Residual stress determination with acoustic birefringence in slightly anisotropic polymers.
Residual stresses have been discussed for a long time since they are critical for the mechanical behavior of polymers and composites and can cause warpage, delamination, and so on , This is important for material processing, especially for the composite fabrication technology, with an increasing desire to better control the internal stresses in products.
A lot of simulations have been done to predict such stresses during composite processing (injection molding). Kabanemi et al.  did a simulation and experimental validation on a thin injection molded complex shape. They used a thermoviscoelastic model with volume relaxation without taking into account the molecular orientation of the polymer and the possibility of anisotropy. The validation of the model used the hole drilling technique which requires a small thickness (a few mm) for the object.
Due to the inaccessibility of residual stresses, different destructive experimental methods have been proposed as hole drilling methods, layer removal method , and compliance method , In general, we measure the displacements, compute strains, and determine residual stresses.
With a request for nondestructive method, X ray diffraction is a method of choice for crystallized materials (metals). Ultrasonic detection is also nondestructive and it is a promising method for polymers and composites. In a previous work, we investigated the acoustoelastic theory with a thermodynamic explanation (Gruneisen parameter) to assess the internal stresses in a thermoplastic polymer , Here, we propose another method also based on the same theory. We will consider the acoustic birefringence with ultrasonic detection and its application in slightly anisotropic polymers for residual stress characterization. This technique is seldom used for these materials and very few articles can be found on this subject.
Acoustic birefringence is similar to optical birefringence. The optical birefringence is "double refraction": the material plays as a polarization light selector. Different polarization components will have different refraction indices. In contrast, the acoustic birefringence is much more suitable to be called "shear wave splitting" like in the seismic domain  where a shear wave can be splitted into two parts along principal anisotropic axes.
A major advantage of ultrasonic wave is its simplicity of use and the possibility of local and time continuous evaluation. Acoustic birefringence is powerful for the stress and texture study of metals. In the literature relative to acoustic birefringence, there are some confusion among strongly and slightly anisotropic materials, texture in metals, molecular orientation of polymers, and anisotropy for composites. Furthermore, coupling effects exist between these material features mentioned previously and applied, or residual stresses. The residual stress measurement tends to be ambiguous when all these effects contribute to the acoustic birefringence.
Texture in metals causing anisotropy has been widely discussed. Grains arrange with certain crystallographic axes parallel to the direction of rolling. Such "slight anisotropy" will equally contribute to the phase difference occurring during acoustic birefringence as well as applied stresses. For example, stress differences are computed from shear waves (velocities) during a compression of aluminum discs [7). But the author reported the inaccuracy of stress computed because of the texture superposed on the applied stresses . Similar to metals, molecular orientation in polymers will also cause slightly anisotropy  and influence the acoustic birefringence behavior during stress detection. There are few experimental data on the subject. Iwashimizu and Kubomura  conducted a theoretical study for slightly anisotropic materials coupled with applied stresses. He showed that it is possible to separate the influence from material anisotropy from the effect of applied stresses only when one of the principal axes, due to the texture, coincides with the direction of applied stresses. The polarization directions of shear waves rotate largely as the uniform applied stress varies while the principal directions of the stress remain unchanged. On the contrary, this will not occur in ordinary anisotropic materials or isotropic materials.
For strong anisotropic materials, like fiber-reinforced composites, without applied stresses, Hsu et al.  used the acoustic birefringence for layup orientation verification. Solodov et al.  showed it could be an efficient nondestructive method to know the stiffness asymmetry in composites through the output amplitude and phase stiffness, which can serve in addition for crack orientation detection. For stress evaluation, it is much simpler than in slightly anisotropic materials. We can separate directly the influence from material anisotropy and stresses . Hasegawa et al. have done acoustic birefringence on woods [9, 13] in which almost no polarization angle change (material effect) were caused by applied stresses. In this case, the shear wave velocity difference correlated well with shear stress during the four point bending test. It is possible to detect the change of texture along the longitudinal axis of the beam from the computation of the birefringence coefficients at different locations.
A big advantage of acoustic birefringence is that no reference state is required, but we must pay attention to the principal axes of the materials and the principal axes of the applied stresses. Here, we will first review the acoustic birefringence theory for slightly anisotropic materials. Then experiment results will be presented and analyzed.
