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1 The Math behind Elastic Machine Learning Tom Veasey, Software Engineer, Machine Learning Hendrik Muhs, Software Engineer, Machine Learning Elastic 1st March

2 Elastic Stack Security Beats Elasticsearch Alerting Kibana Monitoring Logstash X-Pack X-Pack X-Pack Reporting Graph Machine Learning Single install - deployed with X-Pack Data gravity - analyzes data from the same cluster Contextual - anomalies and data stored together Scalable - jobs distributed across nodes Resilient - handles node failure 2

3 Machine Learning in the Elastic Stack 3

4 Machine Learning in the Elastic Stack 4

5 Machine Learning in the Elastic Stack In a nutshell analysis of time series (e.g. log data) help users understanding their data modeling of the data in order to detect anomalies, predict future values be of operational useful: real-time 5

6 Machine Learning News Big ML upgrade in 6.1 / 6.2 smarter job placement automatic job creation data visualizer population analysis job wizards on-demand forecasting scheduled events 6

7 ES Machine Learning Guided Tour What happens inside the black box What basic concepts do I need to know about? What is a model? How many are out there? From Detection to Projection: Forecasting 7

8 Creating an ML Job from a backstage perspective Data Buckets Data is aggregated into buckets 1 hour raw 10 minutes 8

9 Creating an ML Job from a backstage perspective Data Transformation Functions define how data is transformed (mean, count, sum,...) mean raw max 9

10 ML Model What is stored inside of a model self-contained artifact online approach (does not require (re-)access to the raw data) stores features (e.g. seasonality) as well as condensed historic information evolving: up-to-date to the last received bucket adapts to new data (up to complete re-learning) 10

11 Creating an ML Job from a backstage perspective Models Number of build models depend on detectors and data splits a detector defines fields function a data splits allow individual models per split data splits detectors 11

12 Creating an ML Job from a backstage perspective Models Number of build models depend on detectors and data splits data splits detectors 12

13 Creating an ML Job from a backstage perspective Models Number of build models depend on detectors and data splits data splits detectors 13

14 Creating an ML Job from a backstage perspective Models Number of build models depend on detectors and data splits data splits detectors 14

15 Creating an ML Job from a backstage perspective Models Number of build models depend on detectors and data splits data splits detectors 15

16 The Anatomy of a Model Subtitle What we model and why trend model residual model modelling anomalous periods dealing with change, dealing with outliers 16

17 ML Model for forecast From the past to the future (> 6.1) A ML model describes what is usual use existing models to project into the future (on-demand) provide a visualization of projection can be run at different points in time 17

18 ML Forecast From the past to the future Design goals should not interfere with real-time analysis, runs in parallel low resource usage multi-user, repeatable 18

19 ML Forecast Elasticsearch + X-pack Machine Learning Job Realtime Data Feed Detection Results Forecast Request Forecast Results Request for forecast Take a copy of corresponding models Continue real time processing Concurrently run forecast Forecast results get written back into the ML result index 19

20 Forecast challenges

21 Forecasting goal Accurate time series forecasting for a range of realistic data characteristics with minimum human intervention 21

22 Forecasting challenge: multiscale effects Forecast? Forecast? time 22

23 Forecasting challenge: change points change point 23

24 Time series modelling: handling change Controller Model 24

25 Time series modelling: handling change 25

26 Multiscale effects Reversion to behaviour on a given time frame typical Using one model, even with control of the time window, can t capture this Ensemble + adjust weights based on how far ahead to predict 26

27 Change points Change Detection (BIC) Change model H0 No change Change H1 Time shift m x 1{ change } H2 level shift 27

28 Change points 28

29 Change points 29

30 Change points Need explicit handling of change points Detect and model level shifts, time shifts, etc Roll out multiple possible realisations of the change model to forecast and use these to get expectation and confidence intervals 30

31 Summary Deterministic components of model; to forecast at time simply evaluate at time Maintain ensemble of trends for multiple time scales, i.e. Forecast using weighted average with weight a function of look ahead time, i.e. Detect and model probabilistically change points Roll out multiple possible realisations of the change model to forecast 31

32 Forecasting: the future Tell us your use case Alerting: When do I run out of supplies? Further scalability: Large Jobs with lots of data splits Quality assessment: How good was my forecast? Multivariate: Forecast group of metrics using correlations 32

