Designing Plastic Parts for Assembly Paul A. Tres ISBN 3-446-40321-3 3-446-40321-3
Leseprobe
Weitere Informationen oder Bestellungen unter http://www.hanser.de/3-446-40321-3 sowie http://www.hanser.de/3-446-40321-3 sowie im Buchhandel
3
Strength
of Material for Plastics
3.1
Tensile Strength
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 b oth ends of the ba r. S tress is automatically plotted a gainst the strain. T he a xial 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 a pproximate 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. T he ba r is marked with gauge mark s on either side of the mid point of the n arrow, 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 mark s, represents the unit str ess r esist ance of the plastic material to the pulling or tensile force. T ensile str engt h is
Figure 3-1
Test specimen bar
3 .2
45
C ompr essive S tr ess
The stress (σ-sigma) is expressed in pounds per square inch (psi) or in Mega Pascals (MPa). 1 MPa equals 1 N ewton per square millimeter (N / mm2). T o c onvert psi into MPa, multiply by 0.0069169. To convert MPa into psi, multiply by 144.573.
σ
=
F A
=
TENSILE LOAD
(3.1)
AREA
) a s P s e M r ( t i
Yield po int
Ultimate st r ess
S s
p
Elastic limit
Propor tional limit
Figure
3-2 Typical stress/strain diagram for plastic materials
Str ain
%
3.1.1
Proportional Limit
The proportional relationship of force to elongation, or of stress to str ain, 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 f rom the law of proportionality is called pr oport ional str ess limit ( Figure 3-2).
3.1.2
Elastic Stress Limit
Beyond the proportional stress limit the plastic material exhibits an increase in elong ation at a f aster rate. E last ic str ess limit is the greatest stress a material can withstand without sustaining any permanent strain af ter the load is released ( Figure 3-2).
3.1.3
Yield Stress
Beyond the elastic stress limit, f urther 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 r egion.
46
S tr engt h
of
Mat er ial
for Plast ics
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
the m aximum stress a material takes before f ailure. Beyond the plastic m aterial’s elastic limit, continued pulling causes the specimen to neck down across its width. T his is accompanied b y a f urther acceleration of the a xial elongation (deformation), which is now largely confined within the short necked-down section. The pulling force eventually reaches a maximum value and then f alls rapidly, with little additional elongation of the specimen before f ailure occurs. In f ailing, the specimen test bar break s 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 mark s is measured. The increase in length gives the elongation, expressed in percentage. The cross-section at the point of f ailure 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 m aterial ductility. In structural plastic part design it is essential to ensure that the stresses that would result f rom loading will be within the elastic range. If the elastic limit is exceeded, permanent deformation takes place due to plastic flow or slippage along mole cular slip planes. This will result in permanent plastic deformations. U lt imat e str ess is
3 .2
Compressive Stress
C ompr essive str ess is
the c ompressive force divided b y cross-sectional a rea, me asured in
psi or MPa. It is general practice in pl astic p art design to a ssume that the c ompressive 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 of ten greater than the ultimate tensile strength. In other words, most plastics can withstand more compressive surf ace 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 di ameter. The load is applied to the specimen f rom two directions in axial opposition. The ultimate compressive strength is measured when the specimen f ails by crushing.
3 .2
C ompr essive S tr ess
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 break age) region. The structural analysis of thermoplastic parts is more complex when the material is in compression. Fa ilure develops under the influence of a bending moment that increases as the deflection increases. A plastic part’s geometric shape is a significant f actor in its capacity to withstand compressive lo ads.
σ
Figure 3-3
=
P A
=
COMPRESSIVE FORCE AREA
(3.2)
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
str ess 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. S hear
48
S tr engt h
of
Mat er ial
for Plast ics
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 successf ully approximated as 0.75 of the ultimate tensile stress of the m aterial.
τ =
Q A
=
SHEAR LOAD AREA
(3.3)
Q
Q
3.4
Figure
3-4 Shear stress sample specimen example: (a) before; (b) af ter
Torsion Stress
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 f rom 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. T orsional
Figure 3-5
Torsion stress
3.4
T orsion S tr ess
49
When a shaf t is sub jected to a torsional moment or torque, the resulting shear stress is:
τ
=
Mt J
=
Mt r I
(3.4)
Mt is the torsional moment and it is: Mt
=
FR
(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 f rom the center of section to the outer fiber) F load applied
3.5
Elongations
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 r atio of the increase in elong ation by initial dimension of a m aterial. Again, strain is dimensionless. Depending on the nature of the applied load, strains can be tensile, compressive, or shear.
