Mechanical Properties of Materials
The mechanical behavior of materials is the response of the material to external loads. All materials deform in response to loads; however, the specific response of a material depends on its properties, the magnitude and type of load, and the geometry of the element.
Whether the material “fails” under the load conditions depends on the failure criterion. Catastrophic failure of a structural member, resulting in the collapse of the structure, is an obvious material failure.
However, in some cases, the failure is more subtle, but with equally severe consequences. For example, pavement may fail due to excessive roughness at the surface, even though the stress levels are well within the capabilities of the material.
A building may have to be closed due to excessive vibrations by wind or other live loads, although it could be structurally sound. These are examples of functional failures.
Loading Conditions of Materials
One of the considerations in the design of a project is the type of loading that the structure will be subjected to during its design life. The two basic types of loads are static and dynamic. Each type affects the material differently, and frequently the interactions between the load types are important.
Civil engineers encounter both when designing a structure. Static loading implies a sustained loading of the structure over a period of time. Generally, static loads are slowly applied such that no shock or vibration is generated in the structure. Once applied, the static load may remain in place or be removed slowly.
Loads that remain in place for an extended period of time are called sustained (dead) loads. In civil engineering, much of the load the materials must carry is due to the weight of the structure and equipment in the structure.
Loads that generate a shock or vibration in the structure are dynamic loads. Dynamic loads can be classified as periodic, random, or transient, as shown in Figure 1.
A periodic load, such as a harmonic or sinusoidal load, repeats itself with time. For example, rotating equipment in a building can produce a vibratory load. In a random load, the load pattern never repeats, such as that produced by earthquakes.
Transient load, on the other hand, is an impulse load that is applied over a short time interval, after which the vibrations decay until the system returns to a rest condition. For example, bridges must be designed to withstand the transient loads of trucks.
Stress–Strain Relations of Materials
Materials deform in response to loads or forces. In 1678, Robert Hooke published the first findings that documented a linear relationship between the amount of force applied to a member and its deformation. The amount of deformation is proportional to the properties of the material and its dimensions.
The effect of the dimensions can be normalized. Dividing the force by the cross-sectional area of the specimen normalizes the effect of the loaded area. The force per unit area is defined as the stress s in the specimen (i.e., σ = force/area).
Dividing the deformation by the original length is defined as strain ε of the specimen (i.e., ε = change in length/original length). Much useful information about the material can be determined by plotting the stress–strain diagram.
Figure 2 shows typical uniaxial tensile or compressive stress–strain curves for several engineering materials. Figure 2(a) shows a linear stress–strain relationship up to the point where the material fails. Glass and chalk are typical of materials exhibiting this tensile behavior.
Figure 2(b) shows the behavior of steel in tension. Here, a linear relationship is obtained up to a certain point (proportional limit), after which the material deforms without much increase in stress.
On the other hand, aluminum alloys in tension exhibit a linear stress–strain relationship up to the proportional limit, after which a nonlinear relation follows, as illustrated in Figure 2(c).
Figure 2(d) shows a nonlinear relation throughout the whole range. Concrete and other materials exhibit this relationship, although the first portion of the curve for concrete is very close to being linear.
Soft rubber in tension differs from most materials in such a way that it shows an almost linear stress–strain relationship followed by a reverse curve, as shown in Figure 2(e).
Elastic Behavior of Materials
If a material exhibits true elastic behavior, it must have an instantaneous response (deformation) to load, and the material must return to its original shape when the load is removed. Many materials, including most metals, exhibit elastic behavior, at least at low stress levels.
Elastic deformation does not change the arrangement of atoms within the material, but rather it stretches the bonds between atoms. When the load is removed, the atomic bonds return to their original position.
Young observed that different elastic materials have different proportional constants between stress and strain.
For a homogeneous, isotropic, and linear elastic material, the proportional constant between normal stress and normal strain of an axially loaded member is the modulus of elasticity or Young’s modulus, E, and is equal to
E = σ/ε …….(Equation 1)
Where σ is the normal stress and ε is the normal strain.
