39
The strain energy U is equal to the strain energy density integrated over the volume v.
U = U
2
0 dv
(2.6b)
v
The resilience and toughness of the material are measures, respectively, of the strain energy present in the material at the elastic limit or at the fracture point.
RESILIENCE The ability of a material to absorb energy per unit volume without permanent deformation is called its resilience UR (also called modulus of resilience) and is equal to the area under the stress-strain curve up to the elastic limit, shown as the color-shaded area in Figure 2-2 a. Resilience is defined as: 1
R
U = el d = el
S
el
0
2
= 1
S
1 S 2
S el =
el
el
(2.7)
2
E
2 E
2
1 S y
U
R
2 E
where Sel and el represent, respectively, the strength and strain at the elastic limit.
Substitution of Hooke’s law from equation 2.2 expresses the relationship in terms of strength and Young’s modulus. Since the Sel value is seldom available, a reasonable approximation of resilience can be obtained by using the yield strength Sy instead.
This relationship shows that a stiffer material of the same elastic strength is less resilient than a more compliant one. A rubber ball can absorb more energy without permanent deformation than one made of glass.
TOUGHNESS The ability of a material to absorb energy per unit volume without fracture is called its toughness UT (also called modulus of toughness) and is equal to the area under the stress-strain curve up to the fracture point, shown as the entire shaded area in Figure 2-2 a. Toughness is defined as:
* It is interesting to note that one
= f
U
of the toughest and strongest
T
d
materials known is that of spider
0
webs! These tiny arachnids spin a
S +
y Sut
monofilament that has an ultimate
tensile strength of 200 to 300 kpsi
(1380 to 2070 MPa) and 35%
2
f
(2.8)
elongation to fracture! It also can
absorb more energy without
where S
rupture than any fiber known,
ut and f represent, respectively, the ultimate tensile strength and the strain at absorbing 3 times as much energy
fracture. Since an analytical expression for the stress-strain curve is seldom available as Kevlar, the man-made fiber
for actual integration, an approximation of toughness can be obtained by using the av-used for bullet-proof vests.
According to the Boston Globe,
erage of the yield and ultimate strengths and the strain at fracture to calculate an area.
January 18, 2002, researchers in
The units of toughness and resilience are energy per unit volume (in-lb/in3 or joules/
Canada and the U.S. have
synthesized a material with similar
m3). Note that these units are numerically equivalent to psi or Pa.
properties to spider silk in strands
up to 10-ft long with strengths of
1/4 to 1/3 that of natural silk
A ductile material of similar ultimate strength to a brittle one will be much more fiber, “stronger than a steel wire
tough. A sheet-metal automobile body will absorb more energy from a collision through of similar weight,” and one that
has greater elasticity than organic
plastic deformation than will a brittle, fiberglass body.*
silk fiber.
40
MACHINE DESIGN -
An Integrated Approach
IMPACT TESTING Various tests have been devised to measure the ability of materials to withstand impact loading. The Izod and the Charpy tests are two such procedures which involve striking a notched specimen with a pendulum and recording the 2
kinetic energy needed to break the specimen at a particular temperature. While these data do not directly correlate with the area under the stress-strain curve, they nevertheless provide a means to compare the energy absorption capacity of various materials under controlled conditions. Materials handbooks such as those listed in this chapter’s bibliography give data on the impact resistance of various materials.
Fracture Toughness
Fracture toughness Kc (not to be confused with the modulus of toughness defined above) is a material property that defines its ability to resist stress at the tip of a crack.
The fracture toughness of a material is measured by subjecting a standardized, pre-cracked test specimen to cyclical tensile loads until it breaks. Cracks create very high local stress concentrations which cause local yielding (see Section 4.15). The effect of the crack on the local stress is measured by a stress intensity factor K which is defined in Section 5.3. When the stress intensity K reaches the fracture toughness Kc, a sudden fracture occurs with no warning. The study of this failure phenomenon is called fracture mechanics and it is discussed in more detail in Chapters 5 and 6.
Creep and Temperature Effects
The tensile test, while slow, does not last long compared to the length of time an actual machine part may be subjected to constant loading. All materials will, under the right environmental conditions (particularly elevated temperatures), slowly creep (deform) under stress loadings well below the level (yield point) deemed safe in the tensile test.
Ferrous metals tend to have negligible creep at room temperature or below. Their creep rates increase with increasing ambient temperature, usually becoming significant around 30–60% of the material’s absolute melting temperature.
Low-melt-temperature metals such as lead, and many polymers, can exhibit significant creep at room temperature as well as increasing creep rates at higher temperatures.
Creep data for engineering materials are quite sparse due to the expense and time required to develop the experimental data. The machine designer needs to be aware of the creep phenomenon and obtain the latest manufacturer’s data on the selected materials if high ambient temperatures are anticipated or if polymers are specified. The creep phenomenon is more complex than this simple description implies. See the bibliography to this chapter for more complete and detailed information on creep in materials.
It is also important to understand that all material properties are a function of temperature, and published test data are usually generated at room temperature. Increased temperature usually reduces strength. Many materials that are ductile at room temperature can behave as brittle materials at low temperatures. Thus, if your application involves either elevated or low temperatures, you need to seek out relevant material-property data for your operating environment. Material manufacturers are the best source of up-to-date information. Most manufacturers of polymers publish creep data for their materials at various temperatures.