Institute of Fundamental Technological Research

Spatially-resolved modeling of deformation twinning and its interaction with plastic slip is achieved by coupling the phase- field method and crystal plasticity theory. The intricate constitutive relations arising from this coupling render the resulting computational model prone to inefficiencies and lack of robustness. Accordingly, together with the inherent limitations of the phase-field method, these factors may impede the broad applicability of the model. In this paper, our recent phase-field model of coupled twinning and crystal plasticity is the subject of study. We delve into the incremental formulation and computational treatment of the model and run a thorough investigation into its computational performance. We focus specifically on evaluating the efficiency of the finite-element discretization employing various element types, and we examine the impact of mesh density. Since the micromorphic regularization is an important part of the finite-element implementation, the effect of the micromorphic regularization parameter is also studied.

Deformation twinning, Microstructure, Phase-field method, Crystal plasticity, Finite element method

Localization of the stress-induced martensitic phase transformation plays an important role in the fatigue behavior of shape memory alloys (SMAs). The phenomenon of return-point memory that is observed during the subloop deformation of a partially-transformed SMA is a clear manifestation of the interaction between localized phase transformation and degradation of the functional properties. The present study aims to demonstrate this structure–material interaction in the modeling of return-point memory. It seems that this crucial aspect has been overlooked in previous modeling studies. For this purpose, we developed a gradient- enhanced model of pseudoelasticity that incorporates the degradation of functional properties in its constitutive description. The model is employed to reproduce the hierarchical return-point memory in a pseudoelastic NiTi wire under isothermal uniaxial tension with nested subloops. Additionally, a detailed analysis is carried out for NiTi strip with a more complex transformation pattern. Our study highlights the subtle morphological changes of phase transformation under different loading scenarios and the resulting implications for return-point memory.

Shape memory alloys,Phase transformation,Functional degradation ,Propagating instabilities,Subloop deformation,Modeling

A reliable prediction of the pseudoelastic behavior necessitates the involvement of martensite reorientation in the model. This is important not only under non-proportional loading but in general when the phase transformation proceeds in a localized manner, which results in complex local deformation paths. In this work, an advanced model of pseudoelasticity is developed within the incremental energy minimization framework. A novel enhancement of the model over its original version lies in the formulation of a suitable rate-independent dissipation potential that incorporates the dissipation due to martensitic phase transformation and also due to martensite reorientation, thus yielding an accurate description of the inelastic transformation strain. The finite-element implementation of the model relies on the augmented Lagrangian treatment of the non-smooth incremental energy problem. Thanks to the micromorphic regularization, the related complexities are efficiently handled at the local level, leading to a robust finite-element model. Numerical studies highlight the predictive capabilities of the model. The characteristic mechanical behavior of NiTi tube under non-proportional tension– torsion and the intricate transformation evolution under pure bending are effectively captured by the model. Additionally, a detailed analysis is carried out to elucidate the important role of martensite reorientation in promoting the striations of the phase transformation front.

Shape memory alloys,Phase transformation,Martensite reorientation,Strain localization,Finite-element method

Direct experimental characterization of indentation-induced martensitic microstructures in pseudoelastic shape memory alloys (SMAs) is not possible, and thus there is a lack of evidence and understanding regarding the microstructure pattern and related features. To fill this gap, in this work we employ the phase-field method to provide a detailed and systematic analysis of martensitic phase transformation during nanoindentation. A recently-developed finite-element-based computational model is used for this purpose, and a campaign of large-scale 3D simulations is carried out. First, the orientation-dependent indentation response in CuAlNi (a widely studied SMA) is examined. A detailed investigation of the predicted microstructures reveals several interesting features, some of them are consistent with theoretical predictions and some can be (to some extent) justified by experiments other than micro/nanoindentation. The results also highlight the key role of finite-deformation effects and elastic anisotropy of the phases on the model predictions. Next, a detailed study of indentation-induced martensitic transformation in NiTiPd (a potential low-hysteresis SMA) with varying Pd content is carried out. In terms of hysteresis, the results demonstrate the prevailing effect of the transformation volume change over phase compatibility in the conditions imposed by nanoindentation and emphasize on the dominant role of the interfacial energy at small scales. Results of such scope have not been reported so far.

