isogeometric analysis of a dynamic thermo-mechanical phase-field model applied to shape memory...

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Comput Mech DOI 10.1007/s00466-013-0966-0 ORIGINAL PAPER Isogeometric analysis of a dynamic thermo-mechanical phase-field model applied to shape memory alloys R. P. Dhote · H. Gomez · R. N. V. Melnik · J. Zu Received: 11 September 2013 / Accepted: 16 December 2013 © Springer-Verlag Berlin Heidelberg 2014 Abstract This paper focuses on the numerical simula- tion of martensitic transformations in shape memory alloys (SMAs) using a phase-field model. We developed a dynamic thermo-mechanical model for SMAs, using strain based order parameter, having a bi-directional coupling between structural and thermal physics via strain, strain rate and tem- perature. The model involves fourth order spatial derivatives representing a domain wall. We propose an isogeometric analysis numerical formulation for straightforward solu- tion to the fourth order differential equations. We present microstructure evolution under different loading conditions and dynamic loading simulations of the evolved microstruc- tures of SMAs of different geometries to illustrate the flex- ibility, accuracy and stability of our numerical method. The simulation results are in agreement with the numerical and experimental results from the literature. Keywords Shape memory alloys · Twinning · Phase-field · Isogeometric analysis R. P. Dhote (B ) · J. Zu Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, ON M5S-3G8, Canada e-mail: [email protected] H. Gomez Department of Applied Mathematics, University of A Coruña, Campus de Elvina, s/n., 15192 La Coruña, Spain R. N. V. Melnik · R. P. Dhote M 2 NeT Laboratory, Wilfrid Laurier University, 75 University Avenue, Waterloo, ON N2L-3C5, Canada R. N. V. Melnik Universidad Carlos III de Madrid, Avenida de la Universidad 30, E28911 Leganes, Spain 1 Introduction Phase transformations are observed across different length and time scales from phase transitions due to atomic rearrangements to planet evolution. It is a widely studied phenomenon in materials science, mathematics and engineer- ing [10, 33, 37, 42, 43]. The mathematical modeling of phase transformations involves treatment of interfaces, that is, thin transition regions, between two or more phases. Phase-field modeling is a widespread methodology where the interface is treated by a diffused smoothly-varying transition layer. The continuous treatment of interfaces reduces the prob- lem complexity and makes its numerical approximation more tractable. Phase-field modeling is widely used to study phase trans- formations in shape memory alloys (SMAs) at the meso- and nano-scale [9, 31, 33]. SMAs exhibit reconstructive solid- to-solid phase transformations due to symmetry breaking. The high symmetry atomic arrangement (austenite phase) is transformed into the lower symmetry arrangements (marten- sitic variants) from a crystallographic point of view. Phase transformations in SMAs reveal themselves as complex pat- terns of microstructures, such as parallel twins, zig-zag twins, cross twins, wedge and habit plane [7]. These microstruc- tures have different phases separated by an invariant plane. A phase-field model can describe this complex microstruc- ture by using appropriately defined conserved field variables, or order-parameters, that vary continuously across the invari- ant plane. A continuous variation of field variables across the invariant plane often leads to higher order differential terms in the governing equations, which poses significant challenges for the numerical discretization of the model [27]. Different numerical schemes have been proposed to solve higher-order differential equations. The finite difference method has been widely used to solve phase transformations 123

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Page 1: Isogeometric analysis of a dynamic thermo-mechanical phase-field model applied to shape memory alloys

Comput MechDOI 10.1007/s00466-013-0966-0

ORIGINAL PAPER

Isogeometric analysis of a dynamic thermo-mechanical phase-fieldmodel applied to shape memory alloys

R. P. Dhote · H. Gomez · R. N. V. Melnik · J. Zu

Received: 11 September 2013 / Accepted: 16 December 2013© Springer-Verlag Berlin Heidelberg 2014

Abstract This paper focuses on the numerical simula-tion of martensitic transformations in shape memory alloys(SMAs) using a phase-field model. We developed a dynamicthermo-mechanical model for SMAs, using strain basedorder parameter, having a bi-directional coupling betweenstructural and thermal physics via strain, strain rate and tem-perature. The model involves fourth order spatial derivativesrepresenting a domain wall. We propose an isogeometricanalysis numerical formulation for straightforward solu-tion to the fourth order differential equations. We presentmicrostructure evolution under different loading conditionsand dynamic loading simulations of the evolved microstruc-tures of SMAs of different geometries to illustrate the flex-ibility, accuracy and stability of our numerical method. Thesimulation results are in agreement with the numerical andexperimental results from the literature.

Keywords Shape memory alloys · Twinning · Phase-field ·Isogeometric analysis

R. P. Dhote (B) · J. ZuMechanical and Industrial Engineering, University of Toronto,5 King’s College Road, Toronto, ON M5S-3G8, Canadae-mail: [email protected]

H. GomezDepartment of Applied Mathematics, University of A Coruña,Campus de Elvina, s/n., 15192 La Coruña, Spain

R. N. V. Melnik · R. P. DhoteM2NeT Laboratory, Wilfrid Laurier University, 75 UniversityAvenue, Waterloo, ON N2L-3C5, Canada

R. N. V. MelnikUniversidad Carlos III de Madrid, Avenida de la Universidad 30,E28911 Leganes, Spain

1 Introduction

Phase transformations are observed across different lengthand time scales from phase transitions due to atomicrearrangements to planet evolution. It is a widely studiedphenomenon in materials science, mathematics and engineer-ing [10,33,37,42,43]. The mathematical modeling of phasetransformations involves treatment of interfaces, that is, thintransition regions, between two or more phases. Phase-fieldmodeling is a widespread methodology where the interfaceis treated by a diffused smoothly-varying transition layer.The continuous treatment of interfaces reduces the prob-lem complexity and makes its numerical approximation moretractable.

