BME 510 Lecture 5 Fall 19 slides.pdf - Soft Hard Tissue Mechanics Small Deformation Elasticity Einstein indicial notation i=1 2 3 Fx Fi F y F z dealing

BME 510 Lecture 5 Fall 19 slides.pdf - Soft Hard Tissue...

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BME510 1 Soft & Hard Tissue Mechanics - Small Deformation Elasticity z y x i F F F F Einstein indicial notation: i =1, 2, 3 dealing with quantities beyond scalars and vectors, including stress, strain and constitutive coefficients tensors
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BME510 2 The stress tensor: The stress at any point is described by nine components zz zy zx yz yy yx xz xy xx Where only 6 are independent components: ji ij , thus: (tensor symmetry) yx xy zy yz xz zx Normal Stresses: xx 11 ( xx normal stress ) yy 22 ( yy normal stress ) zz 33 ( zz normal stress ) Shear Stresses: xy 12 ( xy shear stress ) xz 13 ( xz shear stress ) yx 21 ( yx shear stress ) yz 23 ( yz shear stress ) zx 31 ( zx shear stress ) zy 32 ( zy shear stress ) i.e., σ yx is stress acting in the x direction on an area (which is defined by the y normal vector) we have: 33 32 31 23 22 21 13 12 11 zz zy zx yz yy yx xz xy xx j i Cauchy stress tensor Each index refers to an axis in the same way as the coordinate system designation, i.e. x =1, y =2, z =3 may be replaces with σ for all stresses 2nd order stress tensor
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BME510 3 For small deformation elasticity (<5%) we can use the small deformation linear strain tensor and the Cauchy stress tensor . Any repeated indices represent a sum of terms containing the repeated indices (aka, dummy indices). For example, hydrostatic pressure is defined as one-third of the sum of normal components of stress: ) ( 3 1 33 22 11 p ) ( 3 1 3 1 33 22 11 ii p the indices ii are repeated - the index i is NOT independent. Therefore, the number of non-repeated (or independent) indices is zero and the quantity is a scalar (tensor of 0th order). Accordingly, it is non-directional (unlike vectors). A traction force boundary condition for an elasticity problem ─ the product between a 2nd order tensor and a vector (a tensor of 1 st order): j ij i n t 3 , 2 , 1 , 3 3 , 2 , 1 , 2 3 , 2 , 1 , 1 3 33 2 32 1 31 3 3 23 2 22 1 21 2 3 13 2 12 1 11 1 j i n n n n j i n n n n j i n n n n j j j j j j σ is the 2 nd order stress tensor and n is the vector normal to the boundary i is the independent index that is transferred on the resulted traction quantity t ( j is a repeated index while i is a non-repeated or independent index - for each i we sum over the j indices)
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