Nonlocal elasticity in shape memory alloys modeled using ... · Nonlinear Dyn (2019) 97:1911–1935...

Nonlinear Dyn (2019) 97:1911–1935https://doi.org/10.1007/s11071-019-04943-5

ORIGINAL PAPER

Nonlocal elasticity in shape memory alloys modeled usingperidynamics for solving dynamic problems

Adam Martowicz · Jakub Bryła ·Wieslaw J. Staszewski · Massimo Ruzzene ·Tadeusz Uhl

Received: 5 June 2018 / Accepted: 6 April 2019 / Published online: 23 April 2019© The Author(s) 2019

Abstract Thework is primarily devoted to the peridy-namic model elaborated for a solid bodymade of shapememory alloys (SMAs). The superelasticity effect istaken into consideration as well as its practical appli-cations. Hence, the numerical simulations, making useof the phenomena of superelasticity, are carried out forthe model of an SMA wire to investigate mechanicalenergy dissipation. The nonlocal peridynamic modelof an SMA component is derived based on the the-ory proposed by Lagoudas—introduced to describe thephenomenon of solid phase transitions that occur inSMA. The results of the conducted experimental work

A. Martowicz (B) · J. Bryła · W. J. Staszewski · T. UhlDepartment of Robotics and Mechatronics, AGHUniversity of Science and Technology, al. A. Mickiewicza30, 30-059 Kraków, Polande-mail: [email protected]://krim.agh.edu.pl/en/

J. Bryłae-mail: [email protected]

W. J. Staszewskie-mail: [email protected]

T. Uhle-mail: [email protected]

M. RuzzeneSchool of Aerospace Engineering, Georgia Institute ofTechnology, 270 Ferst Drive, Atlanta, GA 30332, USAe-mail: [email protected]://www.ae.gatech.edu

M. RuzzeneSchool of Mechanical Engineering, Georgia Institute of Tech-nology, 801 Ferst Drive, Atlanta, GA 30332, USA

are applied by the authors to validate the elaboratedmodel.Moreover, an alternative verification of the peri-dynamicmodel is also performedusing other numericaltool—the finite element code based on the analyticalapproach for modeling SMA, developed by Auricchio.The hysteretic character of the stress–strain relation-ship for the modeled SMA component, which under-goes the superelasticity effect, is shown using quasi-static peridynamic simulations. Finally, the capabil-ity of energy dissipation when cyclic loading for theelaborated nonlocal model is investigated via dynamicsimulations. The numerical results obtained for thestudied case are discussed to show the applicabilityof the presented modeling approach in the field ofstructural dynamics. A particular focus is placed onthe available structural stiffness control functionalityin mechanical systems equipped with SMA and effi-ciency of energy dissipation in SMA-based dampers,which may be effectively applied to suppress mechan-ical vibrations. Both mentioned application areas forSMA are of the authors’ special concern in the design-ing process of the nonlinear supporting structure in gasfoil bearings, which is also discussed in the final partof the paper to highlight the practical aspects of theconducted research.

Keywords Nonlocal model · Peridynamics · Shapememory alloy · Superelasticity · Structural dynamics ·Gas foil bearing

123

http://crossmark.crossref.org/dialog/?doi=10.1007/s11071-019-04943-5&domain=pdf

http://orcid.org/0000-0001-9630-0355

http://orcid.org/0000-0003-3047-498X

http://orcid.org/0000-0003-3071-7427

http://orcid.org/0000-0002-1502-2160

http://orcid.org/0000-0002-4332-3067

http://krim.agh.edu.pl/en/

http://www.ae.gatech.edu

1912 A. Martowicz et al.

1 Introduction

Nonlocal modeling applied to both analytical prob-lem formulations in mechanics and numerical appli-cations within the frame of computational mechanicshas been successfully used for many decades. The ideaof nonlocality was first employed in the sixties of theprevious century to describe and understand the mate-rial behavior observed during experiments. The break-through works of Kröner, Kunin, Eringen and Ede-len [1–3] showed realistic links between experimentaloutcomes and the nonlocality proposed to be consid-ered when modeling internal interactions found in theinvestigated materials. In other words, nonlocality hasarisen as a sort of remedy for the identified deficien-cies of modeling methods. This fact is also of partic-ular importance in dynamics since nonlocal modelingenables efficient control of the dispersion properties ofthe modeled medium [4].

Classically, continuum mechanics refers to locallyformulated governing equations. This means that onlylocal relationships between strains and stresses aretaken into account when dealing with a solid body.Despite undeniable popularity of local elasticity, thisapproach, however, exhibits several drawbacks. Asalready referred to, it is basically incapable of com-pletely describing and, therefore, predicting a realbehavior of a deformable material. This applies espe-cially when considering various physical phenomenarevealing themselves at consecutive geometric scales(lengthscales). In this case, various material proper-ties should be defined at these lengthscales as well,which is not allowed in a local formulation explicitly.However, the unique formulation of a nonlocal elas-ticity leads to the demanded capability of introducingthe required specific lengthscales referring to the rangeof nonlocal interactions within the modeled body [5].In fact, both micro- and macrostructure characteristicsmay contribute in the resultant nonlocal model [6–8].

The most known phenomena that require nonlo-cal elasticity, to allow for more physical description,are wave dispersion and shear bands observed duringstretching [3,8,9]. Other examples explaining the needof nonlocality are provided in [10]. As reported in thecited works, the observed lack of convergence betweenexperiments and theories may be effectively addressedvia nonlocal interactions introduced in amodeled body.As far as dynamics is of concern, the use of nonlocalapproaches enables more accurate simulations of the

phenomenon of wave propagation. Nonlocal elastic-ity leads to lower requirements for model mesh den-sity with its spatial domain discretization. On the onehand, it helps to mitigate numerical dispersion, whichemerges for crude meshes [11–15]. This behavior isobservedwhen the distances between nodes in a numer-icalmodel are comparablewith thewavelengths.More-over, nonlocal elasticity, via long-range interactions,introduces the means for relatively easy formulationof arbitrary shapes for physical dispersion relation-ships [14,16].

Up to now, nonlocal elasticity has been widely usedfor various physical domains, materials and researchfields. First, various types of geometric discontinu-ities (e.g., cracks), that, in general, stand for chal-lenging modeling issues for the classically formulatedlocal elasticity—due to gradient-based equations—canbe effectively addressed by nonlocal approaches. Itshould be noted that various types of geometric discon-tinuities, including those ones originating from mate-rial nonlinearities and boundary conditions, may resultin ambiguity regarding calculations of derivatives.Nonlocality allows for introduction of integral-basedexpressions that help to reduce or even eliminate theabove-mentioned inconvenience. Moreover, via non-locality, more physical behavior of the models maybe assured for growing cracks [17–19]. In fact, morespontaneous crack’s growth is achievable for numeri-cal simulations [20]. The other applications of nonlo-cality deal with vibro-acoustic interaction [21], modelupscaling [22], regularization of boundary value prob-lems [23], impact loading [24], viscoplasticity [25],thermal diffusion [26], thermoelasticity, including boththe earliest and recent papers [27–30] and piezoelec-tricity [31,32]. The theory of generalized continua pro-posed for granular media, which is valid at variousgeometric scales, makes use of nonlocally formulatedmaterial properties [33]. Moreover, nonlocal elasticityhas been successfully applied to model graphene [34,35] and shape memory alloys (SMAs) [36–38].

The only two disadvantages of nonlocal modelingtools, that are worth to be mentioned, are the com-putational costs required and parameters identificationfor nonlocal models. Indeed, more reliable materialdescription may need both greater number of inter-actions between pieces of a modeled body and morecomplicatedmaterialmodelswith the coefficients defy-ing long-range interactions. The former issue refers tomore populated global systemmatrices that needhigher

123

Nonlocal elasticity in shape memory alloys 1913

computational effort for their processing. The latter, inturn, results in more demanding and comprehensiveexperiments to be performed for material data identi-fication. Nowadays, the use of parallel computing inGPU may help to solve the problems partially. How-ever, it should be noted that, whenever required and fea-sible, the coupling between local and nonlocal modelsmay be performed to take advantages of both modelingapproaches [39].

Due to the well-recognized advantages of the com-putational techniques based on nonlocal elasticity, theauthors of the present paper have attempted to applyone of these theories, namely peridynamics, for mod-eling the phenomenon of superelasticity in SMA. Hav-ing introduced the theoretical part of the work, thenumerical outcomes obtained for the elaborated one-dimensional (1-D)model of an SMAwire are presentedgiving an exemplary practical application of the devel-oped approach in the field of structural dynamics.

In detail, the following sections contribute to thepaper. After presenting introductory Sect. 1, which isdevoted to the applications of nonlocal approaches incomputational mechanics, the next three sections pro-vide the fundamentals regardingperidynamics (Sect. 2),SMA (Sect. 3) and the analytical description pro-posed by Lagoudas to model superelasticity in SMA(Sect. 4). Complementarily, other selected knownmod-eling techniques used for SMA are also shortly dis-cussed by the authors. A peridynamic model proposedby the authors of the present work to simulate the phe-nomenon of superelasticity in SMA is described inSect. 5. Next, in Sect. 6, the results of experimentsconducted for an SMA sample are presented to vali-date the elaborated peridynamic model. Moreover, anadditional verification of the peridynamic model isalso addressed using an alternative numerical tool—the finite element (FE) code based on the analyticalapproach for modeling SMA provided by Auricchio.Section 7 presents and discusses the results of thenumerical simulations performed for the peridynamicmodel of a damper made of an SMAwire. Based on theobtained results, practical aspects of the theoretical partof the work are also discussed in Sect. 7, including theissues of structural stiffness control functionality avail-able in mechanical systems equipped with SMA andefficiency of energy dissipation in SMA-based dampersapplied for reduction in mechanical vibrations. A spe-cial authors’ attention is paid on the capability of effi-cient change of the properties of nonlinear support-

ing structure in gas foil bearings (GFBs), which maybe achieved via superelasticity phenomenon, applyingthe proposed nonlocal modeling tools. The last Sect. 8summarizes the paper and draws final conclusions.

2 Fundamentals of peridynamics

Peridynamics is a type of nonlocal modeling approachproposed quite recently by Silling in 2000 [40]. Itassumes coexistence of the pieces in a modeled body(named as particles in the theory of peridynamics),which are linked via both local and nonlocal interac-tions. In case of the so-called bond-based peridynam-ics, the resultant force acting on an actual central parti-cle (localized at the position x) is found based on inte-gration of the contributing interaction forces f over thespecified region—the horizon H , as shown in Fig. 1.