Nanoindentation,Pseudoelasticity,Twinning,Microstructure formation,Phase-field method

We present a comprehensive modeling study on the patterns of propagating instabilities in NiTi tubes under proportional axial–torsion loading. Our study directly refers to the experimental work of Reedlunn et al. (2020), with a particular focus on the unique longitudinal transformation bands that occur in torsion-dominated loading paths. A previously-developed gradient-enhanced model of pseudoelasticity is employed and is adapted to incorporate the residual stresses. In addition, our finite-element setup accounts for the impact of collet grips on the NiTi tubes via a simplified frictional contact model. The results demonstrate the capability of the model in capturing subtle features of the transformation patterns observed in the experiment, including the multi-finger fronts in tension-dominated loading and longitudinal bands in torsion-dominated loading. Our study suggests that the combination of the residual stresses and the collet grips facilitates the formation of longitudinal bands.

Shape memory alloys,Phase transformation,Strain localization,Propagating instabilities,Finite-element method

Needle-like twins are observed experimentally within the transition layer at the martensite–twinned martensite interface. We utilize a phase-field approach to investigate this microstructure. Our goal is to simulate the morphology of the transition layer and to perform a detailed analysis to characterize its interfacial and elastic micro-strain energy. To illustrate the micromechanical framework developed for that purpose, sample computations are carried out for a CuAlNi shape memory alloy undergoing a cubic-to-orthorhombic martensitic transformation. A particular focus of the study is on size-dependent morphology through examining the impact of twin spacing. Additionally, our results reveal that certain twin volume fractions lead to the emergence of twin branching as a way to minimize the total free energy stored in the microstructure.

Microstructure,Martensitic transformation,Transition layer,Phase-field method,Size effects

A finite-strain phase-field model of coupled deformation twinning and crystal plasticity is developed in the paper. Twinning is treated as a displacive transformation characterized by a volume-preserving stretch rather than a simple shear, the latter considered in the conventional approach. It is shown that the two approaches are equivalent in the sharp-interface description, but not in the diffuse-interface description. In the proposed stretch-based kinematics, each pair of conjugate twinning systems is represented by a single twin deformation variant, and thus a single order parameter suffices to consistently describe the two conjugate twinning systems, thereby treating them equally. The model is formulated in the framework of incremental energy minimization, which, upon time discretization, leads to a quasi-optimization problem due to the specific form of the incremental potential within the diffuse interfaces. To facilitate finite-element implementation, a micromorphic formulation of the model is employed. As an application, tensile twinning in HCP magnesium alloys is examined, and a set of comprehensive 2D plane-strain problems is studied to illustrate the features of the proposed approach.

deformation twinning, microstructure, phase-field method, crystal plasticity, magnesium alloy

Large-scale 3D martensitic microstructure evolution problems are studied using a finite-element discretization of a finite-strain phase-field model. The model admits an arbitrary crystallography of transformation and arbitrary elastic anisotropy of the phases, and incorporates Hencky-type elasticity, a penalty-regularized double-obstacle potential, and viscous dissipation. The finite-element discretization of the model is performed in Firedrake and relies on the PETSc solver library. The large systems of linear equations arising are efficiently solved using GMRES and a geometric multigrid preconditioner with a carefully chosen relaxation. The modeling capabilities are illustrated through a 3D simulation of the microstructure evolution in a pseudoelastic CuAlNi single crystal during nano-indentation, with all six orthorhombic martensite variants taken into account. Robustness and a good parallel scaling performance have been demonstrated, with the problem size reaching 150 million degrees of freedom.

