Modeling of materials capable of solid–solid phase transformation. Application to the analytical solution of the semi-infinite mode III crack problem in a phase-changing solid Ziad Moumni a,d,n , Wael Zaki b , Quoc Son Nguyen c , Weihong Zhang d a ENSTA-ParisTech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France b Khalifa University of Science, Technology, and Research P.O. Box 127788, Abu Dhabi, United Arab Emirates c École Polytechnique-ParisTech 91128 Palaiseau Cedex, France d Northwestern Polytechnical University 127 Youyi West Rd, Xi'an, Shaanxi 710072, China article info Article history: Received 17 August 2014 Received in revised form 3 November 2014 Accepted 12 December 2014 Available online 19 December 2014 Keywords: Solid–solid phase change Internal constraints Convexification Phenomenological modeling Homogenization Mode III crack abstract The paper proposes two frameworks for the derivation of constitutive models for solids undergoing phase transformations. Two distinct approaches are considered: the first is based on the assumption that solid phases within the material are finely mixed whereas the second considers the material as a heterogeneous solution of different phase fragments and uses the homogenization theory to derive constitutive relations at the macroscale. For both approaches, the mathematical representation of the material behavior is intentionally kept simple and the derivations are fully developed for ease of future modification and use by the readership. It is shown that in the case of reversible phase transformation, the energy of the material can be obtained by convexifiying the energy functions of the constituent phases. It is further shown that for dissipative phase transformation the material behavior can be made stable by deriving the evolution equations of the state variables from adequately chosen dissipation potentials. Some new results of existence and uniqueness of the solution of boundary value problems of structures undergoing phase change are also given. As an application, a schematic model is derived and used to obtain analytical solutions for the problem of semi-infinite mode III crack in a solid capable of phase transformation. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Phase transformation commonly takes place in solids subjected to processes such as welding, solid phase precipitation, and formation of heterogeneous solid inclusions where it is prompted by thermomechanical solicitations [31]. An interesting example of phase-transforming solids is shape memory alloys (SMAs), where solid-state phase change allows the formation of unusually large recoverable strains [10,25]. Numerous models for SMAs have been developed since the second half of the last century. Some of these models relied on the use of phenomen- ological rules for phase transformation kinetics that were directly inferred from uniaxial experiments, such as the exponential phase transformation rule of Tanaka [35] and the cosine rule of Liang and Rogers [22], and were extended to account for uniaxial and multiaxial loadings with different degrees of complexity [5,17] More recently, several models have been proposed in which the constitutive relations are derived from thermodynamic potentials – often the Helmholtz or Gibbs free energies – taking into account considerations of thermodynamic consistency. In these models, the expression of the thermodynamic potential is constructed from phenomenological considerations [40,41,2,27] or derived using analytical or numerical homogenization techniques [16,28,32,36,21,6,39]. In this paper, we present two simple mathematical frameworks for modeling solid–solid phase transformation. The first framework considers the different solid phases within a representative volume element (RVE) of the material to be finely mixed and uses a rheological model in series to relate the behavior of the RVE to that of the phases. The second framework considers the RVE as a heterogeneous mixture of phases and uses homogenization techni- ques to derive the constitutive equations for the material at the macroscale. For these examples, the energy densities of the different solid phases are considered to be quadratic functions of the local strain and the overall energy density of the material is constructed as the sum of individual phase energies and interaction terms leading to Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics http://dx.doi.org/10.1016/j.ijnonlinmec.2014.12.004 0020-7462/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author at: ENSTA-ParisTech, 828 Boulevard des Maréchaux, 91120 Palaiseau, France. E-mail addresses: ziad.moumni@ensta-paristech.fr (Z. Moumni), wael.zaki@kustar.ac.ae (W. Zaki), zhangwh@nwpu.edu.cn (W. Zhang). International Journal of Non-Linear Mechanics 69 (2015) 146–156