DE10208144B4 - Method for calculating the life of non-proportionally stressed components - Google Patents

Abstract

Method for calculating the lifetime of non-proportionally stressed components by means of a computer, comprising the following steps: a) for a given component geometry (30), a finite element model is created which has a number of finite elements (32) and nodes at their corners; b) the external forces define the boundary conditions for the finite elements concerned (32); c) for a given node and for a given given load time history, a finite time or strain time lapse is determined for each finite element which is a time varying stress tensor or strain tensor; d) first sub-step of determining the lifetime: in each node, a set of planes of cutting planes (Ni) is laid; e) second sub-step of determining the lifetime: the stress tensor of each cutting plane (Ni) of the family of cutting planes (Ni) is transformed into a time-varying voltage vector; f) third sub-step of the determination of the lifetime: from the time-variable voltage vector of each cutting plane, a lifetime is determined and with the life of the other cutting planes of finite ...

Description

1 Introduction

The invention relates to a method for calculating the life of non-proportionally stressed components with the steps specified in the preamble of the independent claim. Non-proportional stress refers to the load due to time-variable forces whose components have different time courses. Of (local) component geometry is the speech, if not an entire component, but only one recognized as sensitive region of a component is examined.

Today's development of vehicles and their components is no longer possible without the help of numerical simulations on the computer. In this case, one no longer settles for the static calculation with the decomposition of the component to be examined into finite elements (hereinafter referred to as "FE"), but instead carries out a calculation of the service life of components. Reliable numerical lifetime estimation can save some trial loops, significantly reducing development time and development costs, which is a key factor in the competition.

However, the lifetime calculation is still fraught with such great uncertainties that one is still far from the complete renunciation of attempts. It is precisely the multiplicity of different factors which influence the service life in terms of the positive as well as the negative that make operational stability an extremely complex topic. The calculation methods offered are correspondingly diverse, but they are all reasonably reliable only within narrow limits.

From Döring, R., u. a .: "Investigations on short-crack progress concept under multi-axial non-proportional loading" in: German Association for Materials Research and Testing e. V., DFG Colloquium 2000 "Lifetime Prediction", pp. 81-96, a concept of the critical cutting plane is known, in which a material damage with the length of a surface crack considered to be critical is correlated. Here it is assumed that the principal stress axes and thus the critical plane under non-proportional stress are usually temporally variable. The critical cutting plane is determined iteratively, whereby the critical cutting plane is ultimately considered to be the one with the shortest lifetime or the highest crack propagation velocity.

Also from Döring, Ralph, u. a: "Material fatigue at non-proportional vibrational stress" in: Thesis wiss Journal of the Bauhaus University Weimar, Issue 3, 2000, pp. 102-113, it is known to capture fatigue by a calculation model for the life prediction on a fracture mechanical basis, which describes the growth of short cracks up to the technical cracking length.

The DE 199 27 941 C1 discloses a method for crack propagation simulation in which, starting from an initial crack, a section through a structure along a crack continuation is calculated and a new crack front is determined by determining tension values by a finite element calculation and applying a crack propagation law. The continued crack is triangulated and the steps mentioned are repeated.

2. Basics

2.1 The Rainflow Counting Method (Fig. 1)

The dynamic loading of a component leads to material fatigue, d. H. the component is "damaged" locally. If this damage exceeds a certain limit, cracking and thus breakage ("failure") of the component can occur as a consequence. The number of load cycles that a component endures until failure is called the "life" of the component. The "total damage" of the component is defined as the reciprocal of the lifetime. Each load cycle generates a "partial damage", which in total results in the total damage (linear damage accumulation according to Palmgren / Miner).

Each dynamic load cycle consists of a loading and unloading phase of the component. Locally on the component, this means that closed loops form in the stress-strain diagram, so-called "hystereses". Such closed hysteresis loops have proven to be relevant to the damage in terms of fatigue strength. They are characterized by amplitude value, mean and frequency of their occurrence. With the aid of the rainflow counting method, these hystereses can be determined from a stress or strain time curve for arbitrary load profiles and optimized for the lifetime calculation Form are saved (rainflow matrix). With simpler periodic load curves one can be content with a conventional counting.