phase-field method, finite-element method, large-scale simulations, shape memory alloys, nano-indentation

We show that the logarithmic (Hencky) strain and its derivatives can be approximated, in a straightforward manner and with a high accuracy, using Padé approximants of the tensor (matrix) logarithm. Accuracy and computational efficiency of the Padé approximants are favourably compared to an alternative approximation method employing the truncated Taylor series. As an application, Hencky-type hyperelasticity models are considered, in which the elastic strain energy is expressed in terms of the Hencky strain, and of our particular interest is the anisotropic energy quadratic in the Hencky strain. Finite-element computations are carried out to examine performance of the Padé approximants of tensor logarithm in Hencky-type hyperelasticity problems. A discussion is also provided on computation of the stress tensor conjugate to the Hencky strain in a general anisotropic case.

logarithmic strain, Padé approximation method, hyperelasticity, anisotropy, finite-element method

This paper is concerned with modelling of propagating instabilities and transformation patterns in NiTi tubes subjected to combined tension–torsion loading. A recently developed gradient-enhanced finite-strain model of pseudoelasticity is employed for this purpose, and respective finite-element computations are carried out. It is shown that the model is capable of representing a number of experimentally observed effects. The major effect, which has not been successfully modelled to date, is that the transformation is inhomogeneous under tension-dominated loading and alters towards a homogeneous transformation as the level of torsion is increased. To capture this effect, the model must deliver a non-monotonic (up-down-up) stress–strain response in tension and a monotonic one in torsion, and this can be achieved if the model includes three features: tension–compression asymmetry, transverse isotropy of the transformation strain, and deformation-dependent hardening/softening response. A detailed study is also carried out regarding the transformation yield locus. The results reveal an ambiguity in determination of the yield locus for tension-dominated loading and hence an ambiguity in determination of the tension–compression asymmetry. This aspect seems to have been overlooked in the literature despite its impact on correct interpretation of experimental results.

shape memory alloys, phase transformation, strain localization, finite-element method

The effect of formation and evolution of stress-induced martensitic microstructures on macroscopic mechanical properties of shape memory alloys in the pseudoelastic regime is investigated with account for size-dependent energy of interfaces. A quantitative relationship is established between the changes in free energy and dissipation on the interfaces at three microstructural scales and the overall mechanical characteristic of the material under tensile loading. The multiscale analysis carried out for a polycrystalline NiTi shape memory alloy has revealed that the interfacial energy storage and dissipation can strongly affect the shape and width of the stress–strain hysteresis loop. The predicted non-monotonic stress–strain response for the material of a selected grain size shows a remarkable similarity to the experimental one extracted from a tensile test of a laminate by Hallai and Kyriakides (2013). By the classical Maxwell construction, the non-monotonic response for a material element results in a commonly observed stress plateau for a tensile specimen, which is associated with the propagation of phase transformation fronts. This behaviour is confirmed with striking accuracy by 3D finite-element computations performed for a macroscopic tensile specimen, in which propagating instability bands are treated explicitly.

microstructures, martensitic transformation, size effects, incremental energy minimization, propagating instabilities

A micromorphic formulation of the phase-field model of martensitic transformation is developed within the incremental energy minimization framework. In contrast to the conventional phase-field formulation, the order parameters are viewed as local variables and the corresponding evolution equations are solved at the material-point level, i.e. at the Gauss points in the finite-element setting. From a computational standpoint, such a treatment is advantageous for complex evolution laws that may lead to computational difficulties if treated globally, as in the conventional phase-field formulation. In the micromorphic formulation, each order parameter is coupled to its micromorphic counterpart governed by a global Helmholtz-type PDE. This coupling ensures that the interfacial energy and related size effects are correctly captured by the model. In this work, the micromorphic approach is applied to a finite-strain multivariant phase-field model that incorporates rate-independent dissipation. The augmented Lagrangian technique is then used to transform the resulting non-smooth incremental minimization problem to a smooth and unconstrained saddle-point problem. Microstructure evolution under nano-indentation is studied to illustrate the approach.

