Inter-fiber breakage criterion according to Puck

The intermediate fiber failure criterion by Puck used to calculate the failure of inter-fiber fraction of unidirectional fiber reinforced composites . This break criterion is a physically sound approach based on the considerations of Christian Otto Mohr and Zvi Hashin .

Preliminary remarks

Stress state on a UD element with a layer level . The position of the - and the - coordinate system is shown. The latter is rotated around the break angle . Possible break planes in the case of an intermediate fiber break are between -90 ° and + 90 ° break angle.

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{\ displaystyle (1,2,3)}

{\ displaystyle (1, x {n}, x {t})}

{\ displaystyle \ theta {fp}}

{\ displaystyle (1, x {t})}

Fiber-plastic composites show a brittle fracture behavior in the event of both fiber breakage and inter- fiber breakage . For this reason, the strength hypotheses for materials with a brittle fracture character, such as the hypotheses of Coulomb and Mohr , are preferable to the flow criteria for ductile materials for fiber-plastic composites .

Puck's intermediate fiber breakage criterion is based on a macromechanical approach. The stresses on the UD layer averaged over fiber and matrix cross-sections and the associated strengths are used to describe the fracture state .

In all previous 3D fracture criteria which determined from the stratified stress analysis were tensions on the solid - coordinate system related. However, in order to set up a physically based breakage condition taking Mohr's hypothesis into account , it was necessary to introduce a coordinate system that can be rotated around the grain direction (1-direction). Depending on the local stress state , the -coordinate system is rotated by an arithmetic operation in relation to the -coordinate system in such a way that the -direction becomes the surface normal of the expected fracture plane and the -direction becomes the surface tangential . The surface normal of the fracture plane (fp) will generally be inclined at an angle of −90 ° << + 90 ° with respect to the layer plane in the event of an intermediate fiber break ; this is exactly the angle by which the coordinate system must be rotated. ${\ displaystyle (1,2,3)}$ ${\ displaystyle (1, x_ {n}, x_ {t})}$ ${\ displaystyle (1, x_ {n}, x_ {t})}$ ${\ displaystyle (1,2,3)}$ ${\ displaystyle x_ {n}}$ ${\ displaystyle x_ {t}}$ ${\ displaystyle \ theta _ {fp}}$

Failure prediction

The task of a break criterion is to determine whether any load (stress state) applied to a UD layer leads to breakage or not. In this context, breakage means the separation of materials due to the action of stress. The physical assumptions about the conditions causing the break can be described using the break hypothesis. The mathematical formulation for the stress and deformation states in which a break occurs is called the break condition.

The experimental proof of when a material separation (break) occurs in a UD layer due to a defined voltage level can be provided by suitable test methods , e.g. B. by tension-compression-torsion test . Using these test methods, the direction-dependent basic strengths of a UD layer can be determined:

Longitudinal tensile strength ${\ displaystyle R _ {\ parallel} ^ {(+)}}$
Longitudinal compressive strength ${\ displaystyle R _ {\ parallel} ^ {(-)}}$
Transverse longitudinal shear strength ${\ displaystyle R _ {\ perp \ parallel}}$
Transverse tensile strength ${\ displaystyle R _ {\ perp} ^ {(+)}}$
Transverse compressive strength . ${\ displaystyle R _ {\ perp} ^ {(-)}}$

Action plane-related break criterion

The plane of action is a cutting plane in which an individually acting stress in a material element is maximal. As expected, the break should take place in this plane.

From a loaded under pressure concrete - specimen , however, it is known that it is not normal to the pressure load - ie not in the plane of action of the pressure - failed, but slides off at an oblique cut. If the UD layer fails due to an intermediate fiber break, the same happens: in the case of a UD layer stressed under transverse pressure, the break plane does not coincide with the effective plane of the stress. So it is not a question of pressure failure , but thrust failure.

Due to this fact Puck introduced a new term, the breaking resistance of the plane of action :

"The breaking resistance of an effective plane is the resistance that a sectional plane opposes to its breakage as a result of a single load acting on it."

To avoid confusion with the above To avoid strength , the breaking resistance is additionally provided with a superscript A (A = action plane). ${\ displaystyle R}$ ${\ displaystyle R ^ {(A)}}$

Flat intermediate fiber break criterion

In the case of thin-walled structures, a level stress state can be assumed, which is why only the stresses in this plane are evaluated with regard to inter-fiber breakage.

Three-dimensional intermediate fiber breakage criterion

This criterion evaluates all 5 stresses that contribute to inter-fiber breakage (cf. the above-mentioned basic strengths).

literature

A. Puck: Strength analysis of fiber-matrix laminates . Hanser, 1996. ISBN 3-446-18194-6
H. Schürmann: Constructing with fiber-plastic composites . Springer Verlag, 2007. ISBN 978-3-540-72189-5
M. Knops: Analysis of Failure in Fiber Polymer Laminates - The Theory of Alfred Puck . Springer, 2008. ISBN 978-3-540-75764-1