Modelling the thermomechanical behaviour of shape memory ... - NOPR

Indian Journal of Engineering & Materials Sciences Vol. 18, February 2011, pp. 15-23

Modelling the thermomechanical behaviour of shape memory polymer materials Rahul P Sinha, Chetan S Jarali* & S Raja Structural Technologies Division, National Aerospace Laboratoires, Bangalore 560 017, India Received 15 September 2010; accepted 10 February 2011 The increasing applications of smart structures in technology development have led to a wide variety of new high performance materials, especially smart polymers. In the present work, a uniaxial model for shape memory polymer (SMP) is analyzed and a consistent form of the constitutive law is proposed. The coefficient of thermal expansion is expressed using the rule of mixtures approach. In this study, a comparison is made between thermal strains computed by using the analogous rule of mixture equation and the empirical relation that has been stated on the basis of experimental findings. The inconsistency in the computation of coefficient of thermal expansion is addressed. Furthermore, the consistent form of the stored strain and thermal strain expressions are also presented using fourth order Runge-Kutta method. A noteworthy difference is observed due to change in temperature. Moreover, the difference between the proposed analytical solutions with the empirical relation is brought out. Finally, the consistent form of the constitutive model with modified CTE and strain expressions for the SMP is presented. Keywords: Shape memory polymer, Shape memory effect, Constitutive model

Modelling the behaviour of shape memory polymer holds the key for using it in structural applications, such as shape memory composites. The stress-straintemperature correlations provide a good insight to understand the material behaviour. The shape memory polymers (SMP), at the very outset are bi-phase materials, consisting of a frozen phase (stiff) and active phase (compliant). SMP is a class of smart material, which shows thermomechanical behaviour under the applied thermomechanical stimulus. More specifically, SMP is able to memorize one or more shapes, which are determined by a network of polymer elasticity. Also, SMP can be stored in temporary shapes by material immobilization, commonly through crystallization1. Such type of behaviour in polymeric materials is also termed as shape memory effect (SME). As compared to conventional polymers, the SMP is able to effectively recover almost all of the residual strain (nearly 400%) because of a micro-Brownian displacement of the atoms in the polymer network, when heated above the glass transition temperature (Tg + 10 to 20°C). This is similar to free recovery behaviour of shape memory alloys (SMA), which can recover up to 8% of residual strains upon heating to a temperature above austenite finish. The stress developed in the SMP to sustain a large strain of 50-100% is nearly 25-50 MPa as _________ *Corresponding author (E-mail: [email protected])

compared to SMA, which develop nearly 500-600 MPa stresses with 6-8% of residual strain. Currently, research is being carried out to develop constitutive models to predict the thermomechanical behaviours of shape memory polymers. In SMP, nonlinear strain appears nearly above 3% of the total stain due to the viscoelastic property. Large stains are recovered during the heating cycle of SMPs. A nonlinear model is proposed wherein a nonlinear term is defined using a power function of stress1. All the model parameters are computed based on experiments and material properties are expressed exponentially as function of temperature. Abranhamson et al.2 proposed a nonlinear model based on the theory of viscoelastoplasticity. All the model parameters are determined experimentally and the solution to the constitutive relation is achieved using Euler’s method of integration. It is found that the thermomechanical behaviour computed by the model is in close accordance with the experimental results. A nonlinear thermomechanical material model of SMP is developed in which the coefficients are expressed using a single exponential function of temperature3. Further, based on the experimental results and the molecular mechanism of shape memory, a threedimensional internal state variable based small-strain constitutive model is proposed4. The model quantifies the storage and release of the stored strain during thermomechanical processes. However, some

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INDIAN J. ENG. MATER. SCI., FEBRUARY 2011

inconsistency in material parameters and the constitutive law is observed. As a result, the material constants are redefined along with the constitutive law5. In the present work, the proposed modifications in the model5 are further validated with the experimental results and shape fixity, shape recovery and recovery stress are computed.

exhibit a fully recoverable linear elastic behaviour, which is similar to pseudoelastic behaviour in SMA. However, small percentage of residual strain may be present in SMPs. If an undeformed polymer is cooled below Tg and then deformed, it will return to its previous undeformed state once the external forces are removed.

