSIaam4CamaaDaaaleaacaWGQbaaba GaaiOkaiabeg7aHbaakiaad2gadaqhaaWcbaGaamyAaaqaaiaacQca cqaHXoqyaaGccaGGPaGaai4laiaaikdaaaa@A11A@

Here, the first term in W ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGxbWaaSbaaSqaaiaadMgacaWGQb aabeaaaaa@35AB@  represents the rotation of the lattice, while the second term is the spin due to lattice shearing.

 

 

10.  In a tensile test oriented for single slip, the crystal rotates so as to align the slip direction with the loading axis.  This rotation is illustrated for an fcc crystal on the inverse pole figure on the right.  Eventually, the crystal rotates far enough to activate a second slip system.  The exact point where this occurs depends on how the crystal hardens; it usually occurs shortly before the loading axis moves out of the standard triangle.  The rotation direction changes after the second slip system becomes active: eventually, the loading axis aligns with the [112] direction. This is a stable orientation, and the crystal continues to deform in double slip without further rotation.

 

11.  The resistance of each slip plane to shearing increases with plastic strain, due to strain hardening.   A typical stress-strain curve for a single crystal that is initially oriented for single slip is illustrated in the figure. The curve is divided into three characteristic regions. Stage I corresponds to the period while a single slip system is active, and has a low hardening rate (due to self hardening); Stage II begins when a second slip system activates, and has a higher hardening rate (due to both self and latent hardening); while Stage III occurs at large strains, and the hardening rate decreases due to dynamic recovery.  The hardening rates in Stages I and II are insensitive to temperature; but the Stage III hardening rate decreases with temperature.

 

12.  Shearing on the α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@  th system increases its own strength g α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbWaaWbaaSqabeaacqaHXoqyaa aaaa@357E@ : this is known as self-hardening.  Shearing on the α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@  th system also increases the strength of all the other slip systems g β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbWaaWbaaSqabeaacqaHYoGyaa aaaa@3580@ , βα MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHYoGycqGHGjsUcqaHXoqyaaa@37CD@ : this is known as latent hardening.  Self-hardening can be measured using single-slip tests.  Latent hardening is often measured by first deforming the material in single slip, then re-loading the specimen to activate a second slip system.   Latent hardening is often quantified by the Latent Hardening Ratio, which specifies the ratio of the strength of the second system to that of the first q αβ = g β / g α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGXbWaaWbaaSqabeaacqaHXoqycq aHYoGyaaGccqGH9aqpcaWGNbWaaWbaaSqabeaacqaHYoGyaaGccaGG VaGaam4zamaaCaaaleqabaGaeqySdegaaaaa@3E68@ .  The details of the hardening behavior of single crystals are very complex, and at present there is no consensus on how best to measure or characterize hardening.

 

13.  Lattice rotation during a tensile test gives rise to a phenomenon known as `geometric softening,’ which plays an important role in shear localization in single crystals.  The term `geometric softening’ refers to the fact that the crystal may rotate so as to increase the resolved shear stress on its active slip system, and therefore lead to a decrease in the tensile flow stress of the crystal.

 

 

 

3.12.3 Kinematic descriptions used in constitutive models of single crystals

 

Let x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaaaaa@327F@  be the position of a material particle in the undeformed crystal. Suppose that the solid is subjected to a displacement field u i ( x k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgaaeqaaO GaaiikaiaadIhadaWgaaWcbaGaam4AaaqabaGccaGGPaaaaa@3860@ , so that the point at x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadIhadaWgaaWcbaGaamyAaaqabaaaaa@327F@  moves to y i = x i + u i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadMhadaWgaaWcbaGaamyAaaqabaGccq GH9aqpcaWG4bWaaSbaaSqaaiaadMgaaeqaaOGaey4kaSIaamyDamaa BaaaleaacaWGPbaabeaaaaa@38A7@ .  Define

 

