Finite strain 3D thermoviscoelastic constitutive model for shape memory polymers.
Shape memory polymers (SMPs) have the capability to retain a temporary shape at low temperature that was created during high temperature deformation. When the material in the temporary shape is heated above a critical temperature, the material recovers its original shape when unconstrained. The "shape storage" and "shape recovery" properties of the polymer are due to a reversible change of state from rubbery to glassy and from glassy to rubbery, respectively. Consequently, the change of state in most SMPs is linked to the glass transition temperature of the material. The mechanisms controlling the thermally induced shape memory effect in polymers are very different from those operating in shape memory metals or ceramics, resulting in differences in observed behaviors. For example, in most SMPs, it is not possible to cycle between the temporary and permanent shape under pure thermal loading, an effect called two-way shape memory in alloy systems. Once the polymer has recovered its original shape under heating, it is necessary to reapply thermomechanical constraint to rememorize the temporary shape. SMPs have a predominantly amorphous structure with either physical or chemical cross-links, resulting in a network structure. In the rubbery state, above the glass transition temperature, [T.sub.g], the cross-linked SMP networks can undergo very large elastic deformations, similar to elastomers, which involve changes in entropy. Below [T.sub.g], SMP networks deform to moderate strains and the stress--strain response is driven by changes in internal energy.
The majority of research on SMPs has focused on experimental observations, physical understanding, and emerging applications [1-10]. Few studies have addressed the constitutive modeling of the thermomechanical shape memory cycle in polymers. Essentially, two approaches have been used to describe the thermomechanical behavior of SMPs. The first approach is based on micromechanics modeling of the material during the change of state. The material is assumed to be "soft" at high temperature (above [T.sub.g]) and "frozen" at low temperature (below [T.sub.g]). During the glass transition, a fraction of the material is in glassy state while the remainder of the material is in rubbery state and a rule of mixtures is applied as a micromechanical constraint . The second approach is based on well-known standard linear viscoelastic approaches commonly used to predict the thermomechanical properties of polymers [12, 13].
Despite the fact that one of the main advantages of SMPs is their large strain deformability, current constitutive models have been developed only in the context of infinitesimal strains [11-14]. To account for large strain deformations, we develop a 3D thermoviscoelastic constitutive model formulated in finite strains. This model is based on the viscoelastic properties of crosslinked SMP networks and is thermodynamically motivated.
In the next section, we will summarize the thermomechanical response of SMPs using the data of Liu et al. . In presenting the data, emphasis is placed on physical interpretation of the energy transfer and stress evolution during a shape memory thermomechanical cycle. In the following section, 3D, finite-strain, and thermodynamically motivated constitutive model is proposed. This section also provides model predictions of the stress and strain change during a classic thermomechanical cycle along with comparison to experimental data from Liu et al. . The article ends with brief conclusions.
THERMOMECHANICAL BEHAVIOR OF SMPS
A heavily crosslinked epoxy network is considered in the experimental studies of Liu et al. . The epoxy material demonstrates a rubbery plateau above [T.sub.g] on a plot of modulus versus temperature, a requirement of a polymer to demonstrate a useful shape memory effect. The elastic properties of the epoxy have been characterized above and below the glass transition temperature, and the material behaves as an isotropic material. The glassy elastic modulus is ~750 MPa, while the rubbery modulus is on the order of 8.8 MPa at [T.sub.h] = 358 K. The thermal expansion coefficient of the polymer was measured to be 9 X [10.sup.-5] [K.sup.-1] below [T.sub.g] and 1.8 X [10.sup.-4] [K.sup.-1] above [T.sub.g].
The shape memory epoxy has been submitted to a 5 step thermomechanical loading cycle outlined below:
* Step 1: Temperature is raised above [T.sub.g].
* Step 2: The material is subjected to a set "pre-strain".
* Step 3: Temperature is decreased below [T.sub.g] (at a cooling rate of 1 K * [min.sup.-1]) while the prestrain is held constant.
* Step 4: Stresses are released by removing the prestrain constraint.
* Step 5: Temperature is raised above [T.sub.g] (at a heating rate of 1 K * [min.sup.-1]) while strain is held constant (the sample size is not allowed to change except for the release of stresses that occurred in Step 4).
Three different prestrain conditions were considered in Step 2: 9.1% uniaxial extension, 9.1% uniaxial compression, or no strain. As above [T.sub.g], the material is in the rubbery state, it can undergo very large elastic deformations with resulting low stresses. Although the applied experimental strains are small, they are sufficient for formulation of the constitutive model, which will then be capable of predicting geometrically nonlinear large strain deformations. Nonlinear deformations resulting from material nonlinearity (e.g. chain alignment) would need to be added to the present framework if they were observed within the imposed deformation limits. Often, SMPs are only deformed within the limits of geometric nonlinearity and not material nonlinearity, since the latter often leads to material damage.
