Chapter 4: Strain

: Deformation

Whenever a force is applied to a body, it will tend to

change the body’s shape and size. These changes are

referred to as deformation, and they may be either highly

visible or practically unnoticeable without the use of

equipment to make precise measurements.

Deformation of a body can also occur when the

temperature of the body is changed.

:Normal strain

The elongation or contraction of a line segment per unit of length is referred

to as normal strain.

n

A

B

Δs

Undeformed body

A

B

Δs’

Deformed body

Average normal strain: s

ssavg

'

As point B is chosen closer and closer to point A, the length of the line

becomes shorter and shorter, such that Δs→0. Also, this causes B’ to

approach A’, such that Δs’→0. Consequently, in the limit the normal strain at

point A and in the direction of n is:

s

ss

nalongAB

'lim

If the normal strain is known, we can use this equation to obtain the

approximate final length of a short line segment in the direction of n after it is

deformed. We have:

ss 1'Hence, when ε is positive the initial line will elongate, whereas if ε is negative

the line contracts.

Units: In SI units, m/m.

strain normal

stress

L

A

P

L

A

P

A

P

2

2

LL

A

P

2

2

Normal Strain

Shear strain:

The change in angle that occurs between two line segments that were originally

perpendicular to one another is referred to as shear strain. This angle is

denoted by γ and is measured in radians (rad).

n

A

B

Δs

Undeformed body

A’

B’

Δs’

Deformed body

C

t

2

C’

'

Hence we define the shear strain at point A that is associated with the

n and t axes as:

talongACnalongAB

nt

'lim2

Notice that if θ’ is smaller than π/2 the shear

strain is positive, whereas if θ’ is larger than

π/2 the shear strain is negative.

Cartesian strain components:

xx 1 yy 1

zz 1

Appropriate angles between the sides, again originally defined by the sides

Δx, Δy, and Δz, are

xy

2

yz

2

xz

2Δx

Δy

Δz 2

2

2

Undeformed element Deformed element

Notice that the normal strains cause a change in volume of the

rectangular element, whereas the shear strains cause a change in its

shape.

In summary, then, the state of strain at a point in a body requires

specifying three normal strains, εx, εy, εz, and three shear strains, γxy,

γyz, γxz.

These strains completely describe the deformation of a rectangular

volume element of material located at the point and oriented so that

its sides are originally parallel to the x, y, z axes.

Problems:

Solution:

1Ex.

Solution:

2Ex.

3Ex.

=?

4Ex.

5Ex.

222 xyc

Shear strain:

222 xyD

radxyD 0116.0