 The deformation gradient and its jacobian

F ij = δ ij + u i x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGgbWaaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iabes7aKnaaBaaaleaacaWGPbGaamOAaaqabaGc cqGHRaWkdaWcaaqaaiabgkGi2kaadwhadaWgaaWcbaGaamyAaaqaba aakeaacqGHciITcaWG4bWaaSbaaSqaaiaadQgaaeqaaaaaaaa@4256@        J=det(F) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGkbGaeyypa0Jaciizaiaacwgaca GG0bGaaiikaiaahAeacaGGPaaaaa@398E@

 The velocity gradient

L ij = u ˙ i y j = F ˙ ik F kj 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0ZaaSaaaeaacqGHciITceWG1bGbaiaadaWgaaWcbaGa amyAaaqabaaakeaacqGHciITcaWG5bWaaSbaaSqaaiaadQgaaeqaaa aakiabg2da9iqadAeagaGaamaaBaaaleaacaWGPbGaam4AaaqabaGc caWGgbWaa0baaSqaaiaadUgacaWGQbaabaGaeyOeI0IaaGymaaaaaa a@43E5@

 The stretch rate and spin

D ij =( L ij + L ji )/2 W ij =( L ij L ji )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaaiikaiaadYeadaWgaaWcbaGaamyAaiaadQgaaeqa aOGaey4kaSIaamitamaaBaaaleaacaWGQbGaamyAaaqabaGccaGGPa Gaai4laiaaikdacaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7 caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVl aaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caWGxbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabg2 da9iaacIcacaWGmbWaaSbaaSqaaiaadMgacaWGQbaabeaakiabgkHi TiaadYeadaWgaaWcbaGaamOAaiaadMgaaeqaaOGaaiykaiaac+caca aIYaaaaa@7036@

 Recall that F ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadAeadaWgaaWcbaGaamyAaiaadQgaae qaaaaa@333C@  relates infinitesimal material fibers d y i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaaSbaaSqaaiaadMgaae qaaaaa@3369@  and d x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG4bWaaSbaaSqaaiaadMgaae qaaaaa@3368@  in the deformed and undeformed solid, respectively, as

d y i = F ij d x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadsgacaWG5bWaaSbaaSqaaiaadMgaae qaaOGaeyypa0JaamOramaaBaaaleaacaWGPbGaamOAaaqabaGccaWG KbGaamiEamaaBaaaleaacaWGQbaabeaaaaa@3A58@

  To decompose the deformation gradient into elastic and plastic parts, we assume that deformation takes place in two stages.  The plastic strain is assumed to shear the lattice, without stretching or rotating it.  The elastic deformation rotates and stretches the lattice. We think of these two events occurring in sequence, with the plastic deformation first, and the stretch and rotation second, giving

d y i = F ij d x j = F ik e F kj p d x j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaamizaiaadMhadaWgaaWcbaGaamyAaa qabaGccqGH9aqpcaWGgbWaaSbaaSqaaiaadMgacaWGQbaabeaakiaa dsgacaWG4bWaaSbaaSqaaiaadQgaaeqaaOGaeyypa0JaamOramaaDa aaleaacaWGPbGaam4AaaqaaiaadwgaaaGccaWGgbWaa0baaSqaaiaa dUgacaWGQbaabaGaamiCaaaakiaadsgacaWG4bWaaSbaaSqaaiaadQ gaaeqaaaaa@4670@

