# Quantification of the evolution of shape storage and recovery in thermally responsive shape memory polymers

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PAMM Proc. Appl. Math. Mech. 10, 333 334 (2010) / DOI 10.1002/pamm.201010159

Quantification of the evolution of shape storage and recovery inthermally responsive shape memory polymers

Holger Steeb1, and Rasa Kazakeviciute-Makovska11 Mechanics-Continuum Mechanics, Ruhr-University Bochum, D-44780 Bochum

This study demonstrates the relevance of strain and stress storage/recovery profiles for the thermomechanical behaviour ofthermally responsive Shape Memory Polymers (SMPs). It is shown how these experimentally determined profiles may be usedfor the development, calibration and validation of continuum-based models describing the shape memory effect in polymericmaterials. The presented methodology applies both to small and finite strain deformation problems under the assumption ofconstant strain and cooling/heating rates.

c 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction and data analysis

Previous investigations of shape memory polymers have primarily focused on the free strain recovery without regardingthe effect of constrain on the evolution of stress recovery. However, when the data from the free strain recovery is usedfor the calibration of constitutive equations and afterwards applied to the prediction of stress recovery, there is generallyobserved a discrepancy between theoretical results and experimental data (see Fig. 1b). This work shows that the development,calibration and validation of continuum-based models describing the shape memory effect in polymeric materials require thatboth unconstrained and constrained strain/stress recovery profiles must be determined in suitable set of experiments. For thecase a notational simplicity, it is sketched here a linear rheological, i.e. one-dimensional setting.The thermomechanical cycle used to characterize shape memory effect in polymers consists of four main steps (cf. [1] and Fig.1a): I) the sample is loaded from the initial state, = 0 and = 0, to the pre-defined strain = m at fixed high temperature = h and the overall stress-strain relation is determined; II) the sample is cooled down while the total strain is held at afixed strain level = m and the stress profile = () is determined. At the end of this step the low temperature = lis reached; III) the sample is unloaded while the temperature is held at = l and the corresponding stress-strain relation isdetermined; IV) the sample is heated at zero stress, = 0, and the function = () representing the unconstrained strainrecovery profile is determined.

temperature

strains

tress

p

m m

r u

th m

u

h

l

g

()

()

()

step I

step IIstep III

step IV

step IVa

a) b)

Fig. 1 Steps in thermomechanical cycles (a) and illustrative comparison of data with model predictions of the stress recovery (b).

The complementary experiment involves the same initial loading, followed by a cooling process at fixed pre-defined strainlevel and an unloading at low temperature. Then instead of heating at zero stress (step IV), the specimen is heated at fixedstrain level u that has occurred in the end of unloading. In this process (step IVa), the stress = () as a function oftemperature is determined. This function represents the constrained stress recovery profile. Thus, the stress profile (),the unconstrained strain recovery profile () and the constrained stress recovery profile () completely characterize theshape memory effect in polymers. Moreover, the analysis of available experimental data shows that there are no generalinterrelationships between the measured strain and stress profiles and therefore all three profiles are strictly needed for thedevelopment of constitutive models, besides the respective stress-strain relations at high and low temperature for differentvalues of pre-strain m.

Corresponding author E-mail: holger.steeb@rub.de, Phone: +49 234 32 23080, Fax: +49 234 32 14229

c 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim

334 Section 6: Material Modelling in Solid Mechanics

2 Calibration of constitutive models and discussion

According to the current viewpoints, the shape memory effect in polymeric materials is due primarily two concurrent pro-cesses: the transition from a rubbery behaviour dominated by entropic behaviour at high temperatures (above the transitiontemperature, either glass transition or melting temperature) to glassy or semicrystalline behaviour dominated by internal en-ergy at low temperatures (below the transition temperature) and the storage of the deformation gained at high temperaturesduring cooling. In the processing mathematical modeling (constitutive theories), these physical mechanisms are accountedfor through two variables, the scalar variable (frozen fraction) describing the volume fraction of the frozen phase (glassy orsemicrystalline) and (in general) a tensorial quantity (stored or frozen strain) describing the deformation or strain that is storedin the material during freezing. Accordingly, the general uniaxial constitutive relation must be of the form

= (,, , s) , (1)

with the frozen fraction and the stored strain s considered as state variables for which appropriate evolution eqs. must bespecified besides the form of the response function . The general constitutive law (1) can now be specified for each step ofthe thermomechanical cycle.In the first step of the thermomechanical cycle , the temperature = h = const and it is generally assumed that = 0and s = 0. During the third step, = l = const and = 1 while ls takes a constant value. In both cases, the generalconstitutive equation (1) yields the strain-stress relations at high and low temperature

= h () (,h, 0, 0) and = l () (,l, 1, ls

), (2)

respectively, which may be determined from standard uniaxial tests at different temperatures.The essence of theoretical modeling of the shape memory effect in polymeric materials lies in the form of the general consti-tutive eq. (1) specified for the steps II, IV and IVa of the thermomechanical cycle

m = (,, , s) , = (0, , , s) , u = (,, , s) (3)

with m and u being given strain values (cf. Fig. 1a). Moreover, numerous phenomenological models for shape memorypolymers proposed in the literature, e.g. [1-3], are based on the assumption that both the frozen fraction and the stored strainare functions of temperature only, i.e. their evolution laws were assumed in the form = () and s = s (), respectively.Under this assumption, the constitutive relations (3) reduce to the form

m = c (,) (,, () , s ()) , (4) = h () (0, , () , s ()) , (5)u = s (,) (,, () , s ()) . (6)

Now, it could be easily observed that the response functions appearing in the constitutive laws (4)-(6) must correctly representthe corresponding measured strain and stress profiles. Specifically, the constitutive law (5) yields the strain as function oftemperature for the unconstrained strain recovery and thus the response function h () corresponds to the measured strainrecovery profile (). Furthermore, for any pre-defined strain level m, the constitutive equation (4) can be solved to givethe stress as a function of temperature, = c (), which represents the step II of the thermomechanical cycle and thus itcorresponds to the experimentally determined stress storage profile (). Finally, the constitutive law (6) can be solved forany value of u thus giving the stress = s () as a function of temperature during the step IVa so that the resulting responsefunction s () corresponds to the measured constrained stress recovery profile ().The presented methodology provides a foundation for the validation and calibration of existing continuum-based models forshape memory polymers. In particular, it should be apparent that if all three measured profiles are used to calibrate specificconstitutive models of the type considered in [1-3], the theoretical predictions will fully coincide with the experimental data.Moreover, for this class of constitutive models, the measured profiles serve for the determination of evolution laws for thefrozen fraction and the stored strain by the method somewhat similar to the one presented in [2]. Further details of thisapproach will be discuss in a forthcoming contribution.

References[1] Y. Liu, K. Gall, M. L. Dunn, A. R. Greenberg and J. Diani, Int. J. of Plasticity 22, 279-313 (2006).[2] Y. C. Chen and D. C. Lagoudas, J. of Mech. and Physics of Solids 56, 1766-1778 (2008).[3] R. D. Siskind, Model development for shape memory polymers. PhD theses (North Carolina State University, 2008), p. 123.

c 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim www.gamm-proceedings.com

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