33 More Questions? Visit us at the AMA 33

34 o

35 Please attribute Elastic with a link to elastic.co Except where otherwise noted, this work is licensed under Creative Commons and the double C in a circle are registered trademarks of Creative Commons in the United States and other countries. Third party marks and brands are the property of their respective holders. 35

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3. Mech. Phys. Solids, 1971, Vol. 19, pp. 389 to 394.
Pergamon Press.
Printed in Great
Britain.
THE A~P~ICA~I~ITY OF SLIP-LINE FIELD THEORY TO CONTAINED ELASTIC-PLASTTC FLOW AROUND A NOTCH By D. J. F. EWING and J. R. GRIFFITHS” Central Electricity Research Laboratories,
(Received
16th March
Leatherhead
1971)
SUMMARY RECENTcalculations by J. R. Griffiths and D. R. J. Owen (1971) on the growth of the elastic-plastic stresses for the plane strain bending of a V-notched bar reveal an interesting phenomenon: the stress maximum lies some way before the elastic-plastic interface, inside the plastic zone. Later calculations have confirmed this effect, for both work-hardening and perfectly-plastic von Mises and Tresca materials. At low applied loads the calculated stresses conflict with plastic slip-line field theory. This result is important, because it means that notch stresses before general yield cannot readily be deduced by etching up plastically-yielded zones. This paper explains the conflict analytically.
1. ZNTRO~XJ~TION GRIFFITHS and OWEN (197I)
have recently given finite-element method calculations of the plane strain elastic-plastic stress distribution in a notched bar loaded in pure bending. The bar geometry chosen was one commonly used in fracture tests. They assumed a slightly work-hardening material obeying the von Mises yield criterion. A
NOTCH
PROFILE
LIPLINE FIELD ASTIC-PLAST!C TERFACE
FIG. 1. Plastic zone at a notch root.
striking feature of their caIculations was the result that the tensile stress a, attained a maximum at a point Q some way within the plastic enclave, as in Fig. I. The phenomenon persisted when the calculations were extended to non-hardening materials, * Now at Department
of Materials Engineering, Monash University, Clayton, Victoria, Australia. 389
D. J. F.
390
EWING
andJ. R. GRIFFITHS
and in particular for a perfectly-plastic Tresca material (unpublished work of A. P. Kfouri, G. C. Nayak, D. R. J. Owen and J. R. Gri%ths). Figure I is drawn for this case (with the same bar geometry as in GRIFFII‘HSand OWEN(197 1): the applied load is O-58 times the theoretical yield-point Ioad. Along QR, the tensile stress decreases. This is surprising, especially for a perfectly-plastic Tresca material, because it conflicts with the predictions of perfectly-plastic plane strain slip-line field theory. According to slip-line theory, the stresses are statically determined (and dependent only on the local notch shape) in that part of the plastic region that is bounded by the slip-lines coming from the ends A,B of the plastically-stressed part of the notch surface (HILL, 1950, pp. 144 et seq., pp. 248 rl seq. ; PRAGERand HODGE,1951, p. 202).
FIG. 2.
aJ) =. 2k.
In Fig. 1, this region includes all of PQR. the whole of PQR to be given by slip-line ment until Q, where the stresses begin to (Fig. 2). The purpose of the present paper is to applicable in the whole of these contained 2. NOTCH STRESS
Thus, one would expect the stresses along field theory. In fact, there is good agreedecrease rather than continue to increase explain why slip-line theory is not in fact plastic regions.
FOR A TRESCA MAERIAL
The explanation of the phenomenon is particularly clearcut for a perfectly-plastic Tresca solid (i.e. one that yields when the difference between the maximum and minimum principal stresses reaches a constant critical value 2k). It is also for this material that the paradox is particularly surprising. Let (rl and (TVbe the in-plane principal stresses; pi. (TVcoincide with (t,. 0,. along PQR. The third principal stress is the through-thickness one, TT_.The condition for slip-line theory to apply is that c,-cr2 = 2k, (11 where k is a constant (the plane strain yield shear stress). Hencky’s equations then follow: these are just the result of combining (I) with the equations of in-plane equiIibrium (HILL, 1950). For a pe~ectly-plastic Tresca solid, (I) is valid in those
391
Slip-line field theory and plastic flow around a notch parts of the plastic
region
where 61 >u,>c2.
(2a)
Geometrically, if we regard 6, as given, c1 and ~~ must lie along of the ‘Tresca hexagon’ drawn in principal stress space (Fig. 2). stress-point lying between A and B:
the segment AB Consider first a
0, > oz > 02.
(2b)