Phase-field modeling is widely used to study phase trans-formations in shape memory alloys (SMAs) at the meso-and nano-scale [9,31,33]. SMAs exhibit reconstructive solid-to-solid phase transformations due to symmetry breaking.The high symmetry atomic arrangement (austenite phase) istransformed into the lower symmetry arrangements (marten-sitic variants) from a crystallographic point of view. Phasetransformations in SMAs reveal themselves as complex pat-terns of microstructures, such as parallel twins, zig-zag twins,cross twins, wedge and habit plane [7]. These microstruc-tures have different phases separated by an invariant plane.A phase-field model can describe this complex microstruc-ture by using appropriately defined conserved field variables,or order-parameters, that vary continuously across the invari-ant plane. A continuous variation of field variables across theinvariant plane often leads to higher order differential terms inthe governing equations, which poses significant challengesfor the numerical discretization of the model [27].

Different numerical schemes have been proposed to solvehigher-order differential equations. The finite differencemethod has been widely used to solve phase transformations

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of SMAs on geometrically simple domains [2,35,46]. Alter-natively, spectral methods have been reported to be efficientin solving phase-field models because they can resolve sharpinterfaces with a moderate number of uniform grid points[11]. Other numerical techniques like the method of lines,the finite volume method or hybrid optimization algorithmshave been proposed to solve phase-field models [45]. Mostof the above methodologies use uniform grid stencils and arenot flexible with complex geometries in real-world applica-tions. In such scenarios, one should resort to the finite elementmethod (FEM), which finds wide applicability across engi-neering disciplines to simulate physics with geometric flexi-bility. The FEM has been extensively used for the variationalformulation of second-order differential operators, where theconforming element has inter-element continuity restricted toC 0. However, the conforming discretization of fourth-orderspatial differential operators requires the basis functions to beat least C 1-continuous across element boundaries. There arefew conforming finite elements (e.g., 2D Argyris element)which support higher order continuity, necessary for solvinghigher-order differential equations. One of the roundaboutways is to use mixed finite elements for spatial discretization,which leads to an increase in degrees of freedom, and, thus,is not an ideal solution. In short, there is a need of a numeri-cal method which can have geometric flexibility and achievehigher-order continuity of the basis functions. We believethat isogeometric analysis (IGA), a recently-proposed com-putational method, can prove to be an effective procedure forsolving higher-order problems on complex geometries.

IGA is an analysis tool based on the Non-UniformRational B-Spline (NURBS) basis, a backbone of modernComputer-Aided Design (CAD) packages. IGA employscomplex NURBS-based representations of the geometry andthe field variables in a variational formulation of the equa-tions. Thus, IGA may open the door to technologies thatdirectly integrate CAD and analysis, simplifying or elimi-nating altogether the mesh-generation bottleneck [6,13,14,28,40,41]. Using IGA, the geometry can be modeled exactlyin many instances and, as a consequence, geometric errors areeliminated. IGA accommodates the classical concepts of h-and p-refinement, but introduces a new procedure, namelyk-refinement that seems to be unique within the geometri-cally flexible numerical methods. In k-refinement the orderof the approximation is elevated, but continuity is likewiseincreased, creating new opportunities for the approximationof problems whose discretization requires smoothness [3–5,16,20,24–26,29,30,38]. Thus, we believe that IGA pro-vides unique attributes in solving higher-order differentialoperators with C 1 or higher order continuity along withhigher accuracy and robustness without a need of usingmixed finite element formulation or non conforming ele-ments. The accuracy and robustness of IGA methodologyhave been reported for solving the phase-field models in the

field of crack propagation [8], spinodal decomposition [32],topology optimization [17], and tumor angiogenesis [44]. Wereported the first use of the IGA for solving the phase-fieldmodels for microstructure evolution in SMAs in [18]. Here,we extend the studies to different geometries and bound-ary conditions, performing physically-relevant simulations(for example, tensile tests) to investigate thermo-mechanicalbehavior of SMA structures as a function of their microstruc-tures.

In this paper, we model the two dimensional square-to-rectangular phase transformations in FePd SMA speci-mens and numerically solve the governing equations usingIGA. The coupled equations of nonlinear thermoelasticityhave been developed by using the phase-field model and theGinzburg–Landau theory. The governing laws are introducedin the IGA framework by using a variational formulation thatcan be simply derived by multiplying the governing equa-tions with a smooth function and integrating by parts twice.Thus, no additional variables are introduced avoiding the useof mixed methods which typically require complicated sta-bility analyses. Using the proposed algorithm, we performseveral numerical studies on SMAs under coupled thermo-mechanical dynamic loading conditions. The examples showthe strong thermo-mechanical coupling of SMAs and illus-trate the flexibility, accuracy and stability of our numericalmethods.

The rest of the paper is organized as follows: Sect. 2describes the derivation of the coupled thermo-mechanicalmodel of square-to-rectangular phase transformations inSMAs. The weak formulation and IGA numerical imple-mentation of the governing equations are described in Sect.3. The above methodology is exemplified with several twodimensional numerical simulations on SMA patches in Sect.4. Finally, the conclusions are given in Sect. 5.

2 Formulation of dynamic coupled thermo-mechanicalmodel for square-to-rectangular phasetransformations in SMAs

The square-to-rectangular phase transformations are a 2Drepresentation of cubic-to-tetragonal phase transformationsobserved in several SMA materials like FePd, InTl or NiAl.The square phase represents high-temperature and high-symmetric austenite phase while rectangles, with a lengthalong two coordinate axes, represent low-temperature low-symmetric martensite variants. The dynamics of SMA ishighly dependent on the temperature. The SMA may exhibitferroelastic, pseudo-elastic and elastic behavior at low, inter-mediate and high temperatures [21]. This wide range of qual-itative behaviors is captured in the simulations by accountingfor the coupling effects between the structural and thermalfields.

