Finite element simulation of a self-healing shape memory...

Mechanics of Materials 38 (2006) 525–537

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Finite element simulation of a self-healing shape memoryalloy composite

D.S. Burton a, X. Gao a,b,c, L.C. Brinson a,b,*

a Department of Mechanical Engineering, Robert R. McCormick School of Engineering and Applied Science, Northwestern University,

2145 Sheridan Road, Evanston, IL 60208-3111, United Statesb Department of Materials Science and Engineering, Northwestern University, 2145 Sheridan Road, Evanston,

IL 60208-3111, United Statesc Currently at Vehicle Development Research Lab, General Motors Research and Development, MC 480-106-256,

30500 Mound Road, Warren, MI 48090-9055, United States

Received 1 March 2004; received in revised form 1 August 2004

Abstract

A self-healing, metal matrix composite reinforced by shape memory alloy wires is simulated using finite element anal-ysis. A one-dimensional constitutive model for SMA behavior is implemented as a user-defined truss element in ABAQUS.The matrix is brittle and a mode I crack is allowed to propagate through the specimen upon loading. During the loadingprocess the wires undergo a martensitic phase transformation, bridging the crack. To heal the composite, simple heating isrequired which reverse transforms the wires and brings the crack faces back into contact. When using pre-strained SMAwires for reinforcement, the reverse transformation of the wires during heating causes a closure force across the crack. Themethod to simulate cracking and ‘‘healing’’ behaviors of the composite allow assessment of the effects of component prop-erties and composite geometry. The results shed light on design of self-healing composites using shape memory alloys.� 2005 Elsevier Ltd. All rights reserved.

Keywords: Shape memory alloy; SMA; Composite; Finite element; Self-healing

1. Introduction

Shape memory alloys (SMAs) have materialproperties which vary with applied stress, tempera-ture, and load history. As such, they are uniquely

0167-6636/$ - see front matter � 2005 Elsevier Ltd. All rights reserved

doi:10.1016/j.mechmat.2005.05.021

* Corresponding author. Address: Department of MechanicalEngineering, Robert R. McCormick School of Engineering andApplied Science, Northwestern University, 2145 Sheridan Road,Evanston, IL 60208-3111, United States. Tel.: +1 847 467 2347;fax: +1 847 510 0540.

E-mail address: [email protected] (L.C. Brinson).

suited for use in composites. For instance, pseudo-elasticity allows SMAs to have extremely large,apparently elastic, strains, thus enabling SMA rein-forcements to prevent fracture of a composite.Other applications exploit the shape memory effectin which SMA components return to a ‘‘remem-bered’’ size or shape when heated, leading tooptions for shape control of the composite.

The shape memory behavior is caused by athermoelastic crystalline phase transformationbetween a high symmetry parent phase (austenite)and a low symmetry product phase (martensite)

.

mailto:[email protected]

Fig. 1. Schematic of pseudoelasticity and shape memory effectbehavior, illustrating microstructural mechanisms and macro-scopic stress–strain response.

526 D.S. Burton et al. / Mechanics of Materials 38 (2006) 525–537

(Funakubo, 1987; Wayman and Duerig, 1990). Thistransformation is reversible and is a function ofboth stress and temperature as indicated in Fig. 1.For a typical NiTi shape memory alloy, the parentphase austenite exists as a cubic crystal lattice struc-ture (Fig. 1a), and upon cooling with no appliedload, it transforms to martensite. This phase has alower symmetry and forms in a self-accommodatedstructure in which different variants of martensiteform together in order to form zero distortion inter-faces (habit planes) with the parent phase. Themartensitic variants are crystallographically twin-related structures, i.e. of identical crystallographicstructure, but with differing spatial orientation.When formed by cooling, all habit plane variants(typically 24 for a given system) appear equallyand there is no macroscopic strain (Fig. 1b). Uponheating, the martensite undergoes a reverse trans-formation to austenite (Fig. 1a).

If the martensitic structure is mechanicallyloaded at constant temperature, the crystallographictwins will reorient to the particular variant favoredby the orientation of the load: this process is calledreorientation and results in a macroscopic strain(Fig. 1c). Further loading in different directionsresults in additional reorientation and changingmacroscopic strain (Fig. 1d). Upon unloading, ifthe temperature is still low, the macroscopic strainremains. However the strain can be recovered uponheating back to the austenite phase, and this load–unload-heat-recovery cycle (path given by Fig. 1a–b–c–a) is often called the shape memory effect(SME).

