ASCII2MathML-SVG: The Cutting of Sand, Clay and Rock

		
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Click on an arrow for explanation.


General cutting equations

The different forces:

The gravitational force:

`FG`

The inertial force:

`FI`

the cohesive force;

`FC`

The adhesive force:

`FA`

The pore pressure forces:

No cavitation

`W1ncav`

`W2ncav`

Cavitation

`W1cav`

`W2cav`

Relations between shear forces and normal forces:

`S1N1`

`S2N2`

Horizontal equilibrium of forces:

`HEQ`

Vertical equilibrium of forces:

`VEQ`

This gives for the unknown grain forces `K_1` and `K_2`:

`K1general`

`K2general`

This gives for the unknown normal forces `N_1` and `N_2`:

`N1general`

`N2general`

The horizontal and vertical forces on the cutting blade can now be calculated according to:

`Fhgeneral`

`Fvgeneral`

The specific energy;

`ESP`


Cutting forces in dry sand

In dry sand only the gravitational force `G`, the inertial force `I`, the internal friction angle `phi` and the external friction angle `delta` play a role.

Horizontal equilibrium of forces:

`HEQdrysand`

Vertical equilibrium of forces:

`VEQdrysand`

This gives for the unknown grain forces `K_1` and `K_2`:

`K1drysand`

`K2drysand`

This gives for the unknown normal forces `N_1` and `N_2`:

`N1drysand`

`N2drysand`

The horizontal and vertical forces on the cutting blade can now be calculated according to:

`Fhdrysand`

`Fvdrysand`

The specific energy;

`ESP`


Cutting forces in water saturated sand

In water saturated sand only the pore pressure forces `W_1` and `W_2`, the internal friction angle `phi` and the external friction angle `delta` play a role.

Horizontal equilibrium of forces:

`HEQwetsand`

Vertical equilibrium of forces:

`VEQwetsand`

This gives for the unknown grain forces `K_1` and `K_2`:

`K1wetsand`

`K2wetsand`

This gives for the unknown normal forces `N_1` and `N_2`:

`N1wetsand`

`N2wetsand`

The horizontal and vertical forces on the cutting blade can now be calculated according to:

`Fhwetsand`

`Fvwetsand`

With given values for the pore pressure forces `W_1` and `W_2` and distinguishing the non-cavitating and the cavitating cutting process, this results in:

No cavitation

`Fhnc`

`Fvnc`

Cavitation

`Fhc`

`Fvc`

The specific energy;

`ESP`


Cutting forces in clay

In clay only the cohesive force `C` and the adhesive force `A` play a role.

Horizontal equilibrium of forces:

`HEQclay`

Vertical equilibrium of forces:

`VEQclay`

This gives for the unknown grain forces `K_1` and `K_2`:

`K1clay`

`K2clay`

This gives for the unknown normal forces `N_1` and `N_2`:

`N1clay`

`N2clay`

The horizontal and vertical forces on the cutting blade can now be calculated according to:

`Fhclay`

`Fvclay`

Conditions

Curling type: `ClayCurlingType`

Tear type: `ClayTearType`

The specific energy;

`ESP`


Cutting forces in rock

In rock only the cohesive force `C`, the internal friction angle `phi` and the external friction angle `delta` play a role.

Horizontal equilibrium of forces:

`HEQrock`

Vertical equilibrium of forces:

`VEQrock`

This gives for the unknown grain forces `K_1` and `K_2`:

`K1rock`

`K2rock`

This gives for the unknown normal forces `N_1` and `N_2`:

`N1rock`

`N2rock`

The horizontal and vertical forces on the cutting blade can now be calculated according to:

`Fhrock`

`Fvrock`

Conditions

Tear type atmospheric: `RockTearType1`

Tear type hydrostatic: `RockTearType2`

The specific energy;

`ESP`


 

List of symbols used

 
`A` The adhesive force exerted by the blade on the layer cut. kN
`c_1, c_2` Coefficients for the cavitating cutting process -
`C` The cohesive force exerted by the situ sand on the layer cut. kN
`d_1, d_2` Coefficients for the cavitating cutting process -
`e` Dilatation -
`E_sp` Specific energy kPa
`F_h` Horizontal cutting force kN
`F_v` Vertical cutting force kN
`F_hydr` Force resulting from hydrostatic pressure kN
`g` Gravitational constant (approximately 9.81 m/sec2) m/sec2
`G` The gravitational force on the layer cut. kN
`h_i` The thickness of the layer cut m
`h_b` The blade height m
`I` The inertial force exerted by the situ sand on the layer cut. kN
`k_i` Initial permeability of the sand m/sec
`k_max` Maximum permeability of the sand m/sec
`k_m` Permeability of the sand m/sec
`K_1` The grain force exerted by the situ sand on the layer cut. kN
`K_2` The grain force exerted by the blade on the layer cut. kN
`N_1` The normal force exerted by the situ sand on the layer cut. kN
`N_2` The normal force exerted by the blade on the layer cut. kN
`Deltap_1` Average pore pressure difference on the shear zone kPa
`Deltap_2` Average pore pressure difference on the blade kPa
`S_1` The shear force exerted by the situ sand on the layer cut. kN
`S_2` The shear force exerted by the blade on the layer cut. kN
`v_c` Cutting velocity m/sec
`w` Width of the blade m
`W_1` The force resulting from pore pressures exerted by the situ sand on the layer cut. kN
`W_2` The force resulting from pore pressures exerted by the blade on the layer cut. kN
`z` Waterdepth m
`alpha` The angle of the blade with the horizontal
`beta` The angle of the shear plane with the horizontal
`phi` The angle of internal friction (sand-sand, clay-clay, rock-rock)
`delta` The angle of external friction (sand-steel, clay-steel, rock-steel)
`tau_a` The adhesive shear strength kPa
`tau_c` The cohesive shear strength kPa
`rho_s` Density of the soil tons/m3
`rho_w` Density of water tons/m3
The original ASCIIMathML and ASCIIsvg scripts have been developed by by Peter Jipsen, Chapman University (jipsen@chapman.edu)
LaTeXMathML has been developed by Douglas Woodall (and extended by Jeff Knisley), based on ASCIIMathML
Image fallback scripts have been developed by David Lippman based on MimeTex and MathTex of John Forkosh.
The version of ASCIIMathML used on this website, is a modified and extended version based on version 2.1 of Peter Jipsen, the new SVG library has been developed by Dr.ir. S.A. Miedema
Other sources: An ASCIIsvg manual by Robert Fant.  An ASCIIsvg manual by Peter Jipsen. An ASCIIMathML manual by James Gray.

Plugins and fonts required (depending on your browser): MIT MathML font packages, MathPlayer, Adobe SVGviewer
Look at: http://www.w3.org/TR/SVG11/ for detailed information about SVG.
Look at: http://www.w3.org/Math/ for detailed information about MathML

Copyright © Dr.ir. S.A. Miedema, Delft University of Technology, Faculty of Mechanical Engineering, Marine Technology & Materials Science
Department of Marine & Transport Technology, The Chair of Dredging Engineering
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