Rainflow counting has been developed to evaluate uniaxial stress or strain conditions. If local stress (strain) and external stress are proportional to each other (elastic range), the Rainflow count can also be applied to the external load (force, moment). Importantly, the Rainflow counting method applies only to scalar quantities (eg uniaxial strain, strain, external stress), but not to vectors or tensors. Multi-axis stress states with components of different line progression lead out of the scope in the generic method.

2.2 The Wöhler curve (Fig. 2)

The basis of almost every fatigue life calculation are so-called "Wöhler" curves, which are recorded on simple, mostly cylindrical samples. The sample is subjected to a dynamic load of constant amplitude (one-step load) and the associated sustainable load change number for different load horizons is entered in a diagram. This is how the "Wöhler" diagram is created.

2.3 The Haigh Diagram (FIG. 3)

In addition to the stress level (stress amplitude), there are a number of other factors influencing the strength behavior of a component. A particularly important is the mean value of a load cycle, the so-called medium voltage, which is superimposed on the voltage amplitude. If there is a positive, ie tensile medium stress, the fatigue strength is reduced. If there is a negative, ie pressure medium, this usually has a favorable influence on the service life. The fatigue strength shifts upwards. The relationship between medium voltage and fatigue strength can be illustrated in a Haigh diagram.

Other significant influencing factors are the type of load and the properties of the material. For example, if a round bar is subjected to torsion, ductile materials (eg steel) will have a much lower fatigue strength (about 1 / √3) than with pure tensile / compressive stress. In the case of brittle materials (eg gray cast iron), however, the fatigue strengths are approximately the same. The ratio of tensile / compression to torsional fatigue strength is therefore a measure of the ductility of a material.

Not only the fatigue strength changes, but also the size and shape of the Haigh diagram. Under axial load can be distinguished between tensile and compressive stress, resulting in the asymmetrical shape of the Haigh diagram. With pure torsional load this is not possible. In this case, therefore, the Haigh diagram must be symmetric to the ordinate.

2.4 Classical damage hypotheses

In simple sample forms usually a uniaxial stress state is present. Accordingly, the life prediction with simple knowledge of the Wöhler curve. In the case of notched samples or even complex components, a multiaxial stress state will generally be established. Unloaded component surfaces and shell structures are usually a flat, ie biaxial, stress state, but in the interior of the component they are spatial, ie three-axis, stress states.

If one wishes to retain the classical concepts of the Wöhler curve and the Haigh diagram for lifetime prediction, then the multiaxial stress state has to be converted into a damage equivalent equivalent stress, since Wöhler curve and Haigh diagram were indeed taken with locally uniaxial stress. Depending on the type of material, various damage hypotheses have been established. For example, in ductile materials, the shape change energy hypothesis is usually applied, which prescribes the use of Von Mises stress as the reference stress. In the case of brittle materials, on the other hand, the hypothesis of the maximum main normal stress has proved itself.

All these hypotheses assume that the principal directions of tension are constant, that is, they are fixed in time. This is the case when all external loads (forces, moments) are proportional to each other. If external loads are not proportional to each other, that is, if there are several uncorrelated forces acting on different points of application of force, local rotation of the main directions of tension occurs. The application of classical damage hypotheses is then no longer permissible. Other methods, which also consider the changes in the direction of the main normal stresses must be used.

3. Prior Art: The Method of "Critical Section Plane"

In the case of nonproportional external loading, the method of the "Critical Section Plane" represents an accepted method of lifetime prediction that is insufficiently adequately secured in terms of testing. The component is locally split along imaginary cutting planes (cut edges) and the stresses acting in such sectional planes are used for the damage calculation. This is done for all cutting planes at a certain angular distance (usually 10 degrees). The local section plane at which the damage accumulation is maximal is considered to be relevant to the damage and thus "critical". The crack, if it occurs at the point of interest, will form along this critical cutting plane.

