STRESS AND STRAIN OF STRUCTURAL MATERIALS DEFINITION AND BASIC INFORMATION

Stress.

Stress is the intensity at a point in a body of the internal forces or components of force that act on a given plane through the point. Stress is expressed in force per unit of area (pounds per square inch, kilograms per square millimeter, etc.).

There are three kinds of stress: tensile, compressive, and shearing.

Flexure involves a combination of tensile and compressive stress. Torsion involves shearing stress. It is customary to compute stress on the basis of the original dimensions of the cross section of the body, though “true stress” in tension or compression is sometimes calculated from the area of the time a given stress exists rather than from the original area.

Strain.

Strain is a measure of the change, due to a force, in the size or shape of a body referred to its original size or shape. Strain is a nondimensional quantity but is frequently expressed in inches per inch, etc.

Under tensile or compressive stress, strain is measured along the dimension under consideration. Shear strain is defined as the tangent of the angular change between two lines originally perpendicular to each other.

Stress-Strain Diagram.

A stress-strain diagram is a diagram plotted with values of stress as ordinates and values of strain as abscissas. Diagrams plotted with values of applied load, moment, or torque as ordinates and with values of deformation, deflection, or angle of twist as abscissas are sometimes referred to as stress-strain diagrams but are more correctly called load-deformation diagrams.

The stress-strain diagram for some materials is affected by the rate of application of the load, by cycles of previous loading, and again by the time during which the load is held constant at specified values; for precise testing, these conditions should be stated definitely in order that the complete significance of any particular diagram may be clearly understood.

Modulus of Elasticity.

The modulus of elasticity is the ratio of stress to corresponding strain below the proportional limit. For many materials, the stress-strain diagram is approximately a straight line below a more or less well-defined stress known as the proportional limit.

Since there are three kinds of stress, there are three moduli of elasticity for a material, that is, the modulus in tension, the modulus in compression, and the modulus in shear.

The value in tension is practically the same, for most ductile metals, as the modulus in compression; the modulus in shear is only about 0.36 to 0.42 of the modulus in tension.

The modulus is expressed in pounds per square inch (or kilograms per square millimeter) and measures the elastic stiffness (the ability to resist elastic deformation under stress) of the material.

POISSON’S RATIO BASIC INFORMATION AND TUTORIAL

POISSON'S RATIO TUTORIALS AND SAMPLE PROBLEM
What Is Poisson's Ratio? Sample Problem And Solution Using Poisson's Ratio


When a homogeneous slender bar is axially loaded, the resulting stress and strain satisfy Hooke’s law, as long as the elastic limit of the material is not exceeded.

In all engineering materials, the elongation produced by an axial tensile force P in the direction of the force is accompanied by a contraction in any transverse direction (Fig. 2.36).† In this section and the following sections (Secs. 2.12 through 2.15), all materials considered will be assumed to be both homogeneous and isotropic, i.e., their mechanical properties will be assumed independent of both position and direction.

It follows that the strain must have the same value for any transverse direction.\ Therefore, for the loading shown in Fig. 2.35 we must have Py 5 Pz. This common value is referred to as the lateral strain.

An important constant for a given material is its Poisson’s ratio, named after the French mathematician Siméon Denis Poisson (1781–1840) and denoted by the Greek letter n (nu). It is defined as

v = - lateral strain / lateral stress.

Sample Problem:


A 500-mm-long, 16-mm-diameter rod made of a homogenous, isotropic material is observed to increase in length by 300 mm, and to decrease in diameter by 2.4 mm when subjected to an axial 12-kN load. Determine the modulus of elasticity and Poisson’s ratio of the material.

Solution:

Click on the image to enlarge and see the solution.

HOOKE’S LAW; MODULUS OF ELASTICITY BASICS AND TUTORIALS

HOOKE’S LAW; MODULUS OF ELASTICITY BASIC INFORMATION
What Is Hooke's Law? How To Apply Hooke's Law?


Most engineering structures are designed to undergo relatively small deformations, involving only the straight-line portion of the corresponding stress-strain diagram. For that initial portion of the diagram, the stress s is directly proportional to the strain P, and we can write

σ = ΞE

This relation is known as Hooke’s law, after Robert Hooke (1635–1703), an English scientist and one of the early founders of applied mechanics. The coefficient E is called the modulus of elasticity of the material involved, or also Young’s modulus, after the English scientist Thomas Young (1773–1829). 

Since the strain P is a dimensionless quantity, the modulus E is expressed in the same units as the stress s, namely in pascals or one of its multiples if SI units are used, and in psi or ksi if U.S. customary units are used.

The largest value of the stress for which Hooke’s law can be used for a given material is known as the proportional limit of that material. In the case of ductile materials possessing a well-defined yield poin, the proportional limit almost coincides with the yield point. 

For other materials, the proportional limit cannot be defined as easily, since it is difficult to determine with accuracy the value of the 63 stress s for which the relation between s and P ceases to be linear. But from this very difficulty we can conclude for such materials that using Hooke’s law for values of the stress slightly larger than the actual proportional limit will not result in any significant error.

Some of the physical properties of structural metals, such as strength, ductility, and corrosion resistance, can be greatly affected by alloying, heat treatment, and the manufacturing process used. For example, we note from the stress-strain diagrams of pure iron and of three different grades of steel (Fig. 2.11) that large variations in the yield strength, ultimate strength, and final strain (ductility) exist among these four metals. 

