Eq Table

Name	Eq
Axial stress	\(\sigma = \frac{F_v}{A}\)
Axial strain	\(\varepsilon = \frac{\Delta l}{l_o}\)
Young’s Modulus	\(E = \frac{\sigma}{\varepsilon}\)
Shear Stress	\(\tau = \frac{F_s}{A}\)
Shear Strain	\(\gamma = \frac{d}{l_o}\)
Shear Modulus	\(G = \frac{\tau}{\gamma}\)
Poisson’s Ratio	\(v_{xy} = -\frac{\varepsilon_y}{\varepsilon_x}\)
Bulk Modulus	\(K = \frac{P}{\Delta V/V}\)

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Axial stress

\[\sigma = \frac{F_v}{A}\]

Where

• \(\sigma\) is axial stress
• \(F_v\) is axial force
• \(A\) is area

Units: [Pa]

Axial strain

\[\varepsilon = \frac{\Delta l}{l_o}\]

Where

• \(\varepsilon\) is axial strain
• \(\Delta l\) is change in length
• \(l_o\) is original length

Units: Dimensionless

Young’s Modulus

\[E = \frac{\sigma}{\varepsilon}\]

Where

• \(E\) is Young’s Modulus
• \(\sigma\) is stress
• \(\varepsilon\) is strain

Units: [Pa]

Shear Stress

\[\tau = \frac{F_s}{A}\]

Where

• \(\tau\) is shear strain
• \(F_s\) is shear force
• \(A\) is area

Units: [Pa]

Shear Strain

\[\gamma = \frac{d}{l_o}\]

Where

• \(\gamma\) is shear strain
• \(d\) is deformation
• \(l_o\) is original length

Units: Dimensionless

Shear Modulus

\[G = \frac{\tau}{\gamma}\]

Where

• \(G\) is shear modulus
• \(\tau\) is shear stress
• \(\gamma\) is shear strain

Units: [Pa]

Poisson’s Ratio

\[v_{xy} = -\frac{\varepsilon_y}{\varepsilon_x}\]

Where

• \(v_{xy}\) is Poisson’s Ratio
• \(\varepsilon_y\) is lateral strain
• \(\varepsilon_x\) is applied strain

Bulk Modulus

\[K = \frac{P}{\Delta V/V}\]

Where

• \(K\) is bulk modulus
• \(P\) is pressure
• \(\Delta V/V\) is volumetric strain

Relationships

Shear Modulus G	Young’s Modulus E	Poisson’s Ratio v	Bulk Modulus K
\(\frac{E}{2(1 + v)}\)	\(2G(1 + v)\)	\(\frac{E - 2G}{2G}\)	\(\frac{GE}{2(3G - E)}\)
\(\frac{3EK}{9K - E}\)	\(\frac{3KG}{3K + G}\)	\(\frac{3K - 2G}{2(3K + G)}\)	\(\frac{2G(1 + v)}{3(1 - 2v)}\)
\(\frac{3K(1 - 2v)}{2(1 + v)}\)	\(3K(1 - 2v)\)	\(\frac{3K - E}{6K}\)	\(\frac{E}{3(1 - 2v)}\)