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First Principles Modelling of Shape Memory Alloys

Part of the book series: Springer Series in Materials Science ((SSMATERIALS,volume 163))

Abstract

This chapter reviews the fundamental material properties of shape memory alloys and thereby sets up the physical situation our modelling is referring to. Laboratory observations reveal scale-specific characteristics the modelling must reflect. Therefore, shape memory alloys are a prime example of cross-scale modelling. We briefly explain the respective modelling approaches to place the method of molecular dynamics simulations into the scope of the scientific frame work.

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Notes

  1. 1.

    The diagram also nicely reveals an inflationary increase in publications in the internet age.

  2. 2.

    Note in 1915, Born was able to prove the Cauchy relations do not apply in the case of nested, many-species lattices owing to the contributions from interactions between the sublattices [50].

  3. 3.

    Proof following I. Müller in three steps:

    1. 1.

      Suppose \({\mathbf{ d} }\Vert \tilde{{\mathbf{ d} }}\), that is \(\tilde{{\mathbf{ d} }}=\alpha \,{\mathbf{ d} }\), where \(\alpha \) is a real scalar. Hence \({\mathbf{ F} }-\tilde{{\mathbf{ F} }}={\mathbf{ d} }\otimes ({\mathbf{ p} }-\alpha \tilde{{\mathbf{ p} }})\) and we conclude that the condition (1.3) holds for \({\mathbf{ a} }={\mathbf{ d} }\) and \({\mathbf{ m} }={\mathbf{ p} }-\alpha \tilde{{\mathbf{ p} }}\).

    2. 2.

      Suppose \({\mathbf{ p} }\Vert \tilde{{\mathbf{ p} }}\), that is \(\tilde{{\mathbf{ p} }}=\beta \,{\mathbf{ p} }\), where \(\beta \) is a real scalar. Hence \({\mathbf{ F} }-\tilde{{\mathbf{ F} }}=({\mathbf{ d} }-\beta \tilde{{\mathbf{ d} }})\otimes {\mathbf{ p} }\) and we conclude that the condition (1.3) holds for \({\mathbf{ a} }={\mathbf{ d} }-\beta \tilde{{\mathbf{ d} }}\) and \({\mathbf{ m} }={\mathbf{ p} }\)

    3. 3.

      Now suppose that the two vectors \({\mathbf{ p} }\) and \(\tilde{{\mathbf{ p} }}\) are not parallel, \(\mathbf{p}\nparallel \tilde{\mathbf{p}}\). Thus the normal vector \({\mathbf{ m} }\) may be represented as a linear combination of \({\mathbf{ p} }\) and \(\tilde{{\mathbf{ p} }}\), \({\mathbf{ m} }=\alpha \,{\mathbf{ p} }+\beta \,\tilde{\mathbf{ p} }\), where \(\alpha ,\beta \) are real scalars. It follows from (1.3) that \({\mathbf{ d} }\otimes {\mathbf{ p} }-\tilde{{\mathbf{ d} }}\otimes \tilde{{\mathbf{ p} }} ={\mathbf{ a} }\otimes (\alpha \,{\mathbf{ p} }+\beta \,\tilde{{\mathbf{ p} }})\), or, after re-arranging \(({\mathbf{ d} }-\alpha \,{\mathbf{ a} })\otimes {\mathbf{ p} }=(\tilde{{\mathbf{ d} }}-\beta \,{\mathbf{ a} })\otimes \tilde{{\mathbf{ p} }}\). This condition implies the vectors \({\mathbf{ d} }\) and \(\tilde{{\mathbf{ d} }}\) must be parallel in this case, since \({\mathbf{ d} }=\alpha \,{\mathbf{ a} }\) and \(\tilde{{\mathbf{ d} }}=\beta \,{\mathbf{ a} }\) are the only choices which meet the condition (1.3). With this result, we may now return to 1, which concludes the proof of the statement (1.4).

  4. 4.

    In thermodynamics, a process is called reversible if the entropy production is zero. In this case the entropy balance (1.8) turns into an equality. In nature no such processes exists, therefore the idea of reversibility must be regarded as an idealisation.

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  1. Faculty of Mechanical Engineering, Institute for Materials, Ruhr University Bochum, 44780, Bochum, Germany

    Oliver Kastner

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Kastner, O. (2012). Preparations. In: First Principles Modelling of Shape Memory Alloys. Springer Series in Materials Science, vol 163. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28619-3_1

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  • DOI: https://doi.org/10.1007/978-3-642-28619-3_1

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