A F
.0 (2.1)
Since the cross section changes during the tensile test, the
ini-Fundamental Mechanical
Properties of Materials
tial unit area, A0, is mostly used; see below. If the force is ap-plied parallel to the axis of a rod-shaped material, as in the ten-sile tester (that is, perpendicular to the faces A0), then is called a tensile stress. If the stress is applied parallel to the faces (as in Figure 2.3), it is termed shear stress, .
Many materials respond to stress by changing their dimen-sions. Under tensile stress, the rod becomes longer in the direc-tion of the applied force (and eventually narrower perpendicular to that axis). The change in longitudinal dimension in response to stress is called strain,, that is:
l l0
l0
l0l, (2.2)
where l0 is the initial length of the rod and l is its final length.
The absolute value of the ratio between the lateral strain (shrinkage) and the longitudinal strain (elongation) is called the Poisson ratio, . Its maximum value is 0.5 (no net volume change). In reality, the Poisson ratio for metals and alloys is gen-erally between 0.27 and 0.35; in plastics (e.g., nylon) it may be as large as 0.4; and for rubbers it is even 0.49, which is near the maximum possible value.
10kg
FIGURE2.1.Schematic representation of a bend test. Note that the convex surface is under tension and the concave surface is under compression. Both stresses are essen-tially parallel to the surface. The bend test is particularly used for brittle materials.
Sample
F
A0 FIGURE2.2.Schematic
repre-sentation of a tensile test equipment. The lower cross-bar is made to move downward and thus ex-tends a force, F, on the test piece whose cross-sectional area is A0. The specimen to be tested is either threaded into the specimen holders or held by a vice grip.
The force is measured in newtons (1 N 1 kg m s2) and the stress is given in N m2or pascal (Pa). (Engineers in the United States occasionally use the pounds per square inch (psi) instead, where 1 psi 6.895 103 Pa and 1 pound 4.448 N. See Ap-pendix II.) The strain is unitless, as can be seen from Eq. (2.2) and is usually given in percent of the original length.
The result of a tensile test is commonly displayed in a stress–strain diagram as schematically depicted in Figure 2.4.
Several important characteristics are immediately evident. Dur-ing the initial stress period, the elongation of the material re-sponds to in a linear fashion; the rod reverts back to its orig-inal length upon relief of the load. This region is called the elastic range. Once the stress exceeds, however, a critical value, called the yield strength, y, some of the deformation of the material becomes permanent. In other words, the yield point separates the elastic region from the plastic range of materials.
Plasticpart
Elastic part
Stress
Yield strength
y
Tensile strength
T
Breaking strength
B
Tension
Compression
Strain Necking
FIGURE2.4.Schematic rep-resentation of a
stress–strain diagram for a ductile material. For ac-tual values of yandT, see Table 2.1 and Figure 2.5.
a
xy
yx
a FIGURE2.3.Distortion of a
cube caused by shear stresses
xyandyx.
This is always important if one wants to know how large an ap-plied stress needs to be in order for plastic deformation of a workpiece to occur. On the other hand, the yield strength pro-vides the limit for how much a structural component can be stressed before unwanted permanent deformation takes place.
As an example, a screwdriver has to have a high yield strength;
otherwise, it will deform upon application of a large twisting force. Characteristic values for the yield strength of different materials are given in Table 2.1 and Figure 2.5.
The highest force (or stress) that a material can sustain is called the tensile strength,1 T (Figure 2.4). At this point, a localized decrease in the cross-sectional area starts to occur. The material is said to undergo necking, as shown in Figure 2.6. Because the cross section is now reduced, a smaller force is needed to con-tinue deformation until eventually the breaking strength,B, is reached (Figure 2.4).
The slope in the elastic part of the stress–strain diagram (Fig-ure 2.4) is defined to be the modulus of elasticity, E, (or Young’s modulus):
E. (2.3)
Equation (2.3) is generally referred to as Hooke’s Law. For shear stress, [see above and Figure 2.3], Hooke’s law is appropriately written as:
1Sometimes called ultimate tensile strength or ultimate tensile stress,UTS. Ultrapure
where is the shear strain a/a tan ⬇ and G is the shear modulus.
The modulus of elasticity is a parameter that reveals how “stiff ” a material is, that is, it expresses the resistance of a material to elastic bending or elastic elongation. Specifically, a material hav-ing a large modulus and, therefore, a large slope in the stress–strain diagram deforms very little upon application of even a high stress. This material is said to have a high stiffness. (For average values, see Table 2.1.) This is always important if one re-quires close tolerances, such as for bearings, to prevent friction.
