This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense.Some numerical examples will demonstrate the capability of the presented finite element formulation.
Myocardial infarction is an increasing health problem worldwide. Because of an under-supply of blood, the cardiomyocytes in the affected region permanently lose their ability to contract. This in turn gradually weakens the overall heart function. A new therapeutic approach based on the injection of a gel into the infarcted area aims to support the healing and to inhibit adverse remodelling that can lead to heart failure. A computational model is the basis for obtaining a better understanding of the heart mechanics, in particular, how myocardial infarction and gel injections affect its pumping performance. A strain invariant-based stored energy function is proposed to account for the passive mechanical behaviour of the model, which also makes provision for active contraction. To incorporate injections an additive homogenization approach is introduced. The numerical framework is developed using an in-house code based on the element-free Galerkin method. The main focus of this contribution is to investigate the influence of gel injections on the mechanics of the left ventricle during the diastolic filling and systolic isovolumetric (isochoric) contraction phases. It is found that gel injections are able to reduce the elevated fibre stresses caused by an infarct.
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