[FIGURE 1 OMITTED]
During Step 3, the stress changes were monitored as a function of temperature, as shown in Fig. 1. One observes that the thermal expansion does not generate appreciable stress in the direction of stretching as long as the material is in the rubbery state. The volume change of the polymer is accommodated by expansion in the transverse directions and variation of the stress due to the thermal contraction in the direction of tension is small due to the low value of the Young's modulus above [T.sub.g]. When the material is in the glassy state, the thermal contraction during cooling induces a positive stress, which increases linearly with the temperature. The comparison between the three applied loading states in Fig. 1 is critical to the formulation of a constitutive model, and has not been previously discussed by Liu et al. . In particular, the stress difference imparted above [T.sub.g], at the beginning of the cooling in Step 3, is identical to the stress offset below [T.sub.g] at the end of the cooling. The uniformity of the stress offset during strain storage (Step 3) is crucial for defining stresses and stress evolution in the model. During cooling and material "freezing", the stresses that come from the prestrain at high temperature, or equivalently from an entropy change, do not vanish because of the material change of state but remain intact. This indicates that contributions to the stress due to entropy change in the rubbery state remain while the material is transforming to a glassy state. This result is consistent with the physics of the material and must be incorporated into modeling efforts. At the molecular level, changes in entropy are caused by alterations of chain conformation. At high temperature, the free volume is large enough to promote conformational chain motions. While temperature decreases, cooperative chain motion become impractical and the chain conformations do not change unless the polymer is deformed past yield at a temperature below [T.sub.g]. The latter situation is beyond the range of application of SMPs, since they are typically not submitted to large deformation below [T.sub.g]. Therefore, while a prestrain is applied above [T.sub.g] and temperature is subsequently decreased below [T.sub.g], the chain conformation remains the one imposed by the prestrain and so the entropy contribution is converted to internal energy stored in the polymer.
[FIGURE 2 OMITTED]
At the end of unloading in Step 4, all three samples have sustained a slight compressive strain relative to the value they were deformed to during Step 2. The sample previously stretched to 9.1% strain was unloaded to a strain of 8.6%. Final strains of -0.4% and of -9.4% are observed after unloading in Step 4 for the samples that have been originally submitted to no applied strain, and a -9.1% compressive strain, respectively. If these contractions were exclusively due to the thermal strains in all three samples, they would be identical. On the contrary, we note that the contractions are 0.3% for the sample initially in deformed in compression, 0.4% for the sample that was initially undeformed, and 0.5% for the sample initially deformed in tension. Another source that contributes to the compressive strains during unloading is the material accommodation to the stresses resulting from the entropy change at high temperature. For the undeformed sample, no stress comes from the entropy change; therefore, the value of 0.4% is due to the thermal contraction only. For the sample that was initially deformed in tension, the positive stress due to the entropy change has to be accommodated by a contraction of the material in the glassy state, which explains a higher contraction for the sample initially in tension compared to the undeformed sample. On the other hand, for the sample initially deformed in compression, the material has to accommodate a compressive stress and thus expands. This explains the contraction of 0.3% only in the compressed samples.
In Step 5, stresses have been measured at a constant strain (sample size is fixed after unloading in Step 4) as a function of heating and are plotted in Fig. 2. At the beginning of the individual thermal cycles, the materials are in the glassy state and the restricted thermal expansion creates compressive stresses. Then, with increasing temperature, the glass transition occurs. Thermal contributions to the stresses vanish while the values of stress tend toward the expected responses of a material in the rubbery state. The temperatures at which the stresses start to increase depend on the state of stored strain in the material (tensile versus compressive versus no stored strain). This indicates that the glass transition temperature is modified with the stored prestrain of the specimen. The specimen initially deformed in tension has a [T.sub.g] lower than the initially undeformed specimen, which in turn has a lower [T.sub.g] than the sample initially deformed in compression. This phenomenon is similar to the phenomenon observed in physical aging experiments . Actually, the glass transition temperature is known to increase with compaction of chains. Also, it appears in Fig. 2 that the thermal stress at the beginning of the cooling process depends on the specimens' state of strains. As regards physical aging, an increase of the expansion coefficient, caused by the compaction of the chains, has been observed (see discussion in van der Linde et al. . The higher thermal stress measured in the specimen initially deformed in compression is likely due to a higher expansion coefficient caused by chain compaction.
