Crystal Micro-Mechanics
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1 Crystal Micro-Mechanics Lectre Classical definition of stress and strain Heng Nam Han Associate Professor School of Materials Science & Engineering College of Engineering Seol National University Seol 5-744, Korea Tel : Fa : hnhan@sn.ac.kr Homepage :
2 Concept of stress The forces acting within a stressed body are either body forces or srface forces. We assme an eqilibrim state nder body forces or srface forces. F lim A df da
3 Concept of stress Decomposition of stress in the direction of coordinate ais A
4 Concept of stress 4 This constrction atomatically implies the need for 8 stress components to define the state of stress in the volme element.
5 Concept of stress 5
6 Concept of stress 6 M i When we know 6 stress components at a position, we can define the stress tensor at the position in the material.
7 Concept of stress 7 Transformation of Coordinates
8 Transformation of Vector 8 The vector S can be easily resolved into components that are parallel to any set of reference aes.
9 Transformation of Vector 9 The two coordinate systems are related throgh a series of angles. We define them in terms of direction cosines, or or
10 Concept of stress Principal stresses X X X X Stress components acting on the faces of a tetrahedron OABC
11 Principal stresses τ τ y z τ yy τ y yz τ τ zz z yz Seclar or Characteristic Eq. Invariants of stress : I, I, I
12 Principal stresses I II III Stress components acting on the faces of a tetrahedron OABC
13 Eample Find the principal components and principal aes of the stress ij
14 Concept of stress 4 Maimm shear stress
15 Concept of stress 5 HW (de 9/) : Derive the Table and eplain that the maimm shear stress directions bisect the principal stress direction by 45 o.
16 6 Deviatoric stress Mean stress : P I ii III II I zz yy m Hydrostatic stress or pressre Deviatoric stress : m m m zz yz z yz yy y z y zz yz z yz yy y z y ij ij m ij ij δ
17 7 Concept of stress Familiar types of stress state Uniaial tension Uniaial compression Pre shear Hydrostatic press P P P Hydrostatic stress has the same components irrespective of the choice of coordinate aes. by 45 o rotation for o ais
18 8 Deviatoric stress Deviatoric stress is a symmetric second rank tensor.
19 9 Deviatoric stress J J / ij jk ki / ij ij
20 Eqilibrim Eqations for Stress Stress distribtion in an infinitesimal element F j, ij
21 Concept of strain : classical definition (inifinitesimal strain) A C B before deform. after deform. : -dimensional strain
22 Concept of strain M M α M L L J J K α K K -dimensional strain
23 Concept of strain M J M α J M L K α K K L -dimensional strain e e -directional normal γ -directional normal Shear α α
24 4 Concept of strain -dimensional strain e e e e e e e e e e ij
25 Concept of strain 5 (a) (b) Rigid body rotation by ω
26 6 Concept of strain Strain tensor Spin tensor ij ij ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ ϖ
27 Concept of strain 7 e a ij ki a lj e kl a a ij ki lj kl ω a a ω ij ki lj kl I II III
28 Volmetric Strain 8 Initial length of side :,, Finial length of side : ( ), ( ) ( ) V V V V V V ( )( )( ) V m
29 9 Deviatoric strain Mean strain : V ii III II I zz yy m Deviatoric strain : m m m zz yz z yz yy y z y zz yz z yz yy y z y ij ij m ij ij δ
30 Special Types of Strain Plane strain specimen Pre shear
31 Concept of strain Simple shear and pre shear e e e e Simple shear Pre shear e (/) ( e e ) (/) ( e - e )
32 Strain compatibility Plane strain condition General condition Mathematical Theory of Elasticity I.S. Solkolnikoff, P. 5
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