 To decompose the velocity gradient into elastic and plastic parts, note that

L ij = F ˙ ik F kj 1 =( F ˙ ik e F kl p + F ik e F ˙ kl p )( F lm p1 F mj e1 )= F ˙ ik e F kj e1 + F ik e F ˙ kl p F lm p1 F mj e1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JabmOrayaacaWaaSbaaSqaaiaadMgacaWGRbaabeaa kiaadAeadaqhaaWcbaGaam4AaiaadQgaaeaacqGHsislcaaIXaaaaO Gaeyypa0ZaaeWaaeaaceWGgbGbaiaadaqhaaWcbaGaamyAaiaadUga aeaacaWGLbaaaOGaamOramaaDaaaleaacaWGRbGaamiBaaqaaiaadc haaaGccqGHRaWkcaWGgbWaa0baaSqaaiaadMgacaWGRbaabaGaamyz aaaakiqadAeagaGaamaaDaaaleaacaWGRbGaamiBaaqaaiaadchaaa aakiaawIcacaGLPaaadaqadaqaaiaadAeadaqhaaWcbaGaamiBaiaa d2gaaeaacaWGWbGaeyOeI0IaaGymaaaakiaadAeadaqhaaWcbaGaam yBaiaadQgaaeaacaWGLbGaeyOeI0IaaGymaaaaaOGaayjkaiaawMca aiabg2da9iqadAeagaGaamaaDaaaleaacaWGPbGaam4Aaaqaaiaadw gaaaGccaWGgbWaa0baaSqaaiaadUgacaWGQbaabaGaamyzaiabgkHi TiaaigdaaaGccqGHRaWkcaWGgbWaa0baaSqaaiaadMgacaWGRbaaba GaamyzaaaakiqadAeagaGaamaaDaaaleaacaWGRbGaamiBaaqaaiaa dchaaaGccaWGgbWaa0baaSqaaiaadYgacaWGTbaabaGaamiCaiabgk HiTiaaigdaaaGccaWGgbWaa0baaSqaaiaad2gacaWGQbaabaGaamyz aiabgkHiTiaaigdaaaaaaa@78D0@

Thus the velocity gradient contains two terms, one of which involves only measures of elastic deformation, while the other contains measures of plastic deformation.  We use this to decompose L into elastic and plastic parts

L ij = L ij e + L ij p L ij e = F ˙ ik e F kj e1 L ij p = F ik e F ˙ kl p F lm p1 F mj e1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadYeadaWgaaWcbaGaamyAaiaadQgaae qaaOGaeyypa0JaamitamaaDaaaleaacaWGPbGaamOAaaqaaiaadwga aaGccqGHRaWkcaWGmbWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaa aakiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaamitamaaDaaaleaacaWGPbGaamOAaa qaaiaadwgaaaGccqGH9aqpceWGgbGbaiaadaqhaaWcbaGaamyAaiaa dUgaaeaacaWGLbaaaOGaamOramaaDaaaleaacaWGRbGaamOAaaqaai aadwgacqGHsislcaaIXaaaaOGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caWGmbWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaa kiabg2da9iaadAeadaqhaaWcbaGaamyAaiaadUgaaeaacaWGLbaaaO GabmOrayaacaWaa0baaSqaaiaadUgacaWGSbaabaGaamiCaaaakiaa dAeadaqhaaWcbaGaamiBaiaad2gaaeaacaWGWbGaeyOeI0IaaGymaa aakiaadAeadaqhaaWcbaGaamyBaiaadQgaaeaacaWGLbGaeyOeI0Ia aGymaaaaaaa@8903@

 Plastic flow in the crystal occurs by shearing a set of N slip systems.  The slip systems are characterized by unit vectors parallel to slip directions s i α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbWaa0baaSqaaiaadMgaaeaacq aHXoqyaaaaaa@3678@  and slip plane normals m i α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGTbWaa0baaSqaaiaadMgaaeaacq aHXoqyaaaaaa@3672@  in the undeformed solid.  The rate of shear on the α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHXoqyaaa@3465@  th system is denoted by γ ˙ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaGaamaaCaaaleqabaGaeq ySdegaaaaa@3642@ .  The velocity gradient due to this shearing is

F ˙ ik p F kj p1 = α=1 N γ ˙ α s i α m j α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGgbGbaiaadaqhaaWcbaGaamyAai aadUgaaeaacaWGWbaaaOGaamOramaaDaaaleaacaWGRbGaamOAaaqa aiaadchacqGHsislcaaIXaaaaOGaeyypa0ZaaabCaeaacuaHZoWzga GaamaaCaaaleqabaGaeqySdegaaOGaam4CamaaDaaaleaacaWGPbaa baGaeqySdegaaOGaamyBamaaDaaaleaacaWGQbaabaGaeqySdegaaa qaaiabeg7aHjabg2da9iaaigdaaeaacaWGobaaniabggHiLdaaaa@4E81@