Stress–strain diagrams vary appreciably for different materials and conditions. As an example, brittle materials, such as glass, stone, or ceramics have no separate yield strength, tensile strength, or breaking strength. In other words, they possess essentially no plastic (ductile) region and, thus, break already before the yield strength is reached [Figure 2.7(a)]. Brittle materials (e.g., glass) are said to have a very low fracture toughness. As a consequence, tools (hammers, screwdrivers, etc.) should not be manufactured from brittle materials because they may break or cause injuries.
Ductile materials (e.g., many metals) on the other hand, with-stand a large amount of permanent deformation (strain) before they break, as seen in Figure 2.7(a). (Ductility is measured by the amount of permanent elongation or reduction in area, given in percent, that a material has withstood at the moment of fracture.) Many materials essentially display no well-defined yield strength in the stress–strain diagram; that is, the transition be-tween the elastic and plastic regions cannot be readily determined [Figure 2.7(b)]. One therefore defines an offset yield strength at which a certain amount of permanent deformation (for example, Necking F
F FIGURE2.6.Necking of a test sample that was
stressed in a tensile machine.
TABLE2.1.Some mechanical properties of materials
Modulus of Yield Tensile elasticity, strength, strength,
Material E [GPa] y[MPa] T[MPa]
Diamond 1,000 50,000 same
SiC 450 10,000 same
W 406 1000 1510
Cast irons 170–190 230–1030 400–1200
Low carbon steel, 196 180–260 325–485
hot rolled
Carbon steels, water- ⬃200 260–1300 500–1800 quenched and tempered
Fe 196 50 200
Cu 124 60 400
Si 107 — —
10% Sn bronze 100 190 —
SiO2(silica glass) 94 7200 about the same
Au 82 40 220
Al 69 40 200
Soda glass 69 3600 about the same
Concrete 50 25* —
Wood冨冨 to grain 9–16 — 33–50*; 73–121
Pb 14 11 14
Spider drag line 2.8–4.7 — 870–1420
Nylon 3 49–87 60–100
Wood芯 to grain 0.6–1 5* 3–10*; 2–8
Rubbers 0.01–0.1 — 30
PVC 0.003–0.01 45 —
*compression; tensile.
Note: The data listed here are average values. See Chapter 3 for the di-rectionality of certain properties called anisotropy; see also Figure 2.4.) For glasses, see also Table 15.1.
0.2%) has occurred and which can be tolerated for a given ap-plication. A line parallel to the initial segment in the stress–strain curve is constructed at the distance 0.2%. The intersect of this line with the stress–strain curve yields 0.2[Figure 2.7(b)].
Some materials, such as rubber, deform elastically to a large extent, but cease to be linearly elastic after a strain of about 1%.
Other materials (such as iron or low carbon steel) display a sharp yield point, as depicted in Figure 2.7(c). Specifically, as the stress is caused to increase to the upper yield point, no significant plas-tic deformation is encountered. From now on, however, the ma-terial will yield, concomitantly with a drop in the flow stress, (i.e., the stress at which a metal will flow) resulting in a lower yield point and plastic deformation at virtually constant stress [Figure 2.7(c)]. The lower yield point is relatively well defined but
fluc-FIGURE2.7.Schematic representations of stress–strain diagrams for various materials and conditions: (a) brittle (diamond, ceramics, ther-moset polymers) versus ductile (metals, alloys) materials; (b) defini-tion of the offset yield strength; (c) upper and lower yield points ob-served, for example, in iron and low carbon steels; (d) thermoplastic polymer; and (e) variation with temperature.
0.2%
0.2
(b)
y
T
B
(a) Brittle material
(diamond)
Ductile material (Cu)
(e) Low temperature
High temperature
(c) (d)
Upper
yield point
Lower yield point
Nonlinear elastic deformation (viscoelasticity)
Yield strength
Necking Linear elastic
deformation
Plastic deformation Tensile strength Necking
tuates about a fixed stress level. Thus, the yield strength in these cases is defined as the average stress that is associated with the lower yield point. Upon further stressing, the material eventually hardens, which requires the familiar increase in load if additional deformation is desired. The deformation at the lower yield point starts at locations of stress concentrations and manifests itself as discrete bands of deformed material, called Lüders bands, which may cause visible striations on the surface. The deformation oc-curs at the front of these spreading bands until the end of the lower yield point is reached.
