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Identification of the elasto-viscoplastic parameters for a thermoplastic polymer by instrumented indentation
To cite this article: A. Mokhtari and N. Tala Ighil 2018 IOP Conf. Ser.: Mater. Sci. Eng. 461 012057
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Identification of the elasto-viscoplastic parameters for a thermoplastic polymer by instrumented indentation
A. Mokhtari 1 and N. Tala Ighil 1
1 Research Center in Industrial Technologies (CRTI), P.O.Box 64, Cheraga 16014, Algiers, Algeria
a.mokhtari@crti.dz; n.tala-ighil@crti.dz
Abstract. The indentation test is a simple, fast and reliable tool that allows the determination of the materials mechanical properties from experimental load-penetration curves using the inverse computation methods. Through this approach and using the Berkovich indenter, the creep and elasto-plastic properties of the polymers were estimated. Simulations of the elasto- viscoplastic behaviour of the studied polymers under nanoindentation tests were performed. A finite element analysis was carried out to simulate the mechanical behaviour of polymers which can be defined by the Young's modulus E and the parameters (K and n) that describe the materials hardening for large deformations. The obtained functions from the numerical simulations were validated by nanoindentation and compression tests for the studied polymers.
1 Introduction
Thermoplastic polymers have many applications in various industrial sectors due to their good corrosion resistance, low cost, low density, low energy consumption and low thermal conductivity.
These materials are easy to manufacture by different techniques such as injection moulding, compression and thermoforming. Objects of different shapes (tubes, profiles, films, fibres and fabrics) can be made with these polymers. Their mechanical properties are relatively good, especially since they can easily be modified by copolymerization, by polymer blending, or especially by adding fibres.
The theory of indentation developed by Oliver and Pharr in 1992 [1] made possible the calculation of the Young's modulus and the hardness of a material from the discharge curve. This technique has the advantage of being simple to implement and allows testing of small volumes of material. However, obtaining the elastic and plastic properties of the materials is difficult due to the mechanical complexity of these tests. Numerous studies combining numerical simulation and experimental measurements are currently emerging and propose to link the force-displacement curves with the intrinsic properties of materials. Recently, Dao et al. [2] have established, from a finite element analysis, dimensional functions connecting the force-displacement curves measured during the Berkovich (or Vickers) indentation to the Young's modulus and to the metal flow stress. Bucaille et al.
[3] have shown that by using more acute indenters, precision is increased in the determination of plastic parameters.
The authors [2, 3] defined a representative deformation independent of the indented material and related to the geometry of the indenter. It is difficult to faithfully reproduce the viscoelastic and viscoplastic behaviour of polymers with a reasonable number of parameters [4, 5]. In order to use indentation test measurements to characterize a polymer, it is more prudent to use simple two- parameter models (K and n), such as empirical Norton Hoff's model to simulate behaviour of polymers
2
during indentation. On the other hand, their calculation technique is relatively heavy and requires for the experimenter the use of finite element software.
In this paper, a simple and practical method is presented to predict the curing exponent of a material from an inverse analysis. Numerical simulations corresponding to 24 combinations of the parameters (E, v, σy, n) were carried out making it possible to simulate the viscoplastic behaviour of polypropylene (PP) under an instrumented nanoindentation test. Using the results of dimensional analysis and simulations, two dimensionless functions П1 and П2 have been established to link the parameters of the charge-discharge indentation curve to the flow stress σr of the material. The three mechanical properties (E, σy, n) extracted from these functions were examined and validated by nanoindentation and compression tests for the polypropylene polymer PP.
2 Finite element simulation of indentation test
For the finite element simulation of indentation tests, a rigid conical indenter with a half-included tip angle of 70.3° is applied on an infinite half-space. The numerical model used for the nanoindentation test simulation is shown in figure 1. A two-dimensional axisymmetric model included in the commercial finite element code ABAQUS [6], was used. The size of the sample test represents ten times the radius of the contact region, so that the sample can be considered as an infinite half space.
The sample was modelled using a 4-node quadrilateral element (CAX4R) with a bilinear approximation. The discretized rectangular domain is composed of a fine mesh of 4900 elements near the contact region and a coarser mesh of 1031 elements further from the contact region. A convergence analysis was performed to assess the dependence of the results with the numerical model.
The indenter has been modelled as a rigid body and the contact has been modelled as frictionless [3].
Figure 1. Axisymmetric finite element model to simulate the nanoindentation test with a rigid conical indenter.