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2.1 Theory

In this section, we derive the governing equations of SMAdynamics accounting for thermo-mechanical nonlinear cou-pling. We begin by introducing basic notations for the kine-matics of SMAs. Let us call uuu = {u1, u2}T the displace-ment field. We will work on the physical domain � ⊂ R

2,parameterized by Cartesian coordinates xxx = {x1, x2}T . Letus call εεε the Cauchy–Lagrange infinitesimal strain tensor,component-wise defined as εi j = (

ui, j + u j,i)/2, i, j ∈

{1, 2}, where an inferior comma denotes partial differentia-tion (e.g., ui,1 = ∂ui/∂x1). Let us define the following strainmeasures

e1 = (ε11 + ε22) /√

2, e2 = (ε11 − ε22) /√

2,

e3 = (ε12 + ε21) /2, (1)

that we call hydrostatic, deviatoric, and shear strain, respec-tively. To derive the fundamental equations governing thethermo-mechanical coupling of SMAs, we will use thephase-field method and Landau–Ginzburg free energy. Inphase-field modeling, the different phases are distinguishedusing an order parameter. The order parameter for square-to-rectangular phase transformations is the deviatoric strainthat we denote e2. The austenite and two martensite variantsare defined by e2 = 0, and e2 = ±1, respectively.

The free energy for the 2D square-to-rectangular phase-transformations considering the Landau-based potential isdefined as

F =∫

[a1

2e2

1+a3

2e2

3+a2

2τe2

2−a4

4e4

2+a6

6e6

2+kg

2|∇e2|2

]dxxx

=:∫

Fdxxx (2)

where ai , kg and θm are the material constants,τ = (θ − θm)/θm is the dimensionless temperature, θ is thematerial temperature and | · | denotes the Euclidean norm ofa vector living in R

2. Figure 1 shows the plot of free-energydensity F as a function of e2 and θ . When the temperatureis higher than the phase transformation temperature θm , theLandau energy function has a minimum corresponding tothe austenite phase. When the temperature is lower than θm ,the energy has several minima corresponding to the variantsof martensite. When the temperature is near θm , the Landaufree energy has minima corresponding to the austenite andthe variants of martensite.

The kinetic energy K , the energy associated with theexternal body forces B, and the dissipation D are defined,respectively, as

−2−1

01

2

−0.5

0

0.5−0.2

0

0.2

0.4

0.6

0.8

e2

Fig. 1 Free energy functional of square-to-rectangular phase transfor-mations as a function of e2 and θ . When the temperature is higher thanthe phase transformation temperature θm , the Landau energy as a func-tion of e2 has a minimum corresponding to the austenite phase. Whenthe temperature is lower than θm , the energy has several minima cor-responding to the variants of martensite. When the temperature is nearθm , the Landau free energy has minima corresponding to the austeniteand the variants of martensite (color online)

K =∫

ρ

2|uuu|dxxx; B =

fff · uuudxxx; D =∫

η

2∇uuu : ∇uuu dxxx,

(3)

where ρ is the density, fff is the external body load vector, η

is the dissipation constant, uuu indicates the time derivative ofuuu and ∇uuu : ∇uuu = ui, j ui, j (repeated indices indicate sum-mation). The Hamiltonian of the problem can be expressedas

H =t∫

0

(K − F − B − D)dt

:=t∫

0

H(t, xxx,uuu, uuu,∇uuu,∇∇uuu)dxxxdt, (4)

where [0, t] is the time interval of interest. Using calculusof variations to determine the extrema of the functional (4),leads to the momentum balance equations. If we take varia-tions with respect to ui , we obtain the i’th component of themomentum balance equation, which is,

∂ H

∂ui− ∂

∂x1

(∂ H

∂ui,1

)− ∂

∂x2

(∂ H

∂ui,2

)+ ∂

∂x21

(∂ H

∂ui,11

)

+ ∂

∂x1∂x2

(∂ H

∂ui,12

)+ ∂

∂x22

(∂ H

∂ui,22

)− ∂

∂t

(∂ H

∂ u1

)= 0.

(5)

Equation (5) can be rewritten as

ρ∂2ui

∂t2 = σi j, j + σgi + η∇2ui + fi , (6)

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where repeated indices indicate summation, and

σ11 = 1√2

[a1e1 + a2τe2 − a4e3

2 + a6e52

], (7a)

σ12 = 1

2a3e3 = σ21, (7b)

σ22 = 1√2

[a1e1 − a2τe2 + a4e3

2 − a6e52

]. (7c)

The higher-order differential term σσσ g takes the form

σg1 = −kg

2

[u1,1111 + u1,1122

] + kg

2

[u2,1112 + u2,1222

]

= − kg√2∇2

(∂e2

∂x1

), (8a)

σg2 = +kg

2

[u1,1112 + u1,1222

] − kg

2

[u2,1122 + u2,2222

]

= kg√2∇2

(∂e2

∂x2

). (8b)

If we define the differential operator ∇⊥ as ∇⊥ ={∂(·)/∂x1,−∂(·)/∂x2}T , the stressσσσ g can be written in diver-gence form as

σσσ g = ∇ ·(

kg

2∇∇⊥e2

).

To account for the thermo-mechanical coupling in thisframework, we need to define an energy balance equation. Wewill assume that the contribution of σσσ g to the energy balanceis negligible. Under this assumption, the energy equation canbe expressed as (compare with [34]):

ρ∂e

∂t− σσσ : ∇vvv + ∇ · qqq = g, (9)

wherevvv = uuu, e is the internal energy,qqq is the heat flux and g isthe heat supply per unit mass. Following classical thermody-namics, we derive the state variables from a thermodynamicpotential. Let us define the Helmholtz free energy as

Ψ = F − Cvθ ln θ, (10)

where Cv is the specific heat of the material. In order for thetheory to comply with the second law of thermodynamics,we define the internal energy and the heat flux as follows:

e = Ψ − θ∂�

∂θ; qqq = −κ∇θ, (11)

where κ is the thermal conductivity which we assume con-stant. Using these expressions in Eq. (9), dropping the mixedderivatives and the higher-order powers of the strains ei , weobtain the following equation

ρCv

∂θ

∂t= κ∇2θ + a2

θ

θme2

∂e2

∂t+ g. (12)

