Formulations and applications of multiscale modeling for SMA heterogeneous materials 

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SMA composites and their multiscale modeling

Composites are widely used in advanced engineering fields such as aerospace, medical ins-truments, due to their high strength, rich functions and good durability in complicated working conditions (Węcławski et al. [35], Vesely et al. [36], Liu et al. [37], Jochum et al. [38]). For example, carbon fiber reinforced composites present high strength/weight ratio and high modulus, and have been used in wind turbine blades and cars (Murdani et al. [39], Fasana et al. [40]). Along with the development of manufacturing techniques, more and more complicated compo-sites can be manufactured nowadays and present great potential in the future. For example, using 4D printing (Ge et al. [41], Momeni et al. [42], Tibbits and Skylar [43], Ding et al. [44]), Ge et al. [41] designed an active composite origami which can deform automatically when the temperature is changed. This structure is composed by elastic matrix and shape memory polymer fiber.
As aforementioned, SMA are widely used in the design for functional applications. Howe-ver, the high cost of SMAs limits their promotion to a certain degree. Composites consisting of SMAs and other cheaper components could be an alternative choice for designing SMAs appli-cations. Moreover, the combination of different material characteristics may extend the range of the SMAs’ applications for more complex situation as well as leads to structure lightening (Lester et al. [45]), see for example the hybrid SMA woven composite depicted in Fig. 1.4, which presents good impact resistance (Meo et al. [46]). Concretes reinforced by shape memory alloys present good seismic performance and crack self-healing capacity, which provide a wide applica-tion prospect in civil engineering (Halahla et al. [47]). Feng et al. [48] designed a multifunctional composite formed by elastomer embedded with SMA wires. Their composites show good in-plane and out-plane deformation capability and varying stiffness characteristics in different environ-ment temperatures. Li et al. [49] designed a simple soft actuator with SMA and silicon rubber, based on a periodic deformation mechanism. This soft actuator can achieve crawling and grasping by electrical heating methods.
SMA composites subject to thermomechanical load have very complicated behavior. Because it depends on not only the macroscopic load, but also the microscopic material properties, geo-metrical parameters and phase transformations of SMAs. Thus, SMA composites present typical multi-physical and multi-scale behavior. Valuable works using multiscale homogenization me-thods to study the thermomechanical behavior of SMA composites have been presented over the past few decades. The models using self-consistent method (Lagoudas et al. [50], Siredey et al. [51], Patoor et al. [52], Helm and Haupt [53], Marfia [54]), unit cell methods (Damanpack et al. [55], Kawai et al. [56]), asymptotic methods (Chatzigeorgiou et al. [57], Dehaghani et al. [58]) and FE2 methods (Kohlhaas and Klinkel [59], Chatzigeorgiou et al. [60]) can be mentioned as a review. Chatzigeorgiou et al. [57] proposed a model which can simulate the fully coupled thermomechanical response of a multilayered composites by using the asymptotic expansion ho-mogenization technique. Kohlhaas and Klinkel [59] proposed a FE2 scheme to simulate the free, two-way and the constrained SME of randomly oriented and distributed SMA fiber reinforced composites. The pseudo-plasticity at low temperature range and the pseudo-elasticity at high temperature range were also studied. This model considers a 1D element for the SMA fiber and 3D element for the matrix. The 1D formulation allows for easy meshing in the framework of the finite element method. The present model puts emphasis on the proper simulation of the phase transition from martensite to austenite within the SMA—no matter which loading conditions are applied. This allows for the ability to prestress a surrounding matrix and amend brittle material. The SMA fiber material model is embedded into two meshing approaches which are compared. The approaches proof each other by yielding same results. It is shown that the rebar concept has crucial advantages over the conform meshing concept. Dehaghani et al. [58] presented a 3D finite element based hierarchical multiscale analysis framework to solve the macro and micro scale problems simultaneously. The proposed approach is applied to a number of problems invol-ving SMA fiber reinforced composites and porous SMAs to assess its accuracy and effectiveness. The PE response of SMA constituents in micro scale is modeled based on the work by Boyd and Lagoudas [12], Qidwai and Lagoudas [61]. Macro problems are solved under various stress loading/unloading conditions including tension, flexure, and torsion. Detailed illustrations are presented regarding the distribution and concentration of different field variables such as the martensitic volume fraction and the stress state in the micro scale along the austenite to marten-site transformation and the reverse path. Due to lack of plane symmetry, 2D general plane strain models cannot accurately predict the microscopic behavior of porous materials. Hence, the effect of shape of 3D ellipsoid pores on the mechanical response of the porous SMA is investigated through a detailed parametric study.

