Evolution of multiple Martensite variants in a SMA thick-walled cylinder loaded by internal pressure

International Journal of Solids and Structures 0 0 0 (2018) 1–21

Contents lists available at ScienceDirect

International

Journal

of

Solids

and

Structures

journal homepage: www.elsevier.com/locate/ijsolstr

Evolution

of

multiple

Martensite

variants

in

a

SMA

thick-walled

cylinder

loaded

by

internal

pressure

Enrico

Radi

Dipartimento di Scienze e Metodi dell’Ingegneria, Università di Modena e Reggio Emilia, Via Amendola 2, I-42122 Reggio Emilia, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 10 March 2018 Revised 6 June 2018 Available online xxx Keywords: Analytical solutions Axisymmetric Rings Shape-memory Thermomechanical Cylinder Phase transformation

a

b

s

t

r

a

c

t

Thestressanddeformation fieldsinaSMAringorathick-walledcylinder loadedbyinternalpressure atconstanttemperature(overthestarttemperatureofthemartensitictransformation)aredetermined inclosedformunderplanestressloadingconditions.ThephenomenologicalSMAconstitutivemodel in-corporatesthevolumefractionsofmulti-variantsMartensite,whichareassumedtoevolvelinearlywith theTrescaeffectivestress,accordingtotheassociativeflowruleandthecornerflowrule.Initially,the cylinderiseverywhereinastateofAustenite.Theapplicationofaninternalpressurethentriggersthe martensitictransformationstartingfromtheinnerradiusofthecylinderwallandextendingtowardsthe outerradius.Ifthewallthicknessislargeenough,thetangentialstressmayvanishattheinnerradiusand correspondinglythestressstatemayreachacorneroftheTrescatransformationcondition,thus originat-ingtwodifferentMartensitevariantsaccordingtothecornertransformationrule.Theadmissiblephase partitionswithinthewallthicknessoriginatingduringtheloadingprocesshavebeensystematically in-vestigatedaccordingtotheratiobetweentheouterandinnerradii.Theresultsobtainedheresuggestthat theloadingprocessshouldbeinterruptedsoonafterthecompletemartensitictransformationisachieved attheinnerradiusofthecylindertoavoidpermanentplasticdeformations.

© 2018ElsevierLtd.Allrightsreserved.

1. Introduction

Due to their reliable performance and convenient installation, SMA couplings, fasteners and joints have been widely used for pneumatic and hydraulic connectors in aircraft and piping systems ( KapganandMelton,1990;Borden,1990,1991;Brinson and Lam-mering, 1993; Wanget al., 2005; Jee etal., 2006), as well as for electrical connectors ( Harrison and Hodgson, 1976). They indeed provide joints of the greatest mechanical and electrical reliabil- ity and integrity, which can be quickly applied or removed. Re- cently, an experimental and numerical study has been carried out to assess the use of SMA rings as pipe couplers in radioactive ar- eas of high-energy particle accelerators, where thermally induced mounting and dismounting operations can be operated remotely ( Niccolietal.,2017).

In order to exploit the full potential of SMA coupling systems, it becomes essential to estimate the stress distribution within these components accurately. Severe stresses may develop during the production stage or the installation of SMA coupling and may cause unexpected mechanical deformation, damage and failure or just loss of efficiency ( Tabesh et al., 2013, 2017). To avoid such technological problems, it becomes necessary to predict the me-

E-mail addresses: enrico.radi@unimore.it , eradi@unimore.it

chanical stresses originated in SMA connections during each step of their production and installation. A complete and detailed un- derstanding of their stress and deformation history will be also useful for improving their design, production and usage conditions. SMA pipe couplings displaying cylindrical geometries are usu- ally predeformed, so that they must be stored and transported at low temperature and then installed by induction heating ( Brook,1983). They are previously expanded by applying an inter- nal pressure at temperature T0 over the start Martensite temper-

ature Ms, thus inducing the martensitic transformation. After the coupling is mounted on the pipe the temperature is increased to recover the residual deformation, taking advantage of the reversal austenitic transformation. As a consequence, the coupling diame- ter decreases and a contact pressure originates between pipe and coupling. In the alternative method proposed by Jeeetal.(2006), SMA coupling and pipe are deformed simultaneously and then the coupling is contracted on heating in order to get the tightness also with a poor shape memory effect.

The problem of a SMA ring used as a pipe connector was ini- tially investigated by BrinsonandLammering(1993)by introduc- ing the simplifying assumption of purely elastic behavior along the radial direction. Later, Birman (1999) considered an infinite SMA plate with a circular hole and obtained a closed form approximate solution by assuming a constant ratio between the radial and tan- https://doi.org/10.1016/j.ijsolstr.2018.06.034

0020-7683/© 2018 Elsevier Ltd. All rights reserved.

2 E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21 gential stresses. Chietal., (2003,2005) provided a detailed exam-

ination of the axisymmetric stress, strain and phase fraction fields in an annular SMA plate. These analyses were restricted to isother- mal loadings and assumed single variant Martensite, so that no unloading process nor reverse transformation to Austenite or neg- ative variant Martensite were taken into consideration. Moreover, the assumption of an effective stress of von Mises type made in those papers necessarily required a numerical approach. In a fur- ther paper ( Chietal.,2007), the same authors proposed numerical algorithms for treating cyclic loading histories and applied them to investigate the loading and unloading process of a SMA annular plate.

The study of the process of constrained recovery has been ini- tially limited to uniaxial examples ( Liang andRogers,1990; Brin-son,1993;LeclercqandLexcellent,1996;KoselandVidenic,2007).