At component surfaces, where most of the initial crack is formed, it is sufficient to consider all cutting planes that are perpendicular to the surface, provided that no external pressure forces are applied. In general, however, all sectional planes whose normal vectors sweep a hemisphere must be considered.

4. Disadvantages of the "Critical Section Plane" Method

• 1) Die Spannung in der Schnittebene erweist sich als vektorielle Größe, bestehend aus i. a. drei unabhängigen Komponenten, einer Normal- und zwei Schubkomponenten. Die Zeitverläufe der Komponenten sind bei äußerer nichtproportionaler Belastung im Allgemeinen unkorreliert, das heisst die Spitze des Vektors beschreibt einen beliebigen Weg im dreidimensionalen Spannungsraum. Das Rainflow-Zählverfahren ist jedoch, nur auf skalare Größen und somit auf diesen allgemeinen Fall nicht anwendbar.
• 2) Die alleinige Heranziehung der Normalkomponente zur Rainflow-Zählung, was heutzutage allgemein praktiziert wird, ist nur für spröde Werkstoffe ausreichend. Solche werden aber für tragende Teile in der Regel nicht verwendet.
• 3) Die Zählung der Schubkomponente, unter anderem nach dem Rainflow-Verfahren, bei duktilen Werkstoffen setzt einen lokal ebenen Spannungszustand voraus. Ansonsten ist die Richtung der Schubkomponente nicht eindeutig definiert. Es ergeben sich durch das Vorhandensein zweier unabhängiger Schubkomponenten dieselben Probleme wie unter 1).
• 4) Der manchmal praktizierte Versuch, aus Normal- und Schubkomponente eine Vergleichsspannung zu bilden, etwa ähnlich einer Von Mises-Spannung, trifft nur in Sonderfällen, etwa bei proportionaler Belastung. Um die Klassierbarkeit einer Vergleichsspannung zu gewährleisten, muss diese mit einem Vorzeichen (z. B. Vorzeichen der Normalkomponente) ausgestattet sein. Das führt im weiteren zu unphysikalischen Sprüngen des Vergleichsspannungsverlaufs, oder die Vergleichsspannung entzieht sich überhaupt jeglicher physikalischer Interpretation (z. B. das Nokleby-Kriterium, welches eine Linearkombination aus Normal- und Schubkomponente darstellt).

Despite its widespread use, however, this process involves some unsolved problems that severely limit its applicability:

• 1) The stress in the section plane proves to be a vectorial quantity, consisting essentially of three independent components, one normal and two thrust components. The time courses of the components are generally uncorrelated with external non-proportional loading, ie the peak of the vector describes any path in the three-dimensional stress space. The Rainflow counting method, however, is not applicable to scalar quantities and thus to this general case.
• 2) The sole use of the normal component for rainflow counting, which is commonly practiced today, is sufficient only for brittle materials. Such are usually not used for load-bearing parts.
• 3) The count of the shear component, among other things according to the Rainflow method, for ductile materials requires a locally flat state of tension. Otherwise, the direction of the thrust component is not clearly defined. The existence of two independent thrust components results in the same problems as under 1).
• 4) The sometimes practiced attempt to make normal and shear components a comparative strain, similar to a Von Mises stress, only applies in special cases, for example proportional load. In order to ensure the classifiability of a reference voltage, it must be provided with a sign (eg sign of the normal component). This leads to further unphysical jumps in the comparison voltage curve, or the comparison voltage evades any physical interpretation at all (eg the Nokleby criterion, which represents a linear combination of normal and shear component).

5. Objective and principles of the invention

As described above, the described methods are not suitable for predicting the lifetime of three-dimensional components with non-proportional loading, as currently occurring in vehicle and engine construction. It should be given a reliable for such components new method for lifetime calculation of non-proportionally stressed components.

This method, which is a further development of the "critical section plane" method and overcomes the limits of its applicability, is called the "critical section plane - critical component" method. In order to be applicable also for non-proportional stresses, among other things also using the classical Rainflow counting method, the following solution is inventively, briefly summarized, gone:
In addition to the determination of the critical cutting plane, a critical stress component or-alternatively-strain component is sought within the critical cutting plane. Since one does not know from the outset which cutting plane is the critical one, this search must be carried out for all cutting planes.