The peridynamic governing equation takes the fol-lowing general form with the spatial component con-sisting of an integral expression

ρu (x, t) =∫H

f(u

(x, t

) − u (x, t) , x − x)dVx

+ b (x, t) (1)

With reference to graphical interpretation given inFig. 1, the following parameters are used in Eq. (1):x, x—position vectors for central and neighboring par-ticles, u (x, t), u

(x, t

)—displacement vectors for cen-

tral and neighboring particles, b (x, t)—external bodyforce volumetric density considered for the currentcentral particle, ρ—mass density of the central par-ticle, dVx—fraction of the neighboring particle’s vol-

Fig. 1 Interactions in a nonlocal peridynamic model specifiedwithin a horizon defined for an actual central particle

123


ume which is covered by the horizon H and thereforethey are considered when calculations of the interac-tion forces f . As already referenced, the function fdefines the pairwise interaction force determining theforce links between central and neighboring particlesobserved within the horizon H .

For further explanation, it is convenient to introducethe vectors ξ and η defying the relative particles’ posi-tion and displacement, respectively, where:

ξ = x − x (2)

η = u(x, t

) − u (x, t) (3)

By definition, the pairwise force f = f (ξ,η) can beconditionally defined as follows [40,41]:

f (ξ,η) ={

e (ξ,η) c (ξ) s (ξ,η) , if ‖ξ‖ ≤ δ

0, otherwise(4)

where e—unit vector pointing the direction of the inter-action force f , c—the so-called micromodulus func-tion, which defines elastic properties of the modeledmaterial, s—strain, δ—radius of the horizon H . Asdemanded, nonzero forces may appear only if the con-dition ‖ξ‖ ≤ δ is satisfied, which means that the forceinteractions localized within the horizon H are takeninto account merely.

A very specific and useful property of peridynamicsis its ability to direct use of macroscale, i.e., engineer-ing, properties of the materials—applying the defini-tion of the micromodulus function c—e.g., Young’smoduli, irrespectively from the lengthscales consid-ered. Hence, resultant material properties may be intro-duced to carry out calculations for various geomet-ric scales providing efficient tools for multiscale stud-ies [22].

In case of a two-dimensional (2-D) model built witha homogeneous and isotropic material, an exemplaryform of the definition formicromodulus function cmaybe as follows:

c = 6E

πδ3 (1 − ν) T(5)

where the elastic properties of the material are: E—Young’s modulus and ν—Poisson’s ratio. The param-eter T declares the thickness of the model.

As shown, the integral-based formulation of govern-ing equation used in peridynamics requires a numberof interactions to be taken into account over the defined

horizon. This fact inevitably leads to the increased com-putational effort comparing to the locally formulatedapproaches. However, the issue of peridynamic mod-els’ convergence was already subject to studies. Therelationships between particles’ distances and horizondiameters as well as their absolute values were alreadyproposed for variousmodeling cases [41]. Even thoughnatural consequence of an application of the peridy-namic approach is the introduction nonlocal interac-tions, a locally formulated solution of a given problemmay be easily found as well. A gradual decrease in thehorizon radius δ allows for the expected convergence.

Sinceperidynamicsmakes useof an integral descrip-tion regarding spatial domain, it can relatively easilysolve the problems involving various types of geo-metric discontinuities, e.g., fatigue cracks (consider-ing the research field of damage modeling in general),boundaries of grains, interconnection layers and inter-faceswith considerable impedancemismatches.All theabove-mentioned cases, if improperly handled, maycause significant numerical issues. The overview onvarious applications of peridynamics may be foundin [19–21,35,42–44]. Hence, the present work shouldbe considered as the extension of the previous investi-gations made on peridynamic capabilities, in which thephenomenon of superelasticity in SMA is of particularconcern.

3 Fundamentals of SMA

SMA is one of the most popular and widely appliedgroups of smart materials exhibiting extraordinaryphysical properties. These properties result from thethermomechanical phenomena observed forphase transformations (phase transitions) in SMA.SMA characterizes reversible transformations of theircrystal structures, i.e., they exhibit solid-state phasechanges at nanoscale, which specifically allow tomem-orize arbitrarily given geometric shapes at macroscaleas well as withstand relatively high elastic defor-mations [45]. The former capability of memorizingeither one or two geometric shapes originates from thetwo-way martensitic phase transition, i.e., martensite–austenite phase transition. The latter functionality ofSMA, in turn, reflects the phenomenon of superelastic-ity (also known as pseudoelasticity).

In case of shape memory property, the applied ther-mal activation—via either cooling or heating—leads

123


Fig. 2 Contribution of austenite in an SMA sample fortemperature-activated reversible martensitic phase transition—exemplary plot for hysteretic behavior

to forward and backward transitions of the martensiticphase, respectively, as demanded. This phenomenon isillustrated in Fig. 2, where the changing contribution ofaustenite phase for varying temperatures is presented.As shown, SMA is characterized by four specific tem-peratures. These temperatures, i.e., As , A f , Ms andM f , are the material properties being dependent onthe alloy’s ingredients used. They, respectively, standfor the temperatures at which solid phases, austeniteand martensite, are generated via mutual transitions.The indexes ‘s’ and ‘ f ’ denote begin and finish of therespective transition, which means either forward orbackward martensitic phase transition. It must be men-tioned that the four characteristic temperatures dependon the applied mechanical stress. These dependenciesare not linear, which is particularly seen for the tem-peratures As and A f for small stresses.

Activation of martensitic phase transitions leadsto the demanded rebuilding of the crystal structure,which is required for changes in the geometric shapesobservable at macroscale. Depending on the capabili-ties achieved via shape training procedures, SMA mayexhibit either the so-called one- or two-way memoryeffects. These effects reflect the capabilities of remem-bering either one or two different geometric shapes,respectively.

It should be noted that martensitic phase transitionsare spontaneous and subject to hysteretic behavior. Infact, the number of possible intermediate geometricshapes achieved by evolving SMA sample is infinite,even though the demanded, i.e., memorized, geome-tries finally appear. Thermal loads lead to the specific,

Fig. 3 Exemplary course of solid phase transitions and shaperecovery in an SMA sample during temperature-activated one-way memory effect

unrepeatable gradual changes of the crystal structurewhile phase transitions occurs that may be consideredas random at macroscale.

For the sake of clarity, the phenomenon of one-waymemory effect is briefly described in the following.Consecutive solid phase transitions are illustrated inFig. 3. As shown, the effect of memorizing a geo-metric shape in SMA reflects the existence of the twodifferent types of crystal structures for the consideredaustenite andmartensite solid phases. There coexist thecubic crystal structure for austenite and the rhomboidalone for martensite. Additionally, the rhomboidal struc-ture may take two forms, namely either undeformed ordeformed one, depending on the external load applica-tion history.

The one-way memory effect can be observed withina repeatable cycle, which contains the following steps(enumerated in Fig. 3): (1) mechanical deformation ofan SMA sample at constant temperature when rhom-boidal crystal structures in martensite exhibit defor-mation, (2) stress release while keeping the deformedshape of SMA, (3) recovery of thememorized shape viathermal load at zero stress when austenite phase gener-ation is activated, (4) cooling down of an SMA sampleleading to the undeformed martensite phase present fora memorized shape.

The second type of the unique phenomena experi-mentally observed in SMA, i.e., superelasticity, isman-ifested via significant elastic deformations, not presentin other metallic materials. Typically, allowable strainsin SMA are within the range 6–8%, which is approxi-mately ten times greater than the values of the respec-

123


tive quantities for steel, aluminum, titanium and cop-per. Due to the attempted scope of the present work,superelasticity is of the special authors’ concern, and itis addressed in detail in the following Sect. 4.

The extraordinary physical characteristics of SMAenable many practical applications. Both memoryeffects and superelasticity are very attractive properties.They have opened new perspectives for applications ofSMA in the structures exhibiting quite rigorous require-ments regarding movable components, available spaceand biocompatibility. Hence, SMA is used in such dis-tant applications as medicine and aerospace [46].

The surprising fact is that there is still significantdeficiency of reliable models of SMA. This motivatedthe authors of the present work to look for some alter-native modeling methods, especially the ones based onnonlocal elasticity, to allow for new capabilities whenreflecting the experimental results, as already refer-enced in Sects. 1 and 2. Even though many papersaddress SMAmodeling, there are still several unsolvedproblems, which have appeared in experiments [47].Some of the observed phenomena are not studied yetsufficiently to allow for a reliable theoretical descrip-tion. Providing with an example, it should be high-lighted that the influence of boundary conditions onthe behavior of SMA and spontaneity of the marten-sitic phase transition is still a missing part [48]. Thephysics of SMA seems complex, which explains whyso many attempts have been made so far toward thedevelopment of new reliable and accurate models [47].

4 Analytical description of superelasticity in SMA

The scope of the present study deals with modeling ofthe phenomenon of superelasticity in SMA. Therefore,more detailed characterization of this effect is providedin the following. The analytical description of the prob-lem constitutes the main part of this section, since it iscrucial to understand the fundamentals of the developedperidynamic model.

As stated before, superelasticity reflects signifi-cant elastic deformations present in SMA after exter-nal stresses are applied. This phenomenon may beobserved at a constant temperature; however, theaustenite phase is required to bemaintained at the ambi-ent temperature before any mechanical load is consid-ered. This means that the temperature during experi-ment should exceed the quantity A f , as shown in Fig. 4.

Fig. 4 Superelasticity effect in SMA—reversible solid phasetransitions observed at a constant temperature.Note that four aux-iliary solid lines symbolically depict the relationships betweenstress and characteristic temperatures. In fact, as already men-tioned in Sect. 3, these relationships are monotonic but not linear

In the presence of gradually increasing mechanicalload, an SMA sample experiences solid phase tran-sitions. This behavior results from the fact that allfour characteristic temperatures A (M) f (s) undergo amonotonic growth when the applied stress increases,which is symbolically visualized in Fig. 4 by aux-iliary four parallel lines. In fact, a complete phasechangewithin the entire body of SMAmay be achievedunder the condition that sufficient level of the exter-nally imposed stresses is assured. The increase in thestresses induces spontaneous austenite to martensitephase change. The superelasticity effect is reversible,and the austenite phase is instantly recreated whenthe loads are released. However, different stress–strainpaths are observed for the two phase transition direc-tions, as visualized in Fig. 5. This dependency uponthe transition direction is represented by a hystereticbehavior of SMA [49].