2.2 Strong form of the boundary-value problem for thephase-field model

Let � ⊂ R2 be an open set in two dimensional space. The

boundary of �, assumed sufficiently smooth (e.g., Lipschitz),is denoted by . We call nnn the unit outward normal to .We assume the boundary to be composed of two comple-mentary parts, such that = u ∪ σ , u ∩ σ = ∅. Theboundary-value problem governing the dynamic evolutionof the SMA can be stated as follows: find the displacementsuuu : � × (0, T ) → R

2, and temperature θ : � × (0, T ) → R

such that

ρ∂2uuu

∂t2 = ∇ · σσσ + σσσ g + η∇2uuu + fff , in � × (0, T ),

(13.1)

cv

∂θ

∂t= κ∇2θ + a2θe2

∂e2

∂t+ g, in � × (0, T ),

(13.2)

uuu = uuu D, on uuu × (0, T ), (13.3)(σσσ + kg

2∇∇⊥e2

)nnn = 0, on σσσ × (0, T ), (13.4)

∇θ · nnn = 0, on × (0, T ), (13.5)

uuu(xxx, 0) = uuu0(xxx), in �, (13.6)

θ(xxx, 0) = θ0(xxx), in �, (13.7)

where uuu D : � → R2 denotes the prescribed displacements,

and uuu0 : � → R2, θ0 : � → R are given functions which

represent the initial displacements and temperature, respec-tively.

2.3 Dimensionless form of the phase-field equations

We rescale Eqs. (13.1) and (13.2) to a dimensionless formby using the following change of variables:

ei = ecei , ui = ecδui , x = δ x,

F = FcF , t = tct, θ = θcθ . (14)

The variables with bar and subscript c are rescaled variablesand constants respectively. The thermo-mechanical field Eqs.(13.1) and (13.2) can now be converted to the dimensionlessform:

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∂2ui

∂ t2 = ∂σi j

∂ x j+ σ

gi + η∇2 ˙ui + fi ,

(summation on j is implied) (15)

∂θ

∂ t= k

(∂2θ

∂ x2 + ∂2θ

∂ y2

)+ χ θ e2

∂ e2

∂ t+ g, (16)

with rescaled constants defined as

ec =√

a4

a6, δ =

√kga6

a24

, tc =√

ρδ2

a6e4c, η = ηtc

ρδ2 ,

k = ktcδ2ρCv

, χ = a2e22√

2ρCvθc, g = gtc

ρCvθc, fi = fi t2

c

ρecδ.

(17)

3 Numerical formulation

We numerically implement the governing equations in a vari-ational form utilizing IGA. We discretize the domain usingC 1-continuous functions essential for the discretization offourth-order PDEs in primal form. The second order accurategeneralized-α method, which accounts for high-frequencydamping, is used for time integration along with an adaptivetime stepping scheme developed by the authors of [23].

3.1 Weak formulation

The weak formulation is derived by multiplying the gov-erning equations with weighting functions {UUU ,�} and trans-forming them by using integration by parts. Initially, we con-sider periodic boundary conditions in all directions. Let Xdenote both the trial solution and weighting function spaces,which are assumed to be identical. The variational formu-lation is stated as follows: find SSS = {uuu, θ} ∈ X such thatB(WWW , SSS) = 0 ∀WWW = {UUU ,�} ∈ X, where

B(WWW , SSS) =∫

[ρUUU · ∂2uuu

∂t2 + �Cv

∂θ

∂t

]d�

+∫

(∇UUU : (σσσ+η∇uuu)−kg

2∇2UUU · ∇⊥e2 − UUU · fff

)d�

+∫

[κ∇� · ∇θ − �

(g + a2

θ

θme2

∂e2

∂t

)]d�. (18)

3.2 Semi-discrete formulation

For the space discretization of Eq. (18), the Galerkin methodis used. We approximate Eq. (18) by the following variationalproblem over the finite element space Xh ⊂ X . Thus, theproblem can be stated as: find SSSh = {uuuh, θh} ∈ Xh such that

∀WWW h = {UUU h,�h} ∈ Xh :

B(WWW h, SSSh) = 0, (19)

with WWW h and SSSh defined as

WWW h = {UUU h,�h},UUU h =nb∑

A=1

UUU A NA,�h =nb∑

A=1

�A NA,

(20.1)

SSSh = {uuuh, θh},uuuh =nb∑

A=1

uuu A NA, θh =nb∑

A=1

θA NA, (20.2)

where the NA’s are the basis functions, and nb is the dimensionof the discrete space. In this work, we define the NA’s usingNURBS basis functions of degree greater than two, whichachieve global C 1-continuity or higher. A NURBS basis isconstructed directly from a B-Spline basis, which is definedby using the Cox-de Boor recursion formula as [15] :

p = 0 : Ni,0(ξ) ={

1 if ξi ≤ ξ < ξi+1

0 otherwise

p > 0 : Ni,p(ξ) = ξ − ξi

ξi+p − ξiNi,p−1(ξ)

+ ξi+p+1 − ξ

ξi+p+1 − ξi+1Ni+1,p−1(ξ) (21)

where p is polynomial order, n is the number of functions inthe basis, and ξi ∈ R is the ith component in a knot vector��� = {ξ1, ξ2, . . . , ξn+p+1}. Using the B-Spline basis we candefine a B-Spline curve living in R

d by linearly combiningthe basis functions as

FFFC (ξ) =n∑

i=1

CCCi Ni,p(ξ), (22)

with the control points CCCi ∈ Rd , i ∈ {1, . . . , n}. A NURBS

curve in Rd is a projective transformation of a B-Spline curve

defined in Rd+1. Let us assume that DDDi ∈ R

2, i ∈ {1, . . . , n}is a set of control points in a two-dimensional space, andlet ωi be a set of positive numbers called weights such that{DDDi , ωi } ∈ R

3. We define the following B-Spline curve livingin R

3 as

FFF D(ξ) =n∑

i=1

{DDDi , ωi }Ni,p(ξ). (23)

The NURBS curve associated to FFF D is defined as

FFF(ξ) =n∑

i=1

DDDi

ωi

ωi Ni,p(ξ)∑n

j=1 ω j N j,p(ξ). (24)

Note that the NURBS curve FFF lives in R2. Using the notation

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CCCi = DDDi

ωi, W (ξ)=

n∑

i=1

ωi Ni,p(ξ), Ri,p(ξ) = ωi Ni,p(ξ)

W (ξ),

(25)

we have that

FFF(ξ) =n∑

i=1

CCCi Ri,p(ξ). (26)

We will call Ri,p, NURBS basis functions. Two- or three-dimensional NURBS basis can be constructed by projectingtensor products of one-dimensional B-Spline bases. We notethat when there are no repeated knots in the knot vector,quadratic NURBS or B-Spline functions achieve global C 1

continuity.