A few polymeric materials, such as nylon, initially display a linear and, subsequently, a nonlinear (viscoelastic) region in the stress–strain diagram [Figure 2.7(d)]. Moreover, beyond the yield strength, a bathtub-shaped curve is obtained, as depicted in Fig-ure 2.7(d).
Stress–strain curves may vary for different temperatures [Fig-ure 2.7(e)]. For example, the yield strength, as well as the tensile strength, and to a lesser degree also the elastic modulus, are of-ten smaller at elevated temperatures. In other words, a metal can be deformed permanently at high temperatures with less effort than at room temperature. This property is exploited by indus-trial rolling mills or by a blacksmith when he shapes red-hot metal items on his anvil. The process is called hot working.
On the other hand, if metals, alloys, or some polymeric mate-rials are cold worked, that is, plastically deformed at ambient tem-peratures, eventually they become less ductile and thus harder and even brittle. This is depicted in Figure 2.8(a), in which a ma-terial is assumed to have been stressed beyond the yield strength.
Upon releasing the stress, the material has been permanently de-formed to a certain degree. Restressing the same material [Fig-ure 2.8(b)] leads to a higher Tand to less ductility. The plastic deformation steps can be repeated several times until eventually
yTB. At this point the workpiece is brittle, similar to a ceramic. Any further attempt of deformation would lead to im-mediate breakage. The material is now work hardened (or strain hardened) to its limit. A coppersmith utilizes cold working (ham-mering) for shaping utensils from copper sheet metal. The strain hardened workpiece can gain renewed ductility, however, by heating it above the recrystallization temperature (which is ap-proximately 0.4 times the absolute melting temperature). For copper, the recrystallization temperature is about 200°C.
The degree of strengthening acquired through cold working is given by the strain hardening rate, which is proportional to the slope of the plastic region in a true stress–true strain curve. This needs some further explanation. The engineering stress and the
engineering strain, as defined in Equations (2.1) and (2.2), are essentially sufficient for most practical purposes. However, as mentioned above, the cross-sectional area of a tensile test spec-imen decreases continuously, particularly during necking. The latter causes a decrease of beyond the tensile strength. A true stress and true strain diagram takes the varying areas into con-sideration (Figure 2.9(a)). One defines the true stress as:
t A
F
i
, (2.5)
where Aiis now the instantaneous cross-sectional area that varies during deformation. The true strain is then:1
t
冕
lli0 dll ln
冢
ll0i冣
ln冢
AA0i冣
. (2.6)1See Problems 8 and 9.
(b) (c)
(a)
0 1 1
y2 y3
2 ' 23 "
y1
y1
' ' " "
y0
y2
permanent deformation permanent
deformation permanent
deformation
FIGURE2.8.Increase of yield strength (and reduction of ductility) by re-peated plastic deformation. (a) Sample is moderately stressed until some plastic deformation has occurred, and then it is unloaded, which yields permanent deformation. (b) The sample is subsequently additionally permanently deformed. Note that the coordinate system has shifted after unloading from 0to1. (c) Limit of plastic deforma-tion is reached after renewed stressing.
Elastic region
Plastic region
slope n=1 lnt
lnt slope n<1
In many cases, and before necking begins, one can approximate the true stress–true strain curve by the following empirical equation:
t K(t)n, (2.7)
where n is the strain hardening exponent (having values of less than unity) and K is another materials constant (called the strength coefficient) which usually amounts to several hundred MPa. Taking the (natural) logarithm of Eq. (2.7) yields:
lnt n ln t ln K, (2.8) which reveals that the strain hardening exponent (or strain hard-ening rate), n, is the slope in the plastic portion of an ln tversus lntdiagram, see Figure 2.9(b).
The tensile test and the resulting stress–strain diagrams have been shown above to provide a comprehensive insight into many of the mechanical properties of materials. For specialized appli-cations, however, a handful of further tests are commonly used.
Some of them will be reviewed briefly below.
The hardness test is nondestructive and fast. A small steel sphere (commonly 10 mm in diameter) is momentarily pressed into the surface of a test piece. The diameter of the indentation is then measured under the microscope, from which the Brinell hard-ness number (BHN) is calculated by taking the applied force and the size of the steel sphere into consideration. The BHN is directly proportional to the tensile strength. (The Rockwell hardness tester uses instead a diamond cone and measures the depth of the in-dentation under a known load whereas the Vickers and Knoop mi-crohardness techniques utilize diamond pyramids as indenters.)