Architected cellular SMAs and their multiscale modeling

Cellular materials are widely used for their high strength-to-weight ratio and high energy absorption performance (Ashby and Gibson [62], Ashby et al. [ 63]). For instance, honeycomb, folded cellular materials and foam are usually used as the core of the sandwich structures for dissipating the kinetic energy, damping or reducing the weight of the structure (Ashby et al. [63], Sarvestani et al. [64], García-Moreno [65], Hangai et al. [66], Strano et al. [67]). However, honeycomb and folded cellular materials have high manufacturing costs and moisture problem, as well as buckling problems (Rashed et al. [68]). The mechanical behaviors of foams are too difficult to be accurately measured for their stochastic cells, which always results in the excessive use of materials to satisfy the safety factor. To overcome these shortcomings, partially-ordered foams allowing limited structural control of the pore and spatial distribution of pore levels, such as metal syntactic foams (Taherishargh et al. [69, 70, 71], Broxtermann et al. [72], Linul et al. [73], Luong et al. [74]) are studied. Furthermore, architected cellular materials with an ordered structure were designed and studied during these years (Rashed et al. [68], Pingle et al. [75], Schaedler et al. [76], Schaedler and Carter [77], Lehmhus et al. [78]), such as the ultralight metallic microlattice, see Fig. 1.5a. Thanks to the highly developed additive manufacturing techniques, such as 3D printing (Ngo et al. [79], Mostafaei et al. [80]) and selective laser melting (Rashed et al. [68], Mehrpouya et al. [81]), the manufacturing of architected cellular materials is no longer impossible, see Fig. 1.5b. Users can design a cellular material with a certain behavior by tuning its cellular parameters, such as the geometry, components, local mechanical properties, etc.
Architected cellular materials’ functionality could be extended by combining the features of various materials, such as SMAs. It is well known that SMAs, such as NiTi, can endure large deformation and recover their initial shape after unloading (see for example the reviews of Lagoudas [2], Patoor et al. [84], Lagoudas et al. [85], Tobushi et al. [86], Cisse et al. [87]). This PE brings high performance to SMA in energy absorption. When the given load reaches a critical level in a superelastic test, SMA will apparently soften due to its inner phase transformation. This behavior enables SMA to absorb the external energy as much as possible and prevents material from crushing or buckling. Such a kind of response is very similar to the ideal response of the cellular material designed by Schaedler et al. [76]. Meanwhile, the hysteresis effects observed in SMA behavior can dissipate a large amount of energy. All mentioned features of SMAs meet the requirements of an architected structure for energy absorption applications very well. In addition, taking advantage of the lightweight and SME, architected SMAs may be designed for advanced applications in aerospace, civil engineering, etc.
To investigate the behavior of architected cellular SMAs, rare, but valuable works have been proposed (Machado et al. [88], Ravari et al. [89], Ashrafi et al. [90]). Machado et al. [88] proposed an experimental and modeling study on the cellular NiTi tube based materials. In order to design and optimize the architected SMA tube materials, the authors studied the effective behavior of the thin-walled NiTi cellular materials by carrying out a study based on experiment and numerical simulation. The influences of SMAs’ material properties and cellular architecture on the effective behavior were investigated. To reduce the high cost of fabrication, Ravari et al. [89] focused mainly on numerical modeling for