Paul A. Tres
Designing
Plastics Parts
for Assembly
Sample Chapter 3:
Strength of Material for Plastics
ISBNs
978-1-56990-401-5
1-56990-401-4
HANSER
Hanser Publishers, Munich • Hanser Publications, Cincinnati
3
Strength of Material for Plastics
3.1
Tensile Strength
Tensile strength is a material’s ability to withstand an axial load.
In an ASTM test of tensile strength, a specimen bar (Figure 3-1) is placed in a tensile
testing machine. Both ends of the specimen are clamped into the machine’s jaws, which
pull both ends of the bar. Stress is automatically plotted against the strain. The axial load
is applied to the specimen when the machine pulls the ends of the specimen bar in
opposite directions at a slow and constant rate of speed. Two different speeds are used:
0.2 in. per minute (5 mm/min) to approximate the material’s behavior in a hand assembly
operation; and 2.0 in. per minute (50 mm/min) to simulate semiautomatic or automatic
assembly procedures. The bar is marked with gauge marks on either side of the mid point
of the narrow, middle portion of the bar.
As the pulling progresses, the specimen bar elongates at a uniform rate that is proportionate to the rate at which the load or pulling force increases. The load, divided by the
cross-sectional area of the specimen within the gage marks, represents the unit stress
resistance of the plastic material to the pulling or tensile force.
Figure 3-1 Test specimen bar
45
3.2 Compressive Stress
The stress (σ-sigma) is expressed in pounds per square inch (psi) or in Mega Pascals
(MPa). 1 MPa equals 1 Newton per square millimeter (N/mm2). To convert psi into MPa,
multiply by 0.0069169. To convert MPa into psi, multiply by 144.573.
F TENSILE LOAD
=
A
AREA
Stress
psi (MPa)
σ=
(3.1)
Yield point
Ultimate stress
Elastic limit
Proportional limit
Figure 3-2 Typical stress/strain diagram
for plastic materials
3.1.1
Strain
%
Proportional Limit
The proportional relationship of force to elongation, or of stress to strain, continues until
the elongation no longer complies with the Hooke’s law of proportionality. The greatest
stress that a plastic material can sustain without any deviation from the law of
proportionality is called proportional stress limit (Figure 3-2).
3.1.2
Elastic Stress Limit
Beyond the proportional stress limit the plastic material exhibits an increase in elongation
at a faster rate. Elastic stress limit is the greatest stress a material can withstand without
sustaining any permanent strain after the load is released (Figure 3-2).
3.1.3
Yield Stress
Beyond the elastic stress limit, further movement of the test machine jaws in opposite
directions causes a permanent elongation or deformation of the specimen. There is a point
beyond which the plastic material stretches briefly without a noticeable increase in load.
This point is known as the yield point. Most unreinforced materials have a distinct yield
point. Reinforced plastic materials exhibit a yield region.
46
Strength of Material for Plastics
It is important to note that the results of this test will vary between individual
specimens of the same material. If ten specimens made out of a reinforced plastic
material were given this test, it is unlikely that two specimens would have the same yield
point. This variance is induced by the bond between the reinforcement and the matrix
material.
3.1.4
Ultimate Stress
Ultimate stress is the maximum stress a material takes before failure.
Beyond the plastic material’s elastic limit, continued pulling causes the specimen to neck
down across its width. This is accompanied by a further acceleration of the axial elongation
(deformation), which is now largely confined within the short necked-down section.
The pulling force eventually reaches a maximum value and then falls rapidly, with little
additional elongation of the specimen before failure occurs. In failing, the specimen test
bar breaks in two within the necking-down portion. The maximum pulling load,
expressed as stress in psi or in N/mm2 of the original cross-sectional area, is the plastic
material’s ultimate tensile strength (σULTIMATE).
The two halves of the specimen are then placed back together, and the distance
between the two marks is measured. The increase in length gives the elongation,
expressed in percentage. The cross-section at the point of failure is measured to obtain
the reduction in area, which is also expressed as percentage. Both the elongation
percentage and the reduction in area percentage suggest the material ductility.
In structural plastic part design it is essential to ensure that the stresses that would
result from loading will be within the elastic range. If the elastic limit is exceeded,
permanent deformation takes place due to plastic flow or slippage along molecular slip
planes. This will result in permanent plastic deformations.
3.2
Compressive Stress
Compressive stress is the compressive force divided by cross-sectional area, measured in
psi or MPa.
