Chapter 2

Strain and Material

Properties
In Chapter 1, our concern was the stresses within a body subject to a system of external
forces. Deformations and strains, which are necessary to an analysis of stress, are also
important quantities in themselves, for they relate to changes in the size and shape of a
body.

The plane strain transformation equations are especially important in experimental

investigations, where normal strains are measured with strain gages or a strain rosette, with each
gage measuring the strain in a different direction.

The stress–strain curves in tension, compression, and shear are discussed. Following this the
relationship between strain and stress under uniaxial, shear, and multiaxial loading conditions.
The measurement of strain and the concept of strain energy are discussed too.
Saint-Venant’s principle, is introduced in this chapter which is extremely useful in the solution of
practical problems.
 Deformation
Let us consider a body subjected to external loading that causes it to take up the
position pictured by the dashed lines, in which A is displaced to A′, B to B′, and so on,
until all the points in the body are displaced to new positions. The displacements of any
two points such as A and B are simply AA′ and BB′, respectively, and may be a
consequence of deformation (straining), rigid-body motion (translation and rotation), or
some combination. The body is said to be strained if the relative positions of points in
the body are altered. If no straining has taken place, displacements AA′ and BB′ are
attributable to rigid-body motion. In the latter case, the distance between A and B
remains fixed; that is, L0 = L. Such displacements are not discussed in this chapter.

N.B. To describe the magnitude and direction of the displacements,

points within the body are located with respect to an appropriate
coordinate reference as, for example, the xyz system.
The two-dimensional case shown in Fig. below the components of displacement of point
A to A′ can be represented by u and v in the x and y coordinate directions, respectively.
In general, the
components of displacement at a point, occurring in the x, y, and z directions, are
denoted by u, v, and w, respectively.
The displacement at every point within the body constitutes the displacement field, u =
u(x, y, z), v = v(x, y, z), and w = w(x, y, z). In this text, mainly small displacements are
considered, a simplification consistent with the magnitude of deformation commonly
found in engineering structures. The strains produced by small deformations are small
compared to unity, and their products (higher-order terms) are neglected. For purposes
of clarity, small displacements with which we are concerned will be shown highly
exaggerated on all diagrams.
 Superposition
The small displacement assumption leads to one of the basic fundamentals of solid
mechanics, called the principle of superposition. This principle is valid whenever the
quantity (stress or
displacement) to be determined is a linear function of the loads that produce it.
In the linear condition, in the linearly elastic material situations, the total quantity owing to
the combined loads acting simultaneously on a member may be obtained by determining
separately the quantity attributable to each load and combining the individual results.

For example, normal stresses caused by axial forces and bending simultaneously may
be obtained by superposition, provided that the combined stresses do not exceed the
proportional limit of the material. Likewise, shearing stresses caused by a torque and a
vertical shear force acting simultaneously in a beam may be treated by superposition.
Clearly, superposition cannot be applied to plastic deformations. The principle of
superposition is employed repeatedly in this text. The motivation for superposition is the
replacement of a complex load configuration by two or more simpler loads.
 Strain Definition
For purposes of defining normal strain, line AB of an axially loaded
member in Fig. below has suffered deformation to become A′B′. The length of AB is Δx.
As shown, points A and B have each been displaced: A an amount u, and B, u + Δu.
Line A B has been displaced by an amount Δu in addition to displacement, the length Δx
has been increased by Δu. Normal strain, the unit change in length, is defined as:
This equation represents the strain at a point.

If the deformation is distributed uniformly over the

original length, the normal strain may be written:

where L, Lo and δ are the final length, the original length,

and the change of length of the member, respectively.
 Plane Strain
We now investigate the case of two-dimensional or plane strain, where all points in the
body,
before and after application of load, remain in the same plane.
Two-dimensional views of an element with edges of unit lengths subjected to plane strain
are shown in three parts in Fig. below.
We note that this element has no normal strain εz and no shearing strains γxz and γyz in
the xz and yz planes, respectively.
 Deformations of an element
Normal Strain
Consider an element with dimensions dx, dy and of unit thickness. The total deformation
is a change in length the sides and a relative rotation.