SMP Shape Recovery Process Shape memory polymers (SMPs) are materials that are rubbery in nature, composed of long, intertwined polymer chains (cross linkages) and whose shape recovery processes are entropy driven. Under tension, the polymer chains stretch to accommodate the deformations. Due to stretching, the number of possible network configurations of a chain reduces resulting in a corresponding decrease in the configurational entropy. The tension in the chain is not due to change in energy, as compared to shape memory alloys but rather due to a change in entropy. The shape recovery process for SMPs begins at a temperature above the glass transition temperature, Tg. At this temperature, the polymer is in a rubbery elastic (active) state and is deformed with an applied stress (or strain). Next, it is cooled below Tg at which the active polymer becomes inelastic (frozen or glassy) with increase in stress. After the applied deformation is removed, the polymer remains in its deformed state because the thermally reversible chains are constrained from recovery below Tg. The resulting deformation creates a residual strain which is completely recovered when the polymer is heated to a high temperature (Th) above Tg. It is important to note that polymers at low deformation levels (compared to what they are capable of sustaining)

Constitutive Modelling of SMP One of the key aspects in modelling the behaviours of SMP for their practical applications is to correctly define and determine the glass transition critical temperature of SMP. During the process of glass transition, an important phenomenon observed is that material properties change very rapidly. Hence, it would be appropriate to define the glass transition temperature as a function of material properties as shown in Fig. 1. In the model reported by Liu et al.4, one of the most fundamental quantities defined is the frozen phase fraction as a function of temperature. The volume fraction of frozen phase, and in other words, its variation with respect to temperature can be used to determine the Tg. The frozen fraction (φ f ) is

Fig. 1―The minimum for the derivative of frozen fraction with respect to temperature occurs at T=346 K whereas the value mentioned in Ref.3 is 343 K

defined using a modified sigmoidal function as:

φf =1−

1 1 + ch (T − Th ) n

… (1)

where parameters ch and n are determined independently for SMP using the method of curve fitting. T and Th are the working temperature and high temperature of the SMP, respectively. The variation in the frozen fraction during cooling and heating is shown in Fig. 2. The model parameters are given in Table 1.

Fig. 2―Variation of frozen fraction, φf, as function of temperature during heating and cooling

SINHA et al.: MODELLING THE THERMOMECHANICAL BEHAVIOUR OF SHAPE MEMORY POLYMER

Table 1―Material properties of the matrix (m)4

εS = ∫

0

Coefficients n = 4; Cf = 2.76105 / K 4 ; Ei = 813 MPa; N = 9.86 ×10−4 αf = 1.0 ×10−4 ; αa = 1.8 x10−4 ; α = (3.16 × 10−4 + 1.42 × 10−6 T ) / K

Constitutive modelling of the SMP is based almost completely on the concept of frozen phase fraction and indirectly on the temperature of the specimen.

φf =

V frz V

φ frz

; φa = 1 − φ f

… (2)

where V , V frz and φa are the total volume of SMP, volume of the frozen phase, and volume fraction of the active phase, respectively. A fundamental assumption made is that stresses in both the phases are equal

σ = σ f φ f + σ a (1 − φ f )

… (3)

The total strain (ε ) is then defined as,

ε = ε f φ f + ε a (1 − φ f )

ε f e ( x ) dφ

where ε f is the strain in the frozen phase, which comprises of three quantities ,viz, (i) Average of frozen entropic strain (ε f e ) which is completely

… (6)

It is important to note that the stored strain is anisotropic in nature and may increase in a particular direction. The thermal strain (ε T ) is defined as: T

ε T = ∫ α (φ f ) dT

… (7)

T0

where T and T0 are the working and reference temperatures, respectively. The coefficient of thermal expansion is written as

α = φ f α f + (1 − φ f )α a

… (8)

where, αf = 0.9 × 10-4/K, αa = 1.8 × 10-4/K. The thermal strain is computed numerically using the Simpson’s 1/3rd rule. The strain (ε m ) corresponding to the mechanical deformation is further evaluated using the following relation

ε m = [(φ f / Ei ) + (1 − φ f ) / Ee ]σ . … (4)

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… (9)

where Ei is the modulus contributing to the internal energetic deformation and Ee is the modulus resulting from the deformation due to change in entropy. Finally, the expression for the total strain is

ε total = ε s + ε T + ε m .