Nagaya and Hirata (1992) developed a simplified model of con- strained recovery in SMA rings that considers only the tangential stress and neglects the contribution of radial stresses to the effec- tive stress. Such an extreme simplification allowed these authors to use a uniaxial constitutive model, though it is reasonably ac- curate only for thin-walled SMA rings. Wangetal.(2005)investi- gated experimentally the effects of the wall thickness and temper- ature range on the reverse transformation behavior of SMA pipe joints. Videnicetal.(2008)presented a mathematical model of bi- axial constrained recovery in a SMA ring. These authors adopted a generalized effective stress of Tresca type, which allows to con- sider unequal response in tension and compression. For the sake of simplicity, they assumed a vanishing stress state in the deformed ring after unloading. Such a supposition is not properly correct, since residual stresses are always present in the SMA material after loading-unloading cycle. Piotrowskietal.(2012)performed a com- bined experimental and finite element analysis of a SMA pipe cou- pler mounted on an instrumented elastic ring. The coupling pres- sure predicted by FEA was in excellent agreement with the mea- sured contact pressure. Mirzaeifaretal.(2012) performed a semi- analytic study of the pseudoelastic response of a thick-walled SMA cylinder subject to internal pressure, under plane stress or plane strain conditions. These authors partitioned the cylinder into a fi- nite number of annular regions and provided closed-form solutions for the equilibrium equations in each annulus. Then, a numerical solution was found by solving the system of nonlinear algebraic equations obtained by enforcing stress continuity at the interface between annular regions. Tabesh et al. (2013) provided a closed form solution for the pseudoelastic response of a SMA thick-walled cylinder subjected to internal pressure under plane stress or plane strain loading conditions, for temperature higher than the finish austenitic transformation Af. They assumed a simplified 2D consti- tutive model for SMA that incorporates the Tresca transformation criterion with associative flow rule and linear transformation law. Moreover, they considered the simplifying assumption that the ax- ial strain is constant within the wall thickness and thus the re- sponse in the axial direction remains elastic. Liuetal.(2013) also performed a similar investigation by considering the effects of a radial temperature gradient. Later, LiuandDu(2014)provided the analytical solution to the problem of isothermal loading of a pseu- doelastic SMA cylinder under external pressure. In these analyses the tangential stress is assumed to be always positive, as indeed it occurs during loading and pseudoelastic unloading if the wall thickness is sufficiently thin. In this case the effective Tresca stress is given by the difference between radial and tangential stresses and, thus, the Martensite transformation occurs with elongation in the circumferential direction. However, the results of these inves- tigations show that the tangential stress at the inner radius de- creases during the loading process, especially for very large wall thickness. If it becomes null, then a corner of the Tresca trans- formation condition is attained and transformed Martensite starts

elongating along the axial direction also, according to the cor- ner flow rule. Namely, two different Martensite variants are pro- duced within the corner region according to the corner flow rule by means of two different lattice shearing mechanisms, which may develop within the planes orthogonal either to the axial direction or to the tangential direction. Therefore, the latter analyses pro- vide correct predictions for sufficiently thin-walled cylinders, but they may be inaccurate for a very thick-walled SMA cylinder under plane stress loading conditions. In this case, a corner region with null tangential stress may take place during the loading process starting from the inner boundary, as it occurs for an elastic-plastic thick-walled cylinder when the Tresca yield condition is adopted ( Koiter,1953;DurbanandKubi,1992;MasriandDurban,2007).

A complete and detailed analytical study of the shape mem- ory effect induced by forward transformations of a very thick SMA cylindrical joint has never been performed by using a con- stitutive model that incorporates the phase fractions of multiple Martensite variants as internal variables, whose transformation oc- curs in agreement with the normality rule on the Tresca sur- face. Most of the simulation available in the literature are com- monly performed by using finite element analysis incorporating 3D constitutive models for the thermomechanical behavior of SMAs ( PopovandLagoudas,2007;Piotrowskietal.,2012,Lagoudasetal., 2012;Zaki,2012, Niccolietal., 2017). The purpose of the present study is to develop a rigorous analytical model with closed-form solutions to predict the stress and displacement fields in a SMA ring or thick-walled cylinder composed of several phases, namely Austenite and two different Martensitic variants, subject to internal pressure at constant temperature over the start temperature of the martensitic transformation Ms. In particular, the forward marten- sitic transformation occurring under proportional axisymmetric loading and plane stress conditions is investigated systematically. The phenomenological SMA constitutive model adopted here as- sumes that the Martensite fractions evolve as linear functions of the Tresca effective stress, in agreement with the constitutive mod- els proposed by Govindjee and Kasper (1997, 1999), Arghavani etal.(2010),MarfiaandRizzoni(2013) for multi-variants Marten- site and by Videnicetal.(2008) and LuigandBruhns (2008)for single variant Martensite. These models are derived from the the- ory of generalized plasticity developed by Lubliner and Auric-chio (1996) and Auricchio andLubliner (1997). However, the oc- curring of the Martensite transformation in SMA according to the corner flow rule of the Tresca criterion has been developed here for the first time, to allow for obtaining closed form solutions to complex 2D or 3D thermomechanical problems. The two Marten- site variants consist in the deformed Martensite shortened in the radial direction and stretched in the circumferential or the axial di- rection, respectively. The activation of one or the other mechanism depends on the stress state according to the normality rule.

Although the general case requires a numerical analysis of two non-linear ODEs, the simplifying assumptions of effective stress in a Tresca form and linear phase transformation kinetic allow us to obtain a closed form solution, also for the regions that are fully transformed to both Martensite variants. Obtaining an analytical solution for these regions is one of the main challenges of the present study. It requires to take into account for the history of loading and the evolution of the volume fractions of Martensite variants within these regions. Despite the theoretical difficulties, analytic solutions have a number of advantages. From them one can see clearly the role played by constitutive and geometrical pa- rameters and thus they allow understanding also more complex problems. Moreover, they provide a reliable evaluation of the resid- ual stresses and strains in the SMA thick-walled cylinder after the isothermal loading process, which can be used also for validating the results of numerical procedures ( Auricchioetal.,2014; Bernar-diniandPence,2016).

E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21 3

Fig. 1. Tresca transformation condition with associative flow rule and corner flow under plane stress loading conditions ( σ3 = 0).