ACOUSTIC BIREFRINGENCE THEORY FOR SLIGHTLY ANISOTROPIC MATERIALS
Iwashimizu et al. established shear wave velocities in case of slightly anisotropic materials. Suppose a shear wave propagates in [X.sub.3] direction with polarization directions in the plane of [X.sub.1] and [X.sub.2] (Fig. 1), it will be splitted into two parts along principal anisotropic axes [X'.sub.1] and [X'.sub.2] with an angle 0to [X.sub.1] in the material investigated. The velocities can be expressed as 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[V.sup.0.sub.3] = [V.sup.0.sub.3] = [square root of ([mu]/[[rho].sup.0])] (3)
C' = [[rho][([V.sup.0.sub.32]).sup.2] - [rho] [([V.sup.0.sub.321).sup.2]]/2([mu] + [r.sub.3]) (4)
and the rotation angle reads:
tan 2[theta] = 2[[epsilon].sub.12]/C' + ([[epsilon].sub.11] - [[epsilon].sub.22]) (5)
where [V.sub.ij] are the shear wave velocities in materials with i indicating the wave propagation direction and j the wave polarization direction. The superscript "0" is for the initial state without applied stresses. [lambda], [mu] are the Lame coefficients and [r.sub.1], [r.sub.2], [r.sub.3] are third order elastic coefficients for an isotropic material. [C'.sub.1313] and [C'.sub.2323] are the additional elastic coefficients for slightly anisotropic material. [phi] is the angle between one of the principal stress direction and [X.sub.1] (the difference between the principal stress directions and principal strain directions is neglected here for slightly anisotropic materials). [[rho].sup.0] is the density, [alpha] and [beta] are the angles of transmitter and receiver relative to [X.sub.1].
If C' = 0, there will be no preferential direction, then the most often used shear velocity corresponds to the case of an isotropic material under unidirectional compression is as shown below :
[[rho].sup.0] [V.sup.2.sub.32] = [mu] + [([lambda] + [r.sub.2])(1 - 2v) + 2 ([mu] + [r.sub.3]) (1 - v) - 2v [mu]] [[epsilon].sub.22] (6)
[[rho].sup.0] [V.sup.2.sub.31] = [mu] + [([lambda] + [r.sub.2])(1 - 2v) - 4v([mu] + [r.sub.3]) - 2v [mu]] [[epsilon].sub.22] (7)
[[epsilon].sub.22] = - [[epsilon].sub.11]/v = - [[epsilon].sub.33]/v (8)
if [[epsilon].sub.22] is small, for example, <1.5%,
[[V.sub.32] - [V.sub.31]]/([V.sup.0.sub.32] + [V.sup.0.sub.31])/2 = B[[epsilon].sub.22] = B' [[sigma].sub.22] (9)
B = ([mu] + [r.sub.3]) (1 + v)/[mu] (10)
B' = B/E (11)
where B and B' are the acoustic birefringence factor related to strain and stress, respectively, v is the Poisson ratio and E is the Young's modulus.
We use an ultrasonic recorder Krautkramer and 2 MHz shear sensors K2KY. Through-transmission mode is used here since a strong attenuation is observed under reflection mode.
Laboratory-made metallic frame is used to fix sensors and prevent the influence of coupling layers.
Initial Anisotropy Consideration
Two thermoplastic polymers are discussed here: acrylic (Altuglass) and polyamide 6 (Ertalon). Since nothing is known about the thermomechanical history of our samples, possible anisotropy is tested first (Fig. 2). The polarization direction is settled in the range [0[degrees] 360[degrees]) with a step of 30[degrees] with the condition [alpha] = [beta]. A nearly perfect isotropic material is found for acrylic. We notice no variation of velocities in this case. For the polyamide 6, there is a slight anisotropy with a maximum amplitude variation of 5 m/s and the polarization directions of the shear waves are around (-30[degrees], 60[degrees]). This situation corresponds to two possibilities: material anisotropy or residual stresses in the polymer tested. We can reasonably suppose that the velocity variation comes from the extrusion process of the block of the polymer. So it is related to slight anisotropy.
Effect of Loading
Our specimens are loaded along [X.sub.2] in compression (Fig. 3). We have done first a uniaxial compression test ([??]=0.008mm/s) and have checked that the behavior is always linear and elastic (loading and unloading) in the range of strain investigated for the two materials.
Shear wave velocities are measured with polarization directions along [X.sub.2] (90[degrees]) for [V.sub.22] and afterward along [X.sub.1] (0[degrees]) for [V.sub.31] with always the condition [alpha] = [beta]. The time of flight measurement is repeated three times in each direction at a constant loading procedure of [0-100 kN] with a 10 kN increment. The results can be shown below.
