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Transverse elastic deformation
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When a torque is applied to the surface of a body, it simultaneously generates an area where the material is compressed and another where it expands, leading to motion perpendicular to the normal vector of the surface. This phenomenon is referred to as transverse deformation.
ID:(2064, 0)
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Parameters
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Calculations
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Calculations
Variable 
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Equations
$ E =2 G (1+ \nu )$
E =2* G *(1+ nu )
$ \tau = G \gamma $
tau = G * gamma
$ w =\displaystyle\frac{1}{2} E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) \displaystyle\frac{1}{2} G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$
w = E *( e_1 ^2+ e_2 ^2+ e_3 ^2)/2 + G *( g_1 ^2+ g_2 ^2+ g_3 ^2)/2
$ w =\displaystyle\frac{1}{2} E ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+ \displaystyle\frac{1}{2 G }( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$
w = E *( s_1 ^2+ s_2 ^2+ s_3 ^2)/2 + G *( t_1 ^2+ t_2 ^2+ t_3 ^2)/2
$ W =\displaystyle\frac{1}{2} V G \gamma ^2$
W = V * G * g^2/2
$ W =\displaystyle\frac{1}{2 G } V \tau ^2$
W = V * t ^2/(2* G )
$ W =\displaystyle\frac{1}{2 E } V ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+\displaystyle\frac{1}{2 G } V ( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$
W = V *( s_1 ^2 + s_2 ^2 + s_3 ^2)/(2* E )+ V * ( t_1 ^2+ t_2 ^2+ t_3 ^2)/(2* G )
$ W =\displaystyle\frac{1}{2} V E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) +\displaystyle\frac{1}{2} V G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$
W =( V * E /2)( e_1 ^2+ e_2 ^2+ e_3 ^2)+( V * G/2)( g_1 ^2+ g_2 ^2+ g_3 ^2)
ID:(15373, 0)
Hooke's Law for the Case Shear
Equation
In the case of shear, the deformation is not associated with expanding or compressing, but with laterally offsetting the faces of a cube. The shear is therefore described by the angle
| $ \tau = G \gamma $ |
where
ID:(3771, 0)
Shear Modulus
Equation
The shear modulus
u
| $ E =2 G (1+ \nu )$ |
where
ID:(3772, 0)
Shear Energy
Equation
In analogy to the strain energy, the shear energy is proportional to the shear angle
| $ W =\displaystyle\frac{1}{2} V G \gamma ^2$ |
ID:(3789, 0)
Energy Shear and Torsion
Equation
Since the strain energy is
$W=\displaystyle\frac{1}{2}VG\gamma^2$
with Hook's law for materials
$\tau=G\gamma$
is obtained:
| $ W =\displaystyle\frac{1}{2 G } V \tau ^2$ |
ID:(3791, 0)
Deformation and Shear Energy
Equation
The ratio of energy
$W=\displaystyle\frac{1}{2}VE\epsilon^2$
and the shear energy with the angle
$W=\displaystyle\frac{1}{2}VG\gamma^2$
can be generalized to the three-dimensional case:
| $ W =\displaystyle\frac{1}{2} V E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) +\displaystyle\frac{1}{2} V G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$ |
where
ID:(3766, 0)
Tension and Torsion Energy
Equation
With the relationship of the energy
$W=\displaystyle\frac{1}{2}VE(\epsilon_1^2+\epsilon_2^2+\epsilon_3^2)$
and Hook's law for continuous material
$\sigma_i=E\epsilon_i$
energy can be written as a function of voltage
| $ W =\displaystyle\frac{1}{2 E } V ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+\displaystyle\frac{1}{2 G } V ( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$ |
where
ID:(3767, 0)
Strain Energy Density and Shear
Equation
Since the energy
| $ W =\displaystyle\frac{1}{2} V E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) +\displaystyle\frac{1}{2} V G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$ |
where
| $ w =\displaystyle\frac{ W }{ V }$ |
so we have:
| $ w =\displaystyle\frac{1}{2} E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) \displaystyle\frac{1}{2} G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$ |
where
ID:(3768, 0)
Energy Density Tension and Torsion
Equation
Since the energy
| $ W =\displaystyle\frac{1}{2 E } V ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+\displaystyle\frac{1}{2 G } V ( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$ |
where
| $ w =\displaystyle\frac{ W }{ V }$ |
so we have:
| $ w =\displaystyle\frac{1}{2} E ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+ \displaystyle\frac{1}{2 G }( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$ |
where
ID:(3769, 0)