Interpretation of Stress-Strain Curves and Mechanical Properties of Materials

Taking a First Look:

Let’s examine the stress-strain diagram shown in Figure 1. Notice that this chart is an arithmetic one.

That means that the length of any increment along either the X-axis (the horizontal one) or the Y-axis

(the vertical one) will be the same as the length of any other increment along the same axis. This is

important to us because it means we can use simple arithmetic to find the mathematical relationships

we need between any two points on the chart.

The vertical or Y-axis of the chart (ordinate) represents the load or force applied to the specimen, with

the full height of the Y-axis equaling the capacity of the load range employed.

The horizontal or X-axis (abscissa) represents the strain (elongation or compression of the specimen

under load). Tinius Olsen strain measuring instruments automatically record strain in fundamental

inches per inch (in./in.) units regardless of the gauge length. For example, using an LS-4%-2A (S-

1000-2A) extensometer, a setting of “B” (500x or 2%) on the magnification or range selector will

produce a magnification of 500x, as shown in Figure 1. This means that each inch of chart on the Xaxis

represents 0.2% (0.002 in./in.) of specimen elongation or extension.

Referring to Figure 1, notice that the load-elongation (stress-strain) curve starts at the lower left as a

straight line. This is because the load increases in direct proportion to the extension of the specimen

during the initial part of the test.

The highest point attained before the line begins to curve is called the Proportional Limit, which is the

maximum point at which extension remains proportional to load (Hooke’s Law). Past this point, the

proportional relationship between load and extension begins to increase more rapidly than load,

thereby producing a curve.

As explained on page 3, the initial straight-line portion of the diagram is used to calculate the Modulus

of the material, and one or more points along the curve are plotted to determine the Yield Strength at

designated of Offset or Extension Under Load (EUL).

In brief, a stress-strain diagram will help you determine: Modulus of Elasticity, Elastic Deformation,

Yield Strength, Proportional Limit, Proof Stress, Uniform Elongation, Total Elongation, Yield Point

Elongation and n-Value.

In addition, stress-strain diagrams can be used to calculate hysteresis (permanently absorbed or lost

energy that occurs during any cycle of loading or unloading when a material is subjected to repeated

loading, e.g. cycling tests in which load is remove before the specimen fails).

Steps in Interpreting a Stress-Strain Curve:

1. Using a sharp pencil and a ruler, carefully draw a straight line through as much of the straight

portion of the curve as possible, extending it from the bottom of the chart (Line A-B in Figure 2).

This straight line is referred to as the Modulus Line, from which the Modulus of Elasticity will be

calculated for the material tested.

The intersection of the lower end of this drawn line with the Y-axis is the true origin of the curve,

regardless of where the actual stress-strain curve begins. If proper care has been taken in the

operation of the recorder and instrument, the drawn line should intersect at the same point of origin

as the recorder-produced line.

2. Mark off the fundamental units of strain in inches per inch along the abscissa from the point of

origin. If not known, the value of each inch of chart for the instrument measuring range used can be

obtained from the appropriate instrumentation table furnished with the extensometer, or listed in the

bulletin on Tinius Olsen strain instrumentation (currently Bulletin 96).

If the “B” or 500x range of an LS-4%-2A (S-1000-2A) extensometer is used, each inch of chart will

represent 0.2% (0.002 in./in.) of strain, and should be noted on the chart paper.

3. Since load on the Y-axis is expressed in pounds, a specimen of any cross-sectional area can be

tested within the limits of the equipment. In order to make the necessary computations, the

pertinent loads must be converted into pounds per square inch (psi or stress) by dividing the load

by the cross-sectional area.

  Determining Yield Strengths

1. The Offset Method: The offset is the horizontal distance between the modulus line and any line

running parallel to it. For example, the line C-D in Figure 2 is “offset” from the modulus line by 0.2%

(0.002 in./in.).

The value of the offset for a given material is usually expressed this way: Yield Strength, 0.1% or

0.2% Offset. What this means is that a certain percentage of the set equals a certain percentage

of the fundamental extension units. For example, “0.2% Offset” means 0.2% of the fundamental

extension units of inches per inch, or 0.002 in./in.

Starting at the origin of the curve, measure off a distance equal to 0.002 in./in. along the X-axis.

Now using that as the origin, draw a line (C-D) parallel to the modulus line. Notice that the line C-D

intersects the stress-strain curve at a certain point (Y in Figure 2). The ordinate of this point (the

amount of stress in psi) is the Yield Strength at 0.2% Offset.

You can use the same method to determine the yield strength at a 0.1% offset by noting the

intersection of the curve and a line drawn parallel to the modulus line with an offset of 0.001 in./in.

2. The Extension Under Load Method: This method involves drawing an ordinate line (that is, a

completely vertical line) from the point on the X-axis where the elongation equals the specified

extension, e.g. Yield Strength = 0.5% Extension.