3.3 Time stepping scheme

LetXXX,XXX, andXXX denote the vector of global degrees of freedomand its first, and second time derivatives, respectively. Let usdefine the following residual vectors:

RRR = {RRRC , RRRT }T , (27.1)

RRRC = {RCAi }, (27.2)

RCAi = B

({NAeeei , 0}, {uuuh, θh}

), (27.3)

RRRT = {RTA}, (27.4)

RTA = B

({0, NA}, {uuuh, θh}

). (27.5)

Given XXXn , XXXn and XXXn and �tn = tn+1 − tn , find XXXn+1, XXXn+1,XXXn+1, XXXn+α f , XXXn+α f , and XXXn+αm , such that

RRRC(XXXn+α f , XXXn+α f , XXXn+αm

)= 0, (28.1)

RRRT(XXXn+α f , XXXn+α f , XXXn+αm

)= 0, (28.2)

XXXn+α f = XXXn + α f(XXXn+1 − XXXn

), (28.3)

XXXn+α f = XXXn + α f

(XXXn+1 − XXXn

), (28.4)

XXXn+αm = XXXn + αm

(XXXn+1 − XXXn

), (28.5)

XXXn+1 = XXXn + �tn[(1 − γ ) XXXn + γ XXXn+1

], (28.6)

XXXn+1 = XXXn + �tnXXXn + (�t)2

2

[(1 − 2β) XXXn + 2βXXXn+1

].

(28.7)

To define a second-order accurate and unconditionally stablemethod, the parameters αm and α f can be defined in terms ofρ∞ ∈ [0, 1], the spectral radius of the amplification matrixas �t → ∞, as follows [15]

αm = 1

2

(3 − ρ∞1 + ρ∞

), α f = 1

1 + ρ∞, (29)

while the parameters γ and β must fulfill the relations

γ = 1

2+ αm − α f , β = 1

4

(1 − α f + αm

)2, (30)

with choice of

αm ≥ α f ≥ 1

2,

for the unconditional stability.This method can be implemented as a two-stage predictor-

multicorrector algorithm as follows:

(i) Predictor stage: Set

XXXn+1,(0) = XXXn, (31.1)

XXXn+1,(0) = γ − 1

γXXXn, (31.2)

XXXn+1,(0) = XXXn + �tnXXXn

+ (�tn)2

2

[(1 − 2β) XXXn + 2βXXXn+1,(0)

],

(31.3)

where the subscript 0 on the left-hand-side quantities is theiteration index of the nonlinear solver.

(ii) Multicorrector stage: Repeat the following steps fori = 1, 2, · · · , imax

1. Evaluate iterates at the α-levels

XXXn+α f ,(i−1) = XXXn + α f(XXXn+1,(i−1) − XXXn

), (32.1)

XXXn+α f ,(i−1) = XXXn + α f

(XXXn+1,(i−1) − XXXn

), (32.2)

XXXn+αm ,(i−1) = XXXn + αm

(XXXn+1,(i−1) − XXXn

). (32.3)

2. Use the solutions at the α-levels to assemble the residualand the tangent matrix of the linear system

KKK (i)�XXXn+1,(i) = −RRR(i). (33)

Solve this linear system using a preconditioned GMRESalgorithm to a specified tolerance.

3. Use XXXn+1,(i) to update the iterates as

XXXn+1,(i) = XXXn+1,(i−1) + �XXXn+1,(i), (34.1)

XXXn+1,(i) = XXXn+1,(i−1) + γ�tnXXXn+1,(i), (34.2)

XXXn+1,(i) = XXXn+1,(i−1) + β (�tn)2 XXXn+1,(i). (34.3)

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This completes one non-linear iteration. Steps 1–3 are tobe repeated until the residual vectors RRRC and RRRT havebeen reduced to a given tolerance.The tangent matrix KKK (i) in Eq. (33) may be computedusing the chain rule as follows,

KKK =∂RRR

(XXXn+α f , XXXn+α f , XXXn+αm

)

∂XXXn+αm

∂XXXn+αm

∂XXXn+1

+∂RRR

(XXXn+α f , XXXn+α f , XXXn+αm

)

∂XXXn+α f

∂XXXn+α f

∂XXXn+1

∂XXXn+1

∂XXXn+1

+∂RRR

(XXXn+α f , XXXn+α f , XXXn+αm

)

∂XXXn+α f

∂XXXn+α f

∂XXXn+1

∂XXXn+1

∂XXXn+1

= αm

∂RRR(XXXn+α f , XXXn+α f , XXXn+αm

)

∂XXXn+αm

+ α f γ�tn∂RRR

(XXXn+α f , XXXn+α f , XXXn+αm

)

∂XXXn+α f

+ α f β (�tn)2

∂RRR(XXXn+α f , XXXn+α f , XXXn+αm

)

∂XXXn+α f

,

(35)

where we have omitted the sub-index (i) for notationalsimplicity.

4 Numerical simulations

To illustrate the implementation and effectiveness of theIGA approach, a number of numerical simulations have beenperformed using the developed dynamic thermo-mechanicalmodel. Most numerical simulations have been performed ona rectangular SMA specimen of � = [0, Lx ] × [0, L y] as

shown in Fig. 2a. Moreover, some numerical simulations, inparticular, on annular and circular domains, are presented toillustrate the geometrical flexibility of the approach. (referto Fig. 2 for domain and boundary nomenclature). The ini-tial and boundary conditions for the experiments have beendescribed in the respective sections. The rescaling constantfor spatial and temporal domain are 1.808 nm, and 1.812 ps,respectively. The Fe70Pd30 material parameters [1] used forthe simulations are as described in Table 1.