If the material is loaded in the austenite state, thematerial can undergo a stress-induced phase trans-

formation to martensite, for which only thosevariants favored by the loading direction form.Consequently, a macroscopic strain is achievedupon loading (direct path given by Fig. 1a–c). Thisstrain is recovered directly upon unloading, as themartensite is not stable at the elevated temperaturewithout the presence of the load; this load–unloadcycle (path given by Fig. 1a–c–a) is often called‘‘pseudoelasticity’’ or ‘‘superelasticity’’.

These complex material behaviors necessitatecomplex macroscopic constitutive laws to capturematerial response to thermomechanical loading.Most of the early models were one-dimensional,including models by Tanaka et al. (1986), Pence(1994), Boyd and Lagoudas (1994), Brinson (1993)and Patoor et al. (1988, 1994). In one-dimensionalmodels it is not necessary to consider all possiblemartensite variants explicitly and consequentlythese models have simpler analytic forms thanthree-dimensional models.

Actual physical systems are naturally three-dimensional although certain structures like wiresor plates can be idealized as one- or two-dimen-sional structures. However current three-dimen-sional models are either not robust enough toadequately model all aspects of complicated shapememory behavior present in true three-dimensionalsituations (Auricchio et al., 1997; Qidwai andLagoudas, 2000; Juhasz et al., 2002) or are too com-putationally intensive to be used for modeling entirestructures (Gao et al., 2000; Patoor et al., 2000). Themodeling approach used in this paper combines aone-dimensional SMA model with a more generalmodel for the rest of the structure to enable the sim-ulation of a composite which is not one-dimensionaland which displays full thermo-mechanical SMAbehavior.

One-dimensional constitutive models for SMAsare particularly relevant because there are manyapplications using SMA wires or ribbons in whichthe SMA material behavior is primarily one-dimen-sional. In many of these applications, the SMAcomponents act mainly by a change in their lengthwith no significant non-uniaxial forces, so a one-dimensional model captures their essential actions.A number of different types of SMA compositesuse SMA wires or ribbons embedded in a matrix.Matrix materials that have been used include poly-mers (Sittner et al., 2002; Sun et al., 2002b), metals(Armstrong, 1996; Song et al., 1999; Yue and Wan,

D.S. Burton et al. / Mechanics of Materials 38 (2006) 525–537 527

2003), plaster (Watanabe et al., 2002), and compos-ites such as fiber–glass-epoxy (Gupta et al., 2003;Hamada et al., 2003) and carbon-epoxy (Leeet al., 1999). The SMA components can be usedactively or passively. Active applications use theSMA behavior to control the shape of the compo-site structure (Brinson et al., 1997; Oh et al., 2001;Lind and Doumanidis, 2003), or its vibrational(Brinson and Lammering, 1993; Aoki and Shimam-oto, 2003; Gupta et al., 2003) or buckling response(Birman, 1997; Lee and Lee, 2000). Passive applica-tions use the SMA components to change the com-posite�s stiffness (Boyd and Lagoudas, 1994; Guptaet al., 2003), to strengthen the composite (Songet al., 1999), and to close or repair cracks (Shimam-oto et al., 2001; Araki et al., 2002; Wang, 2002;Hamada et al., 2003).

The response of SMA composites has been simu-lated using a variety of methods, including micro-mechanical methods (Kawai et al., 1999; Arakiet al., 2002) and finite element analysis. In the finiteelement approach, one method is to develop specialcomposite elements. This has been done for multi-layered composite plates (Lagoudas et al., 1997;Ostachowicz et al., 1998; Zak et al., 2003a) andfor layered beams (Marfia et al., 2003). A more gen-eral way of using finite element analysis for compos-ites models the matrix and reinforcing membersseparately. Several researchers have pursued thisstrategy for a composite beam actuated by SMAwires or ribbons bonded to the matrix (Ghomsheiet al., 2001; Sun et al., 2002a). One recent study per-formed a finite element analysis of the buckling of alaminated composite shell with one-dimensional ele-ments for wires, however the SMA model did notconsider twinned martensite (Lee and Lee, 2000).