 It is convenient to define vectors that describe plastic shearing in the current configuration, as 

s i *α = F ik e s k α m i *α = m k α F ki e1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGZbWaa0baaSqaaiaadMgaaeaaca GGQaGaeqySdegaaOGaeyypa0JaamOramaaDaaaleaacaWGPbGaam4A aaqaaiaadwgaaaGccaWGZbWaa0baaSqaaiaadUgaaeaacqaHXoqyaa GccaaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaamyBamaaDaaaleaacaWGPbaabaGaaiOkaiabeg7aHbaakiabg2da 9iaad2gadaqhaaWcbaGaam4Aaaqaaiabeg7aHbaakiaadAeadaqhaa WcbaGaam4AaiaadMgaaeaacaWGLbGaeyOeI0IaaGymaaaaaaa@66F8@

The former can be interpreted as the slip direction in the deformed solid (note that it is not a unit vector, however), while m i *α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGTbWaa0baaSqaaiaadMgaaeaaca GGQaGaeqySdegaaaaa@3720@  can be interpreted as the slip plane normal in the deformed solid. 

 The plastic part of the velocity gradient can then be expressed in terms of the shearing rates as

L ij p = α=1 N γ ˙ α s i *α m j *α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGmbWaa0baaSqaaiaadMgacaWGQb aabaGaamiCaaaakiabg2da9maaqahabaGafq4SdCMbaiaadaahaaWc beqaaiabeg7aHbaakiaadohadaqhaaWcbaGaamyAaaqaaiaacQcacq aHXoqyaaGccaWGTbWaa0baaSqaaiaadQgaaeaacaGGQaGaeqySdega aaqaaiabeg7aHjabg2da9iaaigdaaeaacaWGobaaniabggHiLdaaaa@4A5B@

 The elastic and plastic parts of the velocity gradient can be decomposed in to symmetric and skew symmetric parts, representing stretching and spin, respectively as

D ij e =( L ij e + L ji e )/2 W ij e =( L ij e L ji e )/2 D ij p =( L ij p + L ji p )/2 W ij p =( L ij p L ji p )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOabaeqabaGaamiramaaDaaaleaacaWGPbGaam OAaaqaaiaadwgaaaGccqGH9aqpcaGGOaGaamitamaaDaaaleaacaWG PbGaamOAaaqaaiaadwgaaaGccqGHRaWkcaWGmbWaa0baaSqaaiaadQ gacaWGPbaabaGaamyzaaaakiaacMcacaGGVaGaaGOmaiaaykW7caaM c8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaayk W7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaadEfadaqhaa WcbaGaamyAaiaadQgaaeaacaWGLbaaaOGaeyypa0JaaiikaiaadYea daqhaaWcbaGaamyAaiaadQgaaeaacaWGLbaaaOGaeyOeI0Iaamitam aaDaaaleaacaWGQbGaamyAaaqaaiaadwgaaaGccaGGPaGaai4laiaa ikdaaeaacaWGebWaa0baaSqaaiaadMgacaWGQbaabaGaamiCaaaaki abg2da9iaacIcacaWGmbWaa0baaSqaaiaadMgacaWGQbaabaGaamiC aaaakiabgUcaRiaadYeadaqhaaWcbaGaamOAaiaadMgaaeaacaWGWb aaaOGaaiykaiaac+cacaaIYaGaaGPaVlaaykW7caaMc8UaaGPaVlaa ykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaG PaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaM c8UaaGPaVlaaykW7caaMc8Uaam4vamaaDaaaleaacaWGPbGaamOAaa qaaiaadchaaaGccqGH9aqpcaGGOaGaamitamaaDaaaleaacaWGPbGa amOAaaqaaiaadchaaaGccqGHsislcaWGmbWaa0baaSqaaiaadQgaca WGPbaabaGaamiCaaaakiaacMcacaGGVaGaaGOmaaaaaa@BB50@

 The plastic stretching and spin can be expressed in terms of the lattice shearing as