As a first approximation, the changes in the glass transition temperature [T.sub.g] and in the thermal expansion coefficient will be neglected in the course of the modeling. This implies that the model will generate the same response for the three specimens during the heating process from the low temperature to [T.sub.g], at which temperature, different stress would be obtained according to the specimen state of strain. For this reason, only one complete cycle will be provided. More accuracy can be readily added to the proposed framework as more quantitative data is gathered at larger strain levels and the effects of strain state on glass transition temperature are even more significant. Finally, one can note that above [T.sub.g], the stress values resulting from the recovery of the sample are consistent with the initial stresses imparted above [T.sub.g] caused exclusively by changes in entropy. This implies that the stored strain and energy can cycle between entropic and enthalpic states without being quantitatively disturbed by the glass transition process.
Above [T.sub.g], polymers exist in a rubbery state, and changes in energy are due almost exclusively to changes in entropy (here we are restricting ourselves to the case of isovolume loading above [T.sub.g], since at constant temperature above [T.sub.g], volume change would result from internal energy change). Below [T.sub.g], in the glassy state, conformational chain motions are restricted and the material primarily undergoes changes in internal energy. We will assume that in both rubbery and glassy states, strains are elastic and no plasticity occurs through mechanisms such as chain slippage and crazing. The experimental results demonstrate that during the cooling process, stresses caused by a prior change of entropy remain. This indicates that a change in entropy must be taken into account in the overall temperature range. It is likely that the entropy change below [T.sub.g] will be very small (almost negligible) but nothing in the material behavior indicates that contributions should theoretically withdraw below [T.sub.g]. Therefore, the entropy change may be defined as a function of the total deformation for all temperatures.
On the other hand, changes in internal energy seem to occur only when changes of entropy are impractical. Hence, changes in internal energy are assumed to be involved in stress evolution only when the overall material is not in the rubbery state. We can thus define a reference state for the internal energy, as a state causing no contribution to the stress from the internal energy. This state is described by the state of the material while passing the glass transition temperature.
During the change of state, which is recognized in a range of temperatures [[T.sub.g] - [delta]T,[T.sub.g] + [delta]T], only part of the material is in the glassy state and contributes to the stress in terms of a change of internal energy. The heterogeneity of the material can be simulated by a partial contribution of the two deformation mechanisms. On a rheological basis, this is performed by decomposition of the deformation into a viscous part and an elastic part. Let us now formalize these remarks as equations to propose a rheological model corresponding to the described behavior.
The Helmholtz free energy [psi] is defined as
[psi] = U - T[eta] (1)
where U is the internal energy, T is the temperature, and [eta] is the entropy. The entropy is defined as a function of the total deformation characterized by the deformation gradient F. For the internal energy contribution, the deformation gradient is assumed to split into a viscous part Fv and an elastic part Fe defined by
F = Fe x Fv (2)
and U is only a function of Fv. Therefore, the Helmholtz free energy is written as
[psi] = U(Fe) - T[eta](F). (3)
On a rheological basis, relations in Eqs. 2 and 3, defining the Helmholtz free energy, are sketched by a Zener model, as shown in Fig. 3.
We will now detail the constitutive equations describing the model and use the model to capture basic thermomechanical shape memory cycles.
[FIGURE 3 OMITTED]
During a thermomechanical load, the total deformation F decomposes into
F = Fm x Fth (4)
where Fm accounts for mechanical deformation and Fth accounts for thermal deformation. Assuming isotropy, Fth is given by
Fth = [alpha](T - [T.sub.0])I (5)
where [alpha] is the thermal expansion coefficient and [T.sub.0] characterizes the reference temperature. Considering the rheological scheme in Fig. 3, the elastic deformation in the entropy and the internal energy branches are defined by
Fe = F x [Fth.sup.-1] in the elastic branch (entropy) (6)
Fe = F x [Fth.sup.-1][Fv.sup.-1] in the viscoelastic branch (internal energy).
Also from scheme 3, the total Cauchy stress is given by
[sigma] = [[sigma].sup.n] + [[sigma].sup.U] (7)
where [[sigma].sup.[eta]] is the portion of the stress due to the entropy change and [[sigma].sup.U] is the portion due to the internal energy change.