… (10)

locked and stored during the pre-deformation strain and does not contribute to the evolution of stress. It quantifies the extent of strain stored in the frozen phase as a function of temperature. (ii) Thermal strain (ε f T ) , (iii) Internal energetic deformation (ε f i ) . The

where ε s is the strain evolved due to cooling and ε T is the strain due to the change in temperature. The constitutive relation is now written as:

strain due to active phase ε a , comprises of two

σ = E (ε total − ε s − ∫ α dT ) .

T

T

quantities, (a) thermal strain (ε a ) , and (b) external stress induced entropic strain (ε a e ) , which is the stress-induced strain in the active phase. The additive composition of the strains in the two phases is written as follows 1 V frz e ε f ( x)dv + φ f ε f i V ∫0 + (1 − φ f )ε a e + φ f ε f T + (1 − φ f )ε aT

ε total =

where α is the coefficient of thermal expansion of SMP. The overall Young’s modulus ( E ) is expressed by using the rule of mixtures, E=

… (5)

where i is associated with the internal energy and e represents the elasticity entropy. Since dV / V = dφ , the stored strain (ε S ) can be defined as:

… (11)

Th

1 (φ f / Ei ) + (1 − φ f ) / Ee

… (12)

The expression for elasticity modulus is analogous to the expression of springs in series. A series arrangement implies that the stresses experienced by both the phases are equal. The frozen phase and active phase are discontinuous and strains in the material are

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not equal. Hence, owing to the discontinuity, it is assumed that both phases experience equal stress. Another assumption is that as the temperature is decreased during cooling, a part of the active strain which is taking part in mechanically induced deformation converts to frozen entropic strain. The entropic strain ε a e (T ) , that develops in the active phase is governed by the Hooke’s law. During cooling with a change of temperature dT , a fraction of the active material dφ will be frozen and transforms into a new fraction of the frozen material. As a result, the current active entropic strain, ε a e ( x(T )) , is fixed as the

stored

strain

ε f e ( x)

ε f e ( x) = ε a e ( x(T )) = ε a e (T ) =

such

that

σ . Hence, Ee (T )

dφ f dε s σ dφ f = ε f e ( x) = = dT dT Ee (T ) dT T    E (ε total − ε s − ∫ α dT )   dφ f Th d ε s σ dφ f  = =  dT Ee dT  Ee (T )  dT      

The boundary condition at T = 358 K is φ f = 0, which implies that α a = 1.9236 × 10−4 / K . From the available model3, a discrepancy is observed and the value of α a at 358 K can be assumed to be a constant (1.8×10-4/K). Another observation is made for T = 273 K with boundary condition φ f = 1 , which clearly implies that α f = 0.7155 × 10−4 / K . Discrepancy is further observed between the calculated value and those presented in the model3. Hence, without loss of generality, if the values of α f and α a are constants, then the boundary conditions are not satisfied. Therefore, it is required to determine the variation of α f and α a as function of temperature. Following Liu et al.4, the values have been recomputed so that the boundary conditions of Eq. (16) are satisfied.

… (13)

… (14)

Equation (14) is a first order single degree differential equation in a non separable variable form. Hence, the numerical technique of Runge-Kutta 4th order is applied to solve this differential equation to obtain the stored strain. The initial condition employed is ε S (Th ) = 0 .