Two plateau regions appear in the stress–strain plot.They represent the processes of gradual and sponta-neous rebuilding crystal structures in SMA observedduring phase transitions in both directions. As a result,when the contributing solid phases swap, the stressesremain nearly constant, whereas the strains change dra-matically. This unique behavior allows for building theSMA-based structures exhibiting nearly constant reac-tion force generation for relatively wide ranges of elas-tic deflections. The most known practical applicationsmaking use of both extraordinary strains allowed inSMA and their functionality of constant force genera-tion are stents, medical staples and dental braces [50].

123


Fig. 5 Hysteretic character for the stress–strain paths observedfor superelasticity effect in SMA

Finally, the ability of effective energy dissipation,whileperforming consecutive cycles within the hysteresisloop, leads to the applications of superelasticity inmechanical dampers [51,52].

When investigating the shape of the hysteresis plotshown in Fig. 5, it should be highlighted that significantstrains allowed in SMAhave basically two sources. Themain reason for that is instantaneous creation of thedeformed variants of martensite phase, i.e., deformedrhomboidal structure, while the stresses increase. Theminor reason originates from the fact that the Young’smodulus for themartensite phase is less than the respec-tive elastic quantity for austenite.

Having introduced the theory given by Lagoudas inthe work [53], the following analytical description forsuperelasticity is provided as the required backgroundfor the developed peridynamic model of SMA.

First, a definition for the total specific Gibbs freeenergy (the total Gibbs free energy per unit mass) Gshould be defined for a polycrystalline SMA. An SMAsample is assumed to consist of a mixture of the twocontributingmartensite and austenite phases.Thequan-tity G equals

G

(¯σ, T, ξ,

¯εt

)= − 1

2ρCijklσijσkl

− 1

ρσij

[αij (T − T0) + ¯

εt]

+ c

[(T − T0) − T ln

(T

T0

)]

− s0T + u0 + 1

ρf (ξ) (6)

The arguments of the parameter G are: ¯σ—second-order Cauchy stress tensor, T—temperature, ξ—

martensitic volume fraction and εt—second-ordertransformation strain tensor. The dimensionless param-eter ξ ∈ [0, 1] defines the contribution of both solidphases, where the value 0 refers to the structure entirelymade of austenite, and 1 declares the martensite phaseas the only existing one in amodel of SMA.An alterna-tive symbolic description for the stress tensor ¯σ is intro-duced in Eq. (6) using the quantities σij and σkl that fol-low the Einstein summation notation. It is used when-ever unambiguous definition for the tensor calculationsmust be provided. The remaining parameters used inEq. (6) are: ρ—the mass density, Cijkl—fourth-orderelastic compliance tensor, αij—second-order thermalexpansion coefficient tensor, T0—reference (ambient)temperature, c—specific heat, s0—specific entropy atthe reference state, u0—specific internal energy at thereference state, f (ξ)—the transformation hardeningfunction. The function f (ξ) stands for elastic strainenergy related to the interactions present between var-ious variants of martensitic phase and the surroundingphase as well as the interactions observed within themartensitic phase.

Since the phenomenon of superelasticity is of theauthors’ concern, Eq. (6) may be rewritten in the fol-lowing form:

G

(¯σ, T, ξ,

¯εt

)= − 1

2ρCijklσijσkl

− 1

ρσij

¯εt − s0T0 + u0 + 1

ρf (ξ)

(7)

where the temperature effects are neglected. It isassumed in Eq. (7) that the case of isothermal phasetransitions is considered, i.e., when T = T0. In an idealcase, phase transitions are activated by mechanicalloads only, and with reference, no influential tempera-ture variation is observed. As experimentally proven bythe authors, formula (7) can be successfully used in ana-lytical/numerical simulations for SMA as long as thephase transitions’ speed is limited adequately to followthe physical behavior of the tested SMA. More specifi-cally, the experimentally identified limited scale of thephenomenon of spontaneous phase changes, which isfound at the limited phase transitions’ speed, allowsfor a reliable linear approximation of the stress–strainpath—as considered in Fig. 10 in Sect. 6, where modelvalidation is described. Moreover, the phenomenon of

123


spontaneous phase transitions is strictly related to thevariation of temperature field. Hence, the required con-dition of isothermal phase transitions in Eq. (7) maybe verified either by direct temperature measurementsor, indirectly, analyzing the shape of the stress–straincurve. The above-stated relation between the stress–strain path and the temperature field was observedusing an infrared camera. It also, eventually, allowedto identify the maximum speed for the extension testperformed for an SMA wire made of Nitinol, whichassures the required constraints regarding the scale ofthe phenomenon of spontaneous phase changes. Thisspeed was found to be approximately 0.05mm/s. Thelength of the used SMA wire equals 133mm. It meansthat the allowed strain rate obtained for nearly isother-mal phase transitions is about 0.00038 s−1. This con-dition also assures sufficient thermal energy flow, i.e.,its exchange with the surrounding area. This preventsfrom other possible source of significant variation ofthe temperature field that should be additionally takeninto account in Eq. (7) for more realistic modeling ofthe stress–strain relationship. Even the experimentallyidentified allowed value of the absolute extensions rateis relatively small, i.e., of the order of tens of microm-eters per second, it still allows for reliable transientsimulations to investigate the superelasticity effect inSMA wires mounted in the supporting layers of GFB,as long as a long-period steady-state operation of GFBunder a constant load is taken into account.

All the material properties declared in Eq. (7) areconsidered as the resultant quantities defined using thefraction ξ . Hence, the properties of the two contributingphases are used in the linear combinations to find thefollowing parameters

ρ = ρ (ξ) = ρA + ξ(ρM − ρA

)= ρA + ξ�ρ (8)

Cijkl = Cijkl (ξ) = CAijkl + ξ

(CM

ijkl − CAijkl

)

= CAijkl + ξ�Cijkl (9)

s0 = s0 (ξ) = s A0 + ξ(sM0 − s A0

)= s A0 + ξ�s0

(10)

u0 = u0 (ξ) = uA0 + ξ

(uM0 − uA

0

)= uA

0 + ξ�u0

(11)

The austenite and martensite phases are denoted usingthe indexes A and M, respectively. For a 1-D model ofan SMA rod undergoing uniaxial tension, Eq. (7) takesthe form

G(σ, T, ξ, εt

) = − 1

2ρCσ 2 − 1

ρσεt

− s0T0 + u0 + 1

ρf (ξ) (12)

where the compliance tensor Cijkl becomes a scalarC1111 = C defined using the inversions of the Young’s

moduli for both phases CA = (E A

)−1and CM =(

EM)−1

, with their contributions calculated accordingto the martensite percentage volume fraction

C = C (ξ) = CA+ξ(CM − CA

)= CA+ξ�C (13)

The resultant strain equals

ε = Cσ + εt (14)

where εt is the transformation strain found based onthe formula

εt = Λξ (15)

The general form of the transformation tensor � isdepended upon both the phase transition direction,identified by the sign of the strain rate ξ , and thestresses. However, for a 1-D model the tensor � maybe eventually assumed a scalar Λ being independentfrom any of the above-mentioned quantities

Λ =√3

2H for ξ > 0 and ξ < 0 (16)

The parameter Λ is formulated based on the SMAmaterial property—the maximum uniaxial transforma-tion strain H . The conditions ξ > 0 and ξ < 0 meanan increase and decrease in the amount of martensitephase in the model, respectively. These conditions ade-quately concern the cases when the stress σ grows anddecreases. It should be noted that, even though the twoindependent cases are initially considered to condition-ally define the value of Λ, namely ξ > 0 and ξ < 0,as found in the applications of both the elaborated FEand peridynamic codes, these cases effectively makea reference to the two overlapping conditions ξ ≥ 0and ξ ≤ 0. In fact, from a practical point of view, thevalue of the strain rate ξ never equals to zero duringsimulations.

Next, the entropy of the model is assured toeither remain constant or increase after introduction

123


of the second law of thermodynamics. The respectiveClausius–Planck inequality is defined

σ εt − ρ∂G

∂ξξ ≥ 0 (17)

After introduction of Eq. (15), Eq. (17) becomes(σΛ − ρ

∂G

∂ξ

)ξ ≥ 0 (18)

Based on Eq. (12) and neglecting change of the massdensity ρ during phase transitions, there may be foundthe partial derivative ∂G

∂ξrequired to be declared in

Eq. (18)

∂G

∂ξ= − 1

2ρ

∂C (ξ)

∂ξσ 2 − 1

ρσ

∂εt (ξ)

∂ξ− ∂s0 (ξ)

∂ξT0

+ ∂u0 (ξ)

∂ξ+ 1

ρ

∂ f (ξ)

∂ξ(19)

After introduction of Eqs. (10–11), (13) and (15) inEq. (19), one may obtain

−ρ∂G

∂ξ= 1

2�Cσ 2+σΛ+ρ�s0T0−ρ�u0− ∂ f (ξ)

∂ξ

(20)

Then, substitution of the expression −ρ ∂G∂ξ

in Eq. (18)by Eq. (20) leads to

(2σΛ + 1

2�Cσ 2 + ρ�s0T0 − ρ�u0 − ∂ f (ξ)

∂ξ

)ξ ≥ 0

(21)

which constitutes the conditional expression

Πξ ≥ 0 (22)

in terms of the thermodynamic force Π .The remaining contributor to the thermodynamic

force Π—the transformation hardening functionf (ξ)—is found as follows [54]:

f (ξ) ={ 1

2ρbMξ2 + (μ1 + μ2) ξ, ξ > 0

12ρb

Aξ2 + (μ1 − μ2) ξ, ξ < 0(23)

which leads to the expression defying the partial deriva-tive ∂ f (ξ)

∂ξ

∂ f (ξ)

∂ξ=

{ρbMξ + μ1 + μ2, ξ > 0ρbAξ + μ1 − μ2, ξ < 0

(24)

The material properties bA, bM , μ1 and μ2 can be cal-culated using the Kuhn–Tucker conditions

bA = −�s0(A f − As

)(25)

bM = −�s0(Ms − M f

)(26)

μ1 = 1

2ρ�s0

(Ms + A f

) − ρ�u0 (27)

μ2 = 1

4ρ�s0

(As − A f − M f + Ms

)(28)

based on the characteristic phase transition tempera-tures As , A f , Ms and M f , as previously explained inSect. 3.