The present article is divided into six sections. The constitutive model in the integrated form is briefly reviewed in Section2. The model is able to describe several phenomena occurring in the ra- dial expansion of a SMA thick-walled cylinder, such as phase trans- formations, Martensite reorientation and multiple lattice shear- ing mechanisms. It keeps the essence of the approach used in

Videnic etal.(2008) and Tabesh etal.(2013), but extends these studies by considering corner transformation flow rule, thus al- lowing for the formation of multi-variant Martensite. Moreover, in the present analysis the axial strain may vary within the wall thickness. The general expressions of the stresses, radial displace- ment and Martensite fraction within all the admissible annular re- gions that may arise within the cylinder during the loading pro- cess, composed of Austenite, Martensite variants and a mixture of such phases, are presented in detail in Section 3. A closed-form solution for the loading process is developed in Section4for each possible phase partitioning within the wall thickness and for any value of internal pressure. The solution is obtained by enforcing full stress and displacement continuity at the interface between annular regions and imposing the boundary conditions at the outer and inner radius of the cylinder. Results are presented in Section5, where the effects of the wall thickness and material parameters on the radial distribution of stresses, radial displacement and Marten- site volume fractions are also discussed. A summary of important results is given in Section6.

2. SMAconstitutivemodel

The equilibrium condition for the in-plane stresses

σ

r and

σ

_θ, which originate in the cylinder under axisymmetric loading condi- tions, writes

σ

θ=r

σ

r+

σ

r, (2.1)

where the apex denotes the derivative with respect to the variable

r. In addition, the strain- displacement compatibility conditions re- quire

ε

r=ur,

ε

θ= ur

r. (2.2)

The phase transformation between Austenite and Martensite is assumed to be governed by the effective stress

σ

e given by the Tresca’s criterion under plane stress conditions (

σ

33=0). Assuming

that the radial stress is always compressive (

σ

r<0) and

σ

r≤

σ

_θ, then Tresca’s criterion ( Fig.1) gives

σ

e=

σ

_θ−

σ

r, for

σ

r≤ 0≤

σ

_θ, (2.3)

Fig. 2. SMA phase diagram for an isothermal loading process.

σ

e=−

σ

r, for

σ

r≤

σ

_θ≤ 0. (2.4)

Two different Martensite variants may originate during the loading process according to the sign of the tangential stresses

σ

_θ, as predicted by the Tresca-like transformation condition ( 2.3) and ( 2.4) together with the associative flow rule ( Fig. 1). The princi- pal Martensite variant is produced for

σ

_{θ≥ 0} (side AB in Fig. 1). It causes shortening in the radial direction, elongation in the tan- gential direction and has no effect on the axial direction. The sec- ondary Martensite variant is produced for

σ

_{θ≤ 0}(side BC in Fig.1). It causes shortening in the radial direction, elongation in the axial direction and has no effect on the tangential direction. The volume fractions of these Martensite variants are denoted by

ξ

_rθ and

ξ

r3,

respectively. Then, the total volume fractions of Martensite is given by the sum

ξ

tot=

ξ

rθ+

ξ

r3.

The threshold stresses

σ

s and

σ

f for the start and finish martensitic transformations are determined by the temperature

T0 according to the simplified phase diagram sketched in Fig. 2,

namely

σ

s=CM

(

T0− Ms

)

,

σ

f=CM



T0− Mf



, (2.5)

being CMthe slope of the martensitic transformation lines. Although the elastic modulus of the SMA varies during the phase transformation between the elastic modulus EA of the austenitic phase and the elastic modulus EM of the martensitic phase, for the sake of simplicity, in the following a constant elas- tic modulus E is considered for the two phases, equal to the mean value between the two elastic moduli. Moreover, the as- sociative flow rule with corner flow is assumed for the trans- formation strain, according to the experimental observations of

Chirani etal. (2003). Then, the constitutive relations holding for the isothermal and proportional loading process of the SMA cylin- der under plane stress conditions are assumed in the integrated form ( GovindjeeandKasper,1997,1999;Videnicetal.,2008; Luig andBruhns2008)

ε

r= 1 E

(

σ

r−

νσ

θ

)

−

ε

L

(

ξ

rθ+

ξ

r3

)

,

ε

θ = 1_E

(

σ

θ−

νσ

r

)

+

ε

L

ξ

rθ, for

σ

r≤ 0and

σ

r<

σ

_θ

ε

3 =−

ν

E

(

σ

θ+

σ

r

)

+

ε

L

ξ

r3. (2.6)

where

ε

L is the maximum residual strain obtained by detwinning multiple variant Martensite, coinciding with the maximum inelas- tic strain attained under uniaxial loading when the solid is com- posed of fully oriented Martensite. Note from Eq. (2.6) that the martensitic transformation induces no volume change, being of shearing type.

Eqs.(2.6) can be derived by integration of the rate constitutive equations and thus hold for proportional loading. A similar deriva- tion has been performed by Panoskaltsis etal.(2004) for a SMA constitutive model based on an effective stress of von Mises type.

4 E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21

Table 1

Constitutive parameters for SMA materials BL and TA.

BL TA T0 [ °K ] 298 340 E [ GPa ] 46.65 85 ν 0.33 0.4 σs [ MPa ] 153 1200 σf [ MPa ] 223 1400 εL 0.067 0.033

Eqs. (2.6) hold also for

σ

_θ= 0, namely when the stress state lays in the corner of the Tresca transformation condition. In this case, the transformation strain is not uniquely defined, in agree- ment with the corner flow theory of elastic- plastic materials ( DurbanandKubi,1992).

In the following, two different sets of constitutive parame- ters will be considered. They correspond to those adopted by

Brinson andLammering (1993) and Tabesh et al.(2013) and are denoted by BL and TA, respectively. In particular, the temperature

T0, the Young’s modulus E and Poisson coefficient

ν

of both phases

(Martensite and Austenite), the threshold stresses

σ

s and

σ

f for start and finish martensitic transformations at temperature T0, and

the maximum residual strain

ε

L are reported in Table1 for both sets. For the sake of simplicity, the same values of the Young’s modulus and Poisson coefficient are chosen for both phases.

