Solid Mechanics

 

To perform mechanical analysis of stents in atherosclerotic arteries, it is obviously important to have an understanding of solid continuum mechanics.  They governing equation that applies to the stent, artery, and plaque is the conservation of linear momentum.  It is this vector equation that must be solved using the finite element method (which will be shown later).  In Cartesian coordinates, this governing equation can be expressed as


where i, j = 1,2,3.  Here, σij is the Cauchy stress tensor, xi refers to the deformed coordinates, ρ is the density of the material, fi is the body force vector, and ai is the acceleration vector.  The acceleration ai can be expressed in terms of the material derivative of the velocity vi as


and the velocity vi can be expressed in terms of the material derivative of the displacement ui as


For small (infinitesimal)  displacements and velocities, it is possible to ignore the nonlinear terms in the material derivatives [6].  However, since the deformation of the stent and arterial wall can be large (much greater than 1%), the theory of nonlinear mechanics becomes necessary.  The deformation gradient tensor and the left Cauchy-Green tensor are important in the study of large deformations.  The deformation gradient tensor Fij is defined as


where i, j = 1,2,3, and ui = xi - Xi (i = 1,2,3) is the displacement vector.  Xi refers to the undeformed coordinates, and xi refers to the deformed coordinates.  Note that δij is called the Kronecker-Delta and it is equal to 1 when i = j and 0 when i ≠ j.  Then the left Cauchy-Green tensor Bij can be defined in terms of Fij as [7]


Often, it is mathematically easier to describe the displacement in terms of the strain tensor.  There are many kinds of finite strains used in nonlinear mechanics.  For instance, the Green’s strain tensor is defined as [7]


The term "finite" refers to large deformations in which nonlinear terms are present.  Furthermore, there are also many kinds of finite stresses that are useful in nonlinear mechanics as well.  The second Piola-Kirchhoff stress tensor Sij is sometimes mathematically more convenient than the Cauchy stress tensor σij.  The former is used in the mathematical description of the material properties; this can be done by using a strain energy function.  For more information on this, please see Material Properties.  The Cauchy stress tensor is used in the conservation of linear momentum equations, as shown above.  The two are related by [7]

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