## MINER’S EQUATION : INTRODUCTION TO CUMULATIVE DAMAGE IN FATIGUE

*MINER’S EQUATION* :Till now, we dealt with stresses that acted on the system in a single stress cycle. In practical analysis, especially in high stress applied tasks, the stress acts in PARTICULAR levels.

*Here, a particular mechanical component gets subjected to different stress levels , A*ND *each stress level acting at different parts of the work cycle. *

So,** MINER developed an EQUATION** in order to solve this problem.

The above equation finds use in these stress cycle cases.

### MINER’S EQUATION : FORMULA AND IT’S STUDY

Suppose we take a component ; subjected to various completely reversed stresses , each stress cycle acting for various no. of work cycles.

**Now, suppose no. of cycles (n _{1},n_{2},n_{3},……) at respective stress levels (σ_{1},σ_{2},σ_{3},……) are unknown.**

*SOLVED NUMERICAL :*

**Q1. The work cycle of a mechanical component subjected to completely reversed bending stresses consists of the following three elements;**

**1) ±350 N/mm ^{2} for 85% of time**

**2) ±400 N/mm ^{2} for 12% of time**

**3) ±500 N/mm ^{2} for 3% of time**

**Material and component is 50C4( S _{ut}=660 N/mm^{2}) and corrected endurance limit of the component =280 N/mm^{2}.**

**Determine the life of the component.**

## THEORIES OF FAILURE (DYNAMIC LOADING):

For static loading , we learnt about 5 Theories of Failure (T.O.F) for designing in complex stress systems.

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But in fluctuating loading stress systems, design is quite different from others.

**Fluctuating loading stresses : **Those stresses which have different magnitude of stresses and also induces in a specific direction.

**σ _{m}=**

**σ**&

_{1 }+σ_{2}/2**σ**where ,

_{v = }σ_{1 }– σ_{2}/2**σ**=

_{1}= maximum stress σ_{2}**minimum stress**

Here, we observed that MEAN STRESS {**σ _{m}**} component shows effect on the fatigue failure of a system when present in combination with alternating component.

So, we defined a set THEORIES OF FAILURE (TOF) for the fatigue loading systems.

The criteria for defining TOF for fatigue loading is mainly focused on S N curve characteristics.

### S N CURVE CHARACTERISTICS STUDY (STATIC AND FATIGUE):

**IN THIS ANALYSIS, we plot MEAN STRESS (σ _{m}) on x-axis (abscissa) and STRESS AMPLITUDE (σ_{a}) on y-axis (ordinate).**

*** If * σ_{a}*=0 ; load is purely static and Criterion is either (

**S**These limits plotted on the abscissa.

_{yt}/σ_{yt}) or (S_{ut}/σ_{ut}).*** If **σ**_{m}=0 ; load is completely reversing and Criterion is Endurance limit (**S _{e}**/

**σ**) .This point plotted on the ordinate.

_{e}*** But when both are present ; * σ_{a}* &

**σ**_{m}and are non zero, The actual failure is impossible to find out. And, the failure usually appears as different scattered points as shown in the figure.

*There exists a border in this graph .This border separates safe region from unsafe region for various combinations of σ_{a} & σ_{m}.*

Different criterions came in existence to derive this safe region characteristics and these all in combination came to be known as “**THEORIES OF FAILURE FOR FATIGUE LOADING”.**

**THEORIES OF FAILURE FOR FATIGUE LOADING** kept for discussion in the next blog.

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