Method for measuring thickness, roughness and interface stiffness of coating layer using ultrasonic phase derivative spectrum
The method for measuring thickness, roughness and interface stiffness of coating layer by using ultrasonic phase derivative spectrum which relates to technical field of the ultrasonic non-destructive testing. The method adopts an ultrasonic detection system which includes a sample of the coating layer, an ultrasonic flat probe, an ultrasonic flaw detector, a XYZ three-dimensional stepping device and a computer. The method derives UPDS (Ultrasonic phase derivative spectrum) when the ultrasonic wave is vertically incident on structure with three-medium and two-interface including coupled medium/coating layer/substrate, to conduct sensitivity analysis to identify the high sensitivity range of ultrasonic phase derivative spectrum to thickness, roughness and interface stiffness of coating layer, and to conduct correlation analysis to eliminate deviation introduced by initial phase of the reference signal and the detection signal, so as to achieve simultaneous quantitative characterization of thickness, roughness as well as interface stiffness of coating layer.
1. A method for non-destructively measuring thickness, roughness and interface stiffness of a coating layer using ultrasonic phase derivative spectrum in industries, performed by a ultrasonic detection system comprising an ultrasonic flat probe, a XYZ three-dimensional stepping device and a processor, wherein the ultrasonic flat probe is configured to measure ultrasonic wave on a coating sample and positioned on the XYZ three-dimensional stepping device, and the processor is configured for:
(1) establishing a spring model for a rough and weakly bonded interface based on phase-screen approximation with boundary conditions of continuous (σ 1 =σ 2 ) interface stress σ and discontinuous (u 1 −u 2 =K n σ) displacement u, and calculating reflection coefficient r 12 and transmission coefficient t 12 when ultrasonic wave is vertically incident:
r
12
=
Z
2
-
Z
1
+
i
(
2
π
f
/
K
n
)
Z
2
Z
1
Z
2
+
Z
1
-
i
(
2
π
f
/
K
n
)
Z
2
Z
1
exp
[
-
8
π
2
q
2
f
2
V
1
2
]
(
1
)
t
12
=
2
Z
2
Z
2
+
Z
1
-
i
(
2
π
f
/
K
n
)
Z
2
Z
1
exp
[
-
V
2
2
V
1
2
8
π
2
Rq
2
f
2
(
V
2
-
V
1
)
2
]
(
2
)
wherein, Z 1 and Z 2 respectively represent acoustic impedance of medium 1 and medium 2 adjacent to the rough and weakly bonded interface, V 1 and V 2 respectively represent sound velocities in the medium 1 and medium 2 adjacent to the rough and weakly bonded interface, i represents imaginary part of a complex number, Kn is stiffness coefficient of a vertical interface, Rq is root mean square deviation of an interface contour, and f represents ultrasonic frequency;
(2) obtaining a reflection echo P R when ultrasonic wave is vertically incident on a three-medium and two-interface structure comprising coupling medium/the coating layer/a substrate based on the reflection coefficient r 12 and the transmission coefficient t 12 of the rough and weakly bonded interface:
P
R
=
r
wc
+
r
cw
exp
(
2
ik
c
d
c
)
1
+
r
cs
r
cw
exp
(
2
ik
c
d
c
)
exp
(
i
(
2
k
w
z
0
+
φ
0
)
)
(
3
)
wherein, φ 0 is initial phase, z 0 is distance from a probe to an interface between the coupling medium and the coating layer, k w is a wave number in the coupling medium along an incident direction of acoustic wave, d c represents a thickness of the coating layer, k c represents wave number of the coating layer, r wc and r cs respectively represent reflection coefficients of the interface between the coupling medium and the coating layer and an interface between the coating layer and the substrate, and i represents imaginary part of the complex number;