2.1.Martensitictransformationduringtheloadingprocess

Assuming that the material at the beginning of the loading pro- cess is fully austenitic, then, Austenite is transformed into Marten- site variants during the loading process. According to the model proposed by GovindjeeandKasper, (1997,1999), the transforma- tion of the variant Martensite fractions can be expressed in the following integrated form that holds for

σ

s ≤

σ

e ≤

σ

_f, with ref- erence to the phase transformation model sketched in Figs.1and

2:

ξ

rθ =

σ

e −

σ

s

σ

f−

σ

s ,

ξ

r3=0, for

σ

r<0<

σ

_θ and

σ

e =

σ

_θ−

σ

r, (2.7)

ξ

rθ+

ξ

r3=

σ

_σ

e −

σ

s f−

σ

s , for

σ

r<

σ

θ=0and

σ

e=−

σ

r, (2.8)

ξ

r3=

σ

e −

σ

s

σ

f−

σ

s −

ξ

0 rθ,

ξ

rθ=

ξ

r0θ, for

σ

r<

σ

_θ<0and

σ

e=−

σ

r, (2.9) where

ξ

0

rθ is the principal Martensite variant at the beginning of the transformation.

The dependence of the threshold stresses on the Martensite fraction as depicted in GovindjeeandKasper(1997,1999) has been neglected here for the sake of conciseness, but it can be easily con- sidered in a more refined investigation, as well as the difference between the elastic moduli of Austenite and Martensite.

2.2.Continuityconditions

Continuity of the radial stress and displacement between two adjacent annular regions undergoing different process of phase transformation must be required. Then, by using ( 2.2) 2 through

the front of separation between two different regions one must re- quire

[

σ

r]=[ur]=[

ε

_θ]=0, (2.10)

where the brackets denote the jump of the function within the brackets. Then from the definition of the effective stress ( 2.3) and ( 2.4) and conditions ( 2.10) it follows

[

σ

e]=



_[

_σ

θ] 0 for

σ

r<0<

σ

_θ for

σ

r<

σ

_θ≤ 0. (2.11) The introduction of ( 2.10) and ( 2.11) in the constitutive relations ( 2.6) 2, ( 2.7) and ( 2.8), implies [

σ

_θ]+E

ε

L[

ξ

rθ]=0, [

σ

e]=



[

ξ

rθ]



σ

f−

σ

s



for

σ

r<0<

σ

_θ [

ξ

rθ+

ξ

r3]



σ

f−

σ

s



for

σ

r<

σ

θ=0. (2.12) If

σ

_θ > 0 then

ξ

r3= 0 and thus Eqs.(2.11) and ( 2.12) imply the

continuity of the tangential stress, effective stress and volume frac- tion of Martensite variants during the loading process, namely

[

σ

_θ]=[

σ

e]=[

ξ

rθ]=[

ξ

r3]=0, (2.13)

whereas if

σ

_θ=0 then Eqs. (2.11) and ( 2.12) imply conditions ( 2.13) and also [

ξ

r3] =0.

Therefore, continuity of the radial stress and displacement ( 2.10) also implies continuity of the tangential stress, effective stress and both Martensite volume fractions. In the following, con- tinuity of the radial and effective stresses will be imposed between contiguous annular regions, rather than continuity of the radial displacement.

3. AdmissibleannularregionswithintheSMAcylinder

Let r_i and ro denote the inner and outer radii of the cylinder wall, respectively. Initially the cylinder is everywhere in a state of Austenite, at temperature T0 higher than the start temperature of

the martensitic transformation Ms ( Fig. 2). A uniform pressure p is applied at ri and gradually increased at constant temperature

T0. Correspondingly, the effective stress

σ

e is a decreasing func- tion of the radius r whose maximum is attained at r_i. Therefore, the martensitic transformation starts therein when the effective stress reaches the threshold stress

σ

s. As the internal pressure is increased, a progressive increase in the volume fraction

ξ

rθ of Martensite occurs, starting from ri. Correspondingly, the front of the start of the Martensite transformation, defined by the radius

rs, moves towards the outer boundary. When the effective stress reaches the threshold stress

σ

_f at r_i, then a second front corre- sponding to the finish of the martensitic transformation originates therein. If the internal pressure is further increased, then the sec- ond front propagates within the wall thickness with radius r_f.

During the axisymmetric expansion, the wall thickness can be partitioned into three main kinds of annular regions: a purely austenitic outer region A whose inner and outer radii are rs and

ro, respectively; an intermediate transforming region in a mixture of Austenite and Martensite variants with inner radius rfand outer radius rs; and a purely martensitic inner region M with inner ra- dius ri and outer radius rf. The intermediate transforming region may be divided in correspondence of the radius rc into an outer

AM region where only the principal Martensite variant

ξ

rθ is pro- duced and an inner corner region C where both Martensite variants

ξ

rθ and

ξ

r3 are produced. If no corner region takes place within

the wall thickness, then, the martensitic inner region M contains the principal variant

ξ

_rθ alone. If a corner region has instead de- veloped, then the martensitic inner region M∗ contains a mixture of both Martensite variants. The number of annular regions within the wall thickness depends on its geometry and internal pressure.

In the following, the general expressions of the stress and dis- placement fields in the admissible annular regions that take place during each step of the isothermal loading process, are obtained in terms of the internal pressure p and radii rs,rc and rfseparating the different annular regions.

E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21 5

3.1. FullyausteniticouterregionA

The austenitic region A occupies the outer part of the SMA

cylinder wall where the martensitic transformation has not started yet, being

σ

e<

σ

s. Therefore, the SMA within this annular region behaves elastically, being

ξ

rθ=

ξ

r3=0, and the stress field within

the austenitic region induced by the tractions

σ

r( rs) =−qs

σ

s ex- changed through the front of start martensitic transformation at rs is predicted by the classical Lamé solution of two-dimensional lin- ear elasticity:

σ

A r

(

r

)

σ

s = r2 s r2 o− r2s



1−r2o r2



qs, (3.1)

σ

A θ

(

r

)

σ

s = r2 s r2 o− r2Ms



1+r2o r2



qs. (3.2)

By using Eqs.(3.1)and ( 3.2) and noting that

σ

_θ>

σ

r, the effec- tive stress ( 2.3) in the cylinder wall turns out to be:

σ

A e

(

r

)

σ

s = 2r2 s r2 o− r2s r2 o r2qs. (3.3)

Under plane stress loading conditions, the radial displacement writes uA r

(

r

)

r = 1−

ν

E r2 s r2 o− rs2



1+1+

ν

1−

ν

r2 o r2



qs

σ

s. (3.4)

Eqs.(3.1)-( 3.4) hold for rs≤ r ≤ r o.

