Wael Zaki

An approach to modeling tensile–compressive asymmetry for martensitic shape memory alloys

In this paper, the asymmetric tensile–compressive behavior of shape memory alloys is modeled based on the mathematical framework of Raniecki and Mroz (2008 Acta Mech. 195 81–102). The framework allows the definition of smooth, non-symmetric, pressure-insensitive yield functions that are used here to incorporate tensile–compressive modeling capabilities into the Zaki–Moumni (ZM) model for shape memory materials. It is found that, despite some increased complexity, the generalized model is capable of producing satisfactory results that agree with uniaxial experimental data taken from the literature.

Non-linear dynamic thermomechanical behaviour of shape memory alloys

The non-linear dynamic thermomechanical behaviour of superelastic shape memory alloys is investigated. To this end, the Zaki–Moumni model, initially developed for quasi-static loading cases, is extended to simulate the uniaxial forced oscillations of a shape memory alloy device. First, the influence of loading rate is accounted for by considering the thermomechanical coupling in the behaviour of NiTi shape memory alloy. Comparisons between simulations and experimental results show good agreement. Then, the forced response of a shape memory alloy device is investigated at resonance. Both isothermal and non-isothermal conditions are studied, as well as non-symmetric tensile-compressive restoring force. In the case of large values of forcing amplitudes, simulation results show that the dynamic response is prone to jumps, bifurcations and chaotic solutions.

Modeling Tensile-Compressive Asymmetry for Superelastic Shape Memory Alloys.

In this article, the Zaki-Moumni (ZM) model for shape memory alloys is extended to account for tensile-compressive asymmetry over a wide temperature range. To this avail, a mathematical framework recently developed by Raniecki and Mróz is utilized to define new yield functions that are sign-sensitive. With respect to the original ZM model, the modifications are essentially made to the expressions of the Helmholtz free energy and of the internal constraints. The model is shown to properly simulate the asymmetric behavior of shape memory alloys both for martensite orientation and pseudoelasticity.

Direct Numerical Determination of the Asymptotic Cyclic Behavior of Pseudoelastic Shape Memory Structures

The design of shape memory alloys (SMAs) structures against fatigue requires the computation of the stabilized mechanical state. The classical computation method, based on a plasticity-like algorithm, requires a step-by-step calculation, leading to prohibitive computation time to reach this stabilized state. To overcome this issue, we propose to extend the direct cyclic method (DCM), for elastoplastic structures, for use with the Zaki-Moumni (ZM) model for SMAs. DCM is a large time increment method in which a periodicity condition is enforced on the state variables. Comparison with the classical incremental approach shows considerable reduction in computation time.

Shakedown based model for high-cycle fatigue of shape memory alloys

The paper presents a high-cycle fatigue criterion for shape memory alloys (SMAs) based on shakedown analysis. The analysis accounts for phase transformation as well as reorientation of martensite variants as possible sources of fatigue damage. In the case of high-cycle fatigue, once the structure has reached an asymptotic state, damage is assumed to become confined at the mesoscopic scale, or the scale of the grain, with no discernable inelasticity at the macroscopic scale. Using a multiscale approach, a high-cycle fatigue criterion analogous to the Dang Van model (Dang Van 1973) for elastoplastic metals is derived for SMAs obeying the Zaki–Moumni model for SMAs (Zaki and Moumni 2007a). For these alloys, a safe domain is established in stress deviator space, consisting of a hypercylinder with axis parallel to the direction of martensite orientation at the mesoscopic scale. Safety with regard to high-cycle fatigue, upon elastic shakedown, is conditioned by the persistence of the macroscopic stress path at every material point within the hypercylinder, whose size depends on the volume fraction of martensite. The proposed criterion computes a fatigue factor at each material point, indicating its degree of safeness with respect to high cycle fatigue.

A simple 1D model with thermomechanical coupling for superelastic SMAs

This paper presents an outline for a new uniaxial model for shape memory alloys that accounts for thermomechanical coupling. The coupling provides an explanation of the dependence of SMA behavior on the loading rate. 1D simulations are carried in Matlab using simple finite-difference discretization of the mechanical and thermal equations.