This paper models a self-healing composite inwhich SMA wires reinforce a more brittle matrix.During loading, as the matrix deforms and a crackpropagates, the SMA the wires span the crack andaccommodate the deformation with elongationdue to either phase transformation from austeniteto detwinned martensite or reorientation of twinnedmartensite. The self-healing characteristic is due tothe shape memory effect in which the SMA wiresreturn to a shorter form (as austenite) when heatedwhich enables the SMA wires to pull cracks in thematrix back together. This type of composite hasbeen realized experimentally by Olson�s group inMaterials Science and Engineering at NorthwesternUniversity (Files, 1997; Forbell et al., 1997; Berni-kowicz, 1998; Kang et al., 2002). In this paper, we

present a modeling strategy that can fully describethis type of self-healing composite and thus providesa tool to optimize material parameters and compos-ite design.

In contrast to the special composite user ele-ments, which would have to be reformulated for dif-ferent composite characteristics, in this paper theSMA wires and elastic–plastic matrix material aremodeled separately. Thus the method can be easilyadapted for different composites: the material prop-erties of the matrix and the wires and the locationsand positions of wires can all be changed indepen-dently without reformulation of the constitutivemodel for the whole composite. Other studies,which used separate elements for the SMA wires,implemented a simpler form of the SMA constitu-tive law which does not permit SMA in the twinnedmartensite state (Lee et al., 1999; Lee and Lee,2000). The work here implements a SMA constitu-tive law with robust kinetics to allow full implemen-tation of the range of SMA behaviors shown inFig. 1.

Some experimental studies have examined thephenomenon of matrix crack formation and closureby SMA wires (Shimamoto et al., 2001; Hamadaet al., 2003), but few studies have attempted to sim-ulate the behavior using numerical methods (Arakiet al., 2002). The work by Araki et al. is performedusing a homogenization micromechanical approachusing eigenstrains for the transformation strainsin the SMA inclusions. The work considers crackbridging, but not healing, and thus differs signifi-cantly from the work here. To the authors� knowl-edge, this work represents the first study ofself-healing SMA composites. In addition, thenumerical implementation of the SMA constitutivelaw exceeds previous work in ability to address thefull space of SMA transformation/reorientationbehavior and has general application to adaptivestiffening, shape control and other SMA compositebehavior.

In the following sections of the paper, we presentfirst the constitutive model used for the SMA ele-ments, then the finite element modeling strategyand results, including specific examples of crackpropagation, and healing.

2. SMA constitutive model

The SMA constitutive model used is based on amodel originally formulated by Tanaka et al.(1986), and modified by Liang and Rogers (1990)


and subsequently by Brinson (1993), Brinson andHuang (1996), Bekker and Brinson (1998). Theone-dimensional model is based on phenomenolo-gical macro-scale constitutive behavior and can bewritten as

r ¼ Eðe� eLnSÞ þHDT ð1Þwhere r is stress and e is strain; nS is the stress-in-duced martensite fraction, E is the elastic modulus,eL is the maximum transformation strain, and H isrelated to the thermal coefficient of expansion forthe material.

Coupled to the constitutive law, a kinetic lawmust be used to determine the martensite volumefraction based on the stress and temperature loadinghistory. The transformation criteria can be repre-sented on an experimentally determined phasediagram shown in Fig. 2. The kinetic law used inthe simulations for this paper includes both twinnedand detwinned martensite variants (Brinson, 1993),as is essential to properly handle the issue ofmartensitic variant reorientation. The kinetics arefurther enhanced to consider the full thermome-chanical history implemented through ‘‘switchingpoints’’ for changes in load path direction (Bekkerand Brinson, 1998). This allows the model toappropriately model hysteresis effects in the phase

Fig. 2. Typical phase diagram for a shape memory alloy. [M],[A], [d] and [t] are regions of transformation between martensiteand austenite. [M] and [d] are associated with formation ofdetwinned martensite (with accompanying macroscopic strain),while [t] is associated with twinned martensite (no accompanyingmacroscopic strain). A thermomechanical loading path, C, isshown with several switching points indicated by solid circles.Loading path tangent, s, and transformation strip distances, q,are indicated. At high r and T, SMAs have a zone of irreversibleplastic deformation which is not shown in this figure. Thecomposite is loaded so that SMA wires do not enter this region asit is not useful for self-healing.

transformation and to capture cyclic or repetitiveloading conditions, under which many kinetic lawsfail. The reader is referred to the previous workfor a full explanation and examples of the algo-rithms. Only a brief synopsis is presented here fortransformation between the [M] and [A] regions.