The entropy changes are based on the theory of rubber elasticity. Hence the entropy function may be defined by a neo-Hookean law :
- [eta]T = [[E.sup.r]/6][T/[T.sub.h]]([I.sub.l] - 3) (8)
where [I.sub.l] is the first invariant of the right Cauchy-Green tensor of the elastic deformation, C = [Fe.sup.T]xFe, and [E.sup.r] is the Young's modulus of the material at [T.sub.h]. Young's modulus is assumed to depend linearly on temperature; therefore, stress dependence on temperature is taken into account for the entropy contribution. The contribution of the entropic energy to hydrostatic deformation can be assumed as negligible and the Cauchy stress derived from Eq. 8 is given by
[[sigma].sup.[eta]] = [[E.sup.r]/3][T/[T.sub.h]]B - p1 (9)
where B = Fex[Fe.sup.T] is the left Cauchy-Green tensor and p is a Lagrange multiplier related to the boundary conditions.
In a finite strain framework, the stress contribution from the internal energy may be defined by
[[sigma].sup.U] = Le[ln(Ve)] (10)
where Le is the fourth-order elastic constant tensor, which can be reasonably considered to be temperature independent. The left stretch tensor, Ve, is obtained from the polar decomposition Fe = VeRe where Re is a rotation tensor. To compute [[sigma].sup.U], it is necessary to know Fe, or equivalently Fv. We will now define the evolution of Fv in light of remarks and observations in made earlier and consistency with the Clausius-Duhem inequality.
When the material is completely in the rubbery state, it has been noted that contributions of internal energy changes to stress evolution are null, which implies
For T [greater than or equal to] [T.sub.g] + [delta]T Fe = 1 [left and right arrow] [dot.Fv] = [dot.F]. (11)
While the material is entirely in the glassy state, applied strain is accommodated by a change of internal energy only at strains below the material yield strain. This is understood in terms of Fv by the relation:
For T [less than or equal to] [T.sub.g] + [delta]T [dot.Fv] = 1. (12)
To define the evolution equations during the change of state, we consider thermodynamic requirements. The Helmholtz free energy must satisfy the Clausius-Duhem inequality:
[1/2]S:[dot.C] - [[rho].sub.0]([dot.[psi]] + T[eta]) - [[q.sub.0]/T]gradT [greater than or equal to] 0 (13)
where S is the second Piola-Kirchoff stress tensor. Introducing
[dot.[psi]] = [[[partial derivative][psi]]/[[partial derivative]T]][dot.T] + [[[partial derivative][psi]]/[[partial derivative]C]][dot.C] + [[[partial derivative][psi]]/[[partial derivative]Fv]][dot.Fv] (14)
and, after some straightforward steps (see Govindjee and Reese , for example) Eq. 13 transforms into
[1/2](S - 2[[[partial derivative]T[eta]]/[[partial derivative]C]] - 2Fv[[[partial derivative]U]/[[partial derivative]C]][Fv.sup.-T]:C + 2[[[partial derivative]U]/[[partial derivative]Ce]]:(CeLv) - [[q.sub.0]/T]grad T [greater than or equal to] 0. (15)
From which one can extract three relations:
(a) Stress-strain relation: S = 2[[[partial derivative]T[eta]]/[[partial derivative]C]] + 2Fv[[[partial derivative]U]/[[partial derivative]C]][Fv.sup.-T]
(b) Mechanical dissipation: 2[[[partial derivative]U]/[[partial derivative]Ce]]:(Ce x Lv) [greater than or equal to] 0
(c) Thermal dissipation: - [[q.sub.0]/T] grad T [greater than or equal to] 0 (16)
where Lv = [dot.Fv][Fv.sup.-1] is the velocity gradient corresponding to Fv. For an isotropic material, Eq. 16b transforms into
(Ce x [[[partial derivative]U]/[[partial derivative]Ce]]):Dv [greater than or equal to] 0 (17)
where Dv is the symmetric part of Lv. For conditions in Eqs. 11 and 12, Eq. 17 is trivially satisfied. For T[member of]([T.sub.g] - [delta]T,[T.sub.g] + [delta]T), we define changes in Fv to satisfy Eq. 17, and
Dv = [1/[zeta]](Ce x [[[partial derivative]U]/[[partial derivative]Ce]]) (18)
is one simple solution. Parameter [zeta] is a parameter of viscosity.
The constitutive equations required to solve a thermomechanical cycle are given by Eqs. 6, 7, 9-12, and 18. Considering that the material elastic constants and thermal expansion coefficients have been estimated above and below [T.sub.g]. The only remaining free parameters are [delta]T and [zeta]. [T.sub.g] and [delta]T can be estimated in Figs. 1 and 2 and [zeta] will be calculated while fitting the material response.
Prediction of Experimental Data
The experimentally measured material properties are presented in Table 1. The theoretical responses corresponding to the thermomechanical cycle described earlier are compared to the experimental data.