D ij p = α=1 N γ ˙ α ( s i *α m j *α + s j *α m i *α )/2 W ij p = α=1 N γ ˙ α ( s i *α m j *α s j *α m i *α )/2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaqhaaWcbaGaamyAaiaadQgaae aacaWGWbaaaOGaeyypa0ZaaabCaeaacuaHZoWzgaGaamaaCaaaleqa baGaeqySdegaaOGaaiikaiaadohadaqhaaWcbaGaamyAaaqaaiaacQ cacqaHXoqyaaGccaWGTbWaa0baaSqaaiaadQgaaeaacaGGQaGaeqyS degaaaqaaiabeg7aHjabg2da9iaaigdaaeaacaWGobaaniabggHiLd GccqGHRaWkcaWGZbWaa0baaSqaaiaadQgaaeaacaGGQaGaeqySdega aOGaamyBamaaDaaaleaacaWGPbaabaGaaiOkaiabeg7aHbaakiaacM cacaGGVaGaaGOmaiaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8Uaam4vamaaDaaaleaaca WGPbGaamOAaaqaaiaadchaaaGccqGH9aqpdaaeWbqaaiqbeo7aNzaa caWaaWbaaSqabeaacqaHXoqyaaGccaGGOaGaam4CamaaDaaaleaaca WGPbaabaGaaiOkaiabeg7aHbaakiaad2gadaqhaaWcbaGaamOAaaqa aiaacQcacqaHXoqyaaaabaGaeqySdeMaeyypa0JaaGymaaqaaiaad6 eaa0GaeyyeIuoakiabgkHiTiaadohadaqhaaWcbaGaamOAaaqaaiaa cQcacqaHXoqyaaGccaWGTbWaa0baaSqaaiaadMgaaeaacaGGQaGaeq ySdegaaOGaaiykaiaac+cacaaIYaaaaa@899E@

 

 

 

3.12.4 Stress measures used crystal plasticity

 

Stress measures that appear in descriptions of crystal plasticity are summarized below:

 

 The Cauchy (“true”) stress represents the force per unit deformed area in the solid and is defined by

n i σ ij = Lim dA0 d P j (n) dA MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGUbWaaSbaaSqaaiaadMgaaeqaaO Gaeq4Wdm3aaSbaaSqaaiaadMgacaWGQbaabeaakiabg2da9maaxaba baGaamitaiaadMgacaWGTbaaleaacaWGKbGaamyqaiabgkziUkaaic daaeqaaOWaaSaaaeaacaWGKbGaamiuamaaDaaaleaacaWGQbaabaGa aiikaiaah6gacaGGPaaaaaGcbaGaamizaiaadgeaaaaaaa@47F6@

 Kirchhoff stress  τ ij =J σ ij MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaqcaaSaaGPaVlabes8a0PWaaSbaaSqaai aadMgacaWGQbaabeaakiabg2da9iaadQeacqaHdpWCdaWgaaWcbaGa amyAaiaadQgaaeqaaaaa@3CA6@

 Material stress for the intermediate configuration   Σ ij =J F ik e1 σ kl F jl e1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeu4Odm1aaSbaaSqaaiaadMgacaWGQb aabeaakiabg2da9iaadQeacaWGgbWaa0baaSqaaiaadMgacaWGRbaa baGaamyzaiabgkHiTiaaigdaaaGccqaHdpWCdaWgaaWcbaGaam4Aai aadYgaaeqaaOGaamOramaaDaaaleaacaWGQbGaamiBaaqaaiaadwga cqGHsislcaaIXaaaaaaa@44F0@

 Resolved shear stress on a slip system   τ α =J m i *α σ ij s j *α MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaGaeqiXdq3aaWbaaSqabeaacqaHXoqyaa GccqGH9aqpcaWGkbGaamyBamaaDaaaleaacaWGPbaabaGaaiOkaiab eg7aHbaakiabeo8aZnaaBaaaleaacaWGPbGaamOAaaqabaGccaWGZb Waa0baaSqaaiaadQgaaeaacaGGQaGaeqySdegaaaaa@42D9@

 Lattice Jaumann rate of Kirchoff stress   τ ij e = d τ ij dt W ik e τ kj + τ ik W kj e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaCbiaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaaqabeaacqGHhis0caWGLbaaaOGaeyypa0ZaaSaa aeaacaWGKbGaeqiXdq3aaSbaaSqaaiaadMgacaWGQbaabeaaaOqaai aadsgacaWG0baaaiabgkHiTiaadEfadaqhaaWcbaGaamyAaiaadUga aeaacaWGLbaaaOGaeqiXdq3aaSbaaSqaaiaadUgacaWGQbaabeaaki abgUcaRiabes8a0naaBaaaleaacaWGPbGaam4AaaqabaGccaWGxbWa a0baaSqaaiaadUgacaWGQbaabaGaamyzaaaaaaa@503D@