It is general practice in plastic part design to assume that the compressive strength of a
plastic material is equal to its tensile strength. This can also apply to some structural
design calculations, where Young’s modulus (modulus of elasticity) in tension is used,
even though the loading is compressive.
The ultimate compressive strength of thermoplastic materials is often greater than the
ultimate tensile strength. In other words, most plastics can withstand more compressive
surface pressure than tensile load.
The compressive test is similar to that of tensile properties. A test specimen is
compressed to rupture between two parallel platens. The test specimen has a cylindrical
shape, measuring 1 in. (25.4 mm) in length and 0.5 in. (12.7 mm) in diameter. The load is
applied to the specimen from two directions in axial opposition. The ultimate
compressive strength is measured when the specimen fails by crushing.
3.2 Compressive Stress
47
A stress/strain diagram is developed during the test, and values are obtained for the
four distinct regions: the proportional region, the elastic region, the yield region, and the
ultimate (or breakage) region.
The structural analysis of thermoplastic parts is more complex when the material is in
compression. Failure develops under the influence of a bending moment that increases as
the deflection increases. A plastic part’s geometric shape is a significant factor in its
capacity to withstand compressive loads.
σ=
P COMPRESSIVE FORCE
=
A
AREA
(3.2)
Figure 3-3 Compressive test specimen
The stress/strain curve in compression is similar to the tensile stress/strain diagram,
except the values of stresses in the compression test are greater for the corresponding
elongation levels. This is because it takes much more compressive stress than tensile
stress to deform a plastic.
3.3
Shear Stress
Shear stress is the shear load divided by the area resisting shear. Tangential to the area,
shear stress is measured in psi or MPa.
There is no recognized standard method of testing for shear strength (τ-tau) of a thermoplastic or thermoset material. Pure shear loads are seldom encountered in structural part
design. Usually, shear stresses develop as a by-product of principal stresses, or where
transverse forces are present.
48
Strength of Material for Plastics
The ultimate shear strength is commonly observed by actually shearing a plastic plaque
in a punch-and-die setup. A ram applies varying pressures to the specimen. The
ram’s speed is kept constant so only the pressures vary. The minimum axial load
that produces a punch-through is recorded. This is used to calculate the ultimate shear
stress.
Exact ultimate shear stress is difficult to assess, but it can be successfully approximated
as 0.75 of the ultimate tensile stress of the material.
τ=
Q SHEAR LOAD
=
A
AREA
(3.3)
Q
Q
3.4
Figure 3-4 Shear stress sample specimen
example: (a) before; (b) after
Torsion Stress
Torsional loading is the application of a force that tends to cause the member to twist
about its axis (Figure 3-5).
Torsion is referred to in terms of torsional moment or torque, which is the product of
the externally applied load and the moment arm. The moment arm represents the distance
from the centerline of rotation to the line of force and perpendicular to it.
The principal deflection caused by torsion is measured by the angle of twist or by the
vertical movement of one side.
Figure 3-5 Torsion stress
3.4 Torsion Stress
49
When a shaft is subjected to a torsional moment or torque, the resulting shear stress is:
τ=
Mt Mt r
=
J
I
(3.4)
Mt is the torsional moment and it is:
Mt = F R
(3.5)
The following notation has been used:
J polar moment of inertia
I moment of inertia
R moment arm
r radius of gyration (distance from the center of section to the outer fiber)
F load applied
3.5
Elongations
Elongation is the deformation of a thermoplastic or thermoset material when a load is
applied at the ends of the specimen test bar in opposite axial direction.
The recorded deformation, depending upon the nature of the applied load (axial, shear
or torsional), can be measured in variation of length or in variation of angle.
Strain is a ratio of the increase in elongation by initial dimension of a material. Again,
strain is dimensionless.
Depending on the nature of the applied load, strains can be tensile, compressive, or
shear.
ε=
3.5.1
ΔL
(%)
L
(3.6)
Tensile Strain
A test specimen bar similar to that described in Section 3.1 is used to determine the
tensile strain. The ultimate tensile strain is determined when the test specimen, being
pulled apart by its ends, elongates. Just before the specimen breaks, the ultimate tensile
strain is recorded.
The elongation of the specimen represents the strain (ε-epsilon) induced in the material,
and is expressed in inches per inch of length (in/in) or in millimeters per millimeter
(mm/mm). This is an adimensional measure. Percent notations such as ε = 3% can also be
used. Figure 3-2 shows stress and strain plotted in a simplified graph.