Modeling and Simulation of the Mechanical Response of Martensitic Shape Memory Alloys

Wael ZakiDepartment of Mechanical EngineeringKhalifa University of Science, Technology, and ResearchAbu Dhabi campus127788 Abu Dhabi, UAEEmail: wael.zaki@kustar.ac.aeThe process of detwinning of martensite in shape memoryalloys involves the deformation of the crystalline lattice ofthe material by twin boundary motion. The amount of max-imum deformation that can be achieved this way is knownto saturate at some point, beyond which further loading willeventually lead to permanent deformation of the material.We present an algorithm for the simulation of martensite re-orientation in shape memory materials subjected to multi-axial loading that may exceed the saturation threshold. Ifthe applied load is still nonproportional beyond this thresh-old, the reorientation strain tensor may continue to evolvewhile its magnitude remains constant. Such evolution can besimulated using a simple strain-based criterion. The com-plete process of martensite reorientation can thus be mod-eled using a set of two yield functions, the first of which isstress-basedandgovernsthe detwinningprocess prior to sat-uration, and the second is strain-based and governs the re-orientation of variants at maximum equivalent reorientationstrain. The model is implemented in a numerical analysiscode. For this purpose, the evolution equations are solvedimplicitly using a Newton-Raphson scheme and the tangentstiffness matrix of the material is determined using a combi-nation of analytical and numerical techniques.NomenclatureK Elastic stiffness tensor.µ,λLame´’s parameters.E Young’s modulus.α Hardening modulus.σ,eStress and strain tensors.s Stress deviator tensor.e

A Model for Iron-Based Shape Memory Alloys Considering Variable Elastic Stiffness and Coupling Between Plasticity and Phase Transformation

The paper presents a new constitutive model for iron-based shape memory alloys (Fe-SMAs) adapted from the ZM model initially proposed for Nitinol by Zaki and Moumni [JMPS2007]. The model introduces nonlinear hardening terms to account for interactions between the grains, martensite variants and slip systems that may exist within a volume element of the material. The expressions used for the hardening terms are similar to those in (Khalil et al. [JIMSS2012]). The equations of the model are derived from the expression of a Helmholtz free energy potential, with complementary loading conditions obtained within the framework of generalized standard materials with internal constraints. A detailed derivation of the implicit algorithm used for the integration of the model is provided and used for numerical simulations that are shown to agree with experimental data.Copyright

Steady State Crack Growth in Shape Memory Alloys

Shape memory alloys experience phase transformation from austenite to martensite around crack tip. When the crack advances, martensitic transformation occurs at the tip and the energy that goes into transformation results in stable crack growth like in the case of plastic deformation. In literature, there are studies on steady-state crack growth in elasto-plastic materials with small scale yielding around crack tip that use stationary movement methods similar to non-local algorithms. In this work, Mode I steady-state crack growth in an edge cracked Nitinol plate is modeled using a non-local stationary movement method. The Zaki-Moumni (ZM) constitutive model is utilized for this purpose. The model is implemented in ABAQUS by means of a user-defined material subroutine (UMAT) to determine transformation zones around the crack tip. Steady-state crack growth is first simulated without considering reverse transformation to calculate the effect of transformation on stress distribution in the wake region, then reverse transformation is taken into account. Stress distribution and transformation regions calculated for both cases are compared to results obtained for the case of a static crack.Copyright

High-Cycle Fatigue Criterion for Shape Memory Alloys Based on Shakedown Theory

Based on a recently developed shakedown theory for non-smooth nonlinear materials, we derive a criterion for high-cycle fatigue in shape memory alloys (SMAs). The fatigue criterion takes into account phase transformation as well as reorientation of martensite variants as the source of fatigue damage. The mathematical derivation of the criterion is based on the requirement of elastic shakedown for a given structure to achieve unlimited fatigue endurance. Elastic shakedown is defined as an asymptotic state in which damage due to time-varying load becomes confined at the mesoscopic scale, or the scale of the grain, with no discernable inelasticity at the macroscopic scale. From an energy standpoint, elastic shakedown corresponds to a situation where energy dissipation becomes bounded and the response elastic after a certain number of loading cycles. A sufficient condition to achieve this state was established by Melan (1936) [1] and Koiter (1960) [2] for elastoplastic materials and later generalized to hardening plasticity by Nguyen (2003) and to non-smooth non-linear materials by Peigney (2014). The latter formulation is applicable to SMAs obeying the ZM constitutive model (Zaki & Moumni, 2007) and is shown here to allow the derivation of a high-cycle fatigue criterion analogous to the one proposed by Dang Van (1973) for elastoplastic materials. The criterion allows establishing a safe domain in stress deviator space at the mesoscopic scale consisting of a hypercylinder with axis parallel to the direction of martensite orientation. The hypercylinder is delimited along its axis by two transverse hyperplanes representing bounds on admissible stress states consistent with the loading conditions for phase transformation. Safety with regard to high-cycle fatigue, upon elastic shakedown, is conditioned by the persistence of the macroscopic stress path, as the load varies and at every material point, strictly within the hypercylinder. The size of the hypercylinder is shown to strongly depend on the relative amount of martensite present in the SMA.Copyright