Chapter 8: Slip. Introduction

Transcription

1 OHP 1 Mechanica Properties of Materias Chapter 8: Sip Prof. Wenjea J. Tseng ( 曾文甲 ) Department of Materias Engineering Nationa Chung Hsing University wenjea@dragon.nchu.edu.tw Reference: W. F. Hosford (Cambridge, 2010) N. E. Dowing (Pearson, 2007) Macroscopic Introduction Pastic deformation of crystaine materias usuay occurs by sip, which is the siding of panes of atoms over one another by disocation movements. The panes on which sip occurs are caed sip panes and the directions of the shear are the sip directions. The sip panes and directions are characteristic of the crysta structure of materias. Microscopic (see next pages) 1

2 Introduction (microscopic view) Sip direction sipped Sip pane un-sipped Introduction (microscopic view) In microscopic viewpoints, sip is caused by the motion of edge and/or screw disocations. Shear/sip steps abe to be seen macroscopicay when mutipe disocations were moved to the surface ends. 2

3 Shear Bands and Shear Steps Figures iustrate shear/sip bands and shear/sip steps of mutipe disocations upon stress oading. Shear bands Shear steps Sip Systems The sip panes and directions, combined to caed the sip systems, for severa common crystas are summarized in Tabe. The sip directions are the crystaographic directions with the shortest distance between ike atoms or ions and the sip panes are usuay densey packed panes. 3

4 Schmid s Law E. Schmid discovered that if a crysta is stressed, sip begins when shear stress on a sip system reaches a critica vaue, c, often caed the critica resoved shear stress (CRSS). In uniaxia tension, Schmid s aw is written as cos cos where is the ange between the sip direction and the tensie axis, and is the ange between the tensie axis and the sip-pane norma. c m cos cos m : Schmid factor. m max = 0.5 Schmid s Law The Schmid s aw can be shorten to / m where m is the Schmid factor, m = cos cos. A arger m indicates a smaer for sip to occur. 4

5 Schmid s Law In a genera form, the shear stress induced by a uniaxia tension aong x-direction can be found from the stress transformation, i.e. where s are the direction cosines. Therefore, the condition for yieding (i.e., for the sip to occur) under a genera stress state is c ( ( nz dy nx dx nz dy xx nd ny dz ij ny dz nx dx ny dy yz yz 3 yy xx 3 ) ( ) ( nx dz nz dz nx dz ny dy im jn n 1 m 1 nz dx zz mn yy nz dx ( nz dz ) ny dx ) im zx zx jn ( zz mn nx dy ny dx ) xy nx dy ) xy Exampe: Schmid s Law in Mutiaxia Stress State 5

6 Strains Produced by Sip The incrementa strain transformation equations may be used to find the shape change that resuts from sip when the strains are sma, i.e., when the attice rotation are negigibe. For infinitesima strains, With sip on a singe sip system in the d direction and on the n pane, the ony strain term is nd, therefore, In Schmid s notation, this is cos cos m d xx ij xx im jn xn xd mn nd Strains Produced by Sip Simiary, the other strain components, referred to the x, y, and z axes, are yy zz yz zx xy ( ( ( yn yd zn zd zn xd yn zd xn yd yd zd xd zn xn yn ) ) ) [010] 6

7 Remarks For a poycrystaine soid to have appreciabe ductiity, each of its grains must be abe to undergo the same shape change as the entire body. Therefore, each grain in a poycrystaine soid must deform with the same externa strains as the whoe. For an individua grain, five strain components, 1, 2, 12, 23, 31, exist independenty. This indicates that at east 5 independent sip systems are needed for ductiity. This means that if a materia has ess than 5 independent sip systems, a poycrysta soid wi have a imited ductiity uness another deformation mechanism suppies the number of freedom. Sip in FCC Crystas Strain hardening of fcc singe crystas The underying figure shows a typica stress-strain curve for an fcc singe crysta. Sip occurs on a singe pane initiay and the rate of strain hardening is very ow, caed easy gide or stage I. Sip is then observed on other sip systems, resuting in disocation on different sip systems intersect which gives rise to strain hardening in stage II. At stage III, the rate of strain hardening decreases. 7

8 Sip in FCC Crystas Strain hardening of fcc singe crystas (continue) When sip in fcc singe crystas occurs simutaneousy on many sip systems (e.g., <111>, <110>). Easy gide region becomes insignificant and the initia strain hardening rate is rapid, even comparabe with that in poycrysta. Strain hardening of fcc poycrystas No observabe easy-gide region. The shaded region indicates strain hardening in orientations with ony a singe sip system. Stereography (review) The stereographic projection is used to find anguar reations between directions and panes in a crysta. For Cubic structure 8