phase-field method, micromorphic approach, rate-independent dissipation, incremental energy minimization, microstructure, shape-memory alloys

Stress-induced martensitic transformation in polycrystalline NiTi under tension often proceeds through formation and propagation of macroscopic phase transformation fronts, i.e., diffuse interfaces that separate the transformed and untransformed domains. A gradient-enhanced 3D finite-strain model of pseudoelasticity is developed in this work with the aim to describe the related phenomena. The underlying softening response is regularized by enhancing the Helmholtz free energy of a non-gradient model with a gradient term expressed in terms of the martensite volume fraction. To facilitate finite-element implementation, a micromorphic-type regularization is then introduced following the approach developed recently in the 1D small-strain context. The complete evolution problem is formulated within the incremental energy minimization framework, and the resulting non-smooth minimization problem is solved by employing the augmented Lagrangian technique. In order to account for the thermomechanical coupling effects, a general thermomechanical framework, which is consistent with the second law of thermodynamics and considers all related couplings, is also developed. Finite-element simulations of representative 3D problems show that the model is capable of representing the loading-rate effects in a NiTi dog-bone specimen and complex transformation patterns in a NiTi tube under tension. A parametric study is also carried out to investigate the effect of various parameters on the characteristics of the macroscopic transformation front.

phase transformation, softening, strain localization, micromorphic regularization, finite-element method

A finite-strain phase-field model is developed for the analysis of multivariant martensitic transformation during nano-indentation. Variational formulation of the complete evolution problem is developed within the incremental energy minimization framework. Computer implementation is performed based on the finite-element method which allows a natural treatment of the finite-strain formulation and of the contact interactions. A detailed computational study of nano-indentation reveals several interesting effects including the pop-in effect associated with nucleation of martensite and the energy-lowering breakdown of the symmetry of microstructure. The effect of the indenter radius is also examined revealing significant size effects governed by the interfacial energy.

phase-field method, microstructure, shape-memory alloys, nano-indentation, size effects

In recent decades, fiber reinforced polymer (FRP) wrapping has become a common technique to retrofit reinforced concrete (RC) columns. Numerous research works have sought to verify analytically and experimentally its effectiveness in terms of enhancement of axial load bearing capacity and ductility. These studies highlighted that in the case of sharp-cornered sections, the maximum allowable confinement pressure is limited by premature failure at corners and, consequently, stress in the FRP, as well as the distribution of the confinement pressure, is not uniform. The prediction of this phenomenon is not straightforward, and existing theoretical studies propose complex numerical simulations, whereas technical codes provide simplified or empirical relationships for its assessment. This paper presents an analytical model for the evaluation of the effective distribution of confinement pressure in FRP confined concrete members with rounded corners. The model allows considering the interaction with the concrete core and different brittle failure modes, including FRP rupture and debonding. It leads to determining the distribution of the confinement pressure along the section. Results are compared with those achieved by finite-element (FE) analyses and with numerical and experimental data available in the literature. Good agreement is obtained in all cases, showing the reliability of the proposed model.

fiber reinforced polymer (FRP) wrapping, corner radius, confinement pressure, brittle failure

Shape memory alloys, notably NiTi, often exhibit softening pseudoelastic response that results in formation and propagation of Lüders-like bands upon loading, for instance, in uniaxial tension. A common approach to modelling softening and strain localization is to resort to gradient-enhanced formulations that are capable of restoring well-posedness of the boundary-value problem. This approach is also followed in the present paper by introducing a gradient-enhancement into a simple one-dimensional model of pseudoelasticity. In order to facilitate computational treatment, a micromorphic-type regularization of the gradient-enhanced model is subsequently performed. The formulation employs the incremental energy minimization framework that is combined with the augmented Lagrangian treatment of the resulting non-smooth minimization problem. A thermomechanically coupled model is also formulated and implemented in a finite-element code. The effect of the loading rate on the localization pattern in a NiTi wire under tension is studied, and the features predicted by the model show a good agreement with the experimental observations. Aditionally, an analytical solution is provided for a propagating interface (macroscopic transformation front) both for the gradient-enhanced model and for its micromorphic version

martensite, phase transformation, micromorphic model, strain localization, thermomechanical coupling

