Abstract.A major open problem in the mathematical analysis of martensitic phase transformations is the derivation of explicit formulae for the set of recoverable strains and for the relaxed energy of the system. These are governed by the mathematical notion of quasiconvexity. Here we focus on bounds on these quasiconvex hulls and envelopes in the setting of geometrically-linear elasticity. Firstly, we will present mathematical results on triples of transformation strains. This yields further insight into the quasiconvex hull of the twelve transformation strains in cubic-to-monoclinic phase transformations. Secondly, we consider bounds on the energy of such materials based on the so-called energy of mixing thus obtaining a lamination upper bound on the quasiconvex envelope of the energy. Here we present a new algorithm that yields improved upper bounds and allows us to relate numerical results for the lamination upper bound on the energy with theoretical inner bounds on the quasiconvex hull of triples of transformation strains.
In order to enhance the load capacity, gears can be nitrided. The diffusion zone, measurable by the nitriding hardness depth, is considered to be the parameter governing for the high load-bearing capacity of nitrided gears. The wear behavior of gears is mainly determined by the characteristics (phase, porosity and chemical composition) of the compound layer but the influence of the compound layer on the load carrying capacity is not known yet. In this work, nitriding treatments for gears were developed with the aim to create compound layers with varying thickness, composition and properties in order to ensure a maximum load carrying capacity for nitrided gears.
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