3.2. IntermediateregionAMinamixtureofausteniteandprincipal Martensitevariant

Within the region AM undergoing phase transformation to the principal Martensite variant, the volume fraction

ξ

rθ increases lin- early with the effective stress according to relations ( 2.7). A sub- stitution of Eqs. (2.1)and ( 2.3) in the constitutive relations ( 2.6), by using the strain-displacement relations ( 2.2) and the integrated form of the transformation laws ( 2.7) to the principal Martensite variant, yields: ur= 1 E



(

1−

ν

)

σ

r−



ν

+

δ

1−

δ



r

σ

r+₁₋

δ

_{δ σ}

s



, (3.5) ur r = 1 E



(

1−

ν

)

σ

r+ r

σ

 r 1−

δ

−

δ

1−

δ σ

s



, (3.6) where:

δ

= E

ε

L

σ

f−

σ

s+E

ε

L < 1, (3.7)

is a non-dimensional parameter close to 1, being E

ε

L 

σ

f−

σ

s. The introduction of Eq.(3.6)for urin ( 3.5) provides the following linear ODE for the function

σ

r( r):

r2

_σ



r +3r

σ

r− 2

δ σ

s=0. (3.8)

The analytic solution of the linear ODE ( 3.8) for the radial stress in the annular region AM in a mixture of Austenite and principal variant Martensite (

ξ

r3= 0) is:

σ

AM r

(

r

)

σ

s = A2+A1 r2 o r2 −

δ

ln ro r, (3.9)

where A1 and A2 are constants of integration. The introduction of

( 3.9) in Eqs.(2.1), ( 2.3) and ( 3.6) provides the following tangential and effective stresses and the radial displacement

σ

AM θ

(

r

)

σ

s = A2− A1 r2 o r2 +

δ

1− lnro r

, (3.10)

σ

AM e

(

r

)

σ

s =

δ

− 2 A1 r2 o r2, (3.11) uAM r

(

r

)

r =

σ

s E



(

1−

ν

)

A2−

δ

ln ro r

− A1



1+

δ

1−

δ

+

ν



r2 o r2



, (3.12) in terms of the constants A1 and A2. Moreover, from ( 3.11) and

( 2.7), the volume fraction of transformed Martensite turns out to be:

ξ

AM rθ

(

r

)

=− 1−

δ

γ

− 1



1+ 2A1 1−

δ

r2 o r2



, (3.13) where

γ

=

σ

f

σ

s, (3.14) is a non-dimensional material parameter greater than 1.

3.3.CornerregionCinamixtureofAusteniteandbothMartensite variants

If the wall thickness is large enough, then, the tangential stress

σ

θmay vanish at riduring the Martensitic transformation. In this case, an annular corner region C appears starting from ri, where the stress state lays within a corner of the Tresca transformation condition ( Fig.1), in agreement with the problem of a pressurized elastoplastic tube studied by DurbanandKubi(1992). According to ( 2.1), vanishing of

σ

_θ within this region leads to the radial stress

σ

C r

(

r

)

σ

s =−C ro

r, (3.15)

where C is a constant. Then, the constitutive Eqs.(2.6), ( 2.8) and the strain compatibility Eqs.(2.2)imply

ur r =

σ

s E



ν

C ro r +

δ

1−

δ

(

γ

− 1

)

ξ

rθ



, (3.16)

(

r

ξ

rθ

)

+

ξ

rθ+

ξ

r3+ 1−

δ

γ

− 1 C ro

δ

r =0. (3.17)

Considering that

σ

e=−

σ

rfor

σ

_θ=0, the transformation law ( 2.8) for the total Martensitic fraction, namely the sum of

ξ

_rθ and

ξ

r3,

then yields

ξ

rθ+

ξ

r3= 1

γ

− 1

Cro r − 1

, (3.18)

Eqs.(3.17) and ( 3.18) provide the following distribution of the Martensite volume fractions

ξ

rθ and

ξ

r3 in the corner region in a

mixture of Austenite and both Martensite variants

ξ

C rθ

(

r

)

= 1

γ

− 1

1−C ro

δ

r ln r ro+ Dro r

, (3.19)

ξ

C r3

(

r

)

= 1

γ

− 1



C− D+C

δ

ln r ro

_ro r − 2



, (3.20)

where D is a constant of integration. The radial displacement then follows from ( 3.16) 2, ( 2.2) 2and ( 3.19) as

uC r

(

r

)

r =

σ

s E



δ

1−

δ

1+Dro r

+Cro r

η

− 1 1−

δ

ln r ro



. (3.21) Note that a corner region cannot extend till the outer radius of the cylinder wall because the condition

σ

C

r

(

ro

)

= 0 necessarily implies C=0, so that the stress field should be null therein.

6 E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21

3.4.FullymartensiticinnerregionM

If the loading process is continued after that the complete transformation to both Martensite variants is achieved at r_i, then a fully martensitic region with

ξ

rθ+

ξ

r3= 1 appears, starting from

ri. The introduction of Eqs. (2.1) and ( 2.2) in the constitutive Eqs.(2.6)gives: ur= 1 E

(

1−

ν

)

σ

r−

ν

r

σ

r



−

ε

L, (3.22) ur r = 1 E

(

1−

ν

)

σ

r+r

σ

r



+

ε

L

ξ

rθ. (3.23)

where

ξ

rθ≤ 1, Two different situations can be envisaged according to the amount of secondary martensitic variant, namely for

ξ

r3= 0

or

ξ

r3> 0.