 

 

The constitutive equations must specify relationships between these stress measures, and the deformation measures outlined in the preceding section.  In particular, the constitutive equations must relate:

1.      The elastic part of the deformation gradient to stress;

2.      The rate of shearing on each slip system to the resolved shear stress.

 

 

 

3.12.5 Elastic stress-strain relation used in crystal plasticity

 

The relations between stress and the elastic part of the deformation gradient follow the procedure developed for finite strain plasticity in Section 3.8.3.  Only the final results will be repeated here

1.      Define the Lagrangean elastic strain as E ij e =( F ki e F kj e δ ij )/2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadweadaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaOGaeyypa0JaaiikaiaadAeadaqhaaWcbaGaam4Aaiaa dMgaaeaacaWGLbaaaOGaamOramaaDaaaleaacaWGRbGaamOAaaqaai aadwgaaaGccqGHsislcqaH0oazdaWgaaWcbaGaamyAaiaadQgaaeqa aOGaaiykaiaac+cacaaIYaaaaa@4438@

2.      Assume that the material stress is proportional to Lagrange strain, as Σ ij = C ijkl E kl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqqHJoWudaWgaaWcbaGaamyAaiaadQ gaaeqaaOGaeyypa0Jaam4qamaaBaaaleaacaWGPbGaamOAaiaadUga caWGSbaabeaakiaadweadaqhaaWcbaGaam4AaiaadYgaaeaacaWGLb aaaaaa@3FE1@ , where C ijkl MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaWgaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqabaaaaa@351A@  are the components of the elastic stiffness tensor (as defined and tabulated in Section 3.1), for the material with orientation in the undeformed configuration.

3.      The elastic stress-strain law is often expressed in rate form, as follows

τ ij e C ijkl e D kl e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaCbiaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaaqabeaacqGHhis0caWGLbaaaOGaeyisISRaam4q amaaDaaaleaacaWGPbGaamOAaiaadUgacaWGSbaabaGaamyzaaaaki aadseadaqhaaWcbaGaam4AaiaadYgaaeaacaWGLbaaaaaa@426E@

where τ ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8ku c9pgc9q8qqaq=dir=f0=yqaiVgFr0xfr=xfr=xb9adbaqaaeGaciGa biaabeqaaiqabaWaaaGcbaWaaCbiaeaacqaHepaDdaWgaaWcbaGaam yAaiaadQgaaeqaaaqabeaacqGHhis0caWGLbaaaaaa@374B@  is the Jaumann rate of Kirchhoff stress with respect to axes that rotate with the crystal lattice; C ijkl e = F in e F jm e C nmpq F kp e F lq e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadoeadaqhaaWcbaGaamyAaiaadQgaca WGRbGaamiBaaqaaiaadwgaaaGccqGH9aqpcaWGgbWaa0baaSqaaiaa dMgacaWGUbaabaGaamyzaaaakiaadAeadaqhaaWcbaGaamOAaiaad2 gaaeaacaWGLbaaaOGaam4qamaaBaaaleaacaWGUbGaamyBaiaadcha caWGXbaabeaakiaadAeadaqhaaWcbaGaam4AaiaadchaaeaacaWGLb aaaOGaamOramaaDaaaleaacaWGSbGaamyCaaqaaiaadwgaaaaaaa@4B17@  can be thought of as the components of the elastic compliance tensor for material with orientation in the deformed configuration, and D ij e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rkY=vi pgYlH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFH e9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabiGaciaa caqabeaacmqaamaaaOqaaiaadseadaqhaaWcbaGaamyAaiaadQgaae aacaWGLbaaaaaa@3425@  is the elastic stretch rate.

