Degree of deformation

The degree of deformation is a deformation parameter with which the permanent geometric change of a workpiece during the forming process can be recorded. The degree of deformation is used, for example, to calculate the force and work required for the purpose of machine selection. The following properties make the degree of deformation a suitable deformation parameter during forming:

• Representation as a related size,
• the direction of the deformation can be recognized by the sign, namely positive when enlarging ( stretching ) and negative when reducing ( compressing ) the dimensions of the workpiece,
• reversing the transformation results in the same absolute value,
• In the case of step-by-step deformation, the total degree of deformation results from the sum of the individual degrees of deformation of the respective steps.

If the dimensions change in the x-direction of a Cartesian main axis system from an initial dimension to an end dimension , the degree of deformation is defined as: ${\displaystyle x_{0}}$ ${\displaystyle x_{1}}$

${\displaystyle \varphi _{x}=\int _{x_{0}}^{x_{1}}{\frac {1}{x}}\,dx=\ln \!\left({\frac {x_{1}}{x_{0}}}\right)}$.

The degree of deformation is also known as logarithmic elongation , true elongation or Hencky elongation. The elongation as the ratio of change in dimension to initial dimension

${\displaystyle \varepsilon _{x}={\frac {\Delta x}{x_{0}}}={\frac {x_{1}-x_{0}}{x_{0}}}}$

can be converted into the degree of deformation:

${\displaystyle \varphi _{x}=\ln \!\left({\frac {x_{1}}{x_{0}}}\right)=\ln \left({\frac {x_{1}-x_{0}}{x_{0}}}+{\frac {x_{0}}{x_{0}}}\right)=\ln \left(\varepsilon _{x}+1\right)}$.

From the law of constant volume of the length (results for the conversion of a prismatic body ), the width ( ) and the height ( ) to the final dimensions ( , , ), the following equation for the volume ( ) of this body. ${\displaystyle {l_{0}}}$ ${\displaystyle {b_{0}}}$ ${\displaystyle {h_{0}}}$ ${\displaystyle {l_{1}}}$ ${\displaystyle {b_{1}}}$ ${\displaystyle {h_{1}}}$ ${\displaystyle V}$

${\displaystyle V={l_{0}}\cdot {b_{0}}\cdot {h_{0}}={l_{1}}\cdot {b_{1}}\cdot {h_{1}}=const}$

If you rearrange this equation by dividing by the initial dimensions, you get:

${\displaystyle 1={\frac {{l_{1}}\cdot {b_{1}}\cdot {h_{1}}}{{l_{0}}\cdot {b_{0}}\cdot {h_{0}}}}}$

The converted formula is then logarithmized

${\displaystyle 0=\ln \left({\frac {l_{1}}{l_{0}}}\right)+\ln \left({\frac {b_{1}}{b_{0}}}\right)+\ln \left({\frac {h_{1}}{h_{0}}}\right)}$,

it follows from the law of constant volume that the sum of the individual degrees of deformation for length, width and height must always be zero:

${\displaystyle 0=\varphi _{l}+\varphi _{b}+\varphi _{h}}$.

If the sum of the degrees of deformation is zero, one degree of deformation must have a different sign than the other two and at the same time be the one with the greatest absolute value. This applies to all three-dimensional transformations. If there is no deformation in one direction, i.e. if the deformation degree in this direction is zero, then it is a two-dimensional deformation in which the two other deformation degrees are equal in value, but with the opposite sign. The greatest degree of deformation in terms of value ( ) is the degree of deformation, which is indicative of the extent of deformation: ${\displaystyle \phi _{g}}$

${\displaystyle \varphi _{g}=|\varphi _{\mathrm {max} }|={\frac {1}{2}}(|\varphi _{l}|+|\varphi _{b}|+|\varphi _{h}|)}$

With the help of the greatest degree of deformation is relatively easy forming force and deformation work are calculated.

Further deformation parameters

When rolls of sheet the percent dimension ratio in the sheet thickness direction is referred to as degree of reduction , respectively.

literature

• R. Neugebauer (Ed.): Forming and cutting technology. Publishing house scientific scripts, Chemnitz 2005, ISBN 3-937524-35-5 .
• K. Lange (Ed.): Umformtechnik. Volume 1: Basics. Springer-Verlag, Berlin / Heidelberg / New York 2002, ISBN 3-540-43686-3 .
• E. Doege, B.-A. Behrens: Handbook of forming technology . Springer Verlag, Berlin 2006, ISBN 3-540-23441-1 .