The shear velocity curves for polyamide 6 in Fig. 4a show two parts: a linear and a nonlinear part. We have shown the nonlinear part is related to a thermodynamic parameter: the Gruneisen coefficient  and is related to the intrachain vibration. This is the characteristic of semicrystalline thermoplastic polymers. Figure 4b shows a linear relation allowing us to use the Eq. 9 to obtain the birefringence factor [B.sup.experiment.sub.polyamides6] = - 0.14. The birefringence factor [B.sup.theoretic.sub.polyamides6] = -0.073 can also be calculated from a theoretical expression (Eq. 10) containing the elastic constants of the material. The discrepancy between the two comes from the difficulty to obtain experimentally the third order coefficient [r.sub.3]. We applied also the same procedure on an amorphous polymer acrylic in Fig. 5 and the behavior is only linear. The birefringence coefficient [B.sup.experiment.sub.acrylic] = -0.88 and [B.sup.theoretic.sub.acrylic] = - 1.1 can be found from experiment and theoretical computation respectively, which shows a good correlation between each other.
The coupling term K shown in Eq. 12 from Eqs. 1 and 2 can have a further explication:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
If [phi] = 0, the principal direction of material coincides with the principal direction of the stress, then the difference [V.sub.31] - [V.sub.32] between shear wave velocities can be simplified to:
[V.sub.31] - [V.sub.32] = 2 [absolute value of ([mu] + [v.sub.3])C'] (C' + [[epsilon].sub.11] - [[epsilon].sub.22]/C') (13)
[V.sup.0.sub.32] - [V.sup.0.sub.31] = 2 [absolute value of ([mu] + [v.sub.3])C'] (14)
Combining Eqs. 13 and 14, we find that
[absolute value of C'/[[epsilon].sub.11] - [[epsilon].sub.22]] = [absolute value of ([V.sup.0.sub.32] - [V.sup.0.sub.31])/([V.sub.32] - [V.sub.31]) - ([V.sup.0.sub.32] - [V.sup.0.sub.31])] (15)
[absolute value of [V.sub.32] - [V.sub.31]] = 2 [absolute value of [V.sup.0.sub.32] - [V.sup.0.sub.31]] (16)
based on Eq. 15, we have
[absolute value of C'] = [absolute value of [[epsilon].sub.11] - [[epsilon].sub.22] (17)
From Fig. 4b, Eq. 16 corresponds to [[epsilon].sub.22] = -0.009 when [absolute value of [V.sub.32] - [V.sub.31]] = 2 [absolute value of [V.sup.0.sub.32] - [V.sup.0.sub.31]] = 2.72 m/s. In this case, combining with [v.sub.polyamide6] = 0.37, Eqs. 8 and 17, we can identify the constant [absolute value of [C'.sub.polyamide6]] = 0.0123. An estimation can be made from an anisotropic metallic material (Aluminum) under the same velocity variation (([V.sup.0.sub.32] - [V.sup.0.sub.31])/( [V.sup.0.sub.32] + [V.sup.0.sub.31])/2 = 0.12%) by using the elastic data  in Eq. 4:
[absolute value of [C'.sub.aluminum]] = [absolute value of 2700 x [(3000 x [(1 + 0.0012)).sup.2] - [3000.sup.2]]/ 2 x (2.76 x [10.sup.10] - 5.7 x [10.sup.10])] = 9.92 x [10.sup.-4] (18)
We see that under the same velocity variation condition, metals are less influenced by the initial anisotropy (texture) than polymers with [absolute value of [[C'.sub.polyamide6]/[absolute value of [C'.sub.aluminium]] [approximately equal to] 12.
We can plot this function for polyamide 6 according to [phi] (Fig. 6): if [phi] = 0[degrees], there are two linear parts; if [phi] = 30[degrees], there is a nonlinear behavior. So the nonlinear behavior in Fig. 4a is probably due to the material axes and loading axes mismatch.
Effect of Residual Stresses on Acrylic
The study above was a preliminary step for the prediction of the residual stresses by the acoustic birefringence method. Here we want to use the acoustic birefringence to detect and quantify the residual stresses in an amorphous polymer. Struik  has classified the origins of residual stresses in amorphous polymers into three types: deformation, physical aging, and inhomogeneous cooling. He argued that when a network of polymers chains is first balanced with external loading above glass transition temperature and then cooled down, not only molecular orientation is frozen but also the entropic elastic stress. Wang et al.  has successfully frozen different stresses in cured epoxy strips.