 

 

 

 

3.12.6 Plastic stress-strain relation used in crystal plasticity

 

The plastic constitutive equations specify the relationship between the stress on the crystal and slip rates γ ˙ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaGaamaaCaaaleqabaGaeq ySdegaaaaa@3642@  on each slip system.   Here, we outline a simple and widely used approach to doing this, based on the work of Pierce, Asaro and Needleman Acta Metall. 31 1951-1976 (1983).  This model is not the best fit to experimental observations, however MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFKI8=feu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFf ea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaGqaaKqzGfaeaa aaaaaaa8qacaWFtacaaa@37E6@  in particular more sophisticated equations are required to accurately describe latent hardening behavior.

 

 Flow Rule

 

There are many advantages to using a viscoplastic flow rule to predict the slip rates in a single crystal: this avoids having to use an iterative procedure to identify active slip systems, and also helps to stabilize material behavior.  The simplest such flow rule is

γ ˙ α = γ ˙ 0 sign( τ α ) ( | τ α | g α ) m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaGaamaaCaaaleqabaGaeq ySdegaaOGaeyypa0Jafq4SdCMbaiaadaWgaaWcbaGaaGimaaqabaGc caWGZbGaamyAaiaadEgacaWGUbGaaiikaiabes8a0naaCaaaleqaba GaeqySdegaaOGaaiykamaabmaabaWaaSaaaeaadaabdaqaaiabes8a 0naaCaaaleqabaGaeqySdegaaaGccaGLhWUaayjcSdaabaGaam4zam aaCaaaleqabaGaeqySdegaaaaaaOGaayjkaiaawMcaamaaCaaaleqa baGaamyBaaaaaaa@4EE2@

where τ α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacqaHepaDdaahaaWcbeqaaiabeg7aHb aaaaa@3657@  is the resolved shear stress on the slip system, g α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbWaaWbaaSqabeaacqaHXoqyaa aaaa@357E@  is its current strength (which evolves with plastic straining), and γ ˙ 0 ,m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaGaamaaBaaaleaacaaIWa aabeaakiaacYcacaWGTbaaaa@3708@  are material properties.

 

 Hardening rule

 

The hardening rule must specify the relationship between the slip system strengths g α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbWaaWbaaSqabeaacqaHXoqyaa aaaa@357E@  and the plastic strain.  At time t=0 each slip system has the same initial strength g 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbWaaSbaaSqaaiaaicdaaeqaaa aa@3498@ .  Thereafter, the slip systems increase in strength as a result of the plastic shearing according to

g ˙ α = β=1 N h αβ | γ ˙ β | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaaceWGNbGbaiaadaahaaWcbeqaaiabeg 7aHbaakiabg2da9maaqahabaGaamiAamaaBaaaleaacqaHXoqycqaH YoGyaeqaaOWaaqWaaeaacuaHZoWzgaGaamaaCaaaleqabaGaeqOSdi gaaaGccaGLhWUaayjcSdaaleaacqaHYoGycqGH9aqpcaaIXaaabaGa amOtaaqdcqGHris5aaaa@481B@

where h αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGObWaaSbaaSqaaiabeg7aHjabek 7aIbqabaaaaa@371F@  are strain dependent hardening rates.  The hardening rate is approximated as

h αβ = q αβ h( γ ¯ ) h( γ ¯ )= h s +( h s h 0 ) sech 2 { ( h 0 h s g s g 0 ) γ ¯ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakqaabeqaaiaadIgadaWgaaWcbaGaeqySde MaeqOSdigabeaakiabg2da9iaadghadaWgaaWcbaGaeqySdeMaeqOS digabeaakiaadIgacaGGOaGafq4SdCMbaebacaGGPaaabaGaamiAai aacIcacuaHZoWzgaqeaiaacMcacqGH9aqpcaWGObWaaSbaaSqaaiaa dohaaeqaaOGaey4kaSIaaiikaiaadIgadaWgaaWcbaGaam4Caaqaba GccqGHsislcaWGObWaaSbaaSqaaiaaicdaaeqaaOGaaiykaiaaboha caqGLbGaae4yaiaabIgadaahaaWcbeqaaiaabkdaaaGcdaGadaqaam aabmaabaWaaSaaaeaacaWGObWaaSbaaSqaaiaaicdaaeqaaOGaeyOe I0IaamiAamaaBaaaleaacaWGZbaabeaaaOqaaiaadEgadaWgaaWcba Gaam4CaaqabaGccqGHsislcaWGNbWaaSbaaSqaaiaaicdaaeqaaaaa aOGaayjkaiaawMcaaiqbeo7aNzaaraaacaGL7bGaayzFaaaaaaa@62D9@

