CIE4145 : STRESS STRAIN RELATION LECTURE TOPICS
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1 CI445 : STRSS STRAIN RLATION LCTUR TOPICS Stress tensor Stress definition Special stress situations Strain tensor Relative displacements Strain definition Strain tensor 3 Tensor properties Introduction of tensors Transformation rules Mohr s circle Stiffness eample 4 Stress - strain relation General formulation Comparison of Mohr s stress and strain circle for isotropic L-material amples 5 Yieldcriteria Stresses in 3D Von Mises and Tresca models Ir J.W. Welleman 7 januari 7 page nr
2 STRSS STRAIN RLATION STRSS z z z Ir J.W. Welleman 7 januari 7 page nr
3 STRAINS γ γz γ z z ij ui u j + i, j,, z j i Ir J.W. Welleman 7 januari 7 page nr 3
4 STRSSS AND STRAINS z z z z and z z z z Relation between stress and strain? fourth order tensor (998 unknown elements!) Ir J.W. Welleman 7 januari 7 page nr 4
5 Ir J.W. Welleman 7 januari 7 page nr 5 STRSSS AND STRAINS z z z z z z z z z z z z of of 6 STRAINS 6 STRSSS
6 MATRIAL BHAVIOUR 6*6 36 unknown elements smmetrical matri >> unknown elements aeolotropic: material properties differ in all directions orthotropic: material properties differ in three orthogonal directions anisotropic: aeolotropic combined with orthotropic isotropic : material properties are the same in all directions homogeneous : material properties are the same for each point Ir J.W. Welleman 7 januari 7 page nr 6
7 STANDARD UNIAXIAL TST STRAIN Specific change in length d pull-direction : l l l l l + d' d d orthogonal direction : necking : d d d d d const l l necking coefficient or Poisson s ratio A LNGTHNING IN ON DIRCTION CAUSS A SHORTNING IN AN ORTHOGONAL DIRCTION Ir J.W. Welleman 7 januari 7 page nr 7
8 STRAIN - STRSS RLATION Normal strain normal stress z. Ir J.W. Welleman 7 januari 7 page nr 8
9 STRAIN - STRSS RLATION Shear angle - shear stress γ Gγ thus : G Ir J.W. Welleman 7 januari 7 page nr 9
10 RLATION BTWN, G and (, ) RC (, ) (, ) MOHR s circle (, ) CONSTANT SHAR STRSS ON PLANS WITH AN ANGL OF 45 DGRS THUS : PUR SHAR Ir J.W. Welleman 7 januari 7 page nr
11 OBSRVD STRAINS ORIGINAL COORDINAT SYSTM ROTATD COORDINAT SYSTM ( SURFAC OF TH BLOCK ) ( INTRNAL BLOCK UNDR PUR SHAR ) thus : ( ) ( ) + + G G G G Ir J.W. Welleman 7 januari 7 page nr
12 STRAIN CIRCL, G MOHR s strain circle RADIUS OF TH CIRCL IS CONSTANT, THRFOR: +,, G +, + G ( + ) G Ir J.W. Welleman 7 januari 7 page nr
13 Ir J.W. Welleman 7 januari 7 page nr 3 SHAR ij ij ij ij ij G j i G G γ ) ( ) ( with : + + THUS : z z z z ) ( ) ( ) (
14 Ir J.W. Welleman 7 januari 7 page nr 4 ISOTROPIC LINAR LASTIC MATRIAL ( HOOK s LAW ) FLXIBILITY RLATION STRAIN STRSS RLATION 3 ( ) ( ) ( ) z z z z
15 STRAIN STRSS RLATION 4 ISOTROPIC LINAR LASTIC MATRIAL ( HOOK s LAW ) z z ( )( ) + z z STIFFNSS RLATION W WR LOOKING FOR! What about Poisson s ratio?? Ir J.W. Welleman 7 januari 7 page nr 5
16 STRAIN STRSS RLATION 5 PLAN STRSS SITUATION (e.g. - plane)( z z ) ( ) ( + ) ( ) ( + ) G G FLXIBILITY STIFFNSS WITH SHAR MODULUS : G ( + ) Ir J.W. Welleman 7 januari 7 page nr 6
17 Ir J.W. Welleman 7 januari 7 page nr 7 PLAN STRSS SITUATION (e.g. - plane) ( z z ) ( ) + ) ( stiffness : fleibilit : STRAIN STRSS RLATION 5b
18 XAMPL (see strain tensor previous lecture) l D C Displacement field: u 4 4, +,3 4 4 u +,8 material : 5 GPa;, 5 3l Question : Compute the stresses on all surfaces A B l l Ir J.W. Welleman 7 januari 7 page nr 8
19 FROM STRAIN TO STRSS Strains ( ) ( 4 4 ;,3 ;, ) ( ) ( 4 4 ;,8 ;, ) Stresses [N/mm ] ( ) ( ) ; 4; 4 ( ; ) ( ; 4) + ( + ) ( + ) Ir J.W. Welleman 7 januari 7 page nr 9
20 ( ; ) MOHR s circle (STRSS), N/mm ( ; ) (;4) ( ; ) r (4; 4) m // -ais // -ais α DC ( ; ) Ir J.W. Welleman 7 januari 7 page nr
21 COMPAR STRAIN CIRCL WITH STRSS CIRCL FOR A HOMOGNOUS ISOTROPIC MATRIAL HOLDS : PRINCIPL DIRCTIONS FOR STRSSS AND STRAINS AR TH SAM TH LOCATION OF TH D.C. IN BOTH CIRCLS MUST THRFOR B TH SAM ALTRNATIV WAY TO OBTAIN A STRSS CIRCL COMPUT TH PRINCIPAL STRSSS FROM TH PRINCIPAL STRAINS DRAW TH STRSS CIRCL MARK TH D.C. (FROM TH STRAIN CIRCL) Ir J.W. Welleman 7 januari 7 page nr
22 STRSSS ON ALL PLANS [ N/mm ] 7 5 C l D 3l 4 4 COMPUT TH RSULTING FORCS ON ACH PLAN AND CHCK TH QUILIBIRUM! 7 5 A 4 B l l Ir J.W. Welleman 7 januari 7 page nr
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139
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