The plastic strain-increments have the same principal axes as the stresses, and so they can be represented in the same diagram by a vector dep. Assuming, as usual, the ‘normality’ flow rule, dtzp is a vector normal to AB and so d&y = - d&z> 0. So, by plastic incompressibility, the third principal strain-increment de: vanishes. But, since we assume plane strain, the total strain-increment ds, vanishes; so the elastic part d&zmust also vanish. Thus, so long as 0; remains strictly between c1 and c2 (and has so remained throughout the loading at the point in question), the total elastic and plastic strains in the z-direction must vanish. So, by Hooke’s Law, or = v(a1 +a,)
= v(a,+a,>
in those parts of the plastic region where (2b) holds. Now consider the stresses along PQR. If slip-line
(3) field theory applies,
6, = 2k( 1+ 0).
then (4)
0 being the slip-line angle turned through. Equations (2a) (3) (4) are consistent only up to some point (say) P’ along PQ, at which oZ has become equal to aY, so that a, = 2k( 1- v)/( 1- 2v),
I$ = a, = 2/U/(1 - 2v).
(5)
For a circular root, the slip-lines are logarithmic spirals (HILL, 1950, p. 248), and P’ occurs at a distance of [- 1 +exp {v/(1 -2v))] times the root radius from the notch root, e.g. 0.89 root radii when v = O-28. Immediately beyond P’, a, remains equal to 0,. We have reached, and remain at, the vertex B of the Tresca yield locus (Fig. 2). So the vector representing the total plastic strain &p (i.e. the sum of all the previous strain-increments vectors d&P) is no longer constrained to be normal to AB; instead it can lie anywhere in the ‘wedge’ of
DISTANCE FIG. 3.
BELOW
NOTCH-
(xl
Stress distribution below notch.
r)
392
D. J. F. EWINGand J. R. GRIFFITHS
normals through B. For all such previous increments ds; 2 --d&e > 0 so that de: G 0, i.e. E: < 0. Plastic slip is now taking place through the thickness, as well as in the xy-plane. The through-thickness plastic component sets itself equal and opposite to the elastic component, so preserving plane strain, i.e. a, = 0. Thus, beyond P’ on PQ in Fig. 1 the stresses are as sketched in Fig. 3, with cr, rising steadily, being given as before by slip-line field theory, so long as try and 6, remain equal. But in the elastic region and in particular at R on the elastic-plastic interface cfZ is again as given by (3). This implies that cr, drops below oY at some stage before the plastic-elastic interface is reached. Q represents the transition-point, beyond which (1) breaks down. Up to Q along PQR, there has been some plastic strain in the ydirection; but between Q and R all plastic strain is in the xz-pIane. The explanation of the conflict with slip-tine theory is now clear. Only in some sub-enclave AQB does (1) hold (Fig. 4). In the plastic region outside this enclave, sfip-line field theory
ELASTIC-PLA
Fro. 4. The inner enclave in which o1 - o2 = 2k (qualitative only).
fails because a, < ~z_ The position of Q (and so the stress maximum) cannot be predicted analytically. unfortunately, it would be very difficult to determine the position of Q experimentally by etching. Etching normally reveals the whole of the plasticallyyielded zone, without discriminating between directions of slip. First, slip-line theory is exact within Finally, we note two theoretical points. our sub-enclave AQB only because we have assumed the ‘normality’ flow rule for the This flow rule gives the through-thickness plastic straining of our Tresca material. plastic strain enough freedom to preserve plane strain while permitting the stresses to satisfy (I). Had we instead used the Levy-Mises flow rule, equation (I) would be incompatible with plane straining, and so slip-line field theory would have held only approximately, under the same kind of limitations as for the von Mises material discussed below. Secondly, as the applied loading is increased towards general yield, the sub-region AQB expands to contain more and more of those regions that would In such regions, the elastic-plastic deform at yield-point in a rigid-plastic analysis. stresses are converging (not necessarily uniformly) to their rigid-plastic theory values, if work-hardening and geol~etry-change effects are neglected (HILL, 1951, 1956). Convergence is in the sense that the stresses cannot finally settle down to other vaiues. This convergence principle provides a further check on the validity of the elasticplastic calculations. Unfortunately the check is not a strong one, because of the nonuniformity of the stress convergence. However, the applied couple should converge monotonically to its theoretical yield-point value as the strains increase: this provides
Slip-line field theory and plastic flow around a notch
a rigorous check. The relevant rigid-plastic solution here is calculated by EWING (1968) as a special solution.