The polymer rheology has been modelled considering that the total deformation is the sum of the elastic and plastic deformation. The elastic behaviour is modelled by the Young modulus E and Poisson coefficient v. The power law strain-stress curve of the engineering metals in the uni-axial compression state (figure 1) can be assumed as [7]:
{𝜎 = 𝐸𝜀 𝑓𝑜𝑟 𝜀 ≤𝜎𝑦
𝐸
𝜎 = 𝐾𝜀𝑛 𝑓𝑜𝑟 𝜀 ≥𝜎𝐸𝑦 (1) 𝜃 = 70.3° Berkovich indenter tip
PP specimen
Where K is strength coefficient and n is the hardening coefficient. The continuity of the curve in equation 1 requires K=σy(E/σy). Therefore, v, E, n, and σy are independent parameters describing the viscoplastic behaviour of the material. A wide range of material parameters were considered, each parameter takes 4 values and the combination of these parameters leads to 256 combinations.
In order to simplify the problem, some combinations were eliminated by fixing a few parameters (for example, Young's modulus and Poisson's ratio were fixed to 2.189 GPa and 0.4, respectively).
Parametric analysis was conducted for 24 cases only and the hardening coefficient n varies from 0 to 0.5. For a given geometry of a sharp indenter (half-angle of a sharp indenter α is fixed), the loading portion of the curve can be expressed by:
𝑃 = 𝑃(ℎ, 𝐸𝑟, 𝜎𝑟, 𝑛) (2) Where 𝜎𝑟 is called the representative stress. During the indentation charging phase by a Berkovich indenter, the normal force applied to the indenter is proportional to the h squared penetration (h2), that for a homogeneous material and in the context of the Continuum mechanics:
𝐹 = 𝐶ℎ2= 𝜎𝑟ℎ2Π1(𝐸𝑟
𝜎𝑟, 𝑛) (3) Where Π1 is a dimensionless function. The loading curvature C is independent of the indentation depth, but depends on material properties. It has been verified for each simulation that the equation 3 perfectly represents the load-unload curve. The unloading stiffness dF/dh during the unloading step is related to viscoelastic-plastic properties. For fixed half-angle α of a sharp indenter, the unloading stiffness is given by:
𝑆 =𝑑𝐹𝑑ℎ|
ℎ=ℎ𝑚= 𝐸𝑟ℎ𝑚 Π2 (𝜎𝐸𝑟
𝑟, 𝑛,𝐸𝜂
𝑟𝑡) (4)
Where S is the contact stiffness. The effect of creep 𝐸𝜂
𝑟𝑡 can be removed by replacing the measured contact stiffness S, with the true elastic contact stiffness S* [9]. Then, equation 4 is written as:
𝑆∗
𝐸𝑟ℎ𝑚= Π3 (𝜎𝐸𝑟
𝑟, 𝑛) (5) This two dimensionless equations Π1 (load) and Π3 (unload) [2], connect C and S to Young's modulus Er and to the stress σr corresponding to a representative strain εr. This strain is the strain accumulated beyond the elastic limit:
𝜀 = 𝜀𝑦+ 𝜀𝑟 (6) Where εy is the representative strain at the end of the elastic domain.
3 Results
The results of the nano-indentation tests on the PP with the maximum load of 100 mN and for three hold times of 30s, 300s and 600s are shown in figure 2a. The specimens was loaded to 100mN at a rapid rate (4mN.s-1), held at constant peak load for 30s (test 1), 300s (test 2) and 600s (test 3) and unloaded quickly (-4mN.s-1). During unloading, an elastic return of 2350 nm is recorded which corresponds to 44.75% of the maximum displacement of the indenter. It is observed from figure 2 that the polypropylene has a very high capacity to deform under constant load (creep). The analysis of these curves reveals that the increase of the hold time under maximum load Pmax is responsible of the increase in the contact depth hc (figure 2b) and therefore in the projected contact area Ap. This will influence the stiffness S of the polymer and therefore the Young's modulus decreases with the elongation of the creep phase.
4
0 1000 2000 3000 4000 5000 6000
0 10 20 30 40 50 60 70 80 90 100 110
LoadP (mN)
Depth h (m) Test 1 (t1=30s)
Test 2 (t2=300s) Test 3 (t3=600s)
0 100 200 300 400 500 600
5200 5300 5400 5500 5600 5700 5800 5900 6000
Depthh (nm)
Holding time t (s) t3=600s
t2=300s t1=30s
(a) (b)
Figure 2. Evolution of (a) indentation load-displacement (P-h) responses and (b) indentation depth of polypropylene PP for three different holding times.
