BME5101Soft & Hard Tissue Mechanics - Small Deformation ElasticityzyxiFFFFEinstein indicial notation:i=1, 2, 3dealing with quantities beyond scalars and vectors, including stress, strain and constitutive coefficients tensors
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BME5102The stress tensor: The stress at any point is described by nine components zzzyzxyzyyyxxzxyxxWhere only 6 are independent components: jiij, thus: (tensor symmetry) yxxyzyyzxzzxNormal Stresses: xx11(xx normal stress) yy22(yy normal stress) zz33(zz normal stress) Shear Stresses:xy12(xy shear stress) xz13(xz shear stress) yx21(yx shear stress) yz23(yz shear stress) zx31(zx shear stress) zy32(zy shear stress) i.e., σyxis stress acting in the x direction on an area (which is defined by the y normal vector) we have:333231232221131211zzzyzxyzyyyxxzxyxxjiCauchy stress tensorEach index refers to an axis in the same way as the coordinate system designation, i.e. x=1, y=2, z=3may be replaces with σfor all stresses2nd order stress tensor
BME5103•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:)(31332211p)(3131332211iipthe indices iiare repeated - the index iis NOT independent. Therefore, the number of non-repeated (or independent) indices is zero and the quantity is ascalar(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 1storder):jijint3,2,1,33,2,1,23,2,1,1333232131332322212123132121111jinnnnjinnnnjinnnnjjjjjjσis the 2ndorder stress tensor and nis the vector normal to the boundaryiis the independent index that is transferred on the resulted traction quantity t(jis a repeated index while iis a non-repeated or independent index - for each iwe sum over the jindices)