50
Strength of Material for Plastics
Figure 3-6 Tensile specimen loaded showing
dimensional change in length. The difference
between the original length (L) and the
elongated length is ΔL
Figure 3-7 Compressive specimen showing dimensional change in length. L is the original length; P is
the compressive force. ΔL is the dimensional change in length
51
3.5 Elongations
3.5.2
Compressive Strain
The compressive strain test employs a set-up similar to the one described in Section 3.2.
The ultimate compressive strain is measured at the instant just before the test specimen
fails by crushing.
3.5.3
Shear Strain
Shear strain is a measure of the angle of deformation γ – gamma. As is the case with
shear stress, there is also no recognized standard test for shear strain.
Q
Q
Q
Q
γ
Figure 3-8 Shear strain: (a) before; (b) after
3.6
True Stress and Strain vs. Engineering Stress
and Strain
Engineering strain is the ratio of the total deformation over initial length.
Engineering stress is the ratio of the force applied at the end of the test specimen by
initial constant area.
True stress is the ratio of the instantaneous force over instantaneous area.
Formula 3.7 shows that the true stress is a function of engineering stress multiplied by
a factor based on engineering strain.
σ TRUE = σ (1+ ε )
(3.7)
True strain is the ratio of instantaneous deformation over instantaneous length.
Formula 3.8 shows that the true strain is a logarithmic function of engineering strain.
ε TRUE = ln(1+ ε )
(3.8)
52
Strength of Material for Plastics
By using the ultimate strain and stress values, we can easily determine the true ultimate
stress as:
σ ULTIMATETRUE = σ ULTIMATE (1+ εULTMATE )
(3.9)
Similarly, by replacing engineering strain for a given point in Equation 3.8
with engineering ultimate strain, we can easily find the value of the true ultimate strain
as:
εULTIMATETRUE = ln(1+ εULTIMATE )
(3.10)
Both true stress and true strain are required input as material data in a variety of finite
element analyses, where non-linear material analysis is needed.
Initial
area
Reduced
area
Figure 3-9 True stress necking-down effect
3.7
Poisson’s Ratio
Provided the material deformation is within the elastic range, the ratio of lateral to
longitudinal strains is constant and the coefficient is called Poisson’s ratio.
ν=
LATERAL STRAIN
LONGITUDINAL STRAIN
(3.11)
In other words, stretching produces an elastic contraction in the two lateral directions.
If an elastic strain produces no change in volume, the two lateral strains will be equal to
half the tensile strain times –1.
53
3.7 Poisson’s Ratio
b
b
L
ΔL
Figure 3-10 Dimensional change in only two of
three directions
b'
b'
Under a tensile load, a test specimen increases (decreases for a compressive test) in
length by the amount ΔL and decreases in width (increases for a compressive test) by the
amount Δb. The related strains are:
ΔL
L
Δb
ε LATERAL =
b
ε LONGITDINAL =
(3.12)
Poisson’s ratio varies between 0, where no lateral contraction is present, to 0.5 for
which the contraction in width equals the elongation. In practice there are no materials
with Poisson’s ratio 0 or 0.5.
Table 3-1 Typical Poisson’s ratio values for different materials
Material Type
Poisson’s Ratio at 0.2 in./min (5 mm/min) Strain Rate
ABS
Aluminum
Brass
Cast iron
Copper
High density polyethylene
0.4155
0.34
0.37
0.25
0.35
0.35
54
Strength of Material for Plastics
Table 3-1 (Continued)
Material Type
Poisson’s Ratio at 0.2 in./min (5 mm/min) Strain Rate
Lead
Polyamide
Polycarbonate
13% glass reinforced polyamide
Polypropylene
Polysulfone
Steel
0.45
0.38
0.38
0.347
0.431
0.37
0.29
The lateral variation in dimensions during the pull-down test is:
Δ b = b − b′
(3.13)
Therefore, the ratio of lateral dimensional change by the longitudinal dimensional
change is:
Δb
ν= b
ΔL
L
Or, by rewriting, the Poisson’s ratio is:
ν=
3.8
Modulus of Elasticity
3.8.1
Young’s Modulus
ε LATERAL
ε LONGITUDINAL
(3.14)
(3.15)
The Young’s modulus or elastic modulus is typically defined as the slope of the
stress/strain curve at the origin.
The ratio between stress and strain is constant, obeying Hooke’s Law, within the
elasticity range of any material. This ratio is called Young’s modulus and is measured in
MPa or psi.
E=
σ STRESS
=
= CONSTANT
ε STRAIN
(3.16)
Hooke’s Law is generally applicable for most metals, thermoplastics and thermosets,
within the limit of proportionality.