Acrylic is chosen here because of its perfect isotropic property which facilitates our analysis. We first have done twice a ramp at 20 K/min in DSC (NETZSCH) and determined the glass transition temperature. We got the value at the inflexion point [T.sub.g] = 121[degrees]C. Then the Young modulus is measured with an INSTRON compression machine at different temperature. The size of the test sample is 50 mm x 50 mm x 50 mm. The Young modulus-temperature curve is shown below (Fig. 7). We can see that the behavior is a classical one with the glass state at a low temperature, the glass transition state at an intermediate temperature, and the beginning of the leathery state at a high temperature. [E.sup.room.sub.acrylic] = 2.78 GPaat room temperature and [B'.sup.theoretic/sub.acrylic] = -3.95 x [10.sup.-4] [MPa.sup.-1] according to Eq. 11.
"Frozen Stress" Procedure
The stresses can be frozen by deforming the material at raised temperature with a moderate force. In the glassy state, the material behaves as purely elastic (large relaxation time). No frozen stresses can be obtained in this domain. In the rubbery state, the Young modulus is too low to take evident frozen stresses. So we concentrate on the early glass transition period (T = 118[degrees]C) to freeze evident stresses for ultrasonic measurement later.
Blocks with different frozen stresses are made with procedures shown in Fig. 8. Each cube is heated at 118[degrees]C for 1 h in an INSTRON environment chamber with a forced convection to ensure a homogeneous temperature in the sample. Then a constant force (F) is applied to deform the block in compression. After 5 min, the environment chamber is slowly cooled down to room temperature at about 0.5[degrees]C/min. F = 0.5 kN, 5 kN, 10 kN are used here under slow cooling situation and are removed after cooling. One more sample with F = 5 kN is rapidly cooled down (merged in water) after 5 min force keeping.
The material anisotropy is tested for each cube with [alpha] = [beta] = 0[degrees], 90[degrees]. The velocity difference and the length variation ([DELTA]L along [X.sub.2]) before and after deformation can be measured for each sample. The velocity variation [V.sub.32] - [V.sub.31]/([V.sup.0.sub.32] + [V.sup.0.sub.31]/2) is shown in Table 1, and the applied stresses are simply calculated according to Hooke's law that [[sigma].sub.applied] = [E.sup.room.sub.acrylic] ([DELTA]L/50 mm).
In the case of purely isotropic thermoplastic material, we can determine residual stresses in the material. According to Eq. 9, the tensile residual stress values are shown in Table 1, which means the samples tend to remain the initial state after the compression loading. Comparing the applied stresses and the residual stresses, the fast cooling corresponds to evident residual stresses in the sample while the slow cooling corresponds to less stress accumulation inside. If the thermoplastic polymer presents a slightly anisotropy due to molecular orientation, a prior test to check the anisotropy is necessary. When the velocity is measured in the principle axis of material, we are able to compute the residual stresses presented in the material. This will be the subject of a future work.
We used the acoustic birefringence theory to investigate thermoplastic polymers. In the literature, few results can be found for thermoset and thermoplastic polymers. Residual stresses can be obtained with acoustic birefringence theory. Attention must be done on material effects (texture, slight anisotropy) that can induce coupling with residual stresses and complicate the analysis.
Acoustic birefringence allows detecting the slight anisotropy of thermoplastic polymers. In case of an amorphous polymer with no initial anisotropy, we are able to detect and quantify the residual stresses. In case of coupling between anisotropy and residual stresses, we need to do measurement along the principle axis of the material. This will be the objective of a future work.
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Q. Zhu, C. Burtin, C. Binetruy, A. Poitou
GeM (Institut de recherche en genie civil et mecanique), CNRS UMR 6183, Ecole Centrale de Nantes, 44321 Nantes, France
Correspondence to: Q. Zhu; e-mail: firstname.lastname@example.org
TABLE 1. Applied/residual stresses in different samples. Slow cooling Fast cooling Applied force F 0.5 kN 5 kN 10 kN 5 kN [V.sub.32] - [V.sub.31]/ 0% -0.082% -0.202% -0.168% ([V.sup.0.sub.32] + [V.sup.0.sub.31]/2] Applied stresses -0.8 -4.3 -7.2 -4.3 [[sigma].sub.applied] (according to Hooke's law) (MPa) Residual stresses 0 2.1 5.1 4.2 [[sigma].sub.22] (according to Eq. 9) (MPa) FIG. 8. Frozen stress procedure for four samples. Sample reference F(kN) Cooling rate G1 0.5 slow G2 5 slow G3 10 slow G4 5 fast
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|Author:||Zhu, Q.; Burtin, C.; Binetruy, C.; Poitou, A.|
|Publication:||Polymer Engineering and Science|
|Date:||Oct 1, 2015|
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