designing and optimizing SMA cellular lattice structures. The effects of the geometry and cellular imperfections on the effective behavior of the material were investigated by unit cell and multi-cell methods. This work was later developed by Ashrafi et al. [90], who proposed an efficient unit cell model with modified boundary conditions for SMA cellular lattice structures. The SME was also simulated by this model, which had good agreement with the experiment. Cissé et al. [91] proposed an effective constitutive model for architected cellular iron-based shape memory alloys considering the pressure dependency and transformation-plasticity interaction. Zhu et al. [92] realized the accurate 3D reconstitution of porous SMAs and studied their superelastic behavior considering different geometrical parameters.
The above works mainly focused on the effective cellular response in order to represent or predict the behavior of architected SMA structures. Considering the scale separation between the microscopic cellular scale and macroscopic structural scale, however, it is difficult to predict the structural response of a unit cell without certain assumed boundary conditions, because the stress-strain states of the macroscopic structure are usually not uniform and the deformations at the microscopic level could be totally different. Thus, in order to directly simulate the structural responses of architected SMA, more appropriate numerical methods should be used. During the past few decades, multiscale modeling approaches have been developed and widely used (Kanouté et al. [93], Geers et al. [94], El Hachemi et al. [95], Kinvi-Dossou et al. [96]). As one of the most effective multiscale methods, the FE2 method (Feyel [97]) to describe the response of high nonlinear structures using generalized continua shows good performance in various applications, such as fiber buckling (Nezamabadi et al. [98]), composite shells (Cong et al. [99]), rate-dependent response (Tikarrouchine et al. [100 ]) and SMA based fiber/matrix composites (Kohlhaas and Klinkel [59], Chatzigeorgiou et al. [60]). In this approach, both the structural level and the RVE level are simulated by the finite element method (FEM). Two levels are fully coupled and computed simultaneously, where the unknown constitutive behaviors on the structure level are represented by the effective behaviors of homogenized RVEs, and the strain states of the RVEs are given by the associated integration points. In order to make multiscale procedures more generic and easy to use for industries, works to extend them on commercial computing platform, such as finite element software ABAQUS, have been proposed (see Yuan and Fish [101] and Tchalla et al. [102]). A brief review on multiscale homogenization methods is given in Section 1.3. Generic and user-friendly numerical modeling tool is of importance to significantly reduce the modeling cost in the design for architected SMA, as well as SMA composites. Thus, this thesis is about to implement the FE2 framework on ABAQUS with the SMA constitutive model, and develop a generic multi-physical and multiscale tool for the simulation of SMA heteregeneous materials.
The aforementioned structures with microscopic fiber and thin-walled cells are prone to bu-ckling phenomenon which usually leads to structural failure. This mechanism is very complex due to its strong nonlinearity and multiple modes. Thus, efficient multiscale modeling on bu-ckling phenomenon is also concerned in this thesis. As a typical multiscale instability problem, the buckling of long fiber composites under compressive stress is quite complicated, and is taken into account as an example. The modeling for the instability problem of long fiber reinforced composite is briefly introduced in the next section.