Consequently, an overestimation of Young's modulus could occur if the load holding time is short. In this case, the Oliver-Pharr method [1] is used to determine the elastic modulus Er, The Feng and Ngan method [10] can calculated the Young's modulus Er* from the measured true contact stiffness S*, using the following equation:
𝐸𝑟∗(𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑) =√𝜋
2 𝑆∗
𝛽√𝐴𝑝∗ (7) with ℎ𝑐(𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑) = ℎ𝑚𝑎𝑥− 𝜀 ∗ 𝑃𝑚𝑎𝑥∗ (1
𝑆+ |𝑃̇|ℎ̇)
where Ap* is the corrected contact area, S* is the true elastic contact stiffness, ℎ̇ is the creep rate (dh/dt) at the end of holding and 𝑃̇ is the unloading rate (dP/dt) at the beginning of unloading. ε is a constant related to the indenter shape (ε = 0.72 for sharp indenters).
0 100 200 300 400 500 600
2,0 2,1 2,2 2,3 2,4 2,5 2,6
Young's modulusEr (GPa)
Holding time t (s)
Er (Oliver&Pharr) Er* (corrected)
Figure 3. Variation of the corrected Young’s modulus with the holding time.
Figure 3 shows a comparison between the Modulus obtained by the Oliver Pharr method and the corrected modulus obtained by the Feng and Ngan method. It can be noted that the effect of the holding time on the reduced module Er is very important for holding time values less than 230s and less important (small correction) for values exceeding this limit. The Feng and Ngan method permits to stabilize the module for a value of 2.189 GPa.
0 20 40 60 80 100 120 140 20
30 40 50 60 70 80
C/r
Er/r n=0
n=0,1 n=0,2 n=0,3 n=0,4 n=0,5
Figure 4. Variation of the ratio C/σr with the parameter Er/σr for representative strain εr=4.3%.
Figure 4 shows the variation of the ratio C/Er as a function of the parameter Er/σr. It is noted that for the Berkovich pyramid, the dimensionless function Π1 is independent of the hardening coefficient n for a representative strain of 4.3%. The points resulting from the results of the numerical computation are then interpolated by a second degree polynomial according to the parameter Er/σr. The dimensionless function П1 is easy to build and the equation is given by:
𝐶
𝜎𝑟= 0.66 (𝑙𝑛 (𝐸𝜎𝑟
𝑟))
2
+ 20.63 ln (𝐸𝜎𝑟
𝑟) − 36.49 (8)
50 100 150 200 250
2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,0 6,5 7,0 7,5
S*/Erhm
Er/r
Figure 5. Variation of the ratio S*/Erhm with the parameter Er/σr for representative strain εr=4.3%.
In figure 5, the S*/Erhm ratio was plotted against the parameter Er/σr. It can be seen that all the points of the digital data lie approximately on a single curve, which means that the relationship between the S*/Erhm ratio and the Er/σr parameter is insensitive to the hardening coefficient n. The dimensionless function П3 obtained from the curve of figure 5 is given by:
𝑆∗
𝐸𝑟ℎ𝑚= 2.63𝑙𝑛 (𝐸𝑟
𝜎𝑟) − 6.52 (9) Polypropylene has a great ability to deform under a constant load (creep). The Young's modulus thus decreases with the extension of the creep phase. As a result, overvaluation of the mechanical properties could occur if the hold time is short. The method of Feng and Ngan [7] allows correcting the Young modulus calculated by the classic method Oliver and Pharr [1].
6 4 Conclusion
This study allows the determination of the creep and elasto-plastic properties of a thermoplastic polymer. For this, two dimensionless functions have been employed that permit to link the indentation measurements charge-displacement to the intrinsic properties of the studied material. This method was applied to a thermoplastic polymer (polypropylene), performing nanoindentation tests at a constant loading rate. The Young modulus Er can be determined uniquely from a single force-depth curve of loading using the revised Oliver-Pharr method presented by Tang and Ngan [8]. To verify the uniqueness of the inverse analysis to evaluate σr and n, a simple and practical method proposed by Dao et al. [2] was adopted. For a representative strain of 4.3%, the computed stress by indentation is very close to the compression stresses [10]. The possibilities of using multiple indenters of different angles to obtain force-depth relationships should also be investigated both experimentally and theoretically.
References
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