Druyd:Knowledge

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.

>Model

ID:(2064, 0)

Mechanisms

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Knowledge

Code

Concept

Mechanisms

ID:(15372, 0)

Model

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Parameters

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$E$

Modulus of Elasticity

$\nu$

Poisson coefficient

$G$

Shear module

Variables

Symbol

Text

Variable

Value

Units

Calculate

MKS Value

MKS Units

$W$

Deformation energy

$w$

Deformation energy density

$\epsilon_1$

e_1

Deformation of the coordinate

$x$

$\epsilon_2$

e_2

Deformation of the coordinate

$y$

$\epsilon_3$

e_3

Deformation of the coordinate

$z$

$\gamma_3$

gamma_3

Shear angle in the

$xy$ plane

rad

$\gamma_1$

gamma_1

Shear angle in the

$yz$ plane

rad

$\gamma_2$

gamma_2

Shear angle in the

$zx$ plane

rad

$\sigma_1$

sigma_1

Stress on axis

$x$

$\sigma_2$

sigma_2

Stress on axis

$y$

$\sigma_3$

sigma_3

Stress on axis

$z$

$\tau$

tau

Torsion

$\tau_1$

tau_1

Torsion in

$x$ axis

$\tau_2$

tau_2

Torsion in

$y$ axis

$\tau_3$

tau_3

Torsion in

$z$ axis

$\gamma$

gamma

Twist angle

rad

$V$

Volume

m^3

Calculations

Equation

Number

First, select the equation:

, then, select the variable:

Calculations

Symbol

Equation

Solved

Translated

Calculations

Symbol

Equation

Solved

Translated

Variable

Given

Calculate

Target :

Equation

To be used

Equations

Equation

$E =2 G (1+ \nu )$

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

>Top, >Model

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 \gamma with which it is possible to rotate the face perpendicular to the displaced surfaces. In analogy to Hook's law for compression and expansion, we have the relationship between torsion \tau and angle \gamma:

$\tau = G \gamma$

$G$

Shear module

$Pa$

5364

$\tau$

Torsion

$Pa$

5366

$\gamma$

Twist angle

$rad$

5367

where G is the so-called shear modulus.

ID:(3771, 0)

Shear Modulus

Equation

>Top, >Model

The shear modulus G is related to the modulus of elasticity E and Poisson's ratio
u by

$E =2 G (1+ \nu )$

$E$

Modulus of Elasticity

$Pa$

5357

$\nu$

Poisson coefficient

$-$

5365

$G$

Shear module

$Pa$

5364

where G is the so-called shear modulus.

ID:(3772, 0)

Shear Energy

Equation

>Top, >Model

In analogy to the strain energy, the shear energy is proportional to the shear angle \gamma squared, the constant being in this case the shear modulus:

$W =\displaystyle\frac{1}{2} V G \gamma ^2$

$W$

Deformation energy

$J$

5368

$G$

Shear module

$Pa$

5364

$\gamma$

Twist angle

$rad$

5367

$V$

Volume

$m^3$

5226

ID:(3789, 0)

Energy Shear and Torsion

Equation

>Top, >Model

Since the strain energy is

$W=\displaystyle\frac{1}{2}VG\gamma^2$

with Hook's law for materials

$\tau=G\gamma$

is obtained:

$W =\displaystyle\frac{1}{2 G } V \tau ^2$

$W$

Deformation energy

$J$

5368

$G$

Shear module

$Pa$

5364

$\tau$

Torsion

$Pa$

5366

$V$

Volume

$m^3$

5226

ID:(3791, 0)

Deformation and Shear Energy

Equation

>Top, >Model

The ratio of energy W, volume V, modulus of elasticity E and strain \epsilon

$W=\displaystyle\frac{1}{2}VE\epsilon^2$

and the shear energy with the angle \gamma and shear modulus

$W=\displaystyle\frac{1}{2}VG\gamma^2$

can be generalized to the three-dimensional case:

$W =\displaystyle\frac{1}{2} V E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) +\displaystyle\frac{1}{2} V G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$

$W$

Deformation energy

$J$

5368

$\epsilon_1$

Deformation of the coordinate

$x$

$-$

5369

$\epsilon_2$

Deformation of the coordinate

$y$

$-$

5370

$\epsilon_3$

Deformation of the coordinate

$z$

$-$

5371

$E$

Modulus of Elasticity

$Pa$

5357

$\gamma_3$

Shear angle in the

$xy$ plane

$rad$

5374

$\gamma_1$

Shear angle in the

$yz$ plane

$rad$

5372

$\gamma_2$

Shear angle in the

$zx$ plane

$rad$

5373

$G$

Shear module

$Pa$

5364

$V$

Volume

$m^3$

5226

where \epsilon_i represents the deformation in each axis.