E longat ion
ε
3.5.1
=
∆L
L
(%)
(3.6)
Tensile Strain
A test specimen bar similar to that described in Section 3.1 is used to determine the t ensile str ain. The ultimate tensile strain is determined when the test specimen, being pulled apart by its ends, elongates. Just before the specimen break s, 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 / i n) 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
S tr engt h
of
Mat er ial
for Plast ics
Figure
3-6 Tensile specimen loaded showing dimensional change in length. The difference between the original length (L) and the elongated length is ∆L
Figure 3-7
Compressive specimen s howing dimensional change in length. L is the original length; P is the compressive fo rce. ∆ L is the dimensional change in length
3.5
3.5.2
51
E longat ions
Compressive Strain
The compr essive str ain 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 f ails by crushing.
3.5.3
Shear Strain
str ain is a measure of the angle of deformation γ – g amma. As is the case with shear stress, there is also no recognized standard test for shear strain. S hear
Q Q
Q Q
γ
Figure 3-8
3 .6
Shear strain: (a) before; (b) af ter
True Stress and Strain vs. Engineering Stress and Strain
E ngineer ing str ain is
the ratio of the total deformation over initial length. E ngineer ing str ess is the ratio of the force applied at the end of the test specimen by initial constant area. T rue str ess is the ratio of the instantaneous force over instantaneous area. Formula 3.7 shows that the true stress is a f unction of engineering stress multiplied b y a f actor based on engineering strain. = σ (1+ ε )
σ TRUE
(3.7)
T rue str ain is
the ratio of instantaneous deformation over instantaneous length. Formula 3.8 shows that the true strain is a logarithmic f unction of engineering strain. ε
= ln(1 + ε )
TRUE
(3.8)
52
S tr engt h
of
Mat er ial
for Plast ics
By using the ultimate strain and stress values, we can easily determine the true ultimate stress as: σ
= σ ULTIMATE (1+ ε ULTMATE )
ULTIMATETRUE
(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: ε ULTIMATE
TRUE
= ln(1 + ε ULTIMATE )
(3.10)
Both true stress and true strain are required input as material data in a v ariety of finite element analyses, where non-linear material analysis is needed.
Initial
Reduced
ar ea
ar ea
Figure 3-9
3 .7
True stress necking-down effect
Poisson’s Ratio
Provided the material deformation is within the elastic range, the ratio of l ateral to longitudinal strains is constant and the coefficient is called Poisson’s r at io.
ν
=
LATERAL STRAIN LONGITUDINAL STRAIN
(3.11)
In other words, stretching produces an elastic contraction in the two lateral directions. If an elastic str ain produces no change in volume, the two lateral str ains will be equal to half the tensile strain times –1.
53
3.7 Poisson’s Rat io b b
L
∆L
'
b
Figure
3-10 Dimensional change in only two of three directions
'
b
Under a tensile load, a test specimen increases (decreases for a compressive test) in length b y the a mount ∆L and decreases in width (increases for a compressive test) b y the amount ∆b. The related strains are: ε LONGITDINAL
ε LATERAL
=
=
∆L
L ∆b
(3.12)
b
Poisson’s ratio varies between 0, where no l ateral 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
S tr engt h
Table 3-1
of
Mat er ial
for Plast ics
(Continued)
Material Type
Poisson’s Ratio at 0.2 in. / min (5 mm / min) Strain Rate
Lead Polyamide Polycarbonate 13% gl ass 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: ∆ b = b − b′
(3.13)
Therefore, the ratio of l ateral dimensional change by the longitudinal dimensional change is: ∆b ν
= b ∆L
(3.14)
L Or, by rewriting, the Poisson’s ratio is:
ν
=
ε LATERAL
(3.15)
ε LONGITUDINAL
3.8
Modulus
of Elasticity
3.8.1
Young’s Modulus
The Y oung’s mod ulus or elast ic mod ulus 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 r ange of any material. T his r atio is called Young’s modulus a nd is measured in MPa or psi.
E=
σ ε
=
STRESS STRAIN
= CONSTANT
(3.16)
Hooke’s Law is generally applicable for most metals, thermoplastics and thermosets, within the limit of proportionality.