After introduction of Eqs. (24–28) in Eq. (21), thethermodynamic force Π can be found as

Π = 1

2�Cσ 2 + ρ�s0T0 + Π1 (29)

where

Π1 ={√

6Hσ + ρ�s0(Ms − M f

)ξ − 1

4ρ�s0(3Ms + A f + As − M f

), ξ > 0√

6Hσ + ρ�s0(A f − As

)ξ − 1

4ρ�s0(Ms + 3A f − As + M f

), ξ < 0

(30)

Finally, the transformation function Φ is calculatedaccording to the conditionally defined formula

Φ ={

Π − Y, ξ > 0 (austenite → martensite)−Π − Y, ξ < 0 (martensite → austenite)

(31)

which depends on both the thermodynamic force Π

and the parameter Y , used to define the critical valueof the quantity for internal dissipation of energy whilephase transition occurs

Y = 1

4ρ�s0

(Ms + M f − A f − As

)(32)

Taking into account the value of the transformationfunctionΦ as well as condition (22), the following twocases are distinguished to specify the required changeof ξ :

• if the stress σ increases and if Φ(ξ > 0) > 0,then phase transition from austenite to martensite

123


is identified and further growth of the fraction ξ isnecessary; note, ξ cannot take the values greaterthan 1

• if the stress σ decreases and if Φ(ξ < 0) > 0,then phase transition from martensite to austeniteis identified and further reduction in the fraction ξ

is necessary; note, ξ cannot take the values less than0

Based on verification, performed for the above-statedconditions, which refer to the transformation functionΦ and the martensitic volume fraction ξ ∈ [0, 1],static, quasi-static and dynamic simulations may becarried out considering gradual increase or decrease inthe stress and the hysteretic character of the modeledsuperelasticity effect. During simulations, the condi-tion Φ ≤ 0 must be satisfied at any time, irrespectivelyfrom the sign of ξ , to assure that the Clausius–Planckinequality (17) is fulfilled.

The above-introduced approach represents the phe-nomenological modeling technique, specifically mak-ing use of the free energy concept. As shown, it allowsfor representation of both thermally and mechani-cally induced phase transformations. Dedicated modelparameters, e.g., ξ and ξ , assure effective control overthe percentage contribution of martensite and austenitephases as well as the rate of the martensitic transforma-tion. The idea of the use of additional internal variablesin the model of SMA is not a new one [55]; however,it has been continuously widely applied for decades toallow for convenient modeling kinetics of phase trans-formations [53,56]. The authors of the present papertake an advantage of both more physical interpretationof the model behavior offered by the free-energy-basedapproach and the relatively newly reported capabilitiesof a nonlocal modeling via peridynamics.

It should be, however, noted that the above-mentio-ned comprehensive understanding of the state of thesimulated material—in terms of transformation kinet-ics, energy flow and the entropy of the model—requires relatively complicated mathematical descrip-tion and therefore leads to implementation inconve-niences. Alternatively, another group of phenomeno-logical models can be used, which introduce approxi-mate description for macroscopic material behavior.

In the work [57], the authors consider thermome-chanical properties of themodeled SMA to simulate theshape memory effect and superelasticity. The appliedphenomenological approach makes use of a polyno-

mial approximation to build an SMA model, success-fully followed by its application to a biomechanicalsystem (a rod-type prosthesis of a human middle ear),as reported in [58]. In the work [57], nonlinear prop-erties of the modeled SMA material are addressed bythe analytical description based on a fifth-order poly-nomial. Moreover, the authors of the referenced workconfirmed the reliability and usability of the devel-oped first-order approximate SMA model dedicatedto operate at the arbitrarily selected frequency range,close to the resonance conditions. The developedmodelwas verified using FE simulations. Another interest-ing phenomenological approach is presented in [59].The authors of the citedwork propose a straightforwardpolynomial- based definition of the constitutive modelfor SMA. The properties of a single and 2-degree-of-freedom (DOF) oscillators are investigated with theintroduced SMA components, including bifurcationdiagrams. The presented modeling approach allows toconveniently simulate the effects of shape memory andsuperelasticity.

In the following, the theory of peridynamic modelfor SMA is provided based on the previously presentedfundamentals in the introductory sections.

5 Peridynamic model for SMA

Making the reference to both the former authors’ mod-eling approaches used for SMA [51,60,61] and the the-ory of the superelasticity effect provided in Sect. 4, aperidynamic model is introduced in the following.

Considering Eq. (1), the governing equation for a1-D case of a continuum solid body (a rod) takes theform [40]

ρu (x, t) =∫H

f(u

(x, t

) − u (x, t) , x − x)dVx

+ b (x, t) (33)

which may be transformed into a discrete form for thei th DOF

ρuti =∑N

j = −Nj �= 0

f(uti+ j − uti , (i + j) L

− i L) Vj + bi (34)

with the pairwise interaction force f

f(uti+ j − uti , j L

)=

{csi , if | j | L ≤ δ

0, otherwise(35)

123


strain si

si = uti+ j − uti| j | L (36)

and the volume Vj

Vj = βi, j AL (37)

uti , uti+ j are the displacements at the time step indexed

by t , respectively, for the actual central i th particle andthe particle, which is located within the horizon H atthe j th relative position with respect to the i th particle.The index j takes any discrete value pointing the parti-cle’s location within the horizon of the radius δ = NL ,where L equals the distance between particles. Note,that in the following, the parameter L also becomes�x ,i.e., �x = L , which is more readable in the context ofboth stability and convergence studies. A denotes thearea of the cross section of the modeled rod. The factorβi, j is introduced to correctly determine the value of thevolume for the neighboring particle Vj at the bound-aries of the horizon, and it is determined conditionally

βi, j ={1, j �= N ∧ j �= −N12 , j = N ∨ j = −N

(38)

Next, Eq. (34) evolves into the following form for thei th particle, i.e., for the i th DOF

ρuti =∑N

j = −Nj �= 0

cuti+ j − uti

| j | βi, j A + bi (39)

It should be noted that Eq. (39) takes a general formallowed to be used within the body of a modeled rodexcluding its two ends. In detail, if the horizon radiusδ exceeds the distance between the current i th particleand the location of one of the rod’s ends, Eq. (39) shouldundergo further modification to limit the number ofpermitted values for the j index.

The micromodulus function c takes the followingfundamental, widely applied value for a 1-D case,which stays constant within the horizon of a peridy-namic model [41]

c = 2E A

δ2A(40)

hence

ρuti =∑N

j = −Nj �= 0

2E A

(NL)2

uti+ j − uti| j | βi, j + bi (41)

When multiplying both sides of Eq. (41) by the volumeof the i th particle Vi

Vi = γi AL (42)

we may find

mi uti = 2

∑N

j = −Nj �= 0

βi, jγiE A A

L

uti+ j − uti| j | N 2 + Fi

(43)

or

mi uti = 2

∑Nj = −Nj �= 0

βi, jγi kAuti+ j − uti

| j | N 2 + Fi (44)

where kA is the classically formulated stiffness coeffi-cient (similarly to a 1-D FE formulation) for an initialaustenite phase

kA = E AA

L(45)

mi and Fi are the mass and external force consideredfor the i th particle. Similarly to the parameter βi, j , theauxiliary factor γi assures, in turn, the correct volumefor the current central particle Vi at the ends of the rod,i.e., where i = 1 ∨ i = iM AX . Hence,

γi ={1, i �= 1 ∧ i �= iM AX12 , i = 1 ∨ i = iM AX

(46)

For the sake of simplicity, the case of a uniform rodmade of SMA is considered in the work. Taking intoaccount Eq. (14), which defines the total strain in SMA,as well as Eqs. (13), (15) and (16), one may obtain

ε =(

1

E A+ ξ

(1

EM− 1

E A

))σ +

√3

2Hξ (47)

123


and

ε =((

1

EM− 1

E A

)σ +

√3

2H

)ξ + σ

E A(48)

After introduction of the two expressions, based on thei th particle’s force and displacement Fi and ui , i.e.,σ = Fi/A and ε = ui/L into Eq. (48), the followingrelationship may be found

uiL

=((

1

EM− 1

E A

)FiA

+√3

2H

)ξi + Fi

E A A(49)

which evolves to the form

kAui =((

E A

EM− 1

)Fi +

√3

2HEAA

)ξi + Fi (50)

and, then,

kAui = (αE Fi + F∗

M

)ξi + Fi (51)

αE and F∗M are the auxiliary constant parameters

expressing the relationship between the Young’s mod-uli for the two SMA phases and the resultant phasetransformation force

αE = E A

EM− 1 (52)

F∗M = √

3/2HEAA (53)

Next, Eq. (51) may be rewritten in the form

kAui = F∗Mξi + (αEξi + 1) Fi (54)

and

k∗i ui = F∗

i + Fi (55)

where the two equivalent resultant parameters are pro-posed to be used in a peridynamic model, namely stiff-ness coefficient k∗

i and the modified phase transforma-tion force F∗

i

k∗i = kA

αEξi + 1(56)

F∗i = F∗

M

αEξi + 1ξi (57)

to follow the standard compacted expression for thestatic problem description of the know general formku = F .

Finally, making the reference between Eqs. (55) and(44), an alternative 1-D peridynamic model for SMAis proposed to take the form

mi uti = 2

∑N

j = −Nj �= 0

(βi, jγi

k A

αEξi + 1

uti+ j − uti| j | N 2

)

+ F∗M

αEξi + 1ξi + Fi (58)

where mi , kA, αE , F∗M are the constant parameters and

ξi is the control parameter considered for the i th particleto declare the current contributions of the austenite andmartensite phases in the mentioned particle.

Application of the governing equation (58) allowsfor solution of static, quasi-static and dynamic prob-lems. In all cases, it is required to aggregate the systemof linear equationsmaking use of the parameters k∗

i andF∗i in the global stiffness matrix and the force vector,

respectively. A general flowchart for applications of theproposed peridynamic model is shown in Fig. 6.