In the following sections, we first validate our numericalmethodology by comparing our results with those obtainedby using a mixed formulation and standard C 0 Lagrange ele-ments. Then, we carry out the mesh refinement studies with h-and k-refinements. In all the following simulations, we useuniform knot vectors with no repeated knots. The dynam-ics of SMA specimens under thermo-mechanical loadings isdescribed afterwards. Finally, we study the microstructureevolution in an annulus and circular geometry.

4.1 Comparison of results with Lagrange and B-Splinebases

The results of IGA have been first validated with the resultsfrom the 2D simulations carried out with the commercialComsol Multiphysics software [12]. The mixed finite ele-ment formulation has been used to implement the thermo-mechanical equations in Comsol. The identical problemshave been set up in both approaches, with same initial andboundary conditions on a square geometry with Lx = L y =150 nm. The microstructures were evolved starting from aninitial displacement seed (in dimensionless unit) in the centerof the domain xxxc = {xc

1, xc2}T defined as

u01 = exp(−5 · 10−2|xxx − xxxc|2), u0

2 = −u01, (36)

starting with initial temperature θ0 = 250 K. All the bound-aries of the domain have been constrained in the struc-tural degrees of freedom uuu = 000. The geometry is meshed

(a) (b) (c)

Fig. 2 Domain and boundary nomenclature of (a) rectangular, (b) annular and (c) circular geometry

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Table 1 Material properties ofFe70Pd30

a1 (GPa) a2 (GPa) a3 (GPa) a4 (GPa) a6 (GPa)

140 212 280 17 ×103 30 ×106

kg (N) θm (K) Cv (Jkg−1 K −1) κ (W m−1 K −1) ρ (kg m−3)

3.15 × 10−8 265 350 78 10,000

(a)

0 0.2 0.4 0.6 0.8 1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04Quadratic Lagrange2nd degree B−spline

(b)

Fig. 3 Evolved microstructure deviatoric strain e2 (a) and cutline ofe2 solution for the quadratic Lagrange basis (using the Comsol Multi-physics software), and second degree B-Spline basis (using IGA) along

normalized x , the diagonal O–O′ (b) (red and blue on the left-hand-sidepicture indicate martensite variants and green indicates the austenitephase) (color online)

using 1282 quadratic Lagrange elements in Comsol and1302 second-order C 1-continuous B-Spline functions in theIGA. Note that using 1302 second-order basis functions inIGA produces 1282 knot spans (elements). Note also that,although the number of elements is the same in both cases,the number of global degrees of freedom is approximately 7times smaller in the case of the IGA . We have used a con-stant time step of 0.25 time units in both cases. Figure 3ashows the microstructure evolution at the static equilibrium.The figures shows self-accommodated twinned microstruc-tures. The microstructures are aligned along the diagonalof the square. The evolved microstructures are identical atthe scale of the plot. We plot cut lines of the deviatoricstrain order parameter e2 across x , the normalized diagonalO–O′, shown in Fig. 3b. The solutions have been sampledacross the lines and plotted using a piecewise linear interpo-lation. The maximum error recorded in the results is less than1 %. The difference in the solution in Comsol and IGA canbe attributed to the different ways of implementation of thesolvers. The results indicate that our implementation is cor-rect in the IGA. The energy and average temperature evo-lutions during the microstructure formation using IGA areshown in Fig. 4. The results of the study show the effec-tiveness of IGA approach to obtain the correct solutionswith far less degrees of freedom than standard Lagrangeelements.

4.2 Mesh refinement studies

In this section, we study the sensitivity of the solution to amesh size using the h- and k-refinements. Within the geomet-ric flexible methods, k-refinement is a unique feature of theIGA that achieves better approximability increasing the orderand the global continuity of the basis functions simultane-ously. Our examples show how highly smooth basis functionscan approximate sharp layers in the solution accurately andstably. The simulations have been carried out on the squaredomain with Lx = L y = 90 nm using B-Splines of degreep = 2, p = 3, and p = 4, and global continuity C p−1. Foreach case, we employ meshes composed of 322, 642, and1282 elements. We use periodic boundary conditions and thesame initial condition as described in Sect. 4.1.

We are interested here in microstructure evolution i.e., thespatial variation of the deviatoric strain e2. The fully evolvedmicrostructure has accommodated martenstic twins whichare periodic on boundaries. The deviatoric strains are plot-ted at the same time instant for the coarsest mesh (seconddegree 322 element mesh) and the finest mesh (fourth degree1282 element mesh) in Fig. 5. The meshes with other basisfunctions approximate sharp layers accurately.

The maximum error with the coarsest mesh is less than2 % with respect to the fine mesh indicating that with theIGA, the good results can be obtained even on the coarsest

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0 0.5 1 1.5 2 2.5−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Time (ns)

Ene

rgy

(a)

0 0.5 1 1.5 2 2.5−0.08

−0.075

−0.07

−0.065

−0.06

−0.055

−0.05

−0.045

−0.04

−0.035

−0.03

Time (ns)

(b)

Fig. 4 Plot of dimensionless (a) energy and (b) temperature τ evolution with time for the numerical example described in Sect. 4.1

Fig. 5 Mesh refinementstudies. The plot shows thedeviatoric strain e2 results forthe coarsest and finest meshes(red and blue indicate martensitevariants and green indicates theaustenite phase) (color online)

(a) Quadratic 322 (b) Quartic 1282

mesh. The cut line of e2 and its error for meshes with 32 B-Spline basis and different continuity, with respect to the finemesh are plotted along x , the normalized diagonal O–O′, inFig. 6.

4.3 Body and thermal loadings

To explore the importance of thermo-mechanical modelingin SMAs, we conduct a dynamic loading on a square domainwith Lx = L y = 250 nm. The body and thermal loads areapplied in the domain as

fff = fff 0 sin(π t/tt ), g = g0 sin(π t/tt ), (37)

where fff 0, g0 are the mechanical load and thermal load actingon the body, and tt is the total time. The sinusoidal load hasbeen applied with f 0

1 = 0.5, f 02 = 0, and g0 = −0.25

in the dimensionless units for tt = 2 ns. Three differentsimulations have been conducted with body load, thermalload, and combined body and thermal loads.

