Abstract
Numerical modeling of implants and specimens made from trabecular structures can be difficult and time-consuming. Trabecular structures are characterized as spatial truss structures composed of beams. A detailed discretization using the finite element method usually leads to a large number of degrees of freedom. It is attributed to the effort of creating a very fine mesh to capture the geometry of beams of the structure as accurately as possible. This contribution presents a numerical homogenization as one of the possible methods of trabecular structures modeling. The proposed approach is based on a multi-scale analysis, where the whole specimen is assumed to be homogeneous at a macro-level with assigned effective properties derived from an independent homogenization problem at a meso-level. Therein, the trabecular structure is seen as a porous or two-component medium with the metal structure and voids filled with the air or bone tissue at the meso-level. This corresponds to a two-level finite element homogenization scheme. The specimen is discretized by a reasonable coarse mesh at the macro-level, called the macro-scale problem, while the actual microstructure represented by a periodic unit cell is discretized with sufficient accuracy, called the meso-scale problem. Such a procedure was already applied to modeling of composite materials or masonry structures. The application of this multi-scale analysis is illustrated by a numerical simulation of laboratory compression tests of trabecular specimens.
Highlights
The scientific and technical development of new materials and structures is significantly supported by additive manufacturing methods rapidly developing in recent years, which allow for manufacturing products of complex shapes and geometry
The proposed approach is based on a multi-scale analysis, where the whole specimen is assumed to be homogeneous at a macro-level with assigned effective properties derived from an independent homogenization problem at a meso-level
Therein, the trabecular structure is seen as a porous or two-component medium with the metal structure and voids filled with the air or bone tissue at the meso-level
Summary
The scientific and technical development of new materials and structures is significantly supported by additive manufacturing methods rapidly developing in recent years, which allow for manufacturing products of complex shapes and geometry. The main advantages of trabecular structures are seen in the enlargement of the contact area between implants and bone tissue and in the possibility to modify mechanical properties, which helps to eliminate some problems of conventional implants, e.g., the stress shielding effect This complex morphology allows the ingrowth of bone cells, guarantees the better stability of implants in bones [1]. Therein, the trabecular structure is seen as a porous or two-component medium with the metal structure and voids filled with the air or bone tissue at the meso-level This corresponds to a two-level finite element homogenization scheme, where the specimen is discretized by a reasonable coarse mesh at the macro-level (called the macro-scale problem), while the actual microstructure in terms of a certain representative volume element (RVE) is discretized with sufficient accuracy (called the meso-scale problem) [2]. Such an RVE is usually presented in the form of a periodic unit cell (PUC) or a statistically equivalent periodic unit cell (SEPUC) constructed, such as to match the real meso-structure as close as possible [3]
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