Data-driven multiscale homogenization methods

Multiscale simulation methods for heterogeneous materials have been intensively developed during the past decades. Based on homogenization theory, a series of analytical methods, such as the mean-field methods (Perdahcıoğlu and Geijselaers [151], Wu et al. [152]), were proposed to study the effective behavior of microscopic heterogeneities. Later, numerical approaches, such as the FE2 method (Feyel [97], Kouznetsova et al. [117], Nezamabadi et al. [124], Miehe et al. [156], Hughes et al. [161], Kodjo et al. [162]), asymptotic homogenization method (Dehaghani et al. [58], Boso et al. [158], Fish et al. [159]), mechanics of structure genome (Liu and Yu [163], Liu et al. [164], Rouf et al. [165]), coarse-graining technique (Budarapu et al. [166]) and multiscale modeling softwares (Tchalla et al. [102], Talebi et al. [167]), were developed to conduct structural analyses by the finite element method. During the past years, the accuracy of homogenization methods made great progresses. However, the computation cost still remains one of the main issues. To guarantee high accuracy within an online simulation for complex heterogeneous material, the pieces of information from different scales needs to be all correlated together. For example, in the framework of the classical FE2 method, the information over each macroscopic integration point is supposed to be updated at each iteration, which requires tremendous repetitive computations over associated RVE (van Tuijl et al. [126]).
Many works were devoted to address this issue by improving the computation efficiency. The empirical interpolation method (van Tuijl et al. [126], Hernández et al. [127]), the empirical cubature method (Hernandez et al. [168]), POD (Yvonnet et al. [125], Yvonnet and He [169]), Fourier’s series method (Huang et al. [130], Attipou et al. [136], Huang et al. [137], Moulinec and Suquet [170], Göküzüm et al. [171], Rambausek et al. [172]), structural elements (Klarmann et al. [173], Hui et al. [174]), specific FE2 computing strategy (Praster et al. [175]) and high performance nonlinear solver (Nezamabadi et al. [98]) contributed to solve this issue during the past 10 years. However, these methods are usually used in the classical multiscale computing framework, in which the problems at different scales have to be solved concurrently. The correlation of scales benefits multiscale computation with high accuracy, but yields a complex multiscale nonlinear system in comparison with single scale problems. On the one hand, the accuracy in different scales will affect each other directly in an online computing process (see chapter 2). On the other hand, convergence issues in a scale will retard and even compromise the solution of the whole problem. Some examples with strong microscopic nonlinear problem, such as fiber buckling (see chapter 3), require very small incremental steps and fine meshes. Thus, structural analysis has to withstand the burden of high computational costs or sacrifice accuracy eventually. To overcome this issue, one of the most promising spirits is the decoupling of the macroscopic and microscopic problems, which can be recalled in most recent works (Le et al. [176], Lu et al. [177], Nguyen-Thanh et al. [178], Yang et al. [179]).
Le et al. [176], Lu et al. [177], Yvonnet et al. [180], Clément et al. [181] first proposed the use of constitutive data to fit the explicit equivalent strain energy density function of heterogeneous materials, and then obtained the offline equivalent constitutive relationship. Compared with the concurrent computing homogenization method, this method does not require real-time exchange of information between macro and micro scales. Once the explicit equivalent strain energy density function w ( » ) is constructed in the offline phase, the equivalent stress and the equivalent elastic constitutive L can be derived without any new meso-scale finite element calculations. It greatly improves the online calculation efficiency of heterogeneous materials. It is worth pointing out that the above method is limited to non-dissipative materials, while the classical method of concurrent computational homogenization is not limited by this. In addition, the key of this calculation framework is how to approximate the equivalent strain energy density function w(« ) efficiently and accurately. The equivalent strain energy density function w(« ) constructed by the above approximate method is highly dependent on the density of the data and the selection of the interpolation function. The number of sample points increases exponentially with the increase of the problem dimension, and the computational efficiency in high-dimensional problems is not high. Based on the previous work, Le et al. [176] used a neural network algorithm to obtain the equivalent strain energy density function w(« ), and then derived the explicit stress and equivalent constitutive expression L. Compared with the regular grid-based interpolation technique (Yvonnet et al. [180]), this method has the following two advantages : (1) the amount of sample data required is small and can be random in the interpolation space ; (2) on the premise of ensuring a certain accuracy, the required samplle data does not increase exponentially with the increase of the variable dimension, which greatly reduces the offline calculation cost required for high-dimensional problems.

Scale transition technique and local thermomechanical SMA model

The composite material is described by infinite repeating a RVE, which has periodic boundary conditions and multiphase materials. In that case, the heterogeneous composite is divided into a continuous macroscopic scale and a multiphase microscopic scale (Kouznetsova et al. [117], Feyel and Chaboche [155], Miehe et al. [156]). For instance, the scale separation of SMA/epoxy composite is illustrated in Fig. 2.1. Two models are implemented on ABAQUS which represent the macroscopic structure with a domain and the microscopic structure with a domain ! respectively, see Fig. 2.1. Then, the multiscale problem is solved by using the effective behavior of the microscopic structure to represent the behavior of the corresponding continuous point in the macroscopic structure.