The first step of calculations is performed to param-eterize the model using geometric and material prop-erties. Moreover, the initial and boundary conditionsare defined to introduce the data regarding both fixeddisplacement areas and external loads. Finally, simu-lation data referring to total simulation time, time step(temporal discretization), distances between particles(spatial discretization), range of the horizon and themaximum error for the particle displacement are pro-vided. During the second step, initial calculations areconducted to determine auxiliary constants in a model:masses, volumes, stiffness coefficients, resultant phasetransformation force as well as critical elongations forthe established bonds between particles. If required,initial cracks are introduced into the model by break-ing selected bonds.

Iterative part of the procedure deals with the cal-culation of the particles’ displacements and velocitiesvia solving the matrix equation based on the updatedvalues of external loads. For an SMA model, the con-ditions regarding the transformation function Φi ≤ 0andmartensitic volume fraction ξi ∈ [0, 1] are checkedduring each iteration. If required, the displacements arerepeatedly updated to assure that the above-mentionedparameters are kept within the specified limits, whichguarantees proper hysteretic behavior ofmodeled SMA

123


Fig. 6 Flowchart forapplications of theelaborated peridynamicmodel of SMA

sample.Moreover, a search for the bondswith exceededcritical elongations is carried out in case of grow-ing cracks. The final step (postprocessing) shows theobtained results making use of both numerical andgraphical presentations.

To assess the property of the elaborated numericalperidynamic model for SMA (58), the stability con-dition is determined for explicit formulation, applyingvon Neumann stability analysis. An exemplary repre-sentative case of N = 2 is taken into account. The cen-tral scheme for finite difference (FD) method is usedfor time integration

uti = ut−1i − 2uti + ut+1

i

�t2(59)

For the case when i �= 1 ∧ i �= iM AX and excludingthe external force Fi , Eq. (58) becomes

miut−1i − 2uti + ut+1

i

�t2

= 2∑N

j = −Nj �= 0

(βi, j

k A

αEξi + 1

uti+ j − uti| j | N 2

)

+ F∗M

αEξi + 1ξi (60)


i

�t2

= 2

(k∗i

16

(uti−2 − uti

) + k∗i

4

(uti−1 − uti

)

+ k∗i

4

(uti+1 − uti

) + k∗i

16

(uti+2 − uti

)) + F∗i

(61)

ut−1i − 2uti + ut+1

i

= �t2k∗i

mi

(1

8uti−2 + 1

2uti−1 − 5

4uti

+ 1

2uti+1 + 1

8uti+2

)+ �t2

miF∗i (62)

Hence, the equation for the numerical error takes theform

εt−1i − 2εti + εt+1

i = r1

(1

8εti−2 + 1

2εti−1

− 5

4εti + 1

2εti+1 + 1

8εti+2

)+ r2 (63)

where the axillary parameters r1 and r2 equal

r1 = �t2

mik∗i (64)

r2 = �t2

miF∗i (65)

The considered error includes both the temporal andspatial terms

εt+qi+ j = exp (a (t + q) �t) exp (jκε (i + j)�x) (66)

where j = √−1 is the imaginary unit, a and κε areconstants. Consequently, Eq. (63) becomes

123


exp (−a�t) + exp (a�t)

= 2 + r1

(1

4cos (2r) + cos (r) − 5

4

)

+ r2exp (−at�t) exp (−jir) (67)

where r = κε�x . For the most demanding case of ξi =0, when the velocity for the longitudinal wave reachesthe maximum value cA = √

E A/ρ, for austeniticphase, since E A > EM , the condition for numericalstability becomes

∣∣∣∣1 + �t2

2�x2

(cA

)2 (1

4cos (2r) + cos (r) − 5

4

)∣∣∣∣ ≤ 1

(68)

which finally allows to determine the constraint for thetime step

�t ≤ √2�x

cA(69)

Condition (69) should be normally taken for properselection of �t for a numerical model in case ofexplicit integration. It is especially requiredwhen high-frequency excitations are applied and the wavelengthof the generated elastic wave is comparable with thelength of the model. It should be noted that this is,however, not the case for the numerical model studiedin Sect. 7. Due to the low-frequency excitation (timeperiod equals 30 s to assure isothermal phase transi-tion), small dimensions of the model (its length equals4mm) and limited number of DOFs, i.e., 5, a signifi-cantly higher value of the time step �t = 10ms wasarbitrarily selected with respect to the one specified byEq. (69), i.e., 0.38μs. It should be clearly stated thatthe choice made for �t was valid for the consideredspecific nearly quasi-static simulations carried out forSMA.

Similarly, the convergence of Eq. (61) to the ana-lytical case was confirmed. For an exemplary case ofξi = 0, Eq. (61) becomes


i

�t2

= kA(1

8

(uti−2 − uti

) + 1

2

(uti−1 − uti

)

+ 1

2

(uti+1 − uti

) + 1

8

(uti+2 − uti

))(70)

and then


i

�t2

= kA(1

8uti−2 + 1

2uti−1 − 5

4uti + 1

2uti+1 + 1

8uti+2

)

(71)

After introduction of a standard wave solution

ut+qi+ j = exp (j (ω (t + q)�t − κ (i + j) �x))

= exp (j (ωt�t − κi�x)) exp (j (ωq�t − κ j�x))

(72)

Eq. (71) becomes

mi

�t2(exp (j (−ω�t)) − 2 + exp (j (ω�t)))

= kA( 1

8 exp (j (2κ�x)) + 12 exp (j (κ�x)) − 5

4++ 12 exp (j (−κ�x)) + 1

8 exp (j (−2κ�x))

)

(73)

and thenmi

�t2(2cos (ω�t) − 2)

= kA(1

4cos (2κ�x) + cos (κ�x) − 5

4

)(74)

After introduction of the two fundamental terms ofthe Taylor series for the trigonometric components,Eq. (74) takes the form

mi

�t2

(2

(1 − (ω�t)2

2

)− 2

)

= kA(1

4

(1 − (2κ�x)2

2

)+ 1 − (κ�x)2

2− 5

4

)

(75)

mi

�t2(ω�t)2 = E AA

�x

((2κ�x)2

8+ (κ�x)2

2

)(76)

miω2 = E AA

�x(κ�x)2 (77)

ω2 = κ2 EA

ρ(78)

ω2 =(cA

)2κ2 (79)

which, finally, stands for an analytical form of the dis-persion relation for an isotropic, homogeneous mate-rial.

123


In the following Sect. 6, the results of experimen-tal validation and numerical verification confirm theapplicability of the above-presented theory making useof the stress–strain relationship obtained for the elabo-rated peridynamic code for SMA.

6 Experimental validation and verification usingcommercial FE code

As mentioned earlier, both experimental validationand numerical verification were carried out to confirmthe demanded functionalities of the elaborated peri-dynamic model of SMA. In the performed validationexperiments, an SMA wire made of Nitinol was used.The length and the diameter of thewire are 133mm and1mm, respectively. Figure 7 shows the test stand usedin experiments, equippedwith a fatigue testingmachineInstron 8872 with the maximum load of 10kN. Duringtensile tests, the stress–strain paths for the SMA samplewere registered.

The experimental tests were conducted at roomtemperature 22 ◦C. Since, for the used SMA mate-rial, the characteristic temperature A f equals 10 ◦C(based on the manufacturer data), the condition regard-ing the presence of austenite phase before applicationof mechanical load was satisfied, which is required fortesting the superelasticity effect.

The undertaken tests consisted in the following con-secutive phases: (step 1) an initial 5-s long pause, (step

2) gradual stretching of the SMA wire until the abso-lute extension of 11mm is achieved with the stretch-ing rate 0.05mm/s, (step 3) 30-s long pause, (step 4)compression by 11mm with the stretching release rate0.05mm/s.

For verification reason, the respective three-dimensional (3-D) FEmodel of the SMAwire was cre-ated to generate the stress–strain path using a commer-cial software MSC.Software/Marc, which is shown inFig. 8. It consists of 101080 FEs.

The FE modeling approach is employed since it isa common and convenient verification tool. The sameexperimental data were applied to validate both the FEcode and the elaborated peridynamic model of SMAto assure the correct assessment of the convergencebetween these two approaches during numerical ver-ification.

The theory of superelasticity effect, which is imple-mented in the used commercial FE code, was proposedbyAuricchio and is presented in detail in [62,63]. Eventhough different mathematical formalisms are used byLagoudas and Auricchio in their theoretical works forSMA—including constitutive equation and the defini-tion for the characteristic points in the stress–strainplots—the theory applied in the FE code is in linewith the analytical formulations used in the peridy-namic model provided in Sect. 5. As shown below, theresults convergence was achieved for the superelastic-ity effect, satisfying the condition of isothermal phase

Fig. 7 Fatigue testing machine Instron 8872 with mounted SMA wire used in the validation experiments for the FE and peridynamicmodels

123


Fig. 8 FE model of an SMA wire used for verification of the peridynamic model. The cross-sectional view is shown on the right

transition. In both modeling approaches, various elas-tic moduli are separately considered for martensite andaustenite phases, and the martensite percentage vol-ume fraction ξ is introduced as the control property toassure hysteretic character of the stress–strain relation-ships for SMA. Although different parameters are usedto declare the characteristic points in the stress–straincurve, their conversion relationshipsmay be effectivelyfound via either verification or validation process.

Finally, the stress–strain curve for the tested peridy-namic model of an SMA sample was generated usingthe theory described in Sect. 5. The details of the peri-dynamic model used for validation and verification aredescribed in the following. A generic 1-D peridynamicmodel of a 4-mm-long piece of an SMA wire was cre-ated to investigate the phenomenon of superelasticity.More specifically, the model constitutes the structureof a rod, exhibiting axial loads only. One of the modelends is clamped, whereas the external axial force P isattached at the opposite edge. The external excitationmay be static, quasi-static or dynamic, as demanded.The model is shown in Fig. 9.

