Measurement Methods
and Calculations
to Determine
Internal Deposit Stress
Frank H. Leaman
Specialty Testing & Development, Inc.
York, PA
Methods for Deposit Stress
Determination

Bent Strip (simple beam theory)

Spiral Contractometer
Simple Beam Theory
Where:
Then:
N = Thickness of the plated coating (inches)
T = Thickness of the test strip (inches)
D = Deflection of the strip due to bending (inches)
L = Length of the test section (inches)
E = Modulus of elasticity of the test strip (lb/in2)
I = Moment of inertia of the cross section of test strip about its neutral axis
S = Stress in plated layer (lb/in2)
S = 4E (N+T) D
3NTL
Bent Strip Method
(Initial Approach)


During the application of a coating, one end
of the test piece is held in a fixed position
and the other end is free to move.
It is difficult to measure the value for D.
Bent Strip Method
(Different Approach)



A test piece split into two legs spreads outward due
to the deposit stress
The deflection is easily read by placing the test piece
over a scale
Calculate the deposit stress value by using a simple
formula
Simple Beam Tensile and Compressive
Stress
Tensile
Compressive
Compressive and Tensile Stress
Compressive
Tensile
Stress Evaluation Using the Bent Strip
Method
Test Strip in a
Plating Cell
In-Site 1 Plating Cell
Ideal for small
solution volumes and
lab studies,
particularly when
working with precious
metals
Bent Strip Test Piece Measuring Stand
Stress Evaluation Using the Bent Strip
Method
Bent Strip
Plating Test Cell
Test Strip Plating Cell with Accessories
Typical Deposit Stress Evaluation Plating Set-Up
Nickel Deposit Stress Calculations For
Test Strips Using The Modified Stoney
Formulua
Stoney Formula Modified for M:
= E T² M δ 3 L² t
E = Modulus of elasticity of the substrate = 120,690 kg/cm².
T = Thickness of the substrate in millimeters = 0.05077 mm.
δ = 1/2 the distance between the test strip leg tips in mm.
Example: 0.540 inch spread ÷ 2 x 25.385 mm/inch = 6.85 mm.
= Stress in megapascals, MPa. Note: MPa x 145 = PSI.
L = Length of substrate on which the deposit is applied in mm.
For Deposit Stress Analyzer test strips, this value is 76.2 mm.
t = Deposit average thickness in millimeters.
M = Correction for modulus of elasticity difference between
the deposit and substrate:
M = EDeposit ÷ ESubstrate = 206,900 ÷ 120,690 = 1.714
= E (.05077 mm) ² M ( δ mm) =
3652 mm³ = 82.6 MPa
3(76.2 mm)²(.002538 mm)
44.21 mm³
Deposit Stress in PSI = MPa x 145 = 11,977PSI
Spiral Contractometer Existing Design
•




The test piece is a spiral.
One end of the spiral is held,
other end is free to move.
As the free end moves, a dial
registers the movement in
degrees.
The stress of the coating can
be calculated.
Spiral on an
Existing
Contractometer
Spiral Plated on Existing Type Contractometer for
Target Nickel Deposit Thickness of 500µ” in a
Semi-bright Bath after 20 Seconds Wood’s Nickel Strike
Deposit Location
Thickness, µ”
Outside Surface
Inside Surface
410
85
Deposit stress over a 2 minute
strike =
26.4% less than the New Design
Contractometer result
New Spiral Properties
New design spirals are constructed from 0.010 inch
thick stainless steel and have a precise surface area of
13.57 in2.
Spirals mount on the contractometer in a way that the
entire spiral plates from end to end and deposition of
metal on the inside of spirals is minimal even if they
are void of a masking material.
The average test deposit thickness is 500 microinches.
Nickel Plating Conditions for Spiral
Contractometer Tests
Spiral Material
Stainless Steel
Spiral Surface Area, in2
13.57
Square Feet
0.0942
Amps per square foot
30
Amps
2.90
Stock Thickness, inches
0.010
Avg. Deposit Thickness µ”
500
Plating Time
20M 40S
Solution Temperature
140° ± 1° F
A new geometry solves problems related to an
exposed interior that allows deposition of the
applied deposit to occur on the inside surface.
Interior deposits reverse the type of stress and
reduce calculated results as much as 30%.
Interior masking is critical.
The new design provides masking of the interior
surface by geometry and enables spirals to be plated
tip to tip so the plated surface area is a constant
value.
Other advantages:
Stainless steel inserts
30% glass filled nylon
construction which prevents
thread damage and spiral
slipping
More accurate results
Saves time
Spiral Contractometer Equipment to Determine
Internal Nickel Deposit Stress