α f = 1.2 × 10-4 / K and α a = 1.8 × 10-4 /K

… (17)

In the model of Tobushi et al.3, the average value of thermal coefficient of expansion over short range of temperature is considered. That is, for α f , the range that has been considered is from around 274 K to 291 K whereas for α a , the range is from 350 K to 357 K. This conclusion is purely based on observation of thermal strain graphs and process of curve fitting. The variation in analytically and empirically computed thermal strain is shown in Fig. 3. Next, using the values of α f = 1.2 × 10−4 / K and α a = 1.8 × 10−4 / K the variation in the thermal coefficients for frozen and active phases is compared with respect to temperature

Modelling the Thermal Strain The thermal strain can be computed either through experimental curve fitting or by analytical expression. Therefore, it is possible to define the following equality condition (ε T ) analytical = (ε T )empirical

… (15)

Equation (15) is differentiated on both sides, and ∂ε using the relation α = , the resulting expression is ∂T written as3:

α f φ f + α a (1 − φ f ) = −3.16 x10−4 + 1.42 × 10−6 T … (16)

Fig. 3―Comparison between analytical and empirical thermal strain

SINHA et al.: MODELLING THE THERMOMECHANICAL BEHAVIOUR OF SHAPE MEMORY POLYMER

Fig. 4―Variation of αa with temperature keeping αf constant

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Fig. 5―Variation of αf with temperature keeping αa constant

in Figs 4 and 5, respectively. Finally, using the values of coefficients as given in model3, α a = 1.8×10-4/K and α f = 0.9×10-4/K, the thermal strain is computed analytically and compared with empirical result in Fig. 6. Modelling the Stored Strain The differential form given in Eq. (14) is rewritten in a form that can be solved using method of integration factors as: T

d ε s E dφ f + εs = dT Ee dT

∫ α dT )

E (ε total −

Th

dφ f

Ee

dT

;

… (18) Fig. 6―The variation of analytically derived thermal strain and empirically derived thermal strain

dy Pdx + Py = Q ; I .F . = e ∫ dx T

where P =

E dφ f and Q = Ee dT

∫ α dT )

E (ε total −

Th

dφ f

Ee

dT

.

Therefore, Eq. (18) is rewritten as

ε s (e

E dφ f dx e dT

∫E

) T

T

= ∫ (e Th

E dφ f dx e dT

∫E

E (ε total −

∫ α (φ f ) dT ) h

dφ f

Ee

dT

T

)dT

… (19)

Equation (19) cannot be thus numerical approaches such, the Simpson’s rule expression for α (φ f ) . For

solved analytically and must be employed. As is used to solve the solving this, MATLAB

provides a function “quad” to approximate the integral function to within an error of 10-6 using recursive adaptive Simpson quadrature. After obtaining the solution for thermal strain using numerical method for integration, evaluation of stored strain can be implemented. The stored strain in Eq. (19) is a first order single degree and it is in a non variable separable form of the differential equation, which is solved using the 4th order Runge-Kutta method. Therefore, the problem is redefined using this method by the following equation,

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ε s' (T , ε s (T )) T

E (ε total − ε s − =

ε s (Th ) = 0

∫ α dT )(−nch (Th − T )

n −1

) and

Th

Ee (1 + ch (Th − T ) n )2

… (20)

where, 1 ε s (Tn+1 ) = ε s (Tn ) + h(k1 + k2 + k3 + k4 ); Tn +1 = Tn + h 6

with ‘h’ is the step size of temperature. The slopes k1 to k4 are the slopes of the intervals defined as below, hk h k1 = ε's (Tn , ε s (Tn )); k2 = ε's (Tn + , ε s (Tn ) + 1 ); 2 2 hk2 h ' ' k3 = ε s (Tn + , ε s (Tn ) + ) ; k4 = ε s (Tn + h, ε s (Tn ) + hk3 ) 2 2

… (21) Thus, the incremental value ε s (Tn+1, ε s (T )) is determined by the present value ε s (Tn , ε s (Tn )) plus the product of the size of the interval (h) and an estimated slope (k). The slope is a weighted average of slopes (k1 to k4). In averaging the four slopes, greater weight is given to the slopes at the midpoint 1 such that average slope is k = (k1 + k2 + k3 + k4 ) . 6 The boundary condition available is ε s (Th ) = 0 , which can be used to compute stored strain at any temperature. The idea of recursion can be used efficiently to solve the problem of large number of computations for every step since the value of ε s (Tn ) is used in the computation of ε s (Tn +1 ) . Therefore in the present work it is proposed that for every successive calculation of stored strain, the initial condition can be updated to be that of the value of stored strain at the previous value of temperature. The arguments passed in the function, that returns stored strain, will now include an added term. It is the stored strain computed at the previous temperature ε s (Tn ) , which is used instead of ε s (Th ) = 0 for the computation of ε s (Tn +1 ) .