In the axial tension test, as the material is elongated, there is a reduction of the cross section in the lateral direction. In the axial compression test, the opposite is true.
The ratio of the lateral strain, εl, to the axial strain, εa, is
Poisson’s ratio, v = – εl/εa …….(Equation 2)
Since the axial and lateral strains will always have different signs, the negative sign is used in Equation 2 to make the ratio positive.
Poisson’s ratio has a theoretical range of 0.0 to 0.5, where 0.0 is for a compressible material in which the axial and lateral directions are not affected by each other. The 0.5 value is for a material that does not change its volume when the load is applied. Most solids have Poisson’s ratios between 0.10 and 0.45.
Although Young’s modulus and Poisson’s ratio were defined for the uniaxial stress condition, they are important when describing the three-dimensional stress– strain relationships, as well.
If a homogeneous, isotropic cubical element with linear elastic response is subjected to normal stresses σx, σy, and σz in the three orthogonal directions (as shown in Figure 3), the normal strains εx, εy, and εz can be computed by the generalized Hooke’s law,
Linearity and elasticity should not be confused. A linear material’s stress–strain relationship follows a straight line. An elastic material returns to its original shape when the load is removed and reacts instantaneously to changes in load.
For example, Figure 4(a) represents a linear elastic behavior, while Figure 4(b) represents a nonlinear elastic behavior.
For materials that do not display any linear behavior, such as concrete and soils, determining a Young’s modulus or elastic modulus can be problematical. There are several options for arbitrarily defining the modulus for these materials.
Figure 5 shows four options: the initial tangent, tangent, secant, and chord moduli.
The initial tangent modulus is the slope of the tangent of the stress–strain curve at the origin.
The tangent modulus is the slope of the tangent at a point on the stress–strain curve.
The secant modulus is the slope of a chord drawn between the origin and an arbitrary point on the stress–strain curve.
The chord modulus is the slope of a chord drawn between two points on the stress–strain curve.
The selection of which modulus to use for a nonlinear material depends on the stress or strain level at which the material typically is used.
Also, when determining the tangent, secant, or chord modulus, the stress or strain levels must be defined.
Table 1 shows typical modulus and Poisson’s ratio values for some materials at room temperature.
Note that some materials have a range of modulus values rather than a distinct value. Several factors affect the modulus, such as curing level and proportions of components of concrete or the direction of loading relative to the grain of wood. Elastoplastic Behavior of Materials
For some materials, as the stress applied on the specimen is increased, the strain will proportionally increase up to a point; after this point, the strain will increase with little additional stress.
In this case, the material exhibits linear elastic behavior followed by plastic response. The stress level at which the behavior changes from elastic to plastic is the elastic limit. When the load is removed from the specimen, some of the deformation will be recovered and some of the deformation will remain as seen in Figure 6(a).
Plastic behavior indicates permanent deformation of the specimen so that it does not return to its original shape when the load is removed. This indicates that when the load is applied, the atomic bonds stretch, creating an elastic response; then the atoms actually slip relative to each other. When the load is removed, the atomic slip does not recover; only the atomic stretch is recovered. Several models are used to represent the behavior of materials that exhibit both elastic and plastic responses.
Figure 6(b) shows a linear elastic–perfectly plastic response in which the material exhibits a linear elastic response upon loading, followed by a completely plastic response. If such material is unloaded after it has plasticly deformed, it will rebound in a linear elastic manner and will follow a straight line parallel to the elastic portion, while some permanent deformation will remain.
If the material is loaded again, it will have a linear elastic response followed by plastic response at the same level of stress at which the material was unloaded.
Figure 6(c) shows an elastoplastic response in which the first portion is an elastic response followed by a combined elastic and plastic response.
