Analysis of martensitic microstructures in shape-memory-alloys : A low volume-fraction limit

In this work, we study the limiting behavior of a variational model which arises in the analysis of microstructures at austenite-martensite interfaces in shape-memory alloys. We consider an energy containing an elastic bulk term and a surface term, in which the bulk term is geometrically linearized. Two martensitic phases, one of them with a much smaller volume fraction, form microstructures at a straight interface to an austenite phase. From literature we know a scaling law for the behavior of minimizers which depends on the ratio of volume fraction and prefactor in the surface energy. For a transition regime of this behavior we consider two rescaled energies, one for scalar-valued functions, one for vector-valued functions. The limiting energy in the sense of Gamma-limits for the first functional is derived. This limiting functional is only finite on a subspace of SBV which bears constraints on the direction of the jump set. A key ingredient for the proof of the Gamma-limit is an approximation result for this subspace with respect to the energy. We conjecture the form of the limiting energy for the vector-valued case that is defined on a subspace of SBD which again does only allow some special directions of the jump set. The Gamma-convergence result for this case is still an open problem since we have not yet been able to provide an appropriate density result. We are, however, able to prove the liminf-inequality, a compactness result and recovery sequences for functions of higher regularity. Additionally, we provide a Korn-Poincare-type inequality for the subspace of SBD

Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

We prove that microstructures in shape-memory alloys have a self-similar refinement pattern close to austenite-martensite interfaces, working within the scalar Kohn-Müller model. The latter is based on nonlinear elasticity and includes a singular perturbation representing the energy of the interfaces between martensitic variants. Our results include the case of low-hysteresis materials in which one variant has a small volume fraction. Precisely, we prove asymptotic self-similarity in the sense of strong convergence of blow-ups around points at the austenite-martensite interface. Key ingredients in the proof are pointwise estimates and local energy bounds. This generalizes previous results by one of us to various boundary conditions, arbitrary rectangular domains, and arbitrary volume fractions of the martensitic variants, including the regime in which the energy scales as ε2/3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon ^{2/3}$\end{document} as well as the one where the energy scales as ε1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varepsilon ^{1/2}$\end{document}.Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659Humboldt-Universität zu Berlin (1034)Peer Reviewe

Energy scaling laws for geometrically linear elasticity models for microstructures in shape memory alloys

We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms of the problem parameters, which represent the shape of the nucleus, the quotient of the elastic moduli of the two phases, the surface energy constant, and the volume fraction of the two martensitic variants. We identify several different scaling regimes, which are distinguished either by the exponents in the parameters, or by logarithmic corrections, for which we have matching upper and lower bounds

The lack of retropatellar resurfacing at index surgery is significantly associated with failure in patients following patellofemoral inlay arthroplasty: a multi-center study of more than 260 patients

Purpose!#!To evaluate the clinical outcomes of patients with a minimum 2-year follow-up following contemporary patellofemoral inlay arthroplasty (PFIA) and to identify potential risk factors for failure in a multi-center study.!##!Methods!#!All patients who underwent implantation of PFIA between 09/2009 and 11/2016 at 11 specialized orthopedic referral centers were enrolled in the study and were evaluated retrospectively at a minimum 2-year follow-up. Clinical outcomes included the Western Ontario and McMaster Universities Osteoarthritis Index (WOMAC) score, the Knee Injury and Osteoarthritis Outcome Score (KOOS), the Tegner Scale, the visual analogue scale (VAS) for pain, and subjective patient satisfaction. Pre- and perioperative risk factors were compared among failures and non-failures to determine potential risk factors.!##!Results!#!A total of 263 patients (85% follow-up rate) could be enrolled. The mean age at the time of index surgery was 49 ± 12 years with a mean postoperative follow-up of 45 ± 18 months. The overall failure rate was 11% (28 patients), of which 18% (5 patients) were patients with patella resurfacing at index surgery and 82% (23 patients) were patients without initial patella resurfacing. At final follow-up, 93% of the patients who did not fail were satisfied with the procedure with a mean transformed WOMAC Score of 84.5 ± 14.5 points, a mean KOOS Score of 73.3 ± 17.1 points, a mean Tegner Score of 3.4 ± 1.4 points and a mean VAS pain of 2.4 ± 2.0 points. An increased BMI was significantly correlated with a worse postoperative outcome. Concomitant procedures addressing patellofemoral instability or malalignment, the lack of patellofemoral resurfacing at the index surgery and a high BMI were significantly correlated with failure in our patient cohort.!##!Conclusion!#!Patellofemoral inlay arthroplasty shows high patient satisfaction with good functional outcomes at short-term follow-up and thus can be considered a viable treatment option in young patients suffering from isolated patellofemoral arthritis. Patellar resurfacing at index surgery is recommended to decrease the risk of failure.!##!Level of evidence!#!Retrospective case series, Level IV