9 Sip in FCC Crystas Tensie deformation of fcc crystas For a orientations of fcc crystas within the basic stereographic triange with [100], [110], and [111] corners, the Schmid factor for sip in the [101] direction on the (111) pane is higher than that for any other sip system. The tensie axis ies on the great circe b/t sip direction and the sippane norma with and both equa to 45 o. cos cos c Sip in BCC Crystas The sip direction in bcc metas is aways the direction of cose packing, <111>. Sip has been reported on various panes, {110}, {123}, and {112}. A of these panes contain at east one <111> direction. The underying figure shows the orientation dependence of the Schmid factors for uniaxia tension with <111>penci gide. The basic orientation triange is divided into two regions, with a different <111> sip direction in each. 9

10 Sip in HCP Crystas The most common sip direction of hcp metas is <1120>, which is the direction of cose contact between atoms in the basa pane. The underying figure shows shear stresses required for severa sip systems in Be metas. The stresses required for pyramid sip are much higher than the stresses to cause basa and prism sip and are often high enough to cause fracture. Lattice Rotation in Tension Sip normay causes a gradua attice rotation or orientation change. The figure shows the eongation of a ong singe crysta by sip on a singe sip system. The sip pane and sip direction are represented as being fixed in space with the tensie axis rotating reative to these. Note that in rea tension test, the tensie axis remains vertica and the crysta eements rotate instead. Sip pane Sip direction (a) (b) w/o constraint w/ constraint Tensie axis Dropping normas to sip pane through O Dropping normas to sip direction through O 10

11 Lattice Rotation in Tension In previous figures, sip causes transation of point P parae to the sip direction to a new position P. Points C and C are constructed by extending the sip direction through point O and dropping normas from P and P. Points B and B are constructed by dropping normas from P and P to the sip pane through O. The attice rotation foows a geometrica reation: = cos / cos -cos o / cos o where is the shear strain. In the figure, increases and decreases during tensie extension. The orientation change in tension is hence a simpe rotation of the sip direction toward the tensie axis. decreases Lattice Rotation in Tension for FCC As shown in the figure beow, primary sip in the [110] direction causes the tensie axis to rotate toward the [101]. Once the tensie axis reaches the [100]-[111] symmetry ine; at this point, sip starts on [110](111) which is in the conjugate triange, a simutaneous sip in the [101] and [110] directions causes a rotation toward [211]. For fcc singe crystas 11

12 Lattice Rotation in Compression for FCC For the compression of thin fat singe crysta, the compression axis is rotates toward the sip-pane norma. As shown in the figure, compression causes a rotation of the sip-pane norma toward the compression axis, which is equivaent to a rotation of the compression axis toward the sip-pane norma. For an fcc crysta, the compression axis rotates toward [111] unti it reaches the [100]-[110] boundary, where dupex sip wi cause a net rotation toward [110]. For fcc singe crystas Lattice Rotation in Tension for BCC The tensie axis of a bcc crysta deforming by giding rotates toward the active <111> sip direction. For orientations near [110] and [111] (region A in figure) the tensie axis wi rotate toward [111]. For crysta orientations in the basic triange near [100], the rotation wi be toward [111]. Once the tensie axis enters region A, the rotation wi be toward [111]. When the [100]-[110] boundary is reached, combined sip in the [111] and [111] directions wi cause rotation toward [110]. For bcc singe crysta 12

13 Lattice Rotation in Compression for BCC For giding mechanism, rotation toward the compression axis is equivaent to rotation away from the active sip direction. The figure shows that for bcc singe crystas, orientations initiay in region A wi end up rotating to [111], whereas those initiay in region B wi rotate toward [100]. Texture Formation in Poycrystas In poycrystaine metas, the grains undergo simiar rotation and these ead to crystaographic textures or preferred orientations. The tabe shows the experimentay observed textures deveoped in tensie extension and compression of poycrystas. 13

14 Summary Sip occurs when the shear stress arising from the externa oading exceeds a critica resoved shear stress. At the time, disocations often move aong a particuar sip system (caed the easy gide region). Strain hardening foows when interaction between mutipe disocations begins to prevai. When sip system is unavaiabe for the disocations to move, strain hardening may occur without going through the easy gide region. Sip can be iustrated by the stereographic projection, so as the attice distortion. Mutipe microscopic sips eventuay become macroscopicay visibe, i.e., the shear/sip bands, and the shear/sip steps. 14