Whenever the thermomechanical state of theSMA material enters the shaded areas on the dia-gram of Fig. 2, nS must be updated. The thermome-chanical path can be complex and the kineticalgorithm described here to update nS is able to con-sistently reproduce the path dependence of martens-itic transformation. Note in Fig. 2 that the normalvectors nA and nM represent the directions of trans-formation change in the [A] and [M] strips respec-tively. At any point on the path, s is the vectortangent to C. A transformation occurs wheneverone of the following conditions is satisfied:

ðaÞ s � nA > 0 in ½A�ðbÞ s � nM > 0 in ½M�

ð2Þ

A given loading path C can be subdivided into sev-eral segments (Cn, n = 1,2, . . .) by introducing theswitching points, defined as the points where thedirection or sense of the transformation changes.The switching points are points where C enters orleaves the transformation strip in the direction oftransformation (point A) or points inside the stripwhere the dot product between s and nA or nMchanges sign (points B, C, D). Along the portionsof C between two switching points, nS is eithermonotonically increasing or decreasing (when in atransformation strip and moving in a transforma-tion direction, e.g. between A and B or C and D)or constant (when outside a transformation stripor when moving opposite to the transformationdirection, e.g. between B and C, or along any por-tions of C outside of the segment A–D).

Bekker and Brinson (1998) present three ways toformulate the local kinetic law, one of which is uti-lized here. We first define Yi = Yi(T,r), (i = A,M)as the normalized distance between a given pointon C inside a transformation strip and the startboundary. Yi is given by the following expression

Y i ¼ qi

qi0

ð3Þ

where qi is the distance of point (T,r) from the startboundary, and qi

0 is the width of the strip. Fig. 2illustrates these quantities for the case of the [M]strip.

Fig. 3. Schematic of self-healing composite showing SMAreinforcing wires, propagating crack and healed composite afterheating.


On a given portion Cn between two switchingpoints, nS can then be updated using the appropriateone of the following expressions:

nS ¼ nSjfAðY A � Y A

j Þ ð4Þ

when the point is inside the [A] strip, and

nS ¼ nSj þ ð1� nSjÞfMðYM � YMj Þ ð5Þ

when the point is inside the [M] strip.In the forgoing expressions, quantities marked

with subscript j are the values of the martensite frac-tion and the normalized distance at the previousswitching point relevant to the current transforma-tion, nS is the stress-induced martensite fraction,and f j are interpolation functions. Here cosine inter-polation functions are used, equivalent to thoseoriginally developed by Liang and Rogers (1990):

f AðY A � Y Aj Þ ¼ 1� 1

2f1� cos½pðY A � Y A

j Þ�g ð6Þ

fMðYM � YMj Þ ¼

1

2f1� cos½pðYM � YM

j Þ�g ð7Þ

The kinetic law and the constitutive law presentedhere have been developed and implemented success-fully for one-dimensional SMA response. The Brin-son model has been tested and found to be anexcellent SMA model for modeling SMA wire rein-forced composite structures (Zak et al., 2003b).

3. SMA composite: finite element model

The composite that is modeled consists of a brit-tle metal matrix reinforced by parallel SMA wires.This composite formulation is motivated by a self-healing composite demonstrated by the Olson group(Files, 1997; Forbell et al., 1997). In this prototypecomposite, bonding between SMA wires and matrixwas quite poor such that the wires had to be knot-ted/constrained at the outside edges to keep themfrom slipping out (Files, 1997). Other compositeslack bonding between the wires and matrix due tothe nature of the wire embedding. For example,researchers have fabricated a composite in whichthe wires run in sleeves embedded in the matrix(Sun et al., 2002b). It is noted that bonding betweenSMA wires and composite matrix material has beenproblematic in many cases (Lau et al., 2002). Apaper by Jonnalagadda et al. (1997), examined dif-ferent wire surface treatment strategies to attemptto achieve better bonding in a composite, notingthat debonding of the SMA wire and matrix is com-mon. With this motivation, in the composite model

considered in this paper the wires are not bonded tothe matrix along their length, and are fixed only atthe outside edges of the matrix. The compositemodel is rectangular and clamped on the ends asshown in Fig. 3. It is loaded in the direction of thewires and allowed to crack in a line perpendicularto the wires.