Flexible Constraints Modelling for Applications of SMP In practical applications, SMP is typically predeformed, maintained at low temperature and then utilized in an application environment. For example, in a surgical application, the pre-compressed shapememory polymer coil is taken out from the low temperature storage environment and then inserted into an artery without deforming the artery at this point. During the temperature-controlled recovery process, the SMP is subjected to a flexible constraint from the surrounding tissue, and a reactive compressive stress is produced inside the device. In contrast, when a predeformed suture is used to close a skin laceration a reactive tensile stress in generated. These situations suggest the use of 1-D bi-material models, in which two materials, the pre-deformed shape memory polymer and a constraining material, are assembled in parallel or series as long as the small-strain condition is satisfied. In parallel configuration, the strain in the SMP and constraining material are the same while the stresses generated are equal in magnitude and opposite in sign. In case of serially bound devices, the stresses generated are the same while the strains are opposite in sign but equal in magnitude. In the present analysis, constraining material and SMP are in parallel as shown in Fig. 74. For the predeformed polymer, it possesses a fixed strain ε fix at low temperature Tl. The length which includes the deformation also equals the undeformed length of the constraining material Lc. Therefore, Lc = L (1 + ε fix ) .

… (22)

Next, when the recovery process begins, the strain change in the polymer is ∆ε.

ε total = ε fix + ∆ε

… (23)

A condition is now imposed in the course of analysis, of not including the contribution of thermal strain (ε T ) , then the total strain in the constraining material becomes

ε c = (∆ε ) L / Lc .

… (24)

But L / Lc =1/(1 + ε fix ) so the final expression for strain in the constraining material is

εc =

(ε total − ε fix ) (1 + ε fix )

.

… (25)

SINHA et al.: MODELLING THE THERMOMECHANICAL BEHAVIOUR OF SHAPE MEMORY POLYMER

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Fig. 7―Simplified models for prediction of stress and strain recovery response of pre-deformed shape memory polymer under flexible constraint (darker shaded parts represent SMP while lighter shaded parts represent constraining material)

Using the condition σ SMP = σ constraint , the stress in the SMP and constraining material can be computed using the following relationship.

σ constraint = − Ec (ε total − ε fix )/(1 + ε fix )

… (26)

Results and Discussion The first cycle of thermomechanical loading involves application of stress at a high temperature (T = Th) above the glass transition temperature Tg. During the process, two conditions are used; namely, φ f (Th ) = 0 and ε s (Th ) = 0; ε T (Th ) = 0 . The strain value at the end of the cycle is the desired pre deformation ε pre required for shape fixation. In the second cycle the condition of ε total = ε pre is used as the boundary condition. As the temperature is decreased, the thermal stress increase slowly and after the temperature of the specimen falls below Tg, it further increases at faster rate. This increase in stress is computed and compared for validation in the Fig. 8. On the contrary, the stored strain ε s on the other hand increases rapidly initially and then steadily as the temperature is cooled below Tg. A very good agreement is clearly observed between the present computation and that of the experiment4. The thermal strain, having a negative value exerts a compressive uniaxial force, and therefore a tensile force is required to be applied to maintain the strain uniformity ε total = ε pre . In the third cycle, the temperature is maintained at Tl and the specimen is unloaded completely. During the unloading, the modulus of elasticity has a very high value (750 MPa). Further,

Fig. 8―Stress response of the SMP during cooling under prestrain constraint

the stored strain ε s and thermal strain ε T have a constant value as they depend only on temperature. Therefore, only ε m changes in magnitude. It is in this stage that shape fixation can take place and shape fixation for SMP is achieved by freezing the conformational changes that take place during the predeformation stage. This is mathematically expressed as follows:

ε fix = ε s (Tl ) + ε T (Tl )

… (28)

In the last cycle, the specimen is stress free (boundary condition is σ = 0 ) and ε m does not contribute to the total stress. The variation in temperature produces changes in the value of frozen fraction and therefore, changes in the thermal strain and stored strain is observed.