3.4.1. FullymartensiticregionMcontainingonlytheprincipalvariant

If no corner region takes place within the cylinder wall then

ξ

rθ= 1 and

ξ

r3= 0. In this case, the substitution of Eq.(3.23) for

ur in Eq. (3.22) yields the following linear ODE for the function

σ

r( r): r2

_σ



r +3r

σ

r+2E

ε

L=0. (3.24)

The linear ODE ( 3.24) admits the following solution for

σ

r, which holds in the fully martensitic region M containing the prin- cipal Martensite variant only:

σ

M r

(

r

)

σ

s = B2+B1 r2 o r2+

δ γ

− 1 1−

δ

ln ro r, (3.25)

where B1and B2are constants of integration, being from ( 3.7) and

( 3.14) E

ε

L

σ

s =

δ γ

− 1

1−

δ

. (3.26)

The corresponding equations for the tangential and effec- tive stresses and radial displacement are derived by substituting

Eq.(3.25)in ( 2.1), ( 2.3) and ( 3.23) as:

σ

M θ

(

r

)

σ

s = B2− B1 r2 o r2 −

δ γ

− 1 1−

δ

1− lnro r

, (3.27)

σ

M e

(

r

)

σ

s =−2 B1 r2 o r2 −

δ γ

− 1 1−

δ

, (3.28) uM r

(

r

)

r =

σ

s E



(

1−

ν

)

B2+

δ γ

− 1 1−

δ

ln ro r

− B1

(

1+

ν

)

r2 o r2



, (3.29) which hold for ri≤ r≤ rf.

3.4.2. FullymartensiticregionM∗containingbothvariants

If a corner region C takes place within the cylinder wall, then, both Martensite variants originates therein due to corner flow. In this case, the inner martensitic region M∗is formed by both vari- ants, namely

ξ

_rθ+

ξ

r3= 1. The substitution of Eq.(3.23) for ur in Eq. (3.22) then yields the following linear ODE for the function

σ

r( r): r2

_σ

 r +3r

σ

r+E

ε

L

1+

(

r

ξ

rθ

)





=0. (3.30)

By using ( 2.1) and ( 3.26), the general solution for the stress field satisfying the ODE ( 3.30) is

σ

M r

(

r

)

σ

s =

δ

(

γ

− 1

)

2

(

1−

δ

)



1− r 2 f r2



₁ 2+

ξ

M rθ



rf



+ lnrf r +  rf r

₁ t + t r2

ξ

M rθ

(

t

)

dt



− C1− C2 r2, (3.31)

σ

M θ

(

r

)

σ

s =−

δ

(

γ

− 1

)

2

(

1−

δ

)



1+2

ξ

M rθ

(

r

)

−



1+ r 2 f r2



₁ 2+

ξ

M rθ



rf



− lnrf r − rf r

₁ t − t r2

ξ

M rθ

(

t

)

dt



− C1+ C2 r2, (3.32)

where the function

ξ

M

rθ

(

r

)

denotes the radial distribution of the principal Martensite variant within the M∗ region. Since no evolu- tion of the Martensite variants is expected within this region, then the function

ξ

M

rθ

(

r

)

is defined by the volume fraction of the princi- pal Martensite variant already formed at radius r when the front of the finish martensitic transformation rfwas coinciding with r. The corresponding radial displacement in the fully martensitic region

M∗follows from the substitution of Eq.(3.31)in ( 3.23) as: ur

(

r

)

r =

σ

s E

δ

(

γ

− 1

)

1−

δ



−1−

ξ

M rθ

(

r

)

+



1−

ν

+

(

1+

ν

)

r 2 f r2



₁ 2+

ξ

M rθ



rf



+

(

1−

ν

)

lnrf r +  rf r

₁₋

_ν

t − 1+

ν

r2 t

ξ

M rθ

(

t

)

dt



−

σ

s E



(

1−

ν

)

C1−

(

1+

ν

)

C2 r2



. (3.33)

The integration constants C1 and C2 as well as the function

ξ

M

rθ

(

r

)

must be determined by using the boundary conditions and continuity conditions between adjacent annular regions.

4. Axisymmetricloadingprocess

The unknown parameters introduced in the analytical results for the stresses and radial displacement in the various annular re- gions obtained in the previous section can be calculated by impos- ing the boundary conditions at r_iand roand the continuity condi- tions between adjacent annular regions. Depending on the cylinder geometry and the internal pressure, eleven types of phase parti- tioning of the cylinder wall may occur during the loading process, namely ( A), ( AM,A), ( C,AM,A), ( M∗,C,AM,A), ( C,AM), ( AM), ( M∗,C,AM), ( M,AM), ( M,AM,A), ( M∗,M,AM,) and ( M). The distribution of the ad- missible configurations is sketched in Fig.3and preliminary plot- ted in Fig.4as function of the ratio r_i/ roand normalized internal pressure q=p/

σ

s for both sets of constitutive parameters BL and TA. The stress and displacement distribution within the wall thick- ness for a fully austenitic ( A) cylinder wall may be obtained from the classical linear elastic solution ( 3.1)-( 3.4) for qs= q and rs= ri. The other ten admissible configurations are systematically exam- ined in Sections4.1- 4.10.

4.1. Thick-walledcylindercomposedoftworegions(AM,A)

Let us study first the case of a cylinder wall partitioned in an outer austenitic region A and an inner region AM in a mix- ture of Austenite and principal variant Martensite, with 0 <

ξ

rθ<1 ( Fig.3a). Such a phase distribution occurs for relatively small val- ues of the internal pressure that are not large enough to originate a complete martensitic transformation within the wall thickness. The unknown constants for regions A and AM are A1, A2, qs and the radius rs of the front separating the two regions. In order to define these unknown parameters, the following conditions can be imposed at rsand ri:

σ

AM

r

(

rs

)

=

σ

rA

(

rs

)

,

σ

eA

(

rs

)

=

σ

s,

σ

eAM

(

rs

)

=

σ

s,

σ

AM

r

(

ri

)

=−q

σ

s. (4.1)

E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21 7

Fig. 3. Sketch of the seven admissible distributions of regions A, AM, C, M and M ∗ within the wall thickness.

Fig. 4. Variation of the normalized internal pressures separating the admissible phase distributions within the wall thickness with the ratio between inner and outer radii, during the axisymmetric loading process, for SMA material BL ( a ) and corresponding detail ( b ).

As discussed in Section 2.1, continuity of the radial displace- ment across the interface between two different regions is equiv- alent to continuity of the effective stress and, thus, it is included within conditions ( 4.1) 2, 3.