The microstructure evolution for three simulations areshown in Figs. 7, 8, and 9. Note that the body load acts in

x1 direction only. In the first simulation with body load, thefavorable martensite variant M+ pocket, to the f 0

1 loading,appears at the boundary 1 and accommodated martensitevariant M− pocket on the opposite boundary 4. The secondsimulation, with thermal load, favors the evolution of accom-modated twinned microstructure morphology in a domain.The needle shaped twin, as also observed experimentally[39], is also captured with the developed dynamic thermo-mechanical model. The third simulation with simultaneousapplication of body and thermal loads causes the domainto evolve into a complex microstructure (refer to Fig. 9).The complex microstructure is a result of thermo-mechanicalcoupling in SMAs. The evolution of the average temperaturein all three cases is shown in Fig. 10. These three simulationsshow the strong impact of dynamic thermo-mechanical cou-pling on microstructure evolution, and illustrate the necessityof using coupled thermo-mechanical theories.

4.4 Tensile testing

To understand the thermo-mechanical properties of themicrostructure, we conduct tensile tests on the SMA wire

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0 0.2 0.4 0.6 0.8 1−0.03

−0.02

−0.01

0

0.01

0.02

0.03

0.04 QuadraticCubicQuartic

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−8

−6

−4

−2

0

2

4

6

8

10x 10

−3

QuadraticCubicQuartic

(a) (b)

Fig. 6 Mesh refinement studies. Plot of cutline of the solution of (a) deviatoric strain, and (b) error with respect to finest mesh (fourth degree 128B-Spline basis functions in each direction), along normalized x , the diagonal O–O′, for 32 basis function meshes

(a) t = 0.5 (b) t = 1 (c) t = 1.5 (d) t = 2

Fig. 7 Microstructure evolution under body load at different time instants t (ns) (red and blue indicate martensite variants and green indicates theaustenite phase) (color online)

(a) t = 0.5 (b) t = 1 (c) t = 1.5 (d) t = 2

Fig. 8 Microstructure evolution under thermal load at different time instants t (ns) (red and blue indicate martensite variants and green indicatesthe austenite phase) (color online)

under dynamic loading conditions. The rectangular nanowiresof domain � = [0, 1100] × [0, 220] nm are dynamicallyloaded under three cases starting with initial temperatures245 K (case 1), 265 K (case 2), and 285 K (case 3) correspond-ing to the temperatures below, equal, and above the criticaltemperature θm . The nanowires, initially in austenite phase,

were first quenched at the temperatures and microstructuresare allowed to evolve with constrained boundariesuuu = 000. Thenanowires are evolved to the accommodated twinned marten-ite phase in the case 1 and austenite in the case 2 and case3 as shown in the subplots (a) of Figs. 11, 12, and 13. Next,tensile tests have been carried out on the evolved nanowire

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(a) t = 0.5 (b) t = 1 (c) t = 1.5 (d) t = 2

Fig. 9 Microstructure evolution under simultaneous body and thermal loads at different time instants t (ns) (red and blue indicate martensitevariants and green indicates the austenite phase) (color online)

0 0.5 1 1.5 2−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

Body loadingThermal loadingCombined (body + thermal) loading

Fig. 10 Average temperature τ (dimensionless) evolutions duringbody, thermal, and combined loadings (color online)

specimens separately. The domain is fixed on boundary 1

with uuu = 000 and ramped loading and unloading based dis-placements, equivalent to the strain rate of 3 × 107 s−1, havebeen applied to the boundary 4 in the x1 direction. The highstrain rate is a consequence of model rescaling.

In the case 1, the twinned microstructure is converted intothe detwinned microstructure with the unfavorable marten-site M− converting into the favorable martensite M+. Inthe case 2 and case 3, the austenite is converted into thedetwinned martensite by the movement of habit plane. Figure14 shows the average axial stress-strain and average temper-ature evolution of the three cases. The significant influenceof dynamic loading is observed in the average temperatureevolution due to bi-directional coupling of e2, e2, and θ . Thephase transformation takes place simultaneously along withthe elastic loading under the influence of high strain rate. Dueto the loading in the cases 2 and 3, the austenite is convertedinto the favorable M+ variant causing habit plane to form.The stress-strain curve is stiffer in the case 2 and 3 due to thehigher stiffness of austenite phase of SMAs at higher temper-ature. These observations are in agreement with the earlier

published results based on the mixed formulation, as well ason the experimental observations of SMAs under high strainloading [19,22,36].

4.5 Loading an annulus geometry

In the earlier sections, we conducted the studies on square andrectangular domains. Here we conduct the dynamic loadingstudies on the annulus geometries. The strength of IGA is tomodel complex geometry exactly, thus eliminating geomet-rical errors. The quarter annulus geometry domain � is mod-eled as shown in Fig. 2b, with R1 = 375 nm and R2 = 500nm. The domain is meshed with 64, and 256 quadratic C 1-continuous NURBS based elements along radial, and circum-ferential directions, respectively.

Two sets of simulations have been carried out in twostages each. In the first stage, boundaries were fixed andmicrostructure was allowed to evolve at 240 K (case 1) and275 K (case 2), below and above the transition tempera-ture starting with a random initial condition. The evolvedmicrostructures in both cases are shown in subplots (a) inFigs. 15 and 16. The domain evolves into an accommodatedmicrostructure in case 1. It is observed that the width ofmartensitic twins are not uniform in the center of the domain,which is distinct from the uniform spacing of martensitictwins in a rectangular domain in Sect. 4.4 (refer to Fig. 11a).In the case 2, the domain remains in an austenitic phase.

