Simon R. Eugster

Contrast enhancement with dual energy CT for the assessment of atherosclerosis

A drawback of the commonly used single source computed tomography systems (CT) is that different materials might show very similar attenuation at any selected radiation energy. However, the assessment of atherosclerosis requires good differentiation between vessel lumen, calcium, adipose, and surrounding tissue. Dual energy CT (DECT) simultaneously measures attenuations at two energies and therefore can improve the differentiation to some extent. A tissue cancelation and enhancement algorithm for dual energy data was already proposed in 1981 and evaluated on experimental settings with a stationary X-ray source. For this study, we adapted this algorithm for DECT and propose its usage as a pre-processing step for the assessment of atherosclerosis. On clinical DECT patient data and with fixed parameters we could show a simultaneous contrast enhancement between 8% and 67% among all targeted tissues.

Augmented Nonlinear Beam Theories

Augmented nonlinear beams are beams whose constrained position field and insofar whose cross section deformations are more involved than those in the classical theory. In classical theories the deformation of the cross sections are described by six generalized position functions, whose dual kinetic quantities are resultant contact forces and contact couples. Since the balance of linear and angular momentum hold six equations, it is also possible to derive the equations of motion of an induced theory for classical beams from the balance of linear and angular momentum, cf. [1]. Assuming more complex deformation states of the cross sections using more than six generalized position functions, as for instance to describe in-plane deformation or out-of-plane warping, more complex and counterintuitive generalized resultant contact forces do appear. The postulation of the correct intrinsic equations requires much mechanical intuition. Hence, we determine the equations of motion of the nonlinear two-director Cosserat beam and the nonlinear Saint–Venant beam in a concise way in the sense of induced beam theories. In Sect. 8.1, we introduce the nonlinear Cosserat beam which is intensely discussed in Naghdi [2] and Rubin [3]. In Sect. 8.2, we treat a beam theory with out-of-plane warping, derived by Danielson and Hodges [4] in its static version as an induced theory. A dynamical version of the Saint–Venant beam is obtained in Simo and Vu-Quoc [5] as an intrinsic theory. In accordance to Saint–Venants solution of a linear elastic body under torsion, who has recognized the effect of warping fields, we call this theory Saint–Venant beam theory.

Classical Plane Linearized Beam Theories

We speak of a classical plane linearized beam as being a classical linearized beam fulfilling the following assumptions. The cross section geometry remains the same for all cross sections, the motion is restricted to a plane, the reference configuration is straight and the material of the continuous body is described by a linear elastic material law. These assumptions on the motion of the beam and material law enable us to formulate statements which are not easily accessible for a more general configuration of a beam. One key point is that we are able to arrive at a fully induced beam theory where the integration of the stress distributions over the cross sections can be performed analytically. Hence, we recognize relations between the generalized internal forces and the three-dimensional stress field of the Euclidean space. This allows to apply concepts of the theory of strength of materials to beams which is of vital importance to solve engineering problems. In order to achieve such a connection, we restate the generalized internal forces for the plane linearized beam. The restriction to small displacements allows us to start from the internal virtual work formulated with the linearized strain. Afterwards, we proceed in a similar way as in the previous chapters. We state the constrained position field of the beam and apply it to the virtual work which leads us consequently to the boundary value problem of the beam. Using the solutions of the boundary value problem and non-admissible virtual displacements, it is possible to access in a further step the constraint stresses of the beam which guarantee the restricted kinematics of the beam. The outline of the chapter is as follows. In Sect. 7.1 we repeat the principle of d’Alembert–Lagrange for linear elasticity and introduce an elastic constitutive law for the impressed stresses. In Sects. 7.2–7.4 the equations of motion and the plane stress distribution of the plane linearized Timoshenko, Euler–Bernoulli and Kirchhoff beam are determined.

Classical Linearized Beam Theories

In many engineering applications beams are so stiff, that only small deformations with respect to a reference configuration occur. Thus, a linear beam theory is preferred which simplifies the problem drastically. Using the nonlinear beam theory from the previous chapter, such a linear beam theory is obtained in a straight forward manner by the linearization around a reference configuration. This chapter presents the process of linearization of a nonlinear theory at the example of classical beam theories, whose results are best-known, cf. [1]. In Sect. 6.1 the kinematical quantities of Chap. 5 are linearized around a reference state. Subsequently, in Sect. 6.2, the virtual work contributions and its corresponding differential equations are stated in its linearized form. Finally, Sects. 6.3–6.5 discuss the elastic constitutive laws of the linearized Timoshenko, Euler–Bernoulli and Kirchhoff beam theory.

Classical Nonlinear Beam Theories

Classical nonlinear beams from the point of view of an induced theory are continuous bodies with a constrained position field which are described by the motion of a centerline and the motion of plane rigid cross sections attached to every point at the centerline. This restricted kinematics allows to determine resultant forces at each cross section and to reduce the equations of motion of a three-dimensional continuous body to a partial differential equation with only one spatial variable. The present chapter is partly based on the publication of Eugster et al. [1]. First, in Sect. 5.1, the kinematical assumptions are stated. Subsequently, in Sect. 5.2, the virtual work contributions of the internal forces, the inertia forces and the external forces are reformulated by the application of the restricted kinematics to the virtual work of the continuous body. In Sects. 5.3–5.5 we present the generalized constitutive laws of the geometrically nonlinear and elastic theories of Timoshenko, Euler–Bernoulli and Kirchhoff in the form of a semi-induced beam theory. Lastly, Sect. 5.6 closes the chapter with a concise literature survey of numerical implementations of nonlinear classical beam theories.