For this purpose, for each section plane, all directions are considered at certain angular intervals (eg between 1 and 10 degrees) whose directional vectors lie with their tip in a hemisphere. These direction vectors of stress components must not be confused with the normal vectors of the cutting planes. It is assumed that the component is dynamically loaded and thus damaged locally in the considered sectional plane in the direction of each direction vector by the associated stress component, independently of other stress components pointing in other directions. For this purpose, the voltage vector is projected onto each of these directions by means of in-product formation (= scalar product of two vectors) (normal projection of the voltage vector onto the direction vector). This gives time courses for different voltage components in different directions.

A damage calculation can now easily be performed for these time courses, for example by means of a Rainflow count and the WÖHLER and HAIGH diagrams. Within a cutting plane, a large number of damages are thus obtained, one result for each calculated direction. Now the direction with the maximum damage is selected for each cutting plane ("critical direction" for each cutting plane). This leaves as many directions as cutting planes, one direction per cutting plane.

In order to determine and definitively define the critical section plane, a second selection is now made of all remaining critical directions (one to each section plane) that which gives the greatest accumulated damage. This direction with the maximum accumulated damage and its cutting plane is considered finally critical and thus relevant to the damage.

With this method, the above disadvantages of the prior art are eliminated. The practical implementation is conceivably simple, because of the large number of operations to be performed, of course, only with the help of a computer.

In a further refinement of the method according to the invention-the basic idea of which makes it possible-it is possible within a sectional plane for each load direction to assign very specific fatigue-strength properties of the respective material. A significant advantage is that, with the help of the polar angle α between the slice plane normal and the direction of the voltage vector, the three basic stress types of the cutting plane, train, thrust and pressure, can be distinguished. Every other direction of stress is a combination of these basic load types.

The method according to the invention is thus even suitable for taking account of the anisotropic material properties in general, and in particular in the case of surface-treated components, for example the different fatigue strengths for varying tensile / compressive and shear stress. In each direction, not only different fatigue strengths can be assigned in follow-up, but even complete Wöhler curves, which were determined approximately experimentally.

Even larger are the differences in the influence of the medium voltage. For example, a fluid pressure (usually) has a much more favorable influence on the service life than a tension medium tension, and this in turn has a more favorable effect than a shear stress (gray cast iron excepted). It can also be considered favorable influences, which helps avoid over-dimensioning of the component.

With this distinction, each material can be characterized by a two-dimensional surface representing the fatigue strength as a function of the polar angle and of the mean stress, which corresponds to a generalized Haigh diagram. In are exemplified (as a rough approximation) the surfaces for a structural steel. Within the ranges pull-thrust (between 0 and 90 degrees) and thrust-pressure (between 90 and 180 degrees) is linearly interpolated. Of course, any other type of interpolation may be used, and the intermediate points could in principle also be determined experimentally.

In the following the invention will be described and explained with reference to figures. They show:

1 : schematically a "rainflow count",

2 such as a WÖHLER curve,

3 for example, a HAIGH diagram,

4 : an extended HAIGH diagram,

5 a component to which the method according to the invention is applicable, and its subdivision into finite elements,

6 : Excerpts two finite elements of the component of 5 .

7 : an infinitesimal small particle from the finite element of the 6 , with stress components forming the stress tensor, independent of cutting planes,

8th : a cutting plane with voltage vector and direction vector.

1 explains in preparation the principle of "rainflow counting", so called because the rain flow on a roof is used to illustrate the method by which the stress-relevant full hysteresis loops are counted. The voltage is plotted on the horizontal ordinate axis, the vertical axis is the time axis, it points downwards in the direction of the raindrops. For counting, the rule is that a hysteresis loop (= full swing cycle) is formed when the underlying roof is the same size or larger than the one above it.