Thermomechanical formulation of SMA constitutive behavior

The adopted 3D SMA constitutive model, proposed by Chemisky et al. [1], is implemented in ABAQUS via UMAT. This model can describe the mechanisms of the phase transformation, the martensite reorientation and the twin accommodation.
Assuming that the behavior of a microscopic RVE containing martensite and austenite can represent the total thermomechanical behavior of SMAs. The total strain without considering the plastic or viscoplastic strain could be decomposed with the following formulation :  » = « e + « th + « tr + « tw; (2.14).
where « e represents the elastic strain, « th stands for the thermal expansion strain, « tr denotes the inelastic strain related to martensitic transformation and « tw stands for the inelastic strain related to the twin accommodation mechanism between martensite variants.
Both the martensite and austenite are assumed to be isotropic with the same thermo-elastic constants. Thus, the elastic strain « e and the thermal expansion strain « th can be formulated as : « e = S : ; (2.15).
« th = (T Tref ); (2.16).
where the isotropic fourth order tensor S, called the compliance tensor, and the isotropic ten-sor describing the thermal expansion are introduced. The reference temperature Tref is the temperature with zero thermal expansion strain.

Table of contents :

Chapitre 1 State of art 
1.1 SMA heterogeneous materials and their modeling
1.1.1 SMAs and their constitutive modeling
1.1.2 SMA composites and their multiscale modeling
1.1.3 Architected cellular SMAs and their multiscale modeling
1.2 Instability phenomenon of long fiber reinforced composites
1.2.1 Microscopic numerical models
1.2.2 Multiscale homogenization models and their nonlinear solver
1.3 Multiscale homogenization methods
1.3.1 Sequential multiscale homogenization methods
1.3.2 Integrated multiscale homogenization methods
1.3.3 Data-driven multiscale homogenization methods
1.3.4 Material-genome-driven multiscale homogenization method
1.3.5 Structural-genome-driven multiscale homogenization method
1.4 Conclusion
Chapitre 2 Formulations and applications of multiscale modeling for SMA heterogeneous materials 
2.1 Scale transition technique and local thermomechanical SMA model
2.1.1 FE2 scale transition technique
2.1.2 Thermomechanical formulation of SMA constitutive behavior
2.2 SMA fiber reinforced composite
2.2.1 Pseudo-elasticity
2.2.2 Shape memory effect
2.2.3 Comments on the computational resources
2.3 Architected cellular SMA
2.3.1 Cellular response
2.3.2 Structural response
2.4 Conclusion
Chapitre 3 Multiscale modeling for the instability of long fiber reinforced composites
3.1 Modeling
3.1.1 Macroscopic scale
3.1.2 Microscopic scale
3.1.3 Formulation of the ANM
3.2 Numerical results
3.2.1 Validation for the multiscale model
3.2.2 Microscopic instability modes
3.2.3 Computation efficiency
3.2.4 Macro-micro coupled instabilities
3.3 Conclusion
Chapitre 4 Data-driven multiscale modeling methods 
4.1 Formulation of data-driven FE2
4.1.1 Classical FE2
4.1.2 Data-driven computing and scale decoupling
4.2 Validation and application of data-driven FE2
4.2.1 Convergence analysis
4.2.2 Inelastic composite plate
4.2.3 Computational cost
4.2.4 Fiber reinforced plate
4.3 Formulation of SGD computing
4.3.1 SGD method
4.3.2 Structure-genome database prepared via FE2 technique
4.4 Validation and application of SGD computing
4.4.1 Validation
4.4.2 Thin composite beam
4.5 Conclusion
Chapitre 5 Conclusion and perspectives 
Appendix A. Asymptotic numerical method
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