393
for the bar geometry in question case of GREEN’S (1953) general
3. NOTCH STRESSESFOR A VON MISES MATERIAL The situation for a perfectly-plastic von Mises material is less clearcut, but is qualitatively similar: cz is again the intermediate principal stress only within a subenclave of the plastic region. So there is no reason to reject the finite-element calculations when they show a maximum stress below that expected from slip-line theory at loads below general yield. In any case, in contrast to the Tresca material, slip-line field theory is exact for a von Mises material only for an elastically incompressible material (with v = 4) or in the limit of large plastic as compared to elastic strains (HILL, 1950). To summarize the argument : with the von Mises material the appropriate ‘normality’ flow rule is the LCvy-Mises one, so that the plastic strain-increment de: is proportional to the out-of-plane stress deviator (I=- f(ol + cz + a,); but E: remains of elastic order because of the plane strain, while the in-plane plastic strains EP, ~5 are not restricted; so oz converges towards &(ol +a,) as the in-plane plastic strains increase. And oz = +(crl + c2) is the necessary and sufficient condition for (1) to be compatible with the von Mises yield condition: (al-a2j2+(02-~j)2+(cr3-~1)2
= 6k2
with u3 = a, in plane strain. Now, at the elastic-plastic interface, (3) must hold; so slip-line field theory certainly fails (for v # 3) near the elastic-plastic interface. Elsewhere it holds increasingly well as the in-plane plastic strains increase (again in agreement with Hill’s convergence principle). 4. CONCLUSIONS (i) We have explained analytically the Griffiths-Owen observation that slip-line field theory applies only in part of the plastic region and that the maximum tensile stress is attained inside the plastic zone surrounding the notch, not at its edge. The reason is essentially that, near the elastic-plastic interface, through-thickness plastic strains develop comparable to the in-plane ones. Before general yield the stress maximum cannot be predicted analytically: it is sensitive to the value of Poisson’s ratio at low applied loads. (ii) This result is important, because it means that a popular way of estimating plastic stresses experimentally-by etching a cross-section to see the extent of the plastic zone, and then applying slip-line theory to deduce the plastic stresses-is not applicable without qualification.
ACKNOWLEDGMENTS The writers are indebted to G. C. Nayak for permission to refer to his unpublished calculations upon which are based Figures 1, 3 and 4. The work was carried out at the Central Electricity Research Laboratories and the paper is published by permission of the Central Electricity Generating Board. 27
394
D. J. F. EWING and J. R. GRIFFITHS REFERENCES
EWING, D. J. F. GREEN, A. P. GRIFFITHS,J. R. and OWEN, D. R. J. HILL, R.
1968 1953 1971 1950
PRAGER,W. and HODGE, P. G., JR.
1951 1956 1951
J. Mech. Phys. Solids 16, 205. Qu. J. Mech. Appl. Math. 6, 223. J. Mech. Phys. Solids 19, 419. The Mathematical Theory of Plasticity. Oxford University Press. Phil. Msg. 42, 868. J. Mech. Phys. Solids 5, 66. Theory of Perfectly-Plastic Solids. Wiley, New York.


BOOK EXCERPT:

Habits That Bend Don't Break Why do so many sincere attempts to build good habits fail? We try our best to be consistent, but some days are better than others. Inevitably, we fail when 'life happens,' because each day we try to hit the same targets regardless of the situation. How, then, can we make our habits more resilient to the turbulence of life? By making them elastic.Most people associate 'elastic' with yoga pants and rubber bands. But the word also means 'resilient' - the ability to withstand pressure. Elastic materials are far more durable than rigid and brittle ones, which will shatter under the slightest pressure. The same is true for habits.Traditional habits are unchanging: the same behavior is done at the same time to the same level every day. They work well until the pressures of modern life break their rigid and brittle shell. Elastic habits are fluid: they can change their form and intensity to suit each unique day. They survive busy, tired, bad days. They thrive in better days. If you're tired of the repetitive and exhausting grind to develop good habits, it's time give your habits the refreshing superpower of elasticity. Read Elastic Habits now, and you'll soon discover the life-changing difference of good habits that adapt to your day.

Product Details :

Genre: Health & Fitness
Author: Stephen Guise
Publisher: Selective Entertainment LLC
Release: 2019-11-20
File: 238 Pages
ISBN-13: 0996435476

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