ID:(3766, 0)

Tension and Torsion Energy

Equation

>Top, >Model

With the relationship of the energy W, the volume V, the modulus of elasticity E and deformations \epsilon_i

$W=\displaystyle\frac{1}{2}VE(\epsilon_1^2+\epsilon_2^2+\epsilon_3^2)$

and Hook's law for continuous material

$\sigma_i=E\epsilon_i$

energy can be written as a function of voltage

$W =\displaystyle\frac{1}{2 E } V ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+\displaystyle\frac{1}{2 G } V ( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$

$W$

Deformation energy

$J$

5368

$E$

Modulus of Elasticity

$Pa$

5357

$G$

Shear module

$Pa$

5364

$\sigma_1$

Stress on axis

$x$

$Pa$

5380

$\sigma_2$

Stress on axis

$y$

$Pa$

5381

$\sigma_3$

Stress on axis

$z$

$Pa$

5382

$\tau_1$

Torsion in

$x$ axis

$Pa$

5383

$\tau_2$

Torsion in

$y$ axis

$Pa$

5384

$\tau_3$

Torsion in

$z$ axis

$Pa$

5385

$V$

Volume

$m^3$

5226

where \epsilon_i represents the deformation in each axis.

ID:(3767, 0)

Strain Energy Density and Shear

Equation

>Top, >Model

Since the energy W is

$W =\displaystyle\frac{1}{2} V E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) +\displaystyle\frac{1}{2} V G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$

where V is the volume, E the modulus of elasticity and \epsilon_i the strain, the energy density can be calculated

$w =\displaystyle\frac{ W }{ V }$

so we have:

$w =\displaystyle\frac{1}{2} E ( \epsilon_1 ^2+ \epsilon_2 ^2+ \epsilon_3 ^2) \displaystyle\frac{1}{2} G ( \gamma_1 ^2+ \gamma_2 ^2+ \gamma_3 ^2)$

$W$

Deformation energy density

$J/m^3$

5375

$\epsilon_1$

Deformation of the coordinate

$x$

$-$

5369

$\epsilon_2$

Deformation of the coordinate

$y$

$-$

5370

$\epsilon_3$

Deformation of the coordinate

$z$

$-$

5371

$E$

Modulus of Elasticity

$Pa$

5357

$\gamma_3$

Shear angle in the

$xy$ plane

$rad$

5374

$\gamma_1$

Shear angle in the

$yz$ plane

$rad$

5372

$\gamma_2$

Shear angle in the

$zx$ plane

$rad$

5373

$G$

Shear module

$Pa$

5364

where \epsilon_i represents the deformation in each axis.

ID:(3768, 0)

Energy Density Tension and Torsion

Equation

>Top, >Model

Since the energy W is

$W =\displaystyle\frac{1}{2 E } V ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+\displaystyle\frac{1}{2 G } V ( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$

where V is the volume, E the modulus of elasticity and \sigma_i the tension, the energy density can be calculated

$w =\displaystyle\frac{ W }{ V }$

so we have:

$w =\displaystyle\frac{1}{2} E ( \sigma_1 ^2+ \sigma_2 ^2+ \sigma_3 ^2)+ \displaystyle\frac{1}{2 G }( \tau_1 ^2+ \tau_2 ^2+ \tau_3 ^2)$

$W$

Deformation energy density

$J/m^3$

5375

$E$

Modulus of Elasticity

$Pa$

5357

$G$

Shear module

$Pa$

5364

$\sigma_1$

Stress on axis

$x$

$Pa$

5380

$\sigma_2$

Stress on axis

$y$

$Pa$

5381

$\sigma_3$

Stress on axis

$z$

$Pa$

5382

$\tau_1$

Torsion in

$x$ axis

$Pa$

5383

$\tau_2$

Torsion in

$y$ axis

$Pa$

5384

$\tau_3$

Torsion in

$z$ axis

$Pa$

5385

where \epsilon_i represents the deformation in each axis.

ID:(3769, 0)