Spiral Contractometer with calibration weights, support stand and
spiral test pieces. Container 4” diameter and 10” height for nickel
strike anode basket and bath (1750 ml)
Titanium Mesh Anode Basket 3.5” outside and 2.25” inside diameter,
8” high with support contact tabs and cover for Wood’s nickel strike
Titanium Mesh Anode Basket 5” outside and 4” inside diameter with
support contact tabs and cover for the plating bath
Nickel anode buttons to fill the anode baskets
Pyrex beaker 4000 ml for a nickel plating bath
Support stand – designed to perfectly center over beaker
Magnetic stirrer hot plate, 115 volt
Digital temperature Controller pre-wired with probe to control ± 10 F
Power Supply constant current, constant voltage, 0-5 amps, 0-30 volts
Magnetic stirrer hot plate, 115 volt
Contractometer Stand, Anode Basket & Beaker
Contractometer Plating Set-Up
Data Recording for Spiral Contractometer Tests
Deposit weight in grams:
Kc degrees:
Kt degrees:
Degrees deflection caused by the deposit:
Spiral weight in grams:
Deposit weight in grams by subraction:
Deposit thickness in microinches:
Average Deposit Thickness Calculation in Inches
T=
W
= Inches
D (87.55 cm2) (2.54 cm/inch)
W = Grams of nickel
D = Density of nickel = 8.90 g/cm3, and
T = Deposit thickness in inches
For the new spirals plated on the new design contractometers,
the constant spiral plated surface area is 13.57 in2 and the
following shortened formula applies:
T=
W
= Inches
1979.2
Calculating Spiral Contractometer Nickel
Deposit Stress On Helices
Stress = 13.02 (D) (M) x 1 + 30,000,050 GPa (d) = PSI
w (d)
28,600,000 (t)
D = Degrees caused by the deposit,
M = Modulus of Elasticity of the deposit ÷ that of the substrate
= 206,897 ÷ 198,186 = 1.044 for nickel deposits over
Specialty Testing helices that are 0.010 inch thick,
w = degrees Kt from helix calibration if the stress is tensile or
degrees Kc if the stress is compressive,
d = Deposit thickness in inches = 0.000536 in, and
t = Helix thickness in inches = 0.010 in.
Calculation :
S = 13.02 (26) (1.04897) x 1+ 30,000,050 (d) = 21,202 PSI
33 (0.000536)
28,600,000 (0.010 in)
Modulus of Elasticity Values
Stock Material
ES*
Stock Thickness, in
Metal
ED**
Cadmium
31,720
Chromium 248,280
Cobalt
206,897
Copper
117,240
Gold
74,480
Nickel
206,900
Platinum
146,900
Rhodium 289,650
Tin
59,310
Silver
75,860
Zinc
82,760
Cu-Fe Alloy
120,690
0.0020
0.263
2.06
1.72
0.971
0.617
1.71
1.22
2.40
0.49
0.629
0.686
Ni –Fe Alloy
144,830
0.0015
Values for M***
0.219
1.71
1.43
0.810
0.514
1.42
1.02
2.00
0.41
0.524
0.571
Ni-Fe Alloy
179,310
0.0010
0.177
1.39
1.15
0.654
0.415
1.14
0.819
1.62
0.33
0.423
0.462
Pure Ni
206,900
0.0010
0.153
1.20
1.00
0.567
0.360
1.00
0.710
1.400
0.29
0.367
0.400
ES*, modulus of elasticity of substrate material in the Stoney Formula.
ED**, Modulus of elasticity of deposit for use in modified Deposit Stress Analyzer and