The tested model is built of 5 particles. Their cross-sectional area A equals 1mm2. The distance betweenparticles and the horizon radius is, respectively, L =1mm and δ = 2mm. An equivalent global stiffness

matrix K is aggregated for the model—making thereference to the components of the derived governingequation (58) to visualize the nonlocal elastic interac-tions in the peridynamic model. The matrixK takes theform

K =

⎛⎜⎜⎜⎜⎜⎜⎜⎝

516k

∗1 − 1

4k∗1 − 1

16k∗1 0 0

− 14k

∗2

78k

∗2 − 1

2k∗2 − 1

8k∗2 0

− 18k

∗3 − 1

2k∗3

54k

∗3 − 1

2k∗3 − 1

8k∗3

0 − 18k

∗4 − 1

2k∗4

78k

∗4 − 1

4k∗4

0 0 − 116k

∗5 − 1

4k∗5

516k

∗5

⎞⎟⎟⎟⎟⎟⎟⎟⎠

(80)

where the resultant stiffness coefficients k∗i = k∗

i (ξi )

may be found according to formula (56). Similarly,the force vector F for the modeled system is asfollows

F =

⎡⎢⎢⎢⎢⎣

−P − F∗2

P + F∗2 − P − F∗

3P + F∗

3 − P − F∗4

P + F∗4 − P − F∗

5P + F∗

5

⎤⎥⎥⎥⎥⎦ (81)

123


Fig. 9 Tested 1-Dperidynamic model of anSMA wire used fornumerical simulations of thesuperelasticity phenomenon

Hence,

F =

⎡⎢⎢⎢⎢⎣

−P − F∗2

F∗2 − F∗

3F∗3 − F∗

4F∗4 − F∗

5P + F∗

5

⎤⎥⎥⎥⎥⎦ (82)

where the resultant modified phase transformationforces F∗

i = F∗i (ξi ) are found according to formula

(57).The global mass matrix M takes a standard diago-

nal form populated with the mass of the consecutiveparticles mi

M = diag (m1, . . . ,m5) (83)

where

mi = ργi AL (84)

The coefficients γi are found using Eq. (46). After fixa-tion of the first particle, the final form for the equivalentmatrix governing equation for the peridynamic modelbecomes

M′u + K′u = F′ (85)

with:

M′ = diag (m2,m3,m4,m5) (86)

K′ =

⎛⎜⎜⎜⎜⎝

78k

∗2 − 1

2k∗2 − 1

8k∗2 0

− 12k

∗3

54k

∗3 − 1

2k∗3 − 1

8k∗3

− 18k

∗4 − 1

2k∗4

78k

∗4 − 1

4k∗4

0 − 116k

∗5 − 1

4k∗5

516k

∗5

⎞⎟⎟⎟⎟⎠ (87)

F′ =

⎡⎢⎢⎣F∗2 − F∗

3F∗3 − F∗

4F∗4 − F∗

5P + F∗

5

⎤⎥⎥⎦ (88)

u = [u2 u3 u4 u5

]T(89)

u = [u2 u3 u4 u5

]T(90)

Figure 10 presents the comparison between the resultsobtained using both experiments and numerical sim-ulations for the elaborated peridynamic model afterits validation and verification. The stress–strain curvefor the peridynamic model, which is shown in Fig. 10,is obtained using Eq. (85), excluding the accelerationcomponent M′u to consider a quasi-static case.

The experimental validation procedure has con-firmed the applicability of both the FE and peridy-namic modeling for SMA components. The resultsfor the numerical simulations were generated usingthe models parameterized based on the experimentaldata. The identified Young’s moduli for the austen-ite and martensite phases are E A = 66GPa andEM = 48GPa. In case of the FE model, the four char-acteristic stresses in the Auricchio model were found:σ ASf = 600MPa, σ SA

f = 630MPa, σ ASs = 420MPa

and σ SAs = 350MPa, as well as the uniaxial transfor-

mation strain εL = 0.075. Moreover, a standard valueof 0.3 was assumed for the Poisson ratios νA and νM .Alternatively, for the peridynamic model—making thereference to the Lagoudas model—the following prop-erties were identified: the characteristic phase transi-tion temperatures: As = 2 ◦C, Ms = −29 ◦C, M f =−39 ◦C, and the factors ρ�s0 = −1.27MPa/K andH = 5.24%. The temperature A f and the mass den-sity ρ are assumed to be known and equal 10 ◦C and6450 kg/m3, respectively. As visible in Fig. 10, theused material properties assure correct shape of thehysteretic stress–strain relationship for both numeri-cal approaches. In addition, the relationship betweenthe axial force and total elongation found in the peri-dynamic model of an SMAwire is presented in Fig. 11.

The identified amount of energy dissipated in themodeled SMA wire—originating from the hystereticcharacter of the superelasticity phenomenon—equals0.103 J per single cycle of the simulated tension test.

123


Fig. 10 Comparisonbetween the experimentaland simulation results(quasi-static numericalanalysis) after validationand verification procedurecarried out for the testedperidynamic model. Twovalidation experiments wereconducted taking intoaccount one and two cyclesfor the hysteresis loop in thestress–strain relationship.The curves identified duringexperiments exhibitirregularities resulting fromspontaneous phasechanges—the issue isdiscussed in Sect. 4

Fig. 11 Relationshipbetween the axial force andtotal elongation in theperidynamic model of anSMA wire used to calculatethe amount of dissipatedenergy per a single cycle ofthe simulated tension test

123


The quantity was calculated as the area covered by thehysteresis loop shown in Fig. 11.

7 Numerical case study, application to GFB

The practical applicability of the proposed peridynamicmodeling is confirmed with numerical simulations car-ried out for the model of a mechanical damper made ofa piece of an SMA wire. The studied case is of greatimportance in the field of structural dynamics. Morespecifically, the structures equipped with SMA mayoffer the functionality of effective structural stiffnesscontrol when installed in mechanical systems. More-over, SMA dampers provide means for energy dissi-pation; hence, they may help to reduce mechanicalvibrations. Based on the above-mentioned propertiesof SMA components, a particular authors’ attention ispaid on the capability of efficient control of the charac-teristics of a nonlinear supporting structure mounted inGFB [64]. Figure 12 presents the scheme and the photo-graph of a typical construction of GFB. The construc-tion is compact. Basically, there are no moving partsin GFB apart from the shaft; what makes this type of

journal bearings relatively reliable and robust [65,66].The nonlinear supporting structure of aGFB consists ofthe two types of metallic foils: top foil, which directlycovers the journal, and bump foil, which constitutesan elastic-damping element. The bump foil is mountedto continuously align the journal, air film and innersurface of GFB. As shown in Fig. 12, there are sev-eral options regarding installation of SMAwires on thebearing foils to modify their properties and, therefore,to help maintaining stable operation of GFB.

Below, the elaborated nonlocal peridynamic mod-eling tool is adapted to make a suggestion regardingSMA material, which may be applied to suppress low-frequency and low-amplitude vibrations of the shaftmounted in GFB, via reduction in its transverse dis-placement drift and low-frequency fluctuations. Theanalyzed construction of GFB is considered to be mod-eled properly as long as its long-period steady-stateoperation is taken into account, under a constant load.This condition results from the limit of the present ver-sion of the elaborated peridynamic model, as alreadymentioned in Sect. 4. When satisfied, it allows to main-tain isothermal character of the simulated phase tran-sitions, which is the case, if the externally induced

Fig. 12 Scheme of the structure and a photograph of a typi-cal radial GFB. The magnified part of GFB shows the proposedlocalizations for mounting the SMAwires on both top and bumpfoils. With a standard operation of GFB, the SMA wires are

stretched and compressed. Hence, they exhibit the phenomenonof superelasticity that allows to control the properties of the GFBstructure

123


Table 1 Material properties proposed for the examined SMAdamper

Parameter Value

EM 60GPa

E A 90GPa

Ms −10 ◦CM f −15 ◦CAs 5 ◦CA f 10 ◦Cρ�s0 −0.1MPa/K

H 3.5%

T0 22 ◦Cρ 6450 kg/m3

strain rate does not exceed the experimentally specifiedapproximate value 0.00038 s−1. Further developmentof the peridynamic model extending its functionality isthe subject of future work.

To simulate the phenomenon of energy dissipation,similarly to the case presented in Sect. 6, a peridynamicmodel of an SMA wire is built using 5 particles. Thefollowing values of the selected geometric propertiesapply: cross-sectional area A = 0.1mm2, the distancebetween particles L = 1mm and the horizon radiusδ = 2mm. Table 1 summarizes the material propertiesof the modeled SMA, which are proposed to param-eterize the peridynamic model. Figure 13 shows boththe stress–strain and force–elongation curves obtainedfor the considered material.

To observe energy dissipation in the modeled struc-ture during isothermal phase transition, the arbitrarycourse of the external low-frequency force P is chosen,as shown in Fig. 14. Explicit time integration procedureis used to solve dynamic problems. The central schemefor FDmethod (59) for temporal derivatives is applied.

Hence, the particle displacements for the time stepindexed by (t + 1) are found based on Eq. (85), usingthe formula

ut+1 = �t2(M′)−1

(F′t − K′tut

)+ 2ut − ut−1 (91)

Since the components F′t and K′t depend upon thefractions ξi , each temporal step is performed as an iter-ative procedure to assure that the conditions referringto the transformation function Φi ≤ 0 and martensiticvolume fraction ξi ∈ [0, 1] are satisfied.

The results of peridynamic calculations are visu-alized in Figs. 14, 15 and 16. The generated curvesconsecutively present: the temporal plots for modelelongation, stress and strain, as well as the hysteresisin the stress–strain and force–elongation coordinates.Figure 15 presents the plot of the strain rate to makethe reference to its maximum allowed value, which is0.00038 s−1, as determined in Sect. 4.

The elaborated peridynamicmodel allowed to inves-tigate the superelasticity phenomenon in case of low-frequency and low-amplitude external force excitationin an SMA wire. Thanks to the performed changeof the mechanical properties for the modeled SMAcomponent—dealing with the resultant stiffness atmost—control of the properties of the GFB structure isconsidered to be feasible.

Although a tiny amount of the dissipated energy0.16μJ was identified in the modeled SMA wire fora single cycle of the hysteresis loop, the mentionedenergy dissipation is expected to dramatically increase,in case whenmany pieces of SMAwires are spread cir-cumferentially in the foils (as their structural parts) tocontrol the behavior of GFB. The described applica-tion of SMA wires mounted in the structure of GFB isconsidered as a complementary passive solution withrespect to the active methods applied to control theoperational properties [67]. The choice made on boththe SMAdiameter and its initial tensionmay lead to thespecific ranges of the reaction forces in GFB, at whichthe desired deformation of the order of micrometersand allowed energy dissipation may be achieved in thesupporting structure.