The matrix material properties are similar tothose of the material used in a prototype composite.The material is elastic–plastic with hardening andtemperature-dependent softening. It is brittle atroom temperature, but ductile at high temperature,as indicated in Fig. 4. At low temperature, thematrix fractures at 60 MPa, shown by an ‘‘x’’ onthe plot. The matrix softening at high temperatureenables the crack to fully close when the wires pullthe crack back together. In the prototype compositerealized by the Olson group, the matrix softenedsufficiently upon heating to allow rewelding of crackfaces in contact. The rewelding is not modeled in thecurrent paper. The Poisson ratio for the matrix istaken as 0.3. The thermal expansion effects are smallin comparison to the shape memory effect and thusare removed from the simulation by setting a forboth the matrix and the wires to 0.

The wires have standard NiTi properties takenfrom the literature (Dye, 1990; Liang, 1990; Brinsonand Lammering, 1993) as shown in Table 1. Theoverall model has a cross-section of 30 mm2 and awidth of 6 mm. The model has five evenly spacedreinforcement wires, so that the wires are spaced

Fig. 4. Yield stress–strain relation for the matrix material atroom temperature and elevated temperature. The critical fracturestress is indicated by an ‘‘x’’ on the low temperature plot. Theinset shows the variation of the Young�s modulus changinglinearly between 30 �C and 80 �C. The yield stress also varieslinearly with temperature in the same range. See text forremaining matrix properties.


1 mm apart. In preliminary studies, it was found thatincreasing the density of the wires had the expectedeffect of proportionally increasing the closure forceupon heating. The wires have cross-sections of0.05 mm2, representative of wires with a 0.252 mmdiameter. The initial temperature for the simulationsis midway between the martensite start and austenitestart temperatures, and the wires are initially 100%austenite for the simulations shown.

ABAQUS, a commercial general-purpose finiteelement analysis code, is used to run the simula-tions. The matrix is modeled with linear plane strainelements and the SMA wires are modeled with user-defined 2-node truss elements. The analyses are allquasi-static. For models in which matrix crackingis allowed to occur, there is a small notch in the mid-dle of one edge to initiate the crack. At this notch,the contact surfaces are initially unbonded for thelength of one element edge (0.1 mm for a 60 · 60mesh). In ABAQUS, crack propagation is imple-mented as a contact option where surfaces that areinitially bonded are allowed to debond when a spec-ified fracture criterion is satisfied. In the modelsanalyzed here, a line of potential debonding is

Table 1Material properties for the Nitinol alloy used in the simulations (Dye,

Moduli, density Transformation temperatures Phase d

EA = 67 · 103 MPa Mf = 9 �C CM = 8EM = 26.3 · 103 MPa MS = 18.4 �C CA = 13H = 0 MPa �C�1 AS = 34.5 �C rcrS ¼ 10q = 6448.1 kg m�3 Af = 49.0 �C rcrf ¼ 17

defined to run perpendicular to the wires down themiddle of the specimen due to the symmetry of theproblem. The crack-tip node is allowed to debondwhen the local stress at the crack tip reaches the crit-ical value of 60 MPa. This type of critical stresscriterion is typically used for crack propagation inbrittle materials (ABAQUS/Standard User�s Man-ual, Version 6.3, 2002).

Since the prototype composites were uniformlyheated in an oven for healing, the simulations usea uniform temperature field during the healing pro-cess. Heat transfer in the matrix is not considered.The loading of the composite and the subsequentheating are performed quasi-statically. Thus, latentheat of transformation of the SMA wires is alsonot considered. Given the quasi-static loading con-ditions, the uniform oven heating and the thermallyconductive matrix material, latent heat effects wouldbe very small. It should be noted that if the compos-ite were to be actuated by resistive heating of theSMA wires, heat transfer effects would be criticaland would be essential in simulations. Such an actu-ation process would thus place significant con-straints on geometric spacing of the SMA wires toensure complete crack closure due to the non-uni-form temperature field.

A user element (UEL) was written to calculatethe behavior of SMA wires. Details of the UEL for-mulation can be found elsewhere (Gao et al., inpreparation). The constitutive law described in theprevious section is used and implemented in thisUEL to provide accurate SMA response for condi-tions of martensite reorientation, inverse transfor-mation, and including proper response under smallcompressive loads that might be realized in anapplication. The SMA elements are 2-node trusselements which can be positioned in three-dimen-sional space and here simply run in straight linesdown the composite as shown in Fig. 3.

