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Measurement of coatings' elastic properties by mechanical methods: Part 1. Consideration on experimental errors

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Abstract

A critical analysis of some mechanical methods employed for measuring the elastic properties of coatings is presented. A rational basis for properly choosing test methods, transducers and layer thicknesses is provided. The analyzed methods, usually applied to relatively thick coatings, include four-point bending tests and the resonance technique. General relationships for the evaluation of coatings' elastic properties, derived from the multilayer beam and plate theories, are discussed. On the basis of the error propagation theory, for different combinations of materials and layer thicknesses, the influence of typical experimental errors, test setup parameters and material properties is analyzed, and sensitivity coefficients for relative errors are discussed. The critical experimental variables and their effect on measurement accuracy are highlighted, suggesting suitable conditions for selecting specimen geometry and testing conditions. A procedure for measurement at high temperature is outlined.

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Abbreviations

X, Y, Z :

Cartesian coordinate system

ɛ X, ɛX, ɛX :

normal strains

σ X, σX, σX :

normal stresses

h k :

thickness of thekth layer

1 k :

z-coordinate of thekth layer's centroid

l k-1 :

z-coordinate of the inferior surface of thekth layer

l k :

z-coordinate of the superior surface of thekth layer

g k :

depth coordinate of thekth layer centroid, with reference to the geometric center of the specimen

b :

specimen width

H :

total specimen thickness

K Y :

specimen bending stiffness in theZ-Y plane

E k :

elastic modulus of thekth layer

v k :

Poisson's ratio of thekth layer

J k :

moment of inertia of thekth layer

F X, FY :

applied external force along theX andY directions

M X, MY :

applied bending moment in theZ-X andZ-Y planes

R X, RY :

radii of curvature of the specimen geometrical axis in theZ-X andZ-Y planes

C X, CY :

normal strains of the centroid fiber in theZ-X andZ-Y cross sections

δ:

position, in the beam model, of the neutral axis related to the geometric centroid of the specimen

L :

total specimen length

S :

inner spacing between the loading pins in the FPB test

a :

spacing between the inner and outer loading pins in the FPB test

P :

half load, applied in the FPB test

E s, Ec :

elastic modulus of the substrate and coating materials

h s, hc :

thickness of the substrate and coating layers

ɛ s, ɛs :

measured longitudinal strain on the external coating surface and substrate surface

w :

midsection displacement, related to the fixed external loading pins

λ:

distance of the measuring gage from the neutral axis of the specimen

f :

first bending resonance frequency of the specimen

β:

4.73004

m :

mass of the specimen

ζ:

ratio of the transverse to the longitudinal strain measured on the external specimen surface

F :

ratio of the longitudinal strains measured on the substrate and coating surfaces

R :

ratio of the substrate to the coating thickness

α:

ratio of the coating to the substrate elastic modulus

\(P\varepsilon _{s,c} \) :

slope of the experimental regression line correlating the measured load and strain on the substrate or coating surfaces

P w :

slope of the experimental regression line correlating the measured load and midsection displacement

ϕ:

relationship giving the coating elastic modulus as a function ofn experimental variables

x 1, x2,..., xn :

general experimental variables

e Ec :

relative error affecting the measured coating elastic modulus

e i :

relative error affecting theith experimental variable

C i :

condition number for relative error made in the evaluation of theith experimental variable

C F :

condition number for the relative error made in the evaluation ofF

C R :

condition number for the relative error made in the evaluation ofR

C hs :

condition number for the relative error made in the evaluation ofhs

C hc :

condition number for the relative error made in the evaluation ofhc

C Es :

condition number for the relative error made in the evaluation ofEs

C reg :

condition number for the relative error made in the evaluation of\(P\varepsilon _{s,c} \) orP w

C L :

condition number for the relative error made in the evaluation ofL

C f :

condition number for the relative error made in the evaluation off

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Authors and Affiliations

  1. Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione, University of Pisa, 56126, Pisa, Italy

    M. Beghini (Professor), L. Bertini (Professor) & F. Frendo (Assistant Professor)

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  1. M. Beghini
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  2. L. Bertini
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  3. F. Frendo
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Beghini, M., Bertini, L. & Frendo, F. Measurement of coatings' elastic properties by mechanical methods: Part 1. Consideration on experimental errors. Experimental Mechanics 41, 293–304 (2001). https://doi.org/10.1007/BF02323922

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  • DOI: https://doi.org/10.1007/BF02323922

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