The functionality of the investigated peridynamicmodel allows for simulation sub-loops within the hys-teresis curve defying the material stress–strain rela-tionship. It should be noted that the received resultsare in line with the recently reported outcomes, e.g.,in [68]. In the referenced work, various shapes of inter-nal sub-loops are found for incomplete processes ofphase transformations. The idealized conditional piece-wise description is used to model the hysteresis effectwhich is observed under various external excitation,making use of numerical simulations.

The proposed peridynamic model of SMA is elab-orated to address geometric discontinuities regardingthe structural parts of the modeled GFB, which is ingeneral an issue for other numerical approaches. More-over, a nonlocal problem formulation used in peridy-namics allows to properly model physical dispersion

123


Fig. 13 Stress–strain andforce–elongationrelationships for the testedSMA material, which isused as a damper to performmechanical energydissipation

Fig. 14 Temporal plots ofthe external force appliedduring simulated tensionand the identified modelelongation

123


Fig. 15 Temporal plots ofstress, strain and strain ratefor the peridynamic modelof SMA. The plot of thestrain rate is bounded by theallowed limits forisothermal phase transition,which is ±0.038% s−1

Fig. 16 Hysteretic responseof the SMA model for theapplied sinusoidal force

of the propagating waves, especially in case of lim-ited number of DOFs in the spatial domain being dis-cretized [14]. Finally, even though the planned elab-oration of the peridynamic approach will allow to go

beyond the limit of isothermal phase transitions, thepreliminary results of the simulations carried out forGFB’s component are valuable due to the rigorousrequirement regarding the temperature gradient inGFB

123


that must be kept within specific limits of the order ofseveral ◦C only [69].

8 Summary and final conclusions

The paper is devoted to the theory and applications forthe proposed peridynamic model of SMA. The prelim-inary study deals with 1-D modeling and isothermalphase transitions, which is of the authors’ concern inthe practical case study on the properties of nonlinearsupporting structure in GFB. Themechanism of energydissipation was studied via dynamic simulations. Themodel of an SMA wire exhibited the phenomenon ofsuperelasticity, as demanded.

SMAundergoes unique phase transitions in the pres-ence of mechanical and thermal loads. Hence, manypractical applications of the phenomena observed inSMA are known, including the critical ones as in caseof stents used in medicine. In contrast, there existstill significant deficiencies regarding understandingthe physics of SMA as well as the theoretical descrip-tions of the observed effects. This inconveniencesmoti-vated the authors to propose a peridynamic model forSMA. The obtained results should be considered as astep toward more realistic modeling for SMA.

As reported in the paper, the two approaches weresuccessfully used to develop the experimentally val-idated models of SMA. Namely, the Lagoudas andAuricchio theoretical contributions were employed toboth build the peridynamicmodel and performverifica-tion procedure. The elaborated peridynamic model ofSMA was validated using the experimental data. Theconvergence regarding the hysteretic character of thestress–strain curves was achieved after parameteriza-tion of the model using the experimentally identifiedmaterial properties. The considered conditions regard-ing theClausius–Planck inequality led to the demandednonlinear material behavior. As shown in the work, thesuperelasticity effectwas properlymodeledmaking useof the SMAmodel. The volumetric contributions of themartensite and austenite phases correctly follow theexternally induced stress in the material.

The authors made an attempt of peridynamics appli-cation to SMA due to very specific properties of thismodeling tool. The inherent nonlocality allows formore arbitrariness with respect to the properties of themodeled body, especially in terms of physical disper-sion. Moreover, an integral-based problem formulation

means that model discontinuities (material, geometricnonlinearities) may be relatively easily handled. Theauthors are aware of the existing deficiencies of theproposed modeling approach; therefore, future devel-opment of the peridynamic SMA model is scheduledto introduce all thermal-based components in the gov-erning equation.

Acknowledgements This study was funded by National Sci-ence Center, Poland (Grant No. OPUS 2017/27/B/ST8/01822“Mechanisms of stability loss in high-speed foil bearings—modeling and experimental validation of thermomechanical cou-plings).

Compliance with ethical standards

Conflict of interest The authors declare that they have no con-flict of interest.

Open Access This article is distributed under the terms of theCreative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestri-cted use, distribution, and reproduction in any medium, pro-vided you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made.

References

1. Kröner, E.: Elasticity theory of materials with long rangecohesive forces. Int. J. Solids Struct. 3, 731–742 (1967)

2. Kunin, I.A.: Inhomogeneous elastic medium with non-localinteractions. J. Appl. Mech. Tech. Phys. 8(3), 60–66 (1967)

3. Eringen, A.C., Edelen, D.G.B.: On nonlocal elasticity. Int.J. Eng. Sci. 10, 233–248 (1972)

4. Martowicz, A.: On nonlocal modeling in continuummechanics. Mech. Control 34(2), 41–46 (2015)

5. Eringen, A.C.: Linear theory of nonlocal elasticity and dis-persion plane waves. Int. J. Eng. Sci. 10, 425–435 (1972)

6. Kunin, I.A.: Elastic Media with Microstructure II: Three-Dimensional Models. Springer, Berlin (1983)

7. Chen, Y., Lee, J.D., Eskandarian, A.: Atomistic viewpoint ofthe applicability of microcontinuum theories. Int. J. SolidsStruct. 41, 2085–2097 (2004)

8. Di Paola,M., Pirrotta, A., Zingales,M.:Mechanically-basedapproach to non-local elasticity: variational principles. Int.J. Solids Struct. 47, 539–548 (2010)

9. Kaliski, S., Rymarz, C., Sobczyk, K., Włodarczyk, E.:Vibrations and Waves, Part B: Waves. Studies in AppliedMechanics 30B. Elsevier, Amsterdam, PWN – Polish Sci-entific Publishers, Warsaw (1992)

10. Bobaru, F., Foster, J.T., Geubelle, P.H., Silling, S.A. (eds.):Handbook of Peridynamic Modeling. Series: Advances inApplied Mathematics. CRC Press Taylor & Francis Group,Boca Raton (2017)

123

http://creativecommons.org/licenses/by/4.0/

http://creativecommons.org/licenses/by/4.0/


11. Ghrist, M.L.: High-order finite difference methods for waveequations. PhD, thesis, University of Colorado (2000)

12. Tam, C.K.W., Webb, J.C.: Dispersion–relation-preservingfinite difference schemes for computational acoustics. J.Comput. Phys. 107(2), 262–281 (2011)

13. Yang, D., Tong, P., Deng, X.: A central difference methodwith low numerical dispersion for solving the scalar waveequation. Geophys. Prospect. 60, 885–905 (2012)

14. Martowicz, A., Ruzzene, M., Staszewski, W.J., Rimoli, J.J.,Uhl, T.: Out-of-plane elastic waves in 2D models of solids:a case study for a nonlocal discretization scheme withreduced numerical dispersion. Math. Problems Eng. Arti-cle ID 584081 (2015)

15. Martowicz, A., Staszewski, W.J, Ruzzene, M., Uhl, T.:Nonlocal numerical methods for solving second-orderpartial differential equations. In: Awrejcewicz, J., et al.(eds.) Mathematical and Numerical Aspects of DynamicalSystem Analysis. 14th Conference Dynamical Systems—Theory and Applications—DSTA 2017, Łódz, Poland, 11–14 December 2017, pp. 357–368 (2017)

16. Martowicz, A., Ruzzene, M., Staszewski, W.J., Rimoli, J.J.,Uhl, T.: A nonlocal finite difference scheme for simulationof wave propagation in 2D models with reduced numericaldispersion. In: Proceedings of SPIE 9064, Health Monitor-ing of Structural and Biological Systems 2014, Article ID90640F (2014). https://doi.org/10.1117/12.2045252

17. Rodrıguez-Ferran, A., Morata, I., Huerta, A.: Efficient andreliable nonlocal damage models. Comput. Methods Appl.Mech. Eng. 193, 3431–3455 (2004)

18. Gunzburger, M., Lehoucq, R.B.: A nonlocal vector calcu-lus with application to nonlocal boundary value problems.Multiscale Model. Simul. 8(5), 1581–1598 (2010)

19. Martowicz, A.: Peridynamics for damage modelling andpropagation via numerical simulations. Diagnostyka 18(1),73–78 (2017)

20. Hu, W., Ha, Y.D., Bobaru, F.: Peridynamic model fordynamic fracture in unidirectional fiber-reinforced compos-ites. Comput.MethodsAppl.Mech. Eng. 217–220, 247–261(2012)

21. Martowicz, A., Staszewski, W.J., Ruzzene, M., Uhl, T.:Vibro-acoustic wave interaction in cracked plate modeledwith peridynamics. In: Onate, E., Oliver, X., Huerta, A.(eds.) Proceedings of the 11thWorld Congress on Computa-tional Mechanics WCCM XI and 5th European Conferenceon Computational Mechanics ECCM V and 6th EuropeanConference on Computational Fluid Dynamics ECFD VI,International Center of Numerical Methods Engineering;Barcelona, Spain, 20–25 July 2014, ID p3987, pp. 4021–4027 (2014)

22. Seleson, P., Parks, M.L., Gunzburger, M., Lehoucq, R.B.:Peridynamics as an upscaling of molecular dynamics. J.Multiscale Model. Simul. 8(1), 204–227 (2009)

23. Bazant, Z.P., Jirasek, M.: Nonlocal integral formulations ofplasticity and damage: survey of progress. J. Eng. Mech.128(11), 1119–1149 (2002)

24. Demmie, P.N., Ostoja-Starzewski, M.: Local and nonlocalmaterial models, spatial randomness, and impact loading.Arch. Appl. Mech. 86, 39–58 (2016)

25. Foster, J.T., Silling, S.A., Chen,W.W.: Viscoplasticity usingperidynamics. Int. J. Numer. Methods Eng. 81(10), 1242–1258 (2009)

26. Chen, Z., Bobaru, F.: Selecting the kernel in a peridynamicformulation: a study for transient heat diffusion. Comput.Phys. Commun. 197, 51–60 (2015)

27. Eringen, A.C.: Theory of nonlocal thermoelasticity. Int. J.Eng. Sci. 12(12), 1063–1077 (1974)

28. Balta, F., Suhubi, E.S.: Theory of nonlocal generalised ther-moelasticity. Int. J. Eng. Sci. 15(9–10), 579–588 (1977)