The introduction of ( 3.3) in condition ( 4.1) 2provides the follow-

ing relation between the parameters qsand rs qs= 1 2



1−r2s r2 o



. (4.2)

A substitution of ( 4.2) in ( 3.2)-( 3.4) thus yields the following stress and displacement fields in the outer austenitic region A in terms of rs

σ

A r

(

r

)

σ

s = r2 s 2 r2 o



1−ro2 r2



,

σ

A θ

(

r

)

σ

s = r2 s 2 r2 o



1+ro2 r2



,

8 E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21

σ

A e

(

r

)

σ

s = r2 s r2, uA r

(

r

)

r =

σ

s E r2 s 2r2 o



1−

ν

+

(

1+

ν

)

ro2 r2



, (4.3)

which hold for rs≤ r≤ ro. The introduction of Eqs. (3.2), ( 3.9) and ( 3.11) in conditions ( 4.1) 1, 3 then gives

A1=− 1−

δ

2 r2 s r2 o , A2= 1 2



r2 s r2 o −

δ

−

δ

lnr 2 s r2 o



. (4.4)

Therefore, Eqs.(3.9)–( 3.13) provide the following stresses, dis- placement and volume fraction of primary Martensite within the inner region AM:

σ

AM r

(

r

)

σ

s = 1 2



r2 s r2 o −

δ



1+lnr 2 s r2



−

(

1−

δ

)

r2s r2



,

σ

AM θ

(

r

)

σ

s = 1 2



r2 s r2 o +

δ



1− lnr2s r2



+

(

1−

δ

)

r2s r2



,

σ

AM e

(

r

)

σ

s =

δ

+

(

1−

δ

)

r2s r2, uAM r

(

r

)

r =

σ

s E 1−

ν

2



r2 s r2 o +

1+

ν

1−

ν

+

δ

_r2 s r2 −

δ



1+lnr 2 s r2



,

ξ

AM rθ

(

r

)

= 1−

δ

γ

− 1



r2 s r2 − 1



. (4.5)

for ri≤ r≤ rs. The introduction of ( 4.5) 1in the condition ( 4.1) 4 then

yields the following relation between the normalized internal pres- sure q and the radius rs:

q=1₂



1−rs2 r2 o +

δ

lnr 2 s r2 i −

(

1−

δ

)



1−r2s r2 i



, (4.6)

which hold for ri≤ r s≤ r o. The martensitic transformation starts at

riwhen rs=ri, namely when the normalized pressure ( 4.6) attains the value q0= 1 2



1−r 2 i r2 o



. (4.7)

If the cylinder wall is sufficiently thin, then relation ( 4.6) holds true until the martensitic transformation starts at rowhile the tan- gential stress is tensile within the AM region. . The corresponding internal pressure is p= q1

σ

s, where q1is given by ( 4.6) for rs= ro, namely q1= 1 2



(

1−

δ

)



r2 o r2 i − 1



+

δ

lnr 2 o r2 i



. (4.8)

The normalized pressure q1separates the wall partitions ( AM,A)

and ( AM) under the further condition

σ

AM

θ

(

ri

)

≥ 0, namely for 1+

δ

−

δ

lnr 2 o r2 i +

(

1−

δ

)

r2o r2 i ≥ 0, (4.9)

according to ( 4.5) 2, or equivalently for ri≥ r 3, where

r2 3=−

(

1−

δ

)

r2 o

δ

W0



− 1−δ δ e1+1/δ



, (4.10)

being W0 the principal branch of the Lambert function

( Corless et al., 1996), namely the solution of the equation

W0

(

x

)

eW0(x)=x, with W0( x) ≥ −1 for x≥ −1/e.

4.1.1. Completemartensitictransformationalongtheinneredgeof thering

The complete martensitic transformation is achieved at r_iwhen the condition

σ

AM

e

(

ri

)

=

σ

f is met, namely for rs=

η

riwhere

η

=



γ

−

δ

1−

δ

>1. (4.11)

According to ( 4.6), condition ( 4.11) is achieved under the inter- nal pressure qm= 1 2

γ

+

δ

ln

η

2₋ri ro

η

2

_{. (4.12)}

The result ( 4.12) holds only if the tangential stress is posi- tive everywhere and the martensitic transformation did not start at ro yet, namely for

σ

_θAM

(

ri

)

≥ 0 and rs≤ ro or equivalently for

ra≤ r i≤ r b, where ra=ro

η



δ

ln

η

2₋

γ

_{, r} b= ro

η

. (4.13)

Note that for rings whose geometric ratios satisfy ro=

η

ri the martensitic transformation starts at ro just when the complete martensitic transformation is achieved at ri.

4.1.2. AppearingofacornerregionCatri

If the cylinder wall is thick enough, however, the tangential stress ( 4.5) 2may vanish at ri. In this case, the corner region C ap- pears at ribefore the martensitic transformation starts at ro. This condition occurs for

σ

AM

θ

(

ri

)

= 0 , namely for

δ

lnr 2 s r2 i −

(

1−

δ

)

r2s r2 i =

δ

+rs2 r2 o , (4.14) or equivalently r2 s r2 o =−

δ

r2i

(

1−

δ

)

r2 o+ri2 W0



−

(

1−

δ

)

r2o+ri2

δ

r2 o e



. (4.15)

According to ( 4.6), the corresponding normalized pressure is then qc=

δ

−

(

1−

δ

)

δ

r2 o

(

1−

δ

)

r2 o+r2i W0



−

(

1−

δ

)

ro2+r2i

δ

r2 o e



. (4.16)

The corner region C may disappear for r3<ri<r4, where r3 is

given by ( 4.10) and r2 4=



δ

− 1 +

δ

e2



r2 o, (4.17)

when the normalized pressure reaches the value qd1=

δ

−

(

1−

δ

)

δ

r2 o

(

1−

δ

)

r2 o+r2i W₋₁



−

(

1−

δ

)

r 2 o+r2i

δ

r2 o e



, (4.18)

where W₋₁ is the secondary branch of the Lambert function, namely the solution of the equation W₋₁

(

x

)

eW₋₁(x)_{=x, with}

W₋₁( x) <−1 for x≥−1/ e. Indeed, condition ( 4.14) is satisfied both for the internal pressures qc and qd1, where qc<qd1, so that un-

der both pressures the AM region extends till the inner radius

ri. Therefore, the cylinder wall is formed by an inner region AM in a mixture of phases and an outer austenitic region A both for q0<q<min{ q1, qc} and also for qd1<q<q1 if r3<r<r4 (see

Figs.4and 5).

