Linear-elastic fracture mechanics

The linear elastic fracture mechanics , short LEFM is a concept from the field of fracture mechanics for the theoretical description of the growth of cracks in a component .

Assumptions

A crack spreads in a component because there is particularly high tension at the crack tip. The linear-elastic fracture mechanics assumes that the stress at the crack tip becomes infinitely great. Physically, this is not the case, as plastic deformation occurs when the yield point of the material used is exceeded , which lowers the stress. However, if this plastic zone is small compared to the zone of excessive stress, the linear-elastic fracture mechanics still deliver good results.

Brittle materials, for example, only allow very little plastic deformation, so that the linear-elastic fracture mechanism can be used particularly well for brittle fractures of materials such as ceramics , cast iron or hardened steel . In contrast, flow rupture mechanics are used for materials that fail more ductile . The size of the plastic zone is estimated with the dog bone model . The corresponding ASTM standards provide limit values up to when it is possible to work with the linear-elastic fracture mechanics .

Types of crack stress

Types of stress (modes) of
a crack

In fracture mechanics, a distinction is made between three basic types of stress (modes) through which a crack can be stressed:

Mode I: Stresses that cause the crack flanks to open. As a rule, these are all loads that act normally to the crack front. Examples of this would be a component that is under tensile or bending stress and where the crack runs perpendicular to the normal stress.
Mode II: Stresses that cause the crack flanks to shift in the opposite direction in the direction of crack propagation, mostly caused by a shear load .
Mode III: Stresses that cause the crack flanks to shift across the direction of crack propagation. This type of stress appears z. B. in waves that are under torsional stress and where a crack is perpendicular to the shaft axis.

If all three modes occur together on a crack front, one speaks of a mixed-mode load . Mixed-mode stresses can be caused on the one hand by a multi-axis external load on the component. However, even with uniaxial loading, a mixed-mode state can occur at the crack front if it is at any non- orthogonal angle to the axis of the main normal stress.

Stress intensity factor

The stress intensity factor describes the stress distribution around the crack tip. It can be calculated for common geometries ( test objects , components) using approximation formulas or FEM simulations. The type of crack stress is indicated by specifying the mode in the index, for example the stress intensity factor for mode I. It can be calculated by ${\ displaystyle K}$ ${\ displaystyle K_ {I}}$

{\ displaystyle K_ {I} = \ sigma \ cdot {\ sqrt {\ pi \ cdot a}} \ \ cdot Y}

.

It is

${\ displaystyle \ sigma}$ the global stress, i.e. the stress that is remote from the crack and encompasses the entire component
${\ displaystyle a}$ the crack length
${\ displaystyle Y}$ a correction factor dependent on the component geometry or the specimen geometry.

The stress intensity factor is used by the Irwin-Williams equations (see section Stress distribution at the crack tip) and can be compared with material parameters such as fracture toughness . This predicts whether the crack will stand still, or whether the crack grows and the component can break (see section breakage criterion ).

Stress distribution at the crack tip

The stress distribution in the immediate vicinity of the crack tip can be described with the Irwin- Williams equations (sometimes also known as Sneddon equations ). For this it is assumed that the crack is small compared to the component dimensions. The crack tip represents the origin of a polar coordinate system with the coordinates and . This is the position vector and the angle between the position vector and the extension of the crack in the direction of propagation (also called ligament ). ${\ displaystyle r}$ ${\ displaystyle \ varphi}$ ${\ displaystyle r}$ ${\ displaystyle \ varphi}$

The Irwin-Williams equations describe the voltages , and each with an infinite series . For the area around the crack tip, the so-called K-dominant zone , it is not necessary to specify the entire row, only the first links. With this simplification, the voltages for the (technically most relevant) mode I can be described by: ${\ displaystyle \ sigma _ {x}}$ ${\ displaystyle \ sigma _ {y}}$ ${\ displaystyle \ tau _ {xy}}$

{\ displaystyle \ sigma _ {x} = {\ frac {K_ {I}} {\ sqrt {2 \ pi \ cdot r}}} \ cdot \ cos {\ frac {\ varphi} {2}} \ cdot \ left (1- \ sin {\ frac {\ varphi} {2}} \ cdot \ sin {\ frac {3 \ varphi} {2}} \ right)}

{\ displaystyle \ sigma _ {y} = {\ frac {K_ {I}} {\ sqrt {2 \ pi \ cdot r}}} \ cdot \ cos {\ frac {\ varphi} {2}} \ cdot \ left (1+ \ sin {\ frac {\ varphi} {2}} \ cdot \ sin {\ frac {3 \ varphi} {2}} \ right)}

{\ displaystyle \ tau _ {xy} = {\ frac {K_ {I}} {\ sqrt {2 \ pi \ cdot r}}} \ cdot \ sin {\ frac {\ varphi} {2}} \ cdot \ cos {\ frac {\ varphi} {2}} \ cdot \ cos {\ frac {3 \ varphi} {2}}}

.

Breakage criterion

According to the K-concept, unstable crack growth occurs with pure Mode I loading (catastrophic failure, sudden break) if the load size, i.e. the stress intensity factor in the LEBM area , reaches or exceeds a critical material parameter, here the critical crack toughness ( breakage criterion ) : ${\ displaystyle K_ {I}}$ ${\ displaystyle K_ {Ic},}$

{\ displaystyle K_ {I} \ geq K_ {Ic}.}

The fracture toughness is determined for each material using special fracture mechanical samples, mostly CT samples. This enables the crack growth behavior of a material to be characterized using a test piece and then transferred to the component. This is possible because it is assumed that the crack always grows at the same speed regardless of the component or test specimen geometry, regardless of the size. ${\ displaystyle K_ {I}}$

evaluation

The three phases of crack growth

The crack growth behavior is usually shown by plotting the crack growth speed da / dN (crack growth per load change) over the cyclical stress intensity factor ΔK ₁ in a double-logarithmic diagram . Usually three characteristic areas can be identified that can be used for material comparisons:

A: Crack initiation
B: stable crack growth
C: unstable crack growth (breakage).

The material behavior is better if the curve is shifted to the right or downwards.

literature

HA Richard, M. Sander: Fatigue cracks: Recognize, assess reliably, avoid . 1st edition, Vieweg + Teubner, Wiesbaden, 2009, ISBN 978-3-8348-0292-7 .

Individual evidence

↑ ^a ^b ^c ^d ^e ^f Christoph Broeckmann, Paul Beiss: Materials Science I . Institute for Material Applications in Mechanical Engineering at RWTH Aachen University , Aachen 2014, pp. 88–101.

Web links

Fundamentals of fracture mechanics (accessed October 14, 2019)
Mechanism-oriented fracture mechanical characterization of aviation alloys under realistic stresses (accessed on October 14, 2019)
TENSIONS AND BREAKAGE MECHANICAL PROCESSES IN NORMAL CONCRETE UNDER TRAINING DEMANDS (accessed on October 14, 2019)
Fracture Mechanics (accessed October 14, 2019)
Fracture mechanical investigations on the crack resistance behavior of non-conversion-reinforced ceramics using the example of aluminum oxide (accessed on October 14, 2019)

[broeckmann-1] ↑ ^a ^b ^c ^d ^e ^f Christoph Broeckmann, Paul Beiss: Materials Science I . Institute for Material Applications in Mechanical Engineering at RWTH Aachen University , Aachen 2014, pp. 88–101.