On this page you will find data from samples provided by Prof. Richard Knight (Drexel University) on work sponsored by the National Science Foundation (NSF). Four samples of a carbon steel 1040 substrate coated on one face with either pure nylon or various thicknesses of nylon/silica were studied and compared to an uncoated substrate sample. These samples are referred to as coupons and were nominally 3 x 1 x 1/8 in3 or 76 x 25 x 3 mm3. The purpose of the study was to determine the Young's moduli of the coatings by using the Buzz-o-sonic nondestructive testing system.

The general technique for measuring the Young's moduli of coatings has been documented by J. Schrooten1 et al and is as follows:

  1. Measure the Young's modulus of the substrate
  2. Measure the overall Young's modulus of the substrate + coating
  3. The Young's modulus of the coating is then determined from the known thicknesses of the coating and substrate and from the known Young's moduli of the substrate and substrate + coating.

 

When testing coupons using the impulse excitation technique in flexure mode, the coupons are subjected to bending at very low strains. The coupons can be considered as a beam in bending (figure 1). At any moment in time one face will be under tension, the opposite face will be under compression, and there will be a plane of zero stress. The intersection of this plane with the Y-Z plane is called the neutral axis and is important for calculating moments of inertia (discussed below). If the neutral axis and symmetry axis coincide, as for a coated cylinder for example, then the location of the neutral axis need not be calculated; otherwise it is a function of the coating thickness (lc), the substrate thickness (ls), and the Young's moduli of the coating (Ec) and substrate (Es). In this work where the coupons are coated on one face only, the neutral axis had be determined. From a mechanics viewpoint, the coupons can be considered as composite beams, made up of homogeneous, isotropic, linear elastic materials (figure 1).

Fig 1: Cross-section through Y-Z plane of the composite beam. l denotes the position of the neutral axis in bending

Now, the Young's modulus of the coating can be determined from the following equation2:

E I = Es Is + Ec Ic ..........................1

Where:
E = overall Young's modulus of the coupon
I = overall moment of inertia of the coupon
Es = Young's modulus of the substrate
Is = moment of inertia of the substrate
Ec = Young's modulus of the coating
Ic = moment of inertia of the coating

However, to calculate Is and Ic, the position of the neutral axis with respect to the bottom of the coating, l, needs to be calculated, thus3

..........................2

Now the Young's modulus, E, as determined from the fundamental flexural frequency, f1, is given by4:

..........................3
Where:
E = Young's modulus
L = length of specimen
f1 = fundamental flexural frequency
k = radius of gyration of the cross-section about the neutral axis
m1 = constant5 = 4.73004
= bulk density
T1 = correction factor for shear and rotary inertia and is given by:
 
(µ = Poisson's ratio)
 

Combining equations 1 to 3 (SI units) gives:

..........................4
Where:
ls = substrate thickness (figure 1)
lc = coating thickness (figure 1)
 

So, all that needs to be done to calculate the Young's modulus of the coating is to measure the Young's modulus of the coupon and substrates using Buzz-o-sonic; all other terms in equation 4 can then be determined.

To measure the thicknesses of the coatings (lc) and substrates (ls), the total thicknesses of the coupons were first measured with digital calipers and then the coatings were carefully removed using SiC paper (100 µ and then 30 µ). The substrate thicknesses could then be measured and the coating thicknesses calculated. The coating thickness measurements correspond reasonably with those obtained at Drexel University (table 1).

Equation 4 can most easily be solved using a numerical or iterative method in which various values of Ec are assumed and f1 is compared to the fundamental flexural frequency measured by Buzz-o-sonic. In this case, Mathematica 5.1 from Wolfram Research, Inc. was used to perform the calculations. The data used to solve equation 4 are shown in Table 1 and Table 2.

 

Table 1: Coating Data
Sample
L
(mm)
t
(mm)
lc
(µm)
lc**
(µm)
f1
(Hz)

(Mg.m-3)
E
(GPa)
JK 03035*
75.73
3.26
133.3
131.0
2918
7.431
186.5
JK 03036
75.61
3.27
155.0
130.0
2936
7.424
186.9
JK 03037
75.70
3.19
76.7
73.0
2936
7.601
202.7
JK 03038
75.58
3.25
143.3
121.0
2938
7.481
189.9
*nylon coating only, **measurements from Drexel

 

 

Table 2: Substrate Data
Sample
L
(mm)
t
(mm)
f1
(Hz)

(Mg.m-3)
Es
(GPa)
Poisson's
Ratio
JK 03035
75.60
3.13
 2922
7.725
212.1
0.296
JK 03036
75.60
3.11
 2939
7.775
217.3
0.298
JK 03037
75.63
3.11
 2939
7.793
218.9
0.303
JK 03038
75.54
3.11
 2938
7.774
216.8
0.304
Mean
7.767
216.3
0.300
 

There was significant scatter in the results however, arising from errors in the coating and substrate thickness measurements. These errors are significant here because of the low modulus of the coating and because ls was measured after the coatings were removed. Clearly, it would have been better to measure ls prior to the HVOF coating process.

Nevertheless, by plotting the Young's modulus (E in equation 1) Vs the coating thickness (lc), a clear trend can be seen (figure 2). A good estimate of Ec could then be obtained with a least-squares fit of equation 4 to the data, using a mean value of Es, ls, and

Fig 2: Young's modulus Vs coating thickness. Note the lower value of the pure nylon

According to Hill6, the overall Young's modulus of a two phase material can be estimated from the volume fraction of each phase using an average of the Reuss model and Voigt7 model namely:
..........................5

..........................6

Where the subscripts Reuss and Voigt indicate the model. v is the volume fraction of the material. The subscript indicates the material.

The following values were used to estimate the Young's modulus of the coating using the Hill assumptions:

  1. vnylon = 90%
  2. vsilica = 10%
  3. Enylon = 2.5 GPa (mid-range of published values for nylon which are 2.1-2.8 GPa)
  4. Esilica = 74 GPa

Thus, the expected value of E for the nylon-silica coating is ~ 6.2 GPa . This is close to the value of 5.0 GPa obtained here.

 

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1 J. Schrooten, G. Roebben, and J.A. Helsen, "Young's Modulus of Bioactive Glass Coated Oral Implants: Porosity Corrected Bulk Modulus Versus Resonance Frequency Analysis," Scripta Materials, 41 [10] 1047-1053 (1999).
2 James M. Gere and Stephen P. Timoshenko, "Mechanics of Materials," 2nd Ed., p250, Publ. Brooks/Cole Engineering Division, 1984.
3 Chin-Chen Chiu and Eldon D. Case, "Elastic Modulus Determination of Coating Layers as Applied to Layered Ceramic Composites," Materials Science and Engineering, A132 39-47 (1991)
4 A. Spinner, and W. E. Tefft, "A Method for Determining Mechanical Resonance Frequencies and for Calculating Elastic Moduli from these Frequencies," Proceedings ASTM, 61 1221-1239 (1961)
5 G. Pickett, "Equations for Computing Elastic Constants from Flexural and Torsional Resonant Frequencies of Vibration of Prisms and Cylinders," Proceedings ASTM, 45 846-865 (1945)
6 R. Hill, "The Elastic Behaviour of a Crystalline Aggregate," Proc. Phys. Soc. Lond., A65 349 (1952)
7 R. W. Davidge, " Mechanical Behaviour of Ceramics," p25, Publ. Cambridge University Press (1986)