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The strain recovery during heating cycle is computed and compared with available results in Fig. 9. Again it is worth mentioning that the modifications carried out in redefining the model4 agree well with that of the experiment. The total strain during the heating cycle is computed as follows: T

ε total = ε s + ∫ α dT

… (29)

Th

The value of stored strain cannot be obtained independently in the last phase as there is one expression (Eq. (2)) and two unknowns ε total and ε s . Therefore, the value of stored strain is obtained from the values of ε s for the cooling phase for the corresponding temperature. If different constraints are applied to the specimen during the recovery process, the stress-strain-temperature relations are obtained using these constraints as boundary conditions. The stress and strain recovery for different ratios of Ec and ESMP is computed and compared for validation in Figs 10 and 11, respectively. It is observed that higher the value of elastic modulus for the constraining SMP material, the lower is extent of stress recovery. Similarly, the strain recovery is also significantly influenced as evident from the result (Fig. 11). These changes are due to the proposed modifications adopted in the modeling approach. There are two cases analyzed using the model4. The first case is ε total = ε pre , and second is ε total = ε fix . In the first

Fig. 10―Stress response of pre-tensioned SMP under flexible constraint with various modulus ratios

case, the behaviour computed by SMP should be Fig. 11―Strain response of pre-tensioned SMP under flexible constraint with various modulus ratios

Fig. 9―Strain recovery response during heating for SMP fixed deformation at ε=0.086

Fig. 12―Stress response of the SMP during heating under prestrain constraint ε=0.091

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observed by heating above the glass transition temperature. Next, the plots using different ratios of Ec and ESMP are computed. It is noted that higher the value of elastic modulus for the constraining material, the lower is the extent of stress recovery.

Fig. 13―Stress recovery response during heating for SMP under fixed strain constraint ε=0.086

Fig. 14―Stress-strain-temperature thermomechanical response of SMP at Tg=70°C

different from the one for cooling due to hysteresis in the material, however in the model4, there is no such inclusion for hysteresis and the cooling and heating curves completely overlap each other as shown in Fig. 12. The second case shown in Fig. 13 represents many of the cases for practical applications of SMP where it is constrained at a fixed strain under low temperature. Further, the phenomenon of shape recovery can be used by increasing the temperature. As such, the complete thermomechanical cycle is computed in Fig. 14. A complete recovery of strain is

Conclusions Constitutive model for the shape memory polymer is addressed to predict the uniaxial stress-straintemperature response. The model is simulated in MATLAB using numerical methods and the results obtained are compared with available experimental results. Runge-Kutta’s 4th order method is used to solve the differential equation for stored strain. The inconsistencies in modeling the thermal strain and its effect on stored strain are discussed. A conclusion is drawn that coefficient of thermal expansion is a nonlinear function of temperature contrary to what is proposed in the reference model3. Attempts are made to resolve the shortcomings in the model by using numerical methods. Finally, the analytical expression for thermal strain and stored strain are presented in a consistent form, and results are successfully validated. Acknowledgement The author Chetan S Jarali would like to acknowledge the financial assistance provided by the Council of Scientific and Industrial Research (CSIR), Ministry of Science and Technology, Government of India, through National Aerospace Laboratories (NAL), under Quick Hire Scheme (QHS) as Scientist Fellow (SF). References 1 2 3 4 5

Abrahamson Erik R, Lake Mark S, Munish Nassem A & Gall Ken, J Intell Mater Syst Struct, 14 (2003) 623-632. Mather Patrick T, Luo Xiaofan & Rousseau Ingrid A, Ann Rev Mater Res, 39 (2009) 445-471. Tobushi Hisaaki, Okumura Kayo, Hayashi Shunichi & Ito Norimitsu, Mech Mater, 33 (2001) 545-554. Liu Yiping, Gall Ken, Dunn Martin L, Greenberg Alan R & Diani Julie, Int J Plast, 22 (2006) 279-313. Jarali Chetan S, Raja S & Upadhya A R, Smart Mater Struct, 19 (2010) 105029.