All of them, however, possess the same modulus of elasticity; in other words, their “stiffness,” or ability to resist a deformation within the linear range, is the same. Therefore, if a high-strength steel is substituted for a lower-strength steel in a given structure, and if all dimensions are kept the same, the structure will have an increased load-carrying capacity, but its stiffness will remain unchanged.

For each of the materials considered so far, the relation between normal stress and normal strain, s 5 EP, is independent of the direction of loading. This is because the mechanical properties of each material, including its modulus of elasticity E, are independent of the direction considered. Such materials are said to be isotropic.

Materials whose properties depend upon the direction considered are said to be anisotropic. An important class of anisotropic materials consists of fiberreinforced composite materials. These composite materials are obtained by embedding fibers of a strong, stiff material into a weaker, softer material, referred to as a matrix. 

Typical materials used as fibers are graphite, glass, and polymers, while various types of resins are used as a matrix. Figure 2.12 shows a layer, or lamina, of a composite material consisting of a large number of parallel fibers embedded in a matrix.

An axial load applied to the lamina along the x axis, that is, in a direction parallel to the fibers, will create a normal stress sx in the lamina and a corresponding normal strain Px which will satisfy Hooke’s law as the load is increased and as long as the elastic limit of the lamina is not exceeded. 

Similarly, an axial load applied along the y axis, that is, in a direction perpendicular to the lamina, will create a normal stress sy and a normal strain Py satisfying Hooke’s law, and an axial load applied along the z axis will create a normal stress sz and a normal strain Pz which again satisfy Hooke’s law. 

However, the moduli of elasticity Ex, Ey, and Ez corresponding, respectively, to each of the above loadings will be different. Because the fibers are parallel to the x axis, the lamina will offer a much stronger resistance to a loading directed along the x axis than to a loading directed along the y or z axis, and Ex will be much 

larger than either Ey or Ez.

A flat laminate is obtained by superposing a number of layers or laminas. If the laminate is to be subjected only to an axial load causing tension, the fibers in all layers should have the same orientation as the load in order to obtain the greatest possible strength. But if the laminate may be in compression, the matrix material may not be sufficiently strong to prevent the fibers from kinking or buckling. 

The lateral stability of the laminate may then be increased by positioning some of the layers so that their fibers will be perpendicular to the load. Positioning some layers so that their fibers are oriented at 308,

45 deg, or 60 deg to the load may also be used to increase the resistance of the laminate to in-plane shear. 

STRESS - STRAIN CURVE FOR BRITTLE MATERIALS BASICS AND TUTORIALS

STRESS - STRAIN CURVE FOR BRITTLE MATERIALS BASIC INFORMATION
What Is The Stress-Strain Curve Of Brittle Materials?


Many of the characteristics of a material can be deduced from the tensile test. It is more convenient to compare materials in terms of stresses and strains, rather than loads and extensions of a particular specimen of a material.

The stress at any stage is the ratio of the load of the original cross-sectional area of the test specimen; the strain is the elongation of a unit length of the test specimen.

For stresses up to about 750 MN/m2 the stress-strain curve is linear, showing that the material obeys Hooke’s law in this range; the material is also elastic in this range, and no permanent extensions remain after removal of the stresses.

The ratio of stress to strain for this linear region is usually about 200 GN/m2 for steels; this ratio is known as Young’s modulus and is denoted by E. The strain at the limit of proportionality is of the order 0.003, and is small compared with strains of the order 0.100 at fracture.

FIG 1.5

We note that Young’s modulus has the units of a stress; the value of E defines the constant in the linear relation between stress and strain in the elastic range of the material. We have

for the linear-elastic range. If P is the total tensile load in a bar, A its cross-sectional area, and Lo its length, then

where e is the extension of the length Lo. Thus the expansion is given by

If the material is stressed beyond the linear-elastic range the limit of proportionality is exceeded, and the strains increase non-linearly with the stresses. Moreover, removal of the stress leaves the material with some permanent extension; h range is then bothnon-linear and inelastic.

The maximum stress attained may be of the order of 1500 MNlm’, and the total extension, or elongation, at this stage may be of the order of 10%. The curve of Figure 1.5 is typical of the behaviour of brittle materials as, for example, area characterized by small permanent elongation at the breaking point; in the case of metals this is usually lo%, or less.

When a material is stressed beyond the limit of proportionality and is then unloaded, permanent deformations of the material take place. Suppose the tensile test-specimen of Figure 1.5 is stressed beyond the limit of proportionality, to a point b on the stress-strain diagram. If the stress is now removed, the stress-strain relation follows the curve bc; when the stress is completely removed there is a residual strain given by the intercept Oc on the &-axis.

If the stress is applied again, the stress-strain relation follows the curve cd initially, and finally the curve df to the breaking point. Both the unloading curve bc and the reloading curve cd are approximately parallel to the elastic line Oa; they are curved slightly in opposite directions.

The process of unloading and reloading, bcd, had little or no effect on the stress at the breaking point, the stress-strain curve being interrupted by only a small amount bd, Figure 1.6. The stress-strain curves of brittle materials for tension and compression are usually similar in form, although the stresses at the limit of proportionality and at fracture may be very different for the two loading conditions.

Typical tensile and compressive stress-strain curves for concrete are shown in Figure 1.7; the maximum stress attainable in tension is only about one-tenth of that in compression, although the slopes of the stress strain curves in the region of zero stress are nearly equal.