Materials, even when stressed below the yield strength, still may eventually break if a large number of tension and compres-FIGURE2.9. (a) True stress versus true strain diagram (compare to Figure 2.4). (b) lntversus ln t
diagram.
y
t
t
Plasticregion
Elastic region
T
(a)
(b)
sion cycles are applied. The fatigue test measures the number of bending cycles that need to be applied for a specific load un-til failure occurs. Fatigue plays a potentially devastating role in airplane and automobile parts.
When subjected to a sudden blow, some materials break at a lower stress than that measured using a tensile machine. The im-pact tester investigates the toughness of materials by striking them at the center while fixing both ends. Toughness is defined as the energy (not the force) required to break a material. A heavy pendulum usually is utilized for the blow. The absorbed energy during the breakage is calculated from the difference in pendu-lum height before and after impact.
The creep test measures the continuous and progressive plas-tic deformation of materials at high temperatures while a con-stant stress or a concon-stant load below the room temperature yield strength is applied. The temperature at which creep commences varies widely among materials but is generally above 0.3 times the absolute melting temperature. Lead creeps already at room temperature. We will return to creep in Chapter 6.
Leonardo da Vinci (1452–1519) invented already a wire test-ing device in which sand is poured into a bucket (acttest-ing as ten-sile load) until the wire breaks.
In conclusion, the mechanical properties of materials include ductility, yield strength, elasticity, tensile strength, hardness, toughness under shock, brittleness, fatigue behavior, stiffness, and creep. The question certainly may be raised whether or not it is possible to explain some or all of these diverse properties by one or a few fundamental concepts. We shall attempt to tackle this question in the next chapter.
Problems
2.1. What was the original length of a wire that has been strained by 30%
and whose final length is 1 m?
2.2. The initial diameter of a wire is 2 cm and needs to be reduced to 1 cm. Cal-culate the amount of cold work (re-duction in area in percent) which is necessary.
2.3. Calculate the initial diameter of a wire that has been longitudinally strained by 30% and whose final di-ameter is 0.1 cm. Assume no volume change.
2.4. What force is needed to plastically deform a wire of 2 cm diameter whose yield strength is 40 MPa?
2.5. Calculate the ductility of a wire (that is, its percent area reduction at frac-ture during tensile stressing) whose initial diameter was 1 cm and whose diameter at fracture is 0.8 cm.
2.6. Calculate the true stress at fracture for a metal rod whose engineering frac-ture strength is 450 MPa and whose diameter at fracture was reduced by plastic deformation from 1 to 0.8 cm.
2.7. Calculate the strain hardening expo-nent for a material whose true stress and true strain values are 450 MPa and 15%, respectively. Take K 700 MPa.
2.8. In Eq. (2.6), the relation
t
冕
lli0 dll ln
冢
ll0i冣
ln冢
AA0i冣
is given. Show in mathematical terms for what condition (pertaining to a possible change in volume) this relation is true.
2.9. Show that the true and engineering stress and strain are related by
t(1 ) and
t ln (1 )
for the case when no volume change occurs during deformation, that is, before the onset of necking.
2.10. Compare engineering strain with true strain and engineering stress with true stress for a material whose ini-tial diameter was 2 cm and whose fi-nal diameter at fracture is 1.9 cm. The initial length before plastic deforma-tion was 10 cm. The applied force was
3 104N. Assume no volume change during plastic deformation.
2.11. A metal plate needs to be reduced to a thickness of 4 cm by involving a rolling mill. After rolling, the elastic properties of the material cause the plate to regain some thickness. Cal-culate the needed separation be-tween the two rollers when the yield strength of the material after plastic deformation is 60 MPa and the mod-ulus of elasticity is 124 GPa.
2.12. A cylindrical rod of metal whose ini-tial diameter and length are 20 mm and 1.5 m, respectively, is subjected to a tensile load of 8 104 N. What is the final length of the rod? Is the load stressing the rod beyond its elas-tic range when the yield strength is 300 MPa and the elastic modulus is 180 GPa?
2.13. Calculate the Poisson ratio of a cylin-drical rod that was subjected to a ten-sile load of 3500 N and whose initial diameter was 8 mm. The modulus of elasticity is 65 GPa, and the change in diameter is 2.5 m. Assume that the deformation is entirely elastic.
2.14. Calculate the Poisson ratio for the case where no volume change takes place.
Suggestions for Further Study
See the end of Chapter 3. Further, most textbooks of materials science cover mechanical properties.