Stress and Strain
A rectangular sample of an unknown material has an initial length of 10 cm, and a cross-section of 2.1 cm X 2.1 cm dimensions. The sample is pulled in tension along the length (with the deformation described below all in the elastic region).
- Calculate the strain if the sample is elongated to 10.5 cm.
- With a Poisson’s ratio of 0.27, calculate the width in cm of the sample if the bar is elongated to 10.5 cm under applied stress.
- Calculate the stress required (MPa) to elongate the sample to 10.5 cm if the elastic modulus is 2.10 GPa.
Expand Hint
Strain is defined as change in length per unit length; for pure tension the following apply:
$$$\varepsilon_z=\frac{l-l_0}{l_0}$$$
where
$$\varepsilon$$
is engineering strain,
$$l$$
is the new length, and
$$l_0$$
is the initial length.
Hint 2
$$$\sigma =E\varepsilon$$$
where
$$\sigma$$
is the stress on the cross section,
$$E$$
is the modulus of elasticity, and
$$\varepsilon$$
is the engineering strain.
Strain is defined as change in length per unit length; for pure tension the following apply:
$$$\varepsilon_z=\frac{l-l_0}{l_0}=\frac{10.5cm-10cm}{10cm}=\frac{0.5cm}{10cm}=0.05$$$
where
$$\varepsilon$$
is engineering strain,
$$l$$
is the new length, and
$$l_0$$
is the initial length.
Poisson’s ratio is:
$$$v=\frac{-\varepsilon_x}{\varepsilon_z}$$$
where
$$\varepsilon_x$$
is the lateral strain, and
$$\varepsilon_z$$
is the longitudinal strain. Calculating the sample width if elongated:
$$$0.27=\frac{-\varepsilon_x}{0.05}\rightarrow \varepsilon_x=-0.27(0.05)=-0.0135$$$
To find the new width after elongation, recall that strain is defined as the change in length per unit length:
$$$\varepsilon_x=\frac{w-w_0}{w_0}\rightarrow w=\varepsilon_xw_0+w_0$$$
$$$w=w_0(\varepsilon_x+1)=2.1cm(-0.0135+1)=2.072\: cm$$$
The elastic modulus (also referred as modulus, modulus of elasticity, Young's modulus) describes the relationship between stress and strain during elastic loading. This is captured via Hooke's Law:
$$$\sigma =E\varepsilon$$$
where
$$\sigma$$
is the stress on the cross section,
$$E$$
is the modulus of elasticity, and
$$\varepsilon$$
is the engineering strain.
Finding the stress for sample elongation,
$$$\sigma =2.10\cdot 10^3MPa(0.05)=105\:MPa$$$
- 0.05
- 2.072 cm
- 105 MPa
Time Analysis
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