Fibre-reinforced plastic (FRP) composites are extensively used to retrofit civil structures. However, the quality and the characteristics of the bond between the FRP and the structure are critical to ensure the efficacy of the retrofit. For this reason, effective non-destructive evaluation (NDE) methods are often necessary to assess the bonding conditions. This article presents an ultrasonic technique for detecting defects at the FRP-substrate interface. The technique uses the Akaike Information Criterion, to detect automatically the onset of the ultrasonic signals, and the novel Equivalent Time Lenght (ETL) parameter, to quantify the energy of the propagating ultrasonic signals along the interface between FRP and concrete. The uniqueness of the ETL is that it is not affected by the coupling conditions between the ultrasonic probes and the structure. The proposed NDE technique has been tested numerically by performing 2D Finite-Element analysis and experimentally on reinforced concrete samples. The results show that the method is robust and cost-effective.

CFRP, fibre-reinforced materials, concrete, bonding, non-destructive testing, ultrasonic methods, equivalent time length

The seismic nonlinear response of reinforced concrete structures permits to identify critical zones of an existing structure and to better plan its rehabilitation process. It is obtained by performing finite element analysis using numerical models classifiable into two categories: lumped plasticity models and distributed plasticity models. The present work is devoted to the implementation, in a finite element environment, of an elastoplastic Euler–Bernoulli beam element showing possible slope discontinuities at any position along the beam span, in the framework of a modified lumped plasticity. The differential equation of an Euler–Bernoulli beam element under static loads in presence of multiple discontinuities in the slope function was already solved by Biondi and Caddemi (Int J Solids Struct 42(9):3027–3044, 2005, Eur J Mech A Solids 26(5):789–809, 2007), who also found solutions in closed form. These solutions are now implemented in the new beam element respecting a thermodynamical approach, from which the state equations and flow rules are derived. State equations and flow rules are rewritten in a discrete manner to match up with the Newton–Raphson iterative solutions of the discretized loading process. A classic elastic predictor phase is followed by a plastic corrector phase in the case of activation of the inelastic phenomenon. The corrector phase is based on the evaluation of return bending moments by employing the closest point projection method under the hypothesis of associated plasticity in the bending moment planes of a Bresler’s type activation domain. Shape functions and stiffness matrix for the new element are derived. Numerical examples are furnished to validate the proposed beam element.

Slope discontinuity, Nonlinear pushover analysis, Lumped plasticity, Plastic hinge

A beam/column element allowing the formation of multiple plastic hinges in columns or beams of a reinforced concrete (RC) framed structure is used in this work to show, through an application, its advantages with respect to conventional lumped plasticity models. Slope discontinuities can be located at any position of an Euler-Bernoulli beam span and not at the two extremes only. The model is in fact written in the framework of a modified lumped plasticity theory, and respectful of a thermodynamic approach. Flow rules and state equations are derived invoking the Theorem of maximum dissipation and using a Bresler’s type activation domain. The beam element has already been implemented in a researchoriented code to run nonlinear analyses on 2-D frames. The discretized loading process is separated, at each step, in two phases: a predictor and a corrector phase. Numerical examples highlight how the new finite element permits to run nonlinear analyses avoiding a mesh refinement.

beam element, plastic hinge, lumped plasticity, slope discontinuity, nonlinear FEM analysis

microstructure, martensitic phase transformation, transition layer, phase-field method, size effects

shape memory alloys, branched microstructures, free energy, continuum model

Shape memory alloys, Branched microstructures, Energy minimization, Continuum framework