where h s , h 0 , g s , q αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGObWaaSbaaSqaaiaadohaaeqaaO GaaiilaiaadIgadaWgaaWcbaGaaGimaaqabaGccaGGSaGaam4zamaa BaaaleaacaWGZbaabeaakiaacYcacaWGXbWaaSbaaSqaaiabeg7aHj abek7aIbqabaaaaa@3F4A@  are material properties, and γ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaqeaaaa@3485@  is the total accumulated slip on all slip systems

γ ¯ = 0 t α=1 N | γ ˙ α | dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaqeaiabg2da9maapehaba WaaabCaeaadaabdaqaaiqbeo7aNzaacaWaaWbaaSqabeaacqaHXoqy aaaakiaawEa7caGLiWoaaSqaaiabeg7aHjabg2da9iaaigdaaeaaca WGobaaniabggHiLdaaleaacaaIWaaabaGaamiDaaqdcqGHRiI8aOGa amizaiaadshaaaa@48B0@

The matrix q αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGXbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaaaaa@3728@  controls the latent hardening rate: for an fcc crystal, it is usually taken to have the form

q αβ ={ 1α,β coplanar qotherwise MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGXbWaaSbaaSqaaiabeg7aHjabek 7aIbqabaGccqGH9aqpdaGabaqaauaabeqaceaaaeaacaaIXaGaaGPa VlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8 UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaeqySdeMaaeil aiabek7aIjaabccacaqGJbGaae4BaiaabchacaqGSbGaaeyyaiaab6 gacaqGHbGaaeOCaaqaaiaadghacaaMc8UaaGPaVlaaykW7caaMc8Ua aGPaVlaaykW7caaMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaaykW7ca aMc8UaaGPaVlaaykW7caaMc8UaaGPaVlaab+gacaqG0bGaaeiAaiaa bwgacaqGYbGaae4DaiaabMgacaqGZbGaaeyzaiaaykW7caaMc8UaaG PaVlaaykW7aaaacaGL7baaaaa@870B@

where q is a material property. The slip systems for an fcc crystal are listed in Section 3.12.2: for example, slip systems a 1 , a 2 , a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadggadaWgaaWcbaGaaGOmaaqabaGccaGGSaGaamyyamaa BaaaleaacaaIZaaabeaaaaa@39A4@  are coplanar, while a 1 , b 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGHbWaaSbaaSqaaiaaigdaaeqaaO GaaiilaiaadkgadaWgaaWcbaGaaGymaaqabaaaaa@371B@  non-coplanar.

 

 

 

3.12.7 Representative values for plastic properties of single crystals

 

Elastic properties of single crystals are listed in Sections 3.1.15 and 3.1.16. The plastic properties of single crystals are strongly sensitive to the material’s crystal structure and composition.  For accurate predictions you will need to test the actual material you plan to use.  As a rough guide, representative parameters for a copper single crystal (taken from Wu, Neale and Van der Giessen, Int J plasticity, 12, p.1199, 1996) are listed in the table.

 

 

 

Properties of Cu single crystal

 

γ ˙ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacuaHZoWzgaGaamaaBaaaleaacaaIWa aabeaaaaa@355C@

m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGTbaaaa@33B8@

g 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbWaaSbaaSqaaiaaicdaaeqaaa aa@3498@

g s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGNbWaaSbaaSqaaiaadohaaeqaaa aa@34D6@

h 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGObWaaSbaaSqaaiaaicdaaeqaaa aa@3499@

h s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGObWaaSbaaSqaaiaadohaaeqaaa aa@34D7@

q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaWGXbaaaa@33BC@

0.001 s 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rk0le9 v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaGadeaadaaakeaacaqGZbWaaWbaaSqabeaacqGHsislca aIXaaaaaaa@3591@

10-20

16 MPa

70.4 MPa

132 MPa

8 MPa

1.4