29. Chang, D.M., Wang, B.L.: Surface thermal shock crackingof a semi-infinite medium: a nonlocal analysis. Acta Mech.226, 4139–4147 (2015)

30. Martowicz, A., Kijanka, P., Staszewski, W.J.: A semi-nonlocal numerical approach for modeling of temperature-dependent crack-wave interaction. In: Proceedings of SPIEVolume 9805: Health Monitoring of Structural and Biolog-ical Systems (2016)

31. Eringen, A.C.: Theory of nonlocal piezoelectricity. J. Math.Phys. 25(3), 717–727 (1984)

32. Zhang, L.L., Liu, J.X., Fang, X.Q., Nie, G.Q.: Effects ofsurface piezoelectricity and nonlocal scale on wave propa-gation in piezoelectric nanoplates. Eur. J. Mech. A/Solids46, 22–29 (2014)

33. Ostoja-Starzewski, M.: From random fields to classical orgeneralized continuum models. Procedia IUTAM 6, 31–34(2013)

34. Arash, B., Wang, Q., Liew, K.M.: Wave propagation ingraphene sheets with nonlocal elastic theory via finite ele-ment formulation.Comput.MethodsAppl.Mech. Eng. 223–224, 1–9 (2012)

35. Martowicz, A., Staszewski, W.J., Ruzzene, M., Uhl, T.:Peridynamics as an analysis tool for wave propagation ingraphene nanoribbons. In: Lynch, J.P. (ed.) Proceedings ofSPIE, Volume 9435: Sensors and Smart Structures Tech-nologies for Civil,Mechanical, andAerospace Systems, SanDiego, USA, 2015, Article ID 94350I. https://doi.org/10.1117/12.2084312 (2015)

36. Duval, A., Haboussi, M., Zineb, T.B.: Modeling of SMAsuperelastic behavior with nonlocal approach. Phys. Proce-dia 10, 33–38 (2010)

37. Badnava, H., Kadkhodaei,M.,Mashayekhi,M.: A non-localimplicit gradient-enhanced model for unstable behaviors ofpseudoelastic shape memory alloys in tensile loading. Int. J.Solids Struct. 51(23–24), 4015–4025 (2014)

38. Martowicz, A., Staszewski,W.J, Ruzzene,M., Uhl, T.: Non-local elasticity theory for solving dynamic problems viaperidynamics. In: Awrejcewicz J., et al. (eds.) Mathemat-ical and Numerical Aspects of Dynamical System Anal-ysis. 14th Conference Dynamical Systems—Theory andApplications—DSTA 2017, Łódz, Poland, 11–14 Decem-ber 2017, pp. 345–356 (2017)

39. Han, F., Lubineau, G., Azdoud, Y.: Adaptive couplingbetween damage mechanics and peridynamics: a route forobjective simulation of material degradation up to completefailure. J. Mech. Phys. Solids 94, 453–472 (2016)

40. Silling, S.A.: Reformulation of elasticity theory for discon-tinuities and long-range forces. J. Mech. Phys. Solids 48,175–209 (2000)

41. Bobaru, F., Yang, M., Alves, L.F., Silling, S.A., Askari, E.,Xu, J.: Convergence, adaptive refinement, and scaling in1D peridynamics. Int. J. Numer. Methods Eng. 77, 852–877(2009)

123

https://doi.org/10.1117/12.2045252

https://doi.org/10.1117/12.2084312

https://doi.org/10.1117/12.2084312


42. Madenci, E.,Oterkus, E.: PeridynamicTheory and ItsAppli-cations. Springer, New York (2014)

43. Martowicz,A., Ruzzene,M., Staszewski,W.J., Uhl, T.:Non-local modeling and simulation of wave propagation andcrack growth. In: AIP Conference Proceedings, vol. 1581,pp. 513–520. AIP Publishing (2014)

44. Nishawala, V.V., Ostoja-Starzewski,M., Leamy,M.J., Dem-mie, P.N.: Simulation of elastic wave propagation usingcellular automata and peridynamics, and comparison withexperiments. Wave Motion 60, 73–83 (2016)

45. Auricchio, F., Bonetti, E., Scalet, G., Ubertini, F.: Theo-retical and numerical modeling of shape memory alloysaccounting for multiple phase transformations and marten-site reorientation. Int. J. Plast. 59, 30–54 (2014)

46. Godard, O., Lagoudas, M., Lagoudas, D.: Design of spacesystems using shape memory alloys. In: Proceedings ofSPIE, Smart Structures and Materials, San Diego, CA, vol.5056, pp. 545–558 (2003)

47. Cisse, C., Zaki, W., Zineb, T.B.: A review of modeling tech-niques for advanced effects in shapememory alloy behavior.Smart Mater. Struct. 25(10), 103001 (2016)

48. Bryła, J., Martowicz, A.: Experimental and numericalassessment of the characteristics describing superelastic-ity in shape memory alloys – influence of boundary condi-tions. ITMWeb ofConferences, 2017 15:No. 06007; CMESII—International Conference of Computational Methodsin Engineering Science, Lublin, Poland, 23–25 November2017 (2017)

49. Lagoudas, D.: Shape Memory Alloys: Modeling and Engi-neering Applications. Springer, Berlin (2008)

50. Auricchio, F.,Boatti, E.,ContiM.: SMAbiomedical applica-tions. In: Lecce, L., Concilio, A. (eds.) ShapeMemoryAlloyEngineering, For Aerospace, Structural and BiomedicalApplications. Butterworth Heinemann/Elsevier, pp. 307–341 (2015)

51. Martowicz, A., Bryła, J., Uhl, T.: Uncertainty quantifica-tion for the properties of a structure made of SMA utilis-ing numerical model. In: Proceedings of the Conference onNoise and Vibration Engineering ISMA 2016 & 5th editionof the International Conference on Uncertainly in StructuralDynamics USD 2016, Katholieke University Leuven; Leu-ven, Belgium, 19–21 September 2016, ID 731 (2016)

52. Auricchio, F., Fugazza, D., DesRoches, R.: Numerical andexperimental evaluation of the damping properties of shape-memory alloys. J. Eng. Mater. Technol. 128(3), 312–319(2006)

53. Lagoudas, D.C., Bo, Z.C., Qidwai, M.A.: A unified ther-modynamic constitutive model for SMA and finite elementanalysis of active metal matrix composites. Mech. Compos.Mater. Struct. 3(2), 153–179 (1996)

54. Lagoudas, D.C., Muhammad, Z.B., Qidwai, A., EntchevP.B.: SMA UM: user material subroutine for thermome-chanical constitutivemodel of shapememory alloys. Report,CiteSeerX (2003)

55. Tanaka, K.O., Nagaki, S.O.: A thermomechanical descrip-tion of materials with internal variables in the process ofphase transitions. Ingenieur Archiv 51, 287–299 (1982)

56. Liang, C., Rogers, C.A.: One-dimensional thermomechan-ical constitutive relations for shape memory materials. J.Intell. Mater. Syst. Struct. 1(2), 207–234 (1990)

57. Rusinek, R.,Warminski, J.,Weremczuk, A., Szymanski,M.:Analytical solutions of a nonlinear two degrees of freedommodel of a human middle ear with SMA prosthesis. Int. J.Non-Linear Mech. 98, 163–172 (2018)

58. Rusinek, R., Warminski, J., Szymanski, M., Kecik, K.,Kozik, K.: Dynamics of themiddle ear ossicleswith an SMAprosthesis. Int. J. Mech. Sci. 127, 163–175 (2017)

59. Savi, M.A., Pacheco, P.M.C.L.: Chaos and hyperchaos inshape memory systems. Int. J. Bifurc. Chaos 12(3), 645–657 (2002)

60. Bryła, J., Palenica, P., Martowicz, A.: Modeling aspectsand simulations for properties of shape memory alloys.In: Manka, M. (ed.) Projektowanie mechatroniczne,zagadnienia wybrane, pp. 137–144. Akademia Górniczo-Hutnicza, Katedra Robotyki iMechatroniki, Kraków (2017)

61. Bryła, J.,Martowicz, A.: Shapememorymaterials as controlelements used in a dot Braille actuator.Mech. Control 33(4),83–89 (2014)

62. Auricchio, F., Taylor, R.L.: Shape-memory alloy: modelingand numerical simulations of the finite-strain superelasticbehavior. Comput. Methods Appl. Mech. Eng. 143, 175–194 (1997)

63. Auricchio, F.: A robust integration-algorithm for a finite-strain shape-memory-alloy superelastic model. Int. J. Plast.17, 971–990 (2001)

64. Lubieniecki, M., Roemer, J., Martowicz, A., Woj-ciechowski, K., Uhl, T.: A muli-point measurement methodfor thermal characterization of foil bearings using cus-tomized thermocouples. J. Electron. Mater. 45(3), 1473–1477 (2016)

65. Howard, S.,Dellacorte, C.,Valco,M., Prahl, J.,Heshmat,H.:Steady-state stiffness of foil air journal bearings at elevatedtemperatures. Tribol. Trans. 44(3), 489–493 (2001)

66. Nalepa, K., Pietkiewicz, K., Zywica, P.: Development of thefoil bearing technology. Tech. Sci. 12, 230–240 (2009)

67. Martowicz, A., Lubieniecki, M., Roemer, J., Uhl, T.: Mixedoperationmodeof thermoelectricmodules for thermal gradi-ent reduction in gas foil bearings. In: Proceedings of ICAST2015—26th International Conference on Adaptive Struc-tures Technologies, Kobe, Japan, 14–16 October (2015)

68. Mitura, A., Rusinek, R.: Effect of sub-loops in SMA ear sys-tem. In: AIP Conference Proceedings, vol. 1922, p. 120015.AIP Publishing (2018)

69. Roemer, J., Lubieniecki, M., Martowicz, A. Uhl, T.: Multi-point control method for reduction of thermal gradients infoil bearings based on the application of smart materials.In: Proceedings of the SMART 2015 ECCOMAS ThematicConference on Smart Structures and Materials, Ponta Del-gada, Azores, 3–6 June (2015)

Publisher’s Note Springer Nature remains neutral with regardto jurisdictional claims in published maps and institutional affil-iations.

123

Nonlocal elasticity in shape memory alloys modeled using ... · Nonlinear Dyn (2019) 97:1911–1935...

Documents

Transcript of Nonlocal elasticity in shape memory alloys modeled using ... · Nonlinear Dyn (2019) 97:1911–1935...