(3) performing Fourier transform on the reflection echo P R to obtain real part Re(P R ) and imaginary part Im(P R ) in a frequency domain f, and then derive reflection echo phase spectrum (RPS) on the three-medium and two-interface structure, wherein Vw represents longitudinal wave velocity of the coupling medium and Vc represents longitudinal wave velocity of the coating layer;
RPS
(
f
)
=
arctan
(
Im
(
P
R
)
Re
(
P
R
)
)
=
arctan
(
(
1
-
r
wc
2
)
r
cs
sin
(
4
π
fd
c
/
V
c
)
r
wc
+
(
1
+
r
wc
2
)
r
cs
cos
(
4
π
fd
c
/
V
c
)
+
r
wc
r
cs
2
)
+
2
π
fz
0
V
w
+
φ
0
(
4
)
Wherein:
URCPS
(
f
)
=
arctan
(
(
1
-
r
w
c
2
)
r
cs
sin
(
4
π
fd
c
/
V
c
)
r
wc
+
(
1
+
r
wc
2
)
r
cs
cos
(
4
π
fd
c
/
V
c
)
+
r
wc
r
cs
2
)
(
5
)
(4) calculating first-order derivative of the RPS obtained in step (3) with respect to a frequency variable f, so as to obtain ultrasonic phase derivative spectrum (UPDS);
UPDS
(
f
)
=
dRPS
(
f
)
df
=
dURCPS
(
f
)
df
+
2
π
z
0
V
w
(
6
)
(5) analyzing sensitivity S of the UPDS to a parameter p to be tested by using a sensitivity formula (7), where p represents three parameters which are thickness d c , roughness Rq and interface stiffness K n of the coating layer; and determining high sensitivity ranges [d c-down ˜d c-up ], [Rq down ˜Rq up ] and [K n-down ˜K n-up ] of thickness d c , roughness Rq and interface stiffness K n of the coating layer based on 10% of a sensitivity extreme value S extr ;
S
UPDS
,
p
=
p
∂
UPDS
∂
p
(
7
)
(6) obtaining acoustic velocities V w , V c , V s and densities ρ w , ρ c , ρ s of the coupling medium, the coating layer and the substrate respectively, based on a ultrasonic transit time method and a Archimedean drainage method;
(7) collecting reflection echo P R (t) from the interface between the coupling medium and the coating layer and the interface between the coating layer and the substrate, wherein ultrasonic wave passes through the coupling medium and is incident to the interface between the coupling medium and the coating layer and the interface between the coating layer and the substrate;
(8) performing a Fourier transform on the reflection echo P R (t) collected in step (7) to obtain experimental ultrasonic phase derivative spectrum UPDS exp (f) at a corresponding position of the coating sample;
(9) inverting the thickness d c , roughness Rq and interface stiffness K n of the coating layer by a cross-correlation formula (8); in high sensitivity ranges [d c-down ˜d c-up ], [Rq down ˜Rq up ] and [K n-down ˜K n-up ], assigning continuously varying values to the thickness d c , roughness Rq and interface stiffness K n of the coating layer to obtain theoretical ultrasonic phase derivative spectrum UPDS cal (f), and performing cross-correlation operation between the experimental ultrasonic phase derivative spectrum UPDS exp (f) and the theoretical ultrasonic phase derivative spectrum UPDS exp (f) one by one, wherein values of d c , Rq and K n corresponding to a position of η max (d c , R q , K n ) where correlation coefficient is highest are optimal results for thickness, roughness and interface stiffness of the coating sample:
η
(
d
c
,
Rq
,
K
n
)
=
∑
j
=
1
N
[
❘
"\[LeftBracketingBar]"
UPDS
cal
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
-
❘
"\[LeftBracketingBar]"
UPDS
cal
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
_
]
×
[
❘
"\[LeftBracketingBar]"
UPDS
exp
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
-
❘
"\[LeftBracketingBar]"
UPDS
exp
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
_
]
∑
j
=
1
N
[
❘
"\[LeftBracketingBar]"
UPDS
cal
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
-
❘
"\[LeftBracketingBar]"
UPDS
cal
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
_
]
2
×
[
❘
"\[LeftBracketingBar]"
UPDS
exp
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
-
❘
"\[LeftBracketingBar]"
UPDS
exp
(
f
;
d
c
,
Rq
,
K
n
)
❘
"\[RightBracketingBar]"
_
]
2
_
(
8
)
wherein N represents a number of data points within −6 dB effective frequency band after Fourier transform of reflection echo P R (t), and j in formula (8) represents a jth data point.