To make this determination, locate the point on the abscissa, which is equal to 0.5% (0.005 in./in.)

of extension from the origin of the curve (E in Figure 2). Draw an ordinate line (E-F) from this point

up through the curve. Convert the load value of this point into psi. The stress value is the Yield

Strength at 0.5% (0.005 in./in.) Extension Under Load.

In some cases, the yield strength may be given in other than strain fundamental units, e.g. Yield

Strength = 0.1 in. / 2 in. Extension. In such cases, the limiting extension must first be converted into

fundamental strain units (in./in.) A limiting extension of 0.01 in with a 2 in gauge length is equal to

0.005 in./in. extension.

Young’s Modulus of Elasticity:

The modulus of elasticity (Young’s Modulus) is the ratio of stress in pounds per square inch (psi) to

strain in inches per inch (in./in.) as computed from the modulus line (A-B).

      Modulus (psi) = Stress (psi)/ Strain (in./in.)

Quantum Number

Concept of quantum number:

Using wave mechanics, every electron in an atom is characterized by four parameters
called quantum numbers. The size, shape, and spatial orientation of an electron’s
probability density are specified by three of these quantum numbers. Furthermore,
Bohr energy levels separate into electron subshells, and quantum numbers dictate the
number of states within each subshell. Shells are specified by a principal quantum
number n, which may take on integral values beginning with unity; sometimes these
shells are designated by the letters K, L, M, N, O, and so on, which correspond,
respectively, to n = 1,2,3,4,5.... as indicated in Table . Note also that this quantum
number, and it only, is also associated with the Bohr model.This quantum number
is related to the distance of an electron from the nucleus, or its position.
The second quantum number, l, signifies the subshell, which is denoted by a
lowercase letter—an s, p, d, or f; it is related to the shape of the electron subshell.
In addition, the number of these subshells is restricted by the magnitude of n.
Allowable subshells for the several n values are also presented in Table 2.1. The
number of energy states for each subshell is determined by the third quantum number,
For an s subshell, there is a single energy state, whereas for p, d, and f subshells,
three, five, and seven states exist, respectively. In the absence of
an external magnetic field, the states within each subshell are identical. However,
when a magnetic field is applied these subshell states split, each state assuming a

slightly different energy.

Associated with each electron is a spin moment, which must be oriented either

up or down. Related to this spin moment is the fourth quantum number, for

which two values are possible ( and ), one for each of the spin orientations.

Thus, the Bohr model was further refined by wave mechanics, in which the introduction

of three new quantum numbers gives rise to electron subshells within

each shell.

A complete energy level diagram for the various shells and subshells using the

wave-mechanical model is shown in Figure below. Several features of the diagram are

worth noting. First, the smaller the principal quantum number, the lower the energy

level; for example, the energy of a 1s state is less than that of a 2s state, which in

turn is lower than the 3s. Second, within each shell, the energy of a subshell level increases

with the value of the l quantum number. For example, the energy of a 3d

state is greater than a 3p, which is larger than 3s. Finally, there may be overlap inenergy of a state in one shell with states in an adjacent shell, which is especially true

of d and f states; for example, the energy of a 3d state is greater than that for a 4s.

Definition of Bauschinger Effect

Bauschinger Effect:

The Bauschinger effect (BE) can be observed during tension-compression conditions and is

associated with a decrease of the yield stress when the loading direction is reversed (Fig. 1). Such

behaviour may have different origins: for instance, if due to residual macroscopic stresses, it is

not a true Bauschinger effect. Macroscopic residual stresses may result from heat treatment or

from cold work during manufacture.

Two other causes exist. One of them, the principal cause, is related to the dislocation

structure in the work-hardened metal. As deformation occurs, the dislocations accumulate at

barriers (precipitates, grain boundaries) and form dislocation pile-ups and tangles. Two types of

mechanisms are used to explain BE. First, local back stresses, which oppose the applied stress on

the slip plane, are produced by dislocations pile-ups on slip planes at barriers (pile-up of

dislocations at grain boundaries and Orowan loops around strong precipitates). Back stresses

assist the movement of dislocations in the reverse direction due to their favourable orientation to

the stress axis. Thus, the dislocations can move easily in the reverse direction and the yield

strength of the metal is lowered. Secondly, when the slip direction is reversed, dislocations of

opposite sign may be created at the same source that produced the slip-causing dislocations in the

initial direction. Since dislocations of opposite sign attract and annihilate each other, the net

effect is a further softening of the lattice.

The other cause, known as “composite effect”, is due to zones of different yield

strength inside the material, on a microscopic scale. This case is sometimes referred to as a

pseudo-Bauschinger effect. The Masing or generalized Saint-Venant model describes this

behaviour. Figure 2 shows a rheological model for two phases differing by their flow stress.

Figure shows the resulting behaviour:

Stress-strain curve demonstrating a Bauschinger effect. After load inversion, plastic

deformation sets in at a lower stress (in absolute values) in compression than during the previous

load cycle (stress at the reversal point).

Figure Rheological model of the Masing or generalized Saint-Venant model, for two phases