The top roof is with 1 referred to, the underlying roof 2 protrudes under the roof 1 so it's bigger in this direction, the triangle 3 . 4 . 5 is thus the first hysteresis loop to be counted. The next roof 8th . 9 sticks out on the side of positive tensions to the point 7 Well, it is just wet by the rain. It forms the pentagon 7 . 8th . 9 as the second hysteresis loop to be counted. The triangle 10 is also counted as so-called subhysteresis. Next is the triangle 9 . 11 . 12 to count, because in one direction it is the same size as the one above. Each hysteresis loop has a medium voltage and an amplitude for the further calculation.

The further treatment is based on the WÖHLER curve and the HAIGH diagram, see 2 and 3 in the professional way. In the former, the number of load cycles, here hysteresis loops up to the crack or break on the abscissa, the stress amplitude on the ordinate, both logarithmic. Accordingly, the curve consists of a steeper branch 15 and a flatter branch 16 , Among the former is the zone of fatigue strength, under the second is the fatigue strength. The Haigh diagram ( 3 ), here for steel and for tension / compression, then takes into account the mean stress, which is plotted on the abscissa. The ordinate shows the associated tolerable stress amplitude (fatigue strength). About the train 19 is a mean stress σ _{m associated with} the corresponding voltage amplitude σ _a for the Wöhler diagram. It can be seen that the values on the pressure side (σ _m negative) are more favorable than on the tension side. The Haigh diagram for torsional stress looks different and is symmetrical.

A three-axis, spatial Haigh diagram shows 4 , the three axes are mean stress σ _m , stress amplitude σ _a and polar angle α (see 8th ), in which: α = 0 degrees: pure tensile stress, polyline 20 α = 90 degrees: pure shear stress, polyline 22 α = 180 degrees: pure compressive stress polyline 21 ,

5 shows a finite element (FE) disassembled component 30 to which the method according to the invention is applicable. From the region identified as endangered (geometry) 31 of the component become two finite elements for explanation 32 and their closer environment singled out and in 6 shown greatly enlarged. It is 32 a FE on the surface of the component and 6 also shows an underlying, inside the component lying FE. Two nodes are designated, the others and the subsequent FE only more intimated.

The area of the FE 32 is on the surface, which is why there can only be a biaxial state of tension there. Therefore, for the process of the critical cutting plane, a group of normal cutting planes on the surface is sufficient (for example, in steps of 10 ° over 180 °), that is to say a one-fold multiple plane family. Their normal vector is indicated.

dargestellt, ihre Spitzen liegen in der Kalotte einer Halbkugel.The situation is different with the other node, in which a three-axis state of tension prevails. Therefore, for the method of critical cutting planes, a two-time-multiple plane family N _{i is} necessary. In place of the planes, here are just a few of those normal vectors

shown, their tips are in the dome of a hemisphere.

Weiters ist wieder der Normalenvektor

einer Schnittebene eingezeichnet. Durch vektorielle Multiplikation der Matrix des Spannunstensors

mit dem Normalenvektor

der Schnittebene wird ein Spannungsvektor

erhalten, der bei zeitvariabler Beanspruchung selbst auch zeitvariabel ist. Wenn die Komponenten dieses Spannungsvektors verschiedene Zeitverläufe haben, also nicht mehr proportional sind, ist das bisher beschriebene Verfahren nicht mehr brauchbar. 7 now shows the closest environment of the other node. The stresses acting there σ _xx , σ _yy , σ _zz , τ _xy = τ _yx , τ _yz = τ _zy , and τ _xz = τ _zx form a Sannunstensor

Furthermore, the normal vector is again

drawn a sectional plane. By vectorial multiplication of the matrix of the tension sensor

with the normal vector

the cutting plane becomes a voltage vector

obtained, which is itself time-variable with time-variable stress. If the components of this voltage vector have different time courses, ie are no longer proportional, the method described so far is no longer usable.

gelegt und der Spannungsvektor

auf die Richtungsvektoren

projiziert. Die Spitzen der Richtungsvektoren

beschreiben wieder eine Kugelkalotte 40. Mathematisch ist das die Bildung eines Inproduktes. So entsteht eine zeitvariable skalare Größe in Richtung des jeweiligen Richtungsvektors