If the load is removed after the plastic deformation, the stress–strain relationship will follow a straight line parallel to the elastic portion; consequently, some of the strain in the material will be removed, and the remainder of the strain will be permanent.
Upon reloading, the material again behaves in a linear elastic manner up to the stress level that was attained in the previous stress cycle. After that point the material will follow the original stress–strain curve. Thus, the stress required to cause plastic deformation actually increases. This process is called strain hardening or work hardening.
Strain hardening is beneficial in some cases, since it allows more stress to be applied without permanent deformation. In the production of cold-formed steel framing members, the permanent deformation used in the production process can double the yield strength of the member relative to the original strength of the steel.
Some materials exhibit strain softening, in which plastic deformation causes weakening of the material. Portland cement concrete is a good example of such a material. In this case, plastic deformation causes microcracks at the interface between aggregate and cement paste.
Materials that do not undergo plastic deformation prior to failure, such as concrete, are said to be brittle, whereas materials that display appreciable plastic deformation, such as mild steel, are ductile. Generally, ductile materials are preferred for construction. When a brittle material fails, the structure can collapse in a catastrophic manner.
On the other hand, overloading a ductile material will result in distortions of the structure, but the structure will not necessarily collapse. Thus, the ductile material provides the designer with a margin of safety.
Figure 7(a) demonstrates three concepts of the stress–strain behavior of elastoplastic materials. The lowest point shown on the diagram is the proportional limit, defined as the transition point between linear and nonlinear behavior. The second point is the elastic limit, which is the transition between elastic and plastic behavior.
However, most materials do not display an abrupt change in behavior from elastic to plastic. Rather, there is a gradual, almost imperceptible transition between the behaviors, making it difficult to locate an exact transition point . For this reason, arbitrary methods such as the offset and the extension methods, are used to identify the elastic limit, thereby defining the yield stress (yield strength).
In the offset method, a specified offset is measured on the abscissa, and a line with a slope equal to the initial tangent modulus is drawn through this point. The point where this line intersects the stress–strain curve is the offset yield stress of the material, as seen in Figure 7(a).
Different offsets are used for different materials (Table 2). The extension yield stress is located where a vertical projection, at a specified strain level, intersects the stress–strain curve. Figure 7(b) shows the yield stress corresponding to 0.5% extension.
Viscoelastic Behavior of Materials
To decrease the length of this web-page, content of this section is placed on another web-page. To access that page, please follow the link.
Temperature and Time Effects on Materials
The mechanical behavior of all materials is affected by temperature. Some materials, however, are more susceptible to temperature than others. For example, viscoelastic materials, such as plastics and asphalt, are greatly affected by temperature, even if the temperature is changed by only a few degrees.
Other materials, such as metals or concrete, are less affected by temperatures, especially when they are near ambient temperature. Ferrous metals, including steel, demonstrate a change from ductile to brittle behavior as the temperature drops below the transition temperature. This change from ductile to brittle behavior greatly reduces the toughness of the material.
While this could be determined by evaluating the stress–strain diagram at different temperatures, it is more common to evaluate the toughness of a material with an impact test that measures the energy required to fracture a specimen.
Figure 8 shows how the energy required to fracture a mild steel changes with temperature. The test results seen in Figure 8 were achieved by applying impact forces on bar specimens with a “defect” (a simple V notch) machined into the specimens (ASTM E23).
During World War II, many Liberty ships sank because the steel used in the ships met specifications at ambient temperature, but became brittle in the cold waters of the North Atlantic.
In addition to temperature, some materials, such as viscoelastic materials, are affected by the load duration. The longer the load is applied, the larger is the amount of deformation or creep. In fact, increasing the load duration and increasing the temperature cause similar material responses.
Therefore, temperature and time can be interchanged. This concept is very useful in running some tests. For example, a creep test on an asphalt concrete specimen can be performed with short load durations by increasing the temperature of the material.