Stoney formulas.
M***, modulus of elasticity of deposit ÷ modulus of elasticity of substrate for deposit
stress determinations using the modified Deposit Stress Analyzer and Stoney Formulas.
A Frequent Mistake in Test Procedure
Spiral
1
Deposit Thickness
To Stock Ratio
Stock Thickness, Inches
Deposit Thickness, µ Inches
Minutes Plated
Current Density, ASF
Deposit Stress, PSI
1:20
0.010
500
20
30
14,060
Test Strips
2
1:20
0.002
500
4
30
14,127
3
1:5
0.002
100
20
30
6,865
Note: Extra thick deposits of the harder metals increases the degree of stiffness which
results in lower proportional test strip spread.
Formulas for Bent Strip with One End Stationary
Bent Strip Stress Curve
For the comparison of equations that follow that apply to calculating
the internal deposit stress of applied metallic coatings over various
substrate materials, the value of U = 8.5 units = 0.780 inch will
consistently be used as a basis. It will be noted that the calculated
internal deposit stress values vary from equation to equation,
particularly where the equation fails to address Modulus of Elasticity
differences between the substrate and the applied metallic deposit.
FOR A GIVEN TEST STRIP
U = 8.5 units = 0.780 inch.
δ = U in inches x 25.385 mm/inch ÷ 2 = 9.90 mm.
Z = δ ÷ 4 = 9.90 mm ÷ 4 = 2.475 mm.
R = L² + 4 Z² = 5831 = 294.5 mm
8Z
19.8
Stoney Formula Without and With Modulus of Elasticity
Correction for Modulus of Elasticity Differences
Example: Nickel deposit on copper test strip, U = 8.5 units = 0.780 inch
and δ = U in inches x 25.385 mm/ in ÷ 2 = 9.900 mm.
WITHOUT CORRECTION
σ = 4ET²Z = ET² δ = 91.137 MPa = 13,214.9 PSI
3L²t
3L²t
L = test strip plating length = 76.2mm,
T = Stock thickness = 0.05077mm and
t = Deposit thickness = 0.000075µʺ = 0.001904mm
WITH CORRECTION
M = EDeposit ÷ESubstrate = 206900 ÷120690 = 1.714
σ = ET²δM = 120690(0.05077)²(9.900 mm)(1.714) = 159.16 MPa
3 L² t
3(76.2mm)²(0.001904 mm)
σ = MPa (145 PSI/MPa)
σ = 23,077 PSI
Other Bent Strip Formulas to Determine Internal
Deposit Stress in Applied Metallic Coatings
Barklie and Davies Formula
σ=
ET²
6Rt (1 – t/T)
Heussner, Balden and Morse Formula
σ =
4ET²Z
3t (T + t) L
Brenner and Senderoff Formula
σ = ET(T+ ᵦt)
6Rt
ᵦ=
EDeposit ÷ ESubstrate
BRENNER AND SENDEROF FORMULA
CORRECTED FOR MODULUS 0F
ELASTICITY DIFFERENCES
σ = E T(T+ ᵦ t)
ᵦ = EDeposit ÷ ESubstrate = 1.714
6Rt
σ = 95.538 MPa x 145PSI/MPa = 13,853 PSI
This formula becomes erroneous as differences in modulus of
elasticity values increase. To be correct, this Brenner and
Senderoff formula requires modification as follows:
Corrected Formula:
σ = E T² ᵦ(1.001) ( 1+ ᵦt )
6Rt
ᵦ = 1.714
t = 0.001904 mm
σ = (120690 MPa)(0.05077²mm)(ᵦ)(1.001)(1 + ᵦt)
6(294.5 mm) (0.001904mm)
σ = 159.2 MPa (145 PSI/MPa) = 23,084 PSI.
Note: Corrected Stoney formula result = 23,079 PSI.