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Microstructures by the Numbers

Measuring phases, grains, nodules, pores and inclusions from polished sections, and turning the counts into sound acceptance decisions

How to get trustworthy numbers from a polished section: phase fractions, grain size, particle sizes, porosity, inclusions and microhardness, with their errors.

11 chapters · About 