A time–temperature shift factor is then used to adjust the results for lower temperatures. Viscoelastic materials are affected not only by the duration of the load but also by the rate of load application. If the load is applied at a fast rate, the material is stiffer than if the load is applied at a slow rate.
For example, if a heavy truck moves at a high speed on an asphalt pavement, no permanent deformation may be observed. However, if the same truck is parked on an asphalt pavement on a hot day, some permanent deformations on the pavement surface may be observed.
Work and Energy
When a material is tested, the testing machine is actually generating a force in order to move or deform the specimen. Since work is force times distance, the area under a force–displacement curve is the work done on the specimen.
When the force is divided by the cross-sectional area of the specimen to compute the stress, and the deformation is divided by the length of the specimen to compute the strain, the force–displacement diagram becomes a stress–strain diagram.
However, the area under the stress–strain diagram no longer has the units of work. By manipulating the units of the stress–strain diagram, we can see that the area under the stress–strain diagram equals the work per unit volume of material required to deform or fracture the material. This is a useful concept, for it tells us the energy that is required to deform or fracture the material.
Such information is used for selecting materials to use where energy must be absorbed by the member. The area under the elastic portion of the curve is the modulus of resilience [Figure 9(a)]. The amount of energy required to fracture a specimen is a measure of the toughness of the material, as in Figure 9(b).
As shown in Figure 9(c), a high-strength material is not necessarily a tough material. For instance, increasing the carbon content of steel increases the yield strength but reduces ductility. Therefore, the strength is increased, but the toughness may be reduced.
Failure and Safety
Failure occurs when a member or structure ceases to perform the function for which it was designed. Failure of a structure can take several modes, including fracture, fatigue, general yielding, buckling, and excessive deformation.
Fracture is a common failure mode. A brittle material typically fractures suddenly when the static stress reaches the strength of the material, where the strength is defined as the maximum stress the material can carry.
On the other hand, a ductile material may fracture due to excessive plastic deformation. Many structures, such as bridges, are subjected to repeated loadings, creating stresses that are less than the strength of the material.
Repeated stresses can cause a material to fail or fatigue, at a stress well below the strength of the material. The number of applications a material can withstand depends on the stress level relative to the strength of the material.
As shown in Figure 10, as the stress level decreases, the number of applications before failure increases. Ferrous metals have an apparent endurance limit, or stress level, below which fatigue does not occur. The endurance limit for steels is generally in the range of one-quarter to one-half the ultimate strength.
Another example of a structure that may fail due to fatigue is pavement. Although the stresses applied by traffic are typically much less than the strength of the material, repeated loadings may eventually lead to a loss of the structural integrity of the pavement surface layer, causing fatigue cracks.
Another mode of failure is general yielding. This failure happens in ductile materials, and it spreads throughout the whole structure, which results in a total collapse. Long and slender members subjected to axial compression may fail due to buckling. Although the member is intended to carry axial compressive loads, a small lateral force might be applied, which causes deflection and eventually might cause failure.
Sometimes excessive deformation (elastic or plastic) could be defined as failure, depending on the function of the member. For example, excessive deflections of floors make people uncomfortable and, in an extreme case, may render the building unusable even though it is structurally sound.
To minimize the chance of failure, structures are designed to carry a load greater than the maximum anticipated load. The factor of safety (FS) is defined as the ratio of the stress at failure to the allowable stress for design (maximum anticipated stress):
FS = σfailure/σallowable ……….(Equation 4)
Where σfailure is the failure stress of the material and σallowable is the allowable stress for design.
Typically, a high factor of safety requires a large structural cross section and consequently a higher cost.
The proper value of the factor of safety varies from one structure to another and depends on several factors, including the
- cost of unpredictable failure in lives, dollars, and time,
- variability in material properties,
- degree of accuracy in considering all possible loads applied to the structure, such as earthquakes,
- possible misuse of the structure, such as improperly hanging an object from a truss roof,
- degree of accuracy of considering the proper response of materials during design, such as assuming elastic response although the material might not be perfectly elastic.