In the second stage, the evolved annulus domain is fixedalong the boundary 1 with uuu = 000 and ramped loading andunloading based displacements, equivalent to the strain rateof 3 × 107 s−1, have been applied to the boundary 4 in thedirection x1. In the case 1, the phase transformation startsat the loading boundary 4. The detwinned microstructure isconverted into the favorable M+ phase through the process ofdetwinning during loading, as observed in Fig. 15a–d. At theend of loading (refer to subplot (e)), the annular domain hasM+ and M− and traces of unconverted detwinned marten-site. The microstructure at the end of the unloading is pre-sented in the subplot (f).

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(e) t = 1.42 (f) t = 2

(a) t = 0 (b) t = 0.2

(c) t = 0.48 (d) t = 0.95

Fig. 11 Microstructure evolution in nanowire during loading and unloading at different times t (ns) for case 1 (red and blue indicate martensitevariants and green indicates the austenite phase) (color online)

(e) t = 1.42 (f) t = 2

(a) t = 0 (b) t = 0.2

(c) t = 0.48 (d) t = 0.0.95

Fig. 12 Microstructure evolution in nanowire during loading and unloading at different times t (ns) for case 2 (red and blue indicate martensitevariants and green indicates the austenite phase) (color online)

The second stage of simulation in case 2 is presented inFig. 16. During a loading cycle, the movement of the habitplane causes the austenite to get converted into the M+ favor-able phase as observed in the subplot (b). As the phase trans-formation progresses rearrangements of microstructures areobserved in the subplots (c–e). The microstructure at the endof the unloading is presented in the subplot (f). The averagetemperature evolutions for both simulations are plotted inFig. 18a.

4.6 Loading a circular geometry

Finally, we conduct a loading simulation on a SMA circu-lar domain. The circular domain � , as shown in Fig. 2c, is

constrained at the bottom boundary 1 and loaded at the topboundary 2. The arc lengths of 1 and 2 are chosen as2α = 2β = π/2 radians. A simulation has been carried outby loading the domain with a ramp displacementuuu equivalentto the strain rate of 3×107 s−1 for 1 ns in the negative x2 direc-tion. The initial conditions used are uuu = 000 and θ0 = 240 K.

The microstructure evolution is as shown in Fig. 17. Thephase transformation starts at the 2 boundary and propa-gates as a habit plane between twinned martensites, alignedalong ±π/4 about the vertical central line of the circulargeometry, and an austenite domain as shown in a subplot (a).As the phase transformation progresses (refer to subplot (b)),the whole domain is converted into a complex microstructure,with twins of different widths and small pockets of marten-sitic variants in dots like microstructures, in a sector defined

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(e) t = 1.42 (f) t = 2

(a) t = 0 (b) t = 0.2

(c) t = 0.48 (d) t = 0.95

Fig. 13 Microstructure evolution in nanowire during loading and unloading at different times t (ns) for case 3 (red and blue indicate martensitevariants and green indicates the austenite phase) (color online)

0 0.5 1 1.5 2 2.5−200

0

200

400

600

800

1000

Case 1Case 2Case 3

0 0.5 1 1.5 2−0.05

0

0.05

0.1

0.15

0.2

Case 1Case 2Case 3

(a) (b)

Fig. 14 Evolution of average (a) axial stress–strain and (b) dimensionless temperature τ in the nanowire specimens for three cases (color online)

(a) t = 0 (b) t = 0.18 (c) t = 0.55 (d) t = 0.74 (e) t = 1 (f) t = 2

Fig. 15 Microstructure evolution in an annulus domain at different times t (ns) on loading for case 1 (red and blue indicate martensite variantsand green indicates the austenite phase) (color online)

by 2α, in Fig. 2c. Further, the partially developed microstruc-tures evolve into twins and start to coalesce as seen in sub-plots (c–d). At the end of the loading, as shown in a subplot(e), the twins span till the vertical central line of the circu-lar geometry. The top sector, described by the sector 2β, is

converted into the M+ martensite. It is to be noted that thereexist traces of austenite at 1, due to the constrained bound-ary conditions. The average temperature evolution is plottedin Fig. 18b.

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(a) t = 0 (b) t = 0.18 (c) t = 0.55 (d) t = 0.74 (e) t = 1 (f) t = 2

Fig. 16 Microstructure evolution in an annulus domain at different times t (ns) on loading for case 2 (red and blue indicate martensite variantsand green indicates the austenite phase) (color online)

(a) t = 0.09 (b) t = 0.18 (c) t = 0.55 (d) t = 0.75 (e) t = 1

Fig. 17 Microstructure evolution in a circular domain at different times t (ns) (red and blue indicate martensite variants and green indicates theaustenite phase) (color online)

0 0.5 1 1.5 2

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Case 1Case 2

0 0.2 0.4 0.6 0.8 1−0.095

−0.094

−0.093

−0.092

−0.091

−0.09

−0.089

(a) (b)

Fig. 18 Evolution of average dimensionless temperature τ for (a) annulus and (b) circular geometries (color online)

5 Conclusions

We have developed a numerical formulation of the phase-field model based on the IGA for SMAs. The formulationpermits straightforward treatment of fourth-order spatial dif-ferential terms representing domain walls in SMAs, with-out the use of mixed finite element formulation or non-conforming elements. Several numerical examples have beenpresented with evolution under dynamic loadings. We vali-dated the IGA implementation with the results of quadratic

Lagrange basis element using the commercial finite ele-ment software with maximum error less than 1 %. The h-and k-refinement studies suggested that the maximum errorwith coarsest mesh (quadratic 322) is 2 % with respectto most refined quartic 1282 indicating that the simula-tions can be conducted even on the coarsest mesh withgood accuracy. The dynamic loading studies are in agree-ment with previously published literature on mixed formula-tions using commercial software and available experimentalresults.

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The developed framework should prove to be useful in thestudy of thermo-mechanical dynamics of a realistic 3D SMAspecimen under complex loadings.

Acknowledgments RD and RM have been supported by NSERC andCRC program, Canada, and JZ by NSERC. HG was partially supportedby the European Research Council through the FP7 Ideas Starting Grantprogram (Project # 307201) and by Consellería de Educación e Orde-nación Universitaria (Xunta de Galicia). Their support is gratefullyacknowledged.

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