Some years ago, a family of continuum finite elements based on reduced integration [1], [2], [3] was investigated. Many structural components with different kinds of elastic and inelastic material behaviour were considered and these elements showed accurate results while beeing more efficient than similar three-dimensional formulations based on full integration. The objective of the present contribution is to extend the analysis to damage and fracture. To this end, we present the incorporation of a modified version of a gradient-extended damage model [4] based on the micromorphic approach [5] into solids with only a single integration point. Due to the analogy to fully-coupled thermomechanical problems, we adapt the derivation of a consistent hourglass stabilization from an earlier contribution for multi-field problems [6]. A numerical benchmark problem of quasi-brittle fracture reveals the accuracy and efficiency of the proposed approach. Besides the ability to deliver meshindependent results, the framework is especially suitable for highly constrained situations in which conventional low-order finite elements suffer from well known locking phenomena.The prediction of damage and fracture of solids and structures is a challenging task. Usually, damage is accompanied by softening until final failure occurs. Utilizing conventional 'local' continuum damage models usually leads to a pathological mesh dependence in finite element simulations. One possible remedy is to use a gradient-extended damage model which introduces an internal length into the material. Due to their simplicity and robustness, conventional low-order finite elements are usually employed for the spatial discretization of the governing partial differential equations, even though it is well known that these elements exhibit poor performance in the limit of incompressibility and bending dominated situations. These phenomena are frequently termed volumetric and shear locking. One possible solution is to use the concept of reduced integration with hourglass stabilization. The purpose of this contribution is to utilize a modified version of a 'nonlocal' damage model [4] which is based on the micromorphic approach [5] in conjunction with a single quadrature point finite element formulation similar to [1] in order to obtain the following advantages: (i) a reliable and locking-free element response for the prediction of the damage onset and progress, (ii) an increase of computational efficiency coming from the usage of less elements together with a lower number of integration points and (iii) mesh distortion insensitivity which particularly improves the robustness in regions of highly localized damage.
Reduced integration frameworkIn analogy to an earlier contribution [6], the starting point of our finite element formulation is a two-field functional which is strongly related to the enhanced strain method by [7]. This functional is extended by taking the weak form of the micromorphic balance equation [5], [4] into account. In total, this leads to three unkno...
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