Dog bone model

Schematic representation of the dog bone model with greatly enlarged plastic zones

The dog bone model (engl. Dog-bone model ) is a model of the fracture mechanics , by means of which the plastic zone can be considered at the tip of a crack. It gets its name because of the shape of the plastic zone, which for steel materials is reminiscent of a dog's bone.

background

The linear-elastic fracture mechanics makes the assumption that when a crack propagates, no plastic, but only elastic deformation occurs. The stress in the area of ​​the crack tip is described by the Irwin-Williams equations , which have a pole at zero, i.e. directly at the crack tip : the stress becomes infinitely large. From a physical point of view, this assumption does not make sense, since plastic deformation occurs when the yield point is exceeded , which reduces the stress. This is where the dog bone model comes in and provides a way of describing the shape and diameter of the plastic zone.

Calculation of the size of the plastic zone

The stress distribution in the plastic zone differs depending on the place of observation. There is a level stress state (ESZ) on the load-free surface of the component . In the interior of the component, on the other hand, there is a level strain state (EDZ) at the crack front, since the elongations in the longitudinal direction of the crack front are hindered by the surrounding material. The shape of the plastic zone differs between EDZ and ESZ.

Contour of the plastic zone at the crack for ideal plastic material and . ${\displaystyle \nu =0,3}$

To determine the contour of the plastic zone for crack opening mode I, the simplest case is an ideal plastic material behavior with the initial yield stress. Using the polar coordinates and , the interface of the plastic zone can be determined as follows: ${\displaystyle \sigma _{\mathrm {F} }}$ ${\displaystyle r_{\mathrm {p} }}$ ${\displaystyle \theta }$

${\displaystyle r_{\mathrm {p} }={\frac {1}{2\pi }}\left({\frac {K_{\mathrm {I} }}{\sigma _{\mathrm {F} }}}\right)^{2}\cos ^{2}{\frac {\theta }{2}}\left\{{\begin{array}{ll}\displaystyle 3\sin ^{2}{\frac {\theta }{2}}+1&{\text{: ESZ}}\\\displaystyle 3\sin ^{2}{\frac {\theta }{2}}+(1-2\nu )^{2}&{\text{: EDZ}}\end{array}}\right.}$

Where:

${\displaystyle K_{\mathrm {I} }}$the stress intensity factor (here for mode I )

${\displaystyle \nu }$ the Poisson's ratio.

A round shape is often assumed to simplify the estimation of the size of the plastic zone. The diameter of the plastic zone in this case can be determined by:

${\displaystyle d_{\mathrm {PZ} }={\frac {1}{\pi }}{\biggl (}{\frac {K_{\mathrm {I} }}{R_{\mathrm {p0,2} }}}{\biggr )}^{2}\left\{{\begin{array}{cl}1&{\text{: ESZ}}\\\displaystyle {\frac {1}{3}}&{\text{: EDZ}}\end{array}}\right.}$.

It is

${\displaystyle d_{\mathrm {PZ} }}$ the diameter of the plastic zone

${\displaystyle R_{\mathrm {p0,2} }}$the yield point .

The plastic zone is therefore larger on the component surface than on the inside of the component.

application

With the dog bone model, the applicability of the linear-elastic fracture mechanics can be checked. For this purpose, the size of the plastic zone is first determined, for example using the relationships given above. This value is then compared with the crack length.

• If the plastic zone is negligibly small compared to the crack length, the linear-elastic fracture mechanics can be used.
• If the plastic zone is large compared to the crack length, the often much more complex flow rupture mechanics must be used.

Individual evidence

1. ↑ ^{a b c d} Christoph Broeckmann, Paul Bite: Materials Science I . Institute for Material Applications in Mechanical Engineering at RWTH Aachen University , Aachen 2014, pp. 88–101.
2. ↑ H.-J. Christ: Fundamentals of fracture mechanics. Website of the University of Siegen , accessed on February 5, 2016.
3. ^ ^{A b} Hans-Jürgen Bargel, Günter Schulze (Ed.): Material science. 11th, revised edition. Springer Vieweg, Berlin / Heidelberg 2012, ISBN 978-3-642-17717-0 , pp. 153-154.
4. ↑ M. Kuna: Numerical stress analysis of cracks - finite elements in fracture mechanics. 2nd Edition. Vieweg + Teubner, 2010, ISBN 978-3-8348-1006-9 .