4.2. Thick-walledcylindercomposedofthreeregions(M,AM,A)

For thick rings obeying the condition ro> h ri a fully marten- sitic region appears at ri for q= qm before the outer Austenitic region has disappeared ( Fig. 3h), where qm has been defined in ( 4.12). The fields ( 4.3) and ( 4.5) defined in terms of rshold true in the outer austenitic region A where rs<r<ro, and in the interme- diate region AM where rf<r<rs, respectively. According to ( 4.5) 3,

the achievement of the condition

σ

AM

e ( rf) =

σ

foccurs for

rs=

η

rf. (4.19)

Therefore, by using ( 4.19) the stress and displacement fields ( 4.3) and ( 4.5) in regions A and AM can be written in terms of Please cite this article as: E. Radi, Evolution of multiple Martensite variants in a SMA thick-walled cylinder loaded by internal pressure,

E. Radi / International Journal of Solids and Structures 0 0 0 (2018) 1–21 9

Fig. 5. Variation of the normalized internal pressures separating the admissible phase distributions within the wall thickness with the ratio between inner and outer radii, during the axisymmetric loading process, both for SMA material TA ( a ) and corresponding detail ( b ).

rfinstead of rs. Then, the radial stress and Martensite volume frac- tion at the inner radius rfof the AM region follow from Eqs.(4.5)1, 5

as

σ

AM r



rf



σ

s = 1 2



r2 s r2 o −

γ

−

δ

ln

η

2



,

σ

AM θ



rf



σ

s = 1 2



r2 s r2 o +

γ

−

δ

ln

η

2



, uAM r



rf



r =

σ

s E 1−

ν

2



r2 s r2 o +1+

ν

1−

ν η

2+

δ η

2−

δ

ln

η

2



,

ξ

AM rθ



rf



=1. (4.20)

By using Eqs.(3.25), ( 3.27) and ( 4.20), continuity of the radial and tangential stresses across the interface at r_f, namely

σ

AM

r

(

rf

)

=

σ

M r

(

rf

)

and

σ

_θAM

(

rf

)

=

σ

_θM

(

rf

)

, yields B1=− r2 f 2 r2 o

η

2_, B2= 1 2



_r2 f r2 o

η

2₋

_δ

_ln

_η

2₊

_δ



_η

2_{− 1}



_1+lnr 2 f r2 o



. (4.21)

The introduction of constants B1 and B2in Eqs. (3.25), ( 3.27)-

( 3.29) then provides also the distributions of radial stress and dis- placement within the fully martensitic region M, as functions of

rf. Finally, the relation between the normalized internal pressure

q= −

σ

M

r

(

ri

)

/

σ

s and the radius rf follows from ( 3.27) and ( 4.21) as q=−1 2



η

2



_r2 f r2 o −r 2 f r2 i



+

δ



η

2_{− 1}



_1+lnr 2 f r2 i



−

δ

ln

η

2



. (4.22)

4.2.1. DisappearingoftheouterausteniticregionA

The outer austenitic region A disappears from the outer part of the wall thickness when rs=ro, namely for rf=ro/

η

according to ( 4.19). Then, Eq.(4.22) provides the normalized internal pressure

qbthat makes the outer austenitic region A disappear qb= 1 2



r2 o r2 i − 1



+

δ

₂



η

2_ln

_η

2₋



_η

2_{− 1}



_1+lnr2o r2 i



, (4.23)

which holds for ri≤rb, namely for

σ

eAM

(

ri

)

≥

σ

faccording to ( 4.5) 3

for rs=ro.

4.3.Thick-walledcylindercomposedofthreeregions(C,AM,A)

For thick rings with ri<r4 a corner region appears at ri for

q=qc before the outer Austenitic region has disappeared ( Fig.3e), where qc has been defined in ( 4.16). The fields ( 4.3) and ( 4.5) defined in terms of rs hold true in the outer austenitic region

A, where rs<r<ro, and in the intermediate region AM, where

rc<r<rs, respectively. The vanishing of the tangential stress at the outer radius of the corner region rc, namely

σ

_θAM( rc) =0, then yields the following relation between the radii rsand rc

δ

lnr 2 s r2 c −

(

1−

δ

)

r2s r2 c −r2s r2 o =

δ

, (4.24) or equivalently r2 s r2 c =−

δ

1−

δ

W0



−1−

δ

δ

e 1+ r2 s δr2 o



. (4.25)

Correspondingly, the radial stress and Martensite volume frac- tion at the inner radius of the AM region rc follow from the intro- duction of ( 4.24) and ( 4.25) in Eqs.(4.5)1, 5written for r= rc:

σ

AM r

(

rc

)

σ

s =−

(

1−

δ

)

r2s r2 c −

δ

=

δ



W0



−1−

δ

δ

e 1+ r2s δr2 o



− 1



, (4.26)

ξ

AM rθ

(

rc

)

= 1−

δ

γ

− 1



r2 s r2 c − 1



=− 1

γ

− 1



1−

δ

+

δ

W0



−1−

δ

δ

e1+r 2 s/

(

δr2o

)



. (4.27) According to ( 4.25)–( 4.26), the corner region C appears under the internal pressure qc defined in ( 4.16), namely for rc= ri. By us- ing Eqs.(3.15)and ( 4.26), continuity of the radial stress across the interface at rc, namely

σ

rAM

(

rc

)

=

σ

rC

(

rc

)

, allows to obtain the con- stant C= rc ro



δ

+

(

1−

δ

)

rs2 r2 c



=

δ

rc ro



1− W0



−1−

_δ

δ

e1+ r2 s δr2 o



. (4.28)