First, Steps 2 and 3 have been estimated using parameters in Table 1, and noting that in Figs. 1 and 2, the glass transition seems to take place between 313 and 339 K. Therefore, [T.sub.g] and [delta]T are set equal to 326 and 13 K respectively. Comparisons between the model and the experiments are plotted in Fig. 4. A fairly good correlation is shown in Fig. 4. The experimental results demonstrate a smoother transition, caused by spatial variation in the glass transition behavior due to local heterogeneity in the epoxy network structure. The shape of the gradual transition could be better captured with the present model by using an evolution function for the transition region of another form than the one proposed in Eq. 18. Parameter [zeta] has been fitted for temperature within the glass transition T[member of][[T.sub.g] - [delta]T,[T.sub.g] + [delta]T]. The cooling rate is of [dot.[lambda]] = 1 K x [min.sup.-1]. It is not the study's objective to evaluate the ability of the model to capture the cooling rate effect, since the experimental data was available only at one rate. Therefore, we have made the assumption that [zeta] was cooling rate independent by setting [zeta] = [dot.[lambda]]x[eta], [eta] being constant and equal to [eta] = 100 MPa x [K.sup.-1].
[FIGURE 4 OMITTED]
Good agreement with experiments has been obtained in terms of the strain contraction after stresses are released in Step 3. Comparisons between the theoretical results and the experimental values are given in Table 2.
An in-depth comparison is now made for the sample that has been submitted to a compressive prestrain. The stress is evaluated as a function of temperature change for the 5 loading steps described in the second section using the parameters mentioned here. The thermoviscoelastic model provides a reasonable prediction of the overall material behavior as shown in Fig. 5. Compaction effects on thermal expansion and glass transition temperature are not presently accounted for. For this reason, the same curve is predicted by the model for the three samples below [T.sub.g], which explains the single result presented for the heating process. Also, the predicted shape of the stress evolution during heating and change of state is different than experimental observations. This discrepancy may be corrected by proposing another form of evolution of Fv from Eq. 18. This last equation is of a very simple form, which may be improved by better knowledge of the evolution of the material between glassy and rubbery state.
[FIGURE 5 OMITTED]
The present constitutive model provides a useful framework for predicting the thermomechanical response of SMPs at large strains. The constitutive model captures overall trends in the complicated thermomechanical response of SMPs (Fig. 5). The model is fit to basic material parameters derived from uniaxial stress--strain tests, free thermal expansion tests, and glass transition data. Such data is readily available for various polymers that show shape memory, and can be easily determined for emerging polymer systems. The shape of the stress and strain evolution curves during the thermomechanical cycle is strongly dependent on the evolution equations used in the model. As long as proposed evolution equations satisfy thermodynamic constraints, more sophisticated mathematical equations will improve point-to-point agreement between experimental results and modeling predictions (Fig. 5). The choice of specific evolution equations within the modeling framework depends strongly on the desired accuracy and intended application of the constitutive model. Future work should emphasize comparison between modeling predictions and experimental results for various applied temperature and strain rates and for large strain deformations, variables that are all relevant to emerging applications of SMPs.
A 3D thermoviscoelastic model has been proposed in the context of finite strains to represent the thermomechanical behavior of SMPs. This model is based on thermodynamic considerations and has been motivated by a mechanical understanding of stress--strain--temperature behaviors in SMPs. The model was formulated using the standard thermoviscoelastic theory in finite strains and shows favorable predictions of experimental data. In particular, the model accurately estimates the remaining strain during stress release and provides a reasonable prediction of the stress--temperature curves during constrained thermomechanical recovery of SMPs.
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Laboratoire d'Ingenierie des Materiaux, UMR 8006 CNRS, ENSAM Paris, 151 bd de I'hopital, 75013 Paris, France
Department of Mechanical Engineering, University of Colorado, Boulder, Colorado 80309-427
School of Materials Science and Engineering and George Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
Correspondence to: J. Diani; e-mail: firstname.lastname@example.org
Contract grant sponsor: DGA; contract grant number: ERE-046000011.
TABLE 1. Material parameters. Above [T.sub.g] Below [T.sub.g] Young's modulus (MPa) 8.8 750.0 Expansion coefficient 1.8 x [10.sup.-4] 9.0 x [10.sup.-5] ([K.sup.-1]) TABLE 2. A comparison between the model predictions and the experimental observations of contractions due to the stress release at low temperature. Permanent strain after unloading Experimental Model value prediction 9.1% Prestrain (tension) 8.6% 8.55% 0% Prestrain 0.4% 0.43% -9.1% Prestrain (compression) -9.4% -9.42%
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|Author:||Diani, Julie; Liu, Yiping; Gall, Ken|
|Publication:||Polymer Engineering and Science|
|Date:||Apr 1, 2006|
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