4. Results

In this section, we demonstrate the response ofthe self-healing SMA composite in two primary

1990; Liang, 1990; Liang and Rogers, 1990)

iagram parameters Max residual strain, Poisson�s ratio

MPa �C�1 eL = 0.067.8 MPa �C�1 m = 0.30 MPa0 MPa


configurations. In the first case, no matrix crackingis allowed providing a simple base test case, whichcan be analytically verified. In the second case, thematrix is notched at the edge and a crack is allowedto propagate; self-healing is demonstrated by subse-quent heating. In the examples shown, the SMAstarts as austenitic, but similar results are obtainedwhen starting from twinned martensite.

4.1. Behavior with no cracking

A preliminary case of a specimen without crack-ing was analyzed. In this basic case the model doesnot have any debonding capabilities, so that highloading levels cause transformation from the initialaustenite to stress-induced (detwinned) martensitein the SMA wires and permanent plastic deforma-tion of the matrix. Starting from a stress-free stateof the matrix and the wires, a fixed displacementis applied to one end of the model to load the modelto a strain level of 4.1%.

The loading case is shown schematically in Fig. 5,with (a) being the unloaded configuration and (b)the model at full displacement. Upon loading, thestress in the wires causes transformation ofthe SMA to detwinned martensite and the stress inthe matrix causes plastic deformation of that com-ponent. When the fixed displacement is released,there is elastic recovery of both the matrix and thewires, but the plastic deformation of the matrixremains, and since the initial temperature is belowthe austenite start temperature, the detwinned mar-tensite of the wires is stable and does not reverse

Fig. 5. (a) Initial configuration, (b) after loading causes plasticityin matrix and transformation in wires and (c) after heating causeswires to shrink.

transform back to austenite. The plastic deforma-tion of the matrix is greater than the increase inlength of the wires due to detwinning, so afterthe load is released, the wires are in tension andthe matrix is in compression. The model is then uni-formly heated, and when the temperature isincreased into the austenite transformation zone,the reverse transformation of the wires causes thewires to shrink and apply a compressive force tothe matrix (Fig. 5c). The force of the wires causesa slight shortening of the specimen compared toits length after the initial load was removed.

Although the matrix softens during the heating,the matrix nevertheless resists the compressionresulting in tensile stresses in the wires. Consideringthe path of wires in the phase diagram as shown inFig. 6a, these tensile stresses result in a higher tem-perature being required for reverse transformation.Given the large plastic strain achieved by the matrixmaterial and the number of SMA wires employed,the SMA wires are unable to fully transformback to austenite and cannot reverse the achievedmatrix deformation within their operating range.

Fig. 6. (a) Loading path for uncracked composite overlaid on thephase diagram for the SMA wires. Composite is first loaded andunloaded at low temperature to achieve detwinned martensite inthe wires and plastic deformation of the matrix. Heating theninduces reverse transformation to austenite and increasing stressas the matrix resists the associated deformation. Cooling back toroom temperature results in reformation of martensite. See textfor details. (b) Change in stress-induced martensite volumefraction in response to stress and temperature loading.


Consequently, the wires remain partially elongatedwith some detwinned martensite. When the compos-ite is allowed to cool back to room temperature, thewires pass back through the martensitic transforma-tion strip and thus reform detwinned martensite dueto the tensile stress state as illustrated in Fig. 6b. Asthe wires elongate with this transformation, thestress drops as the temperature decreases, ultimatelylosing the recovered strain from heating, but retain-ing a small compressive clamping force on thematrix material. The loading path in Fig. 6a showsthese transformation and stress effects.

These results are further illustrated in Fig. 7where the response of the wires, matrix and the tem-perature are plotted versus time step. The changes inthe wire stresses here can be clearly correlated withthe associated temperature and martensite volumefraction. Notice that the martensite volume fractiondecreases during heating, but increases again duringthe cooling step and is associated with the drop inclamping stress. This complex material responseunderscores the difficulty of designing with shapememory alloys and the need for effective modelingstrategies to predict and understand materialbehavior.

Fig. 7. Thermomechanical loading time history for uncrackedcomposite. Wires embedded in matrix material start as austeniteand develop martensite as the composite is loaded. Afterunloading, the composite is heated and then cooled back toroom temperature to illustrate shape recovery features of SMAcomposite. See text for detail.