Diese skalaren Größen können in der bekannten und anhand der 1 bis 4 beschriebenen Weise zur Lebensdauerbeziehungsweise Schädigungsrechnung herangezogen werden.Therefore, according to 8th for each sectional plane N _i, a two-fold multiple family of direction vectors

placed and the voltage vector

on the direction vectors

projected. The tips of the direction vectors

again describe a spherical cap 40 , Mathematically, this is the formation of an in-product. This creates a time-variable scalar variable in the direction of the respective direction vector

These scalar quantities can be found in the known and based on the 1 to 4 described manner for life or injury calculation are used.

mit der größten Schädigung zu einer Schnittebene ist die kritische. Nun wird unter den Schnittebenen diejenige mit der größten Schädigung ausgewählt. Das ist die kritische Schnittebene. Die Lebensdauer in dieser ist für die des betreffenden finiten Elementes die maßgebende.

• α Polarwinkel
• σ Spannung
• τ Schubspannung
• 
  Richtungsvektor
• 
  Normalvektor der Schnittebene
• 
  Spannungsvektor
• 
  Spannungstensor

The direction

with the greatest damage to a cutting plane is the critical one. Now, the one with the greatest damage is selected from the cutting planes. That's the critical cutting plane. The lifetime in this is for the relevant finite element, the authoritative.

• α polar angle
• σ tension
• τ shear stress
• 
  direction vector
• 
  Normal vector of the cutting plane
• 
  voltage vector
• 
  stress tensor

Claims (8)

Method for calculating the life of non-proportionally stressed components by means of a computer, comprising the following steps: a) for a given component geometry ( 30 ), a finite element model is created which contains a number of finite elements ( 32 ) and has vertices at its corners; b) the external forces are used to determine the boundary conditions for the finite elements ( 32 ) Are defined; c) for a given node and for a given given load time history, a voltage-time lapse or stretch time lapse is determined for each finite element which is a time-varying stress tensor

or strain tensor; d) first sub-step of determining the lifetime: in each node, a set of planes of cutting planes (N _i ) is placed; e) second sub-step of determining the life: the stress tensor

in each cutting plane (N _i ) the set of cutting planes (N _i ) becomes a time-variable voltage vector

transformed; f) third sub-step of determining the lifetime: from the time-variable voltage vector

At each intersecting plane, a lifetime is determined and compared to the lifetime of the other finite element intersection planes, in which case the shortest life intersection plane is critical; characterized in that g) the time-variable voltage vector

of each section plane (N _i ) to a set of direction vectors

is projected, creating a scalar size; h) for each direction vector

the family is calculated from each of these scalar sizes the life; i) the lifetime to all direction vectors

the family of a section plane (N _i ) is compared, the direction vector

to which the life is a minimum represents the critical direction; j) from all the cutting planes of the family (N _i ) the one is selected in whose critical direction the lifetime is a minimum, wherein the selected cutting plane is the critical cutting plane. A method according to claim 1, characterized in that in the first sub-step of determining the lifetime when it is a finite element in the interior of the component, the set of cutting planes (N _i ) in each node consists of all those cutting planes whose normal vectors

lie in a hemisphere. A method according to claim 1, characterized in that in the second sub-step of determining the lifetime of the transformation of the stress tensors

into a time-variable voltage vector

by multiplication of the matrix of the tensor

with the normal vector

the respective sectional plane (N _i ) takes place. Method according to claim 1, characterized in that in sub-step (g) the family of direction vectors

consists of all directional vectors which are in a hemisphere ( 40 ) lie. A method according to claim 1, characterized in that in the sub-step (h) the calculation of the lifetime to each direction vector

from medium voltage, amplitude and frequency. A method according to claim 5, characterized in that the frequency is determined as a frequency of closed hysteresis loops according to the Rainflow method. A method according to claim 1, characterized in that the lifetime calculation for each direction

each sectional plane taking into account this direction with respect to the direction of the normal vector

the cutting plane corresponding stress type and the material properties is carried out. A method according to claim 7, characterized in that to the polar angle (α) wipe the respective direction vector

and the voltage vector

is used.

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