Two normal or longitudinal strains are:

Shear Strain
A positive sign is applied to elongation; a negative sign, to contraction.
Now consider the change experienced by right angle DAB. We shall assume the angle
αx between AB and A′B′ to be so small as to permit the approximation αx ≈ tan αx. Also,
in view
of the smallness of αx, the normal strain is small, so AB ≈ A′B′. As a consequence of the
aforementioned considerations, αx ≈ ∂v/∂x, where the counterclockwise rotation is
defined as positive. Similar analysis leads to –αy ≈ ∂u/∂y)from new position to axe y
counterclockwise similar from new position to axe x clockwise) . The total angular
change of angle DAB, the angular change between lines in the x and y directions, is
defined as the shearing strain γxy.

N.B. The shear strain is positive when the right angle between
two positive (or negative) axes decreases.
That is, if the angle between +x and +y or –x and –y decreases,
we have positive γxy; otherwise the shear strain is negative.
 Three-Dimensional Strain
In the case of a three-dimensional element, a rectangular prism with sides dx, dy, dz, an
essentially identical analysis leads to the following normal and shearing strains:

The angular change is not different if it is said to occur between the x and y directions or
between the y and x directions; . As we see it before in the stresses. The components
of shearing strain are:
N.B.

The Cartesian coordinate systems, in the deformed body; xyz is, in this instance, a
Eulerian coordinate system (large deformation theory are attempted or applications in
finite elasticity).

The coordinates in the undeformed body, as applicable to equations of stress or strain.

The xyz set is established in the undeformed body. In this case, xyz is referred to as a
Lagrangian coordinate system. (In this case the assumption of small deformation permits
us to regard x, y, and z, the coordinates in the undeformed body)

The strains are indicated as dimensionless quantities. The normal and shearing strains
are also frequently described in terms of units such as inches per inch or micrometers
per meter and radians or microradians, respectively. The strains for engineering
materials in steel is not exceed 0.002, which is equivalent to 2000 × 10–6 or 2000 μ. We
read this as “2000 micros.”
 Example
Plane Strains in a Plate

A 0.8-m by 0.6-m rectangle ABCD is drawn on a thin plate prior to loading. Subsequent
to loading, the deformed geometry is shown by the dashed lines in Fig. below.
Determine the components of plane strain at point A.
Solution
The following approximate version of the strain–displacement relations:

Thus, by setting Δx = 800 mm and Δy = 600 mm, the normal strains are calculated as follow:
 Large Strains
The small deformations or deflections are considered in most applications of this class.
The preceding is consistent with the magnitude of deformations usually found in
engineering practice.
The following more general large or finite strain–displacement relationships are included
here so that the reader may better understand the approximations resulting in the
relations of small-deformation theory.
When displacements are relatively large, the strain components are given in terms of the
square of the element length instead of the length itself.
Transformation of Two-Dimensional Strain
Principal of Two-Dimensional Strain
Transformation of Three-Dimensional Strain
Mohr’s Circle for Plane Strain
The transformation properties of stress and strain are identical, it is
apparent that a Mohr’s circle for strain may be drawn and that the
construction technique does not differ from that of Mohr’s circle for
stress. In Mohr’s circle for strain,
the normal strains are plotted on the horizontal axis, positive to the
right. When the shear strain is positive, the point representing the
x axis strains is plotted a distance γ/2 below the ε line, and the y
axis point a distance γ/2 above the ε line, and vice versa when the
shear strain is negative.
Note that this convention for shear strain, used only in constructing
and reading values from Mohr’s circle, agrees with the convention
employed for stress.
Example 2.3. State of Plane Strain in a Plate
The state of strain at a point on a thin plate is given by εx = 510 μ,
εy = 120 μ, and γxy = 260 μ. Determine, using Mohr’s circle of
strain,
(a) the state of strain associated with axes x′, y′, which make an
angle θ = 30° with the axes x, y (Fig. 2.8a);
(b) the principal strains and directions of the principal axes;
(c) the maximum shear strains and associated normal strains;
(d) display the given data and the results obtained on properly
oriented elements of unit dimensions.
Stress–Strain Diagrams
A ductile metal is capable of substantial elongation prior to failure,
as in a drawing process. The converse applies to brittle materials.
Note that beyond the point labeled “proportional limit” is a point
labeled “yield point” (for most cases these two points are taken as
one). At the yield point, a great deal of deformation occurs while
the applied loading remains essentially constant.
Hooke’s Law and Poisson’s Ratio
E is the slope of the stress–strain diagram in the linearly elastic region.
For most materials, E in compression is the same as that in tension.
G is the shear modulus of elasticity or modulus of rigidity. Like E, G is a
constant for a given material.
In the linearly elastic region, it is found experimentally that lateral
strains, say in the y and z directions, are related by a constant of
proportionality, v, to the axial strain caused by uniaxial stress only
εx = σx/E, in the x direction:
In terms of Young's modulus and Poisson's ratio, Hooke's law for isotropic materials
can then be expressed as

This is the form in which the strain is expressed in terms of the stress tensor in
engineering. The expression in expanded form is

where E is Young's modulus and ν is Poisson's ratio. (See 3-D elasticity).

Derivation of Hooke's law in three dimensions
In matrix form, Hooke's law for isotropic materials can be written as

where γij = 2εij is the engineering shear strain. The inverse relation may be written as
v is known as Poisson’s ratio, v to be measured values, 0.25 to
0.35 for most metals. Extreme cases range from a low of 0.1 (for
some concretes),
to a high of 0.5 (for rubber).
Volume Change
The lateral contraction of a cubic element from a bar in tension may
taken the final volume as:
Example 2.4. Deformation of a Tension Bar
An aluminum alloy bar of circular cross-sectional area A and length
L is subjected to an axial tensile force P (Fig. 2.16). The modulus
of elasticity and Poisson’s ratio of the material are E and v,
respectively. For the bar, determine
(a)the axial deformation;
(b) the change in diameter d; and
(c) the change in volume ΔV.
(d) Evaluate the numerical values of the quantities obtained in (a)
through (c) for the case in which P = 60 kN, d = 25 mm, L = 3 m, E
= 70 GPa, ν = 0.3, and the yield strength σyp = 260 MPa.
Example 2.5. Volume Change of a Metal Block
Calculate the volumetric change of the metal block shown in Fig.
2.18 subjected to uniform pressure p = 160 MPa acting on all
faces. Use E = 210 GPa and ν = 0.3.

A parallelpiped under pressure.

 Strain Rosettes
The normal strain in the material is measured
usingan electrical resistance strain gauge
Example 2.6. Principal Strains on Surface of a
Steel Frame
Strain rosette readings are made at a critical point on the free
surface in a structural steel member. The 60° rosette contains three
wire gages positioned at 0°, 60°, and 120° (Fig. 2.20b). The
readings are

Determine
(a) the in-plane principal strains and stresses and their directions
(b) the true maximum shearing strain. The material properties are
E = 200 GPa and ν = 0.3.
(b)
(c)

(d)
Employing a procedure similar to that used in the preceding
numerical example, it is possible to develop expressions relating
three-element gage outputs of various rosettes to principal strains
and stresses. Table 2.2 provides two typical cases: equations for the
rectangular rosette (θa = 0°, θb = 45°, and θc = 90°, Fig. 2.20b) and the
delta rosette (θa = 0°, θb = 60°, and θc = 120°, Fig. 2.20b). Experimental
stress analysis is facilitated by this kind of compilation.