4.2. Crack propagation simulation

To realize a self-healing composite, it is essentialto achieve a clamping force from the SMA uponheating. Considering the example above, it is clearthat with the matrix undergoing significant plasticdeformation in the vicinity of the propagatingcrack, heating of the composite will not necessarilyprovide sufficient contraction force from the SMAwires upon reverse transformation to achieveclamping forces across the full width of the crack.Thus, in order to guarantee full crack closure, theembedded wires in the composite thus need to bepre-strained to contain some detwinned martensitewhen embedded in the composite.

In the example demonstrated here, the wires arepre-strained to an initial detwinned martensite frac-tion of 0.30 when unloaded to zero stress. The com-posite model containing pre-strained wires is thenloaded by a fixed displacement of one end as shownin Fig. 3. As described above, a small notch defect isplaced in this composite and material allowed todebond to simulate crack propagation when thestress at the crack tip reaches the prescribed failurestress. The critical stress to debond is taken as60 MPa. The SMA wires accommodate the elonga-tion by forming detwinned martensite and bridgethe crack as it propagates. In the simulation pre-sented here, the wires transform to a final fractionof 0.84 detwinned martensite. As the crack propa-gates, the area ahead of the crack tip is highlystressed and undergoes plastic deformation. Thestress contours (Fig. 8) show that the highest stressoccurs in two lobes on either side of the crack tip asexpected.

Contour plots are shown in Fig. 9 for the areaaround the crack for selected snapshots of load his-tory. After the crack propagates across the fullwidth of the composite, the loading boundary con-dition of the end is released to allow the wires torecover elastically. Some residual stress due to theplastic deformation can be seen in Fig. 9b, whichshows the von Mises stress contours.

When the entire model is heated, the wiresreverse transform to austenite causing them toshorten and pull the fractured matrix halves backtogether. The crack faces that were the last to frac-ture are the first to come back into contact becausethat is where the most plastic deformation tookplace (Fig. 9c). With continued heating, the crackbecomes fully closed and the matrix is compressed.At the end of heating, the residual stresses due to

Fig. 8. Contour plot of von Mises stresses on full model after crack propagation begins in the composite. The five SMA wires runhorizontally across the model with both ends denoted as ·. Dashed lines indicate zoomed region near the crack that is further shown astime progresses in Fig. 9.

Fig. 9. Crack propagation simulation: (a) crack halfway through matrix (b) crack fully opened, matrix halves held together by wires (c)heat causes wires to shorten which closes crack (d) crack fully closed and plasticity reversed. SMA wire locations shown in red. (Forinterpretation of reference to colour in this figure, the reader is referred to the web version of this article.)


plastic deformation are almost completely reversed(Fig. 9d).

Similar to the uncracked model, the temperaturemust be raised to well above the austenite finishtemperature to significantly transform the SMAwires to austenite. Fig. 10b illustrates that the mar-tensite volume fraction decreases rapidly prior tothe crack faces coming back in contact. Subse-

quently, the rate of inverse transformation decreasessubstantially due to the resistance of the matrixmaterial. Considering the loading path on the phasediagram, Fig. 10a, the temperature-stress pathmoves diagonally upward through the austenitetransformation zone after the crack faces contact.Thus, additional austenite formation occurs at avery slow rate (Fig. 10b).

Fig. 10. (a) Loading path for cracked composite overlaid on thephase diagram for the SMA wires. SMA wires are pre-strained to30% detwinned martensite before history shown here. Compositeis then loaded/unloaded at low temperature and a crack isallowed to fully propagate across the composite. The entirecomposite is uniformly heated to initiate self-healing and thencooled back to room temperature. See text for details. (b) Reversetransformation from detwinned martensite to austenite withincreasing temperature. Decrease in rate of martensite volumefraction changes after crack faces touch and further recovery ofaustenite requires compression of the matrix.

Fig. 11. Thermomechanical loading time history for compositewith propagating crack. Pre-straining of SMA wires beforeembedding in composite also shown here in initial segment.Remaining time history for mechanical loading of composite andheating/cooling sequence of self-healing. Matrix softening duringheating explicitly shown. Starting with twinned martensite orwith austenite gives nearly identical results except for tempera-ture-induced martensite fraction.


Fig. 11 shows that for the SMA material proper-ties used, the A and M zones intersect at high stressand temperature values. This intersection is physi-cally reasonable and can be seen even at zero stressfor some SMA materials in which MS < AS. A plas-ticity zone would occur at still higher stresses andtemperatures, but this is not shown in the figure.In the simulation, the unloading occurs before thesystem reaches the region of the phase diagram withoverlapping zones.

Fig. 11 shows the change in temperature, matrixyield strength, stress in the wires and martensite vol-ume fractions on the same graph. The plots shownearly identical results for simulations starting fromtwinned martensite compared to starting from aus-tenite, except of course for the temperature-inducedmartensite volume fraction. The slight differences inthe stress in the wires and stress-induced martensitevolume fraction are due to the different Young�smodulus values of martensite and austenite.

The matrix properties allow the material tosoften at higher temperatures as motivated bythe composites created by the Olson group (Files,

1997; Forbell et al., 1997; Bernikowicz, 1998; Kanget al., 2002). This softening enables the wires to fullyclose the crack upon heating and reverse much ofthe localized plasticity formed in the matrix duringthe fracture process.Without sufficient softening, thestrong resistance by the matrix prevents the wiresfrom closing the crack fully as shown in Fig. 12a.The figure also shows that when the crack doesnot fully close, more of the residual stress remainssince the plasticity is not reversed. Fig. 12b explicitlydemonstrates the impact of matrix softening on thecrack opening at the healing temperature: matrixsoftening as shown in Fig. 4 results in completecrack closure, while less softening results in a par-tially open crack and the crack remains open acrossnearly the entire specimen in the case withoutmatrix softening.

The high clamping forces attained by the wires inthis heating stage are also critical in order to helpheal the composite as the matrix partially flowsand rewelds. As is seen in Fig. 10a, during coolingback to room temperature there is an initial increasein stress due to increasing matrix properties, fol-lowed by a decrease in stress as the austenite trans-forms to martensite. As shown in Fig. 11, duringthis cooling process the martensite fraction increases

Fig. 12. (a) Residual stresses and slightly open crack for matrix without softening and (b) effect of matrix softening on crack opening atthe maximum healing temperature. Distance measured from top edge. The shapes of the open crack edges show the lack of plasticdeformation due to the initial notch.

Fig. 13. Total clamping force of all SMA wires shown with temperature history. Significant clamping force generated during heating stageand residual clamping force remains after cooling to room temperature.


slightly, but the wires remain mostly austenitic andare able to exert a clamping force on the matrixmaterial. The clamping force shown in Fig. 13decreases significantly but is maintained.

With stiffer matrix materials, the loading path inFig. 10a becomes nearly parallel to the zone bound-aries, corresponding to an iso-martensite fractionpath. In this case, little or no additional austeniteis formed after the crack faces first touch at the bot-tom of the crack during heating. The resistance toclosing is due to insufficient plastic deformation ofthe matrix during the heating stage to bring theupper faces of the crack back in contact, as seen

in Fig. 12. In addition, although the stress still risesduring heating, the residual clamping force aftercooling is somewhat smaller than for a soft matrix.With use of the simulation tool presented here, wewere able to determine appropriate matrix proper-ties to allow self-healing of a cracked composite asshown in Fig. 9.

5. Conclusions

In this paper, a self-healing SMA composite issimulated via a finite element approach that allowscrack propagation in a brittle matrix material. The


SMA wires are carefully modeled using a one-dimensional SMA constitutive model and imple-mented into user-defined truss elements. Loadingof the composite allows a crack to propagate froman initiation site and the wires bridge the crack asdetwinned martensite forms with the applied load-ing. Self-healing is initiated upon heating whichcauses reverse transformation of the wires, resultingin strain recovery. Results show that the wires areable to pull the crack faces back together to demon-strate healing under certain conditions. Use of pre-strained SMA wires in the composite is critical fora closure force to develop and adequate matrixmaterial softening is required during heating inorder for the crack faces to heal across the full widthof the crack.

An advantage of the simulation method deve-loped here in which separate elements for the wiresand matrix are used is that material properties orcomposite geometry can be changed easily. Otherapproaches in which special composite elementsare developed which combine both SMA and matrixproperties are more application specific. The flexi-bility of the current approach allows the techniqueto be used to explore design issues for a composite.This capability was illustrated here by selecting theappropriate matrix softening characteristics toallow self-healing to occur. This design tool can alsobe applied to general active SMA composite struc-tures to investigate the effects of adaptive stiffening,vibration control and shape control.

Acknowledgments

The authors deeply appreciate helpful andinsightful discussions with Professor Greg Olsonand Michele Manuel as well as Professor BrianMoran during the development of this researchproject. Funding support from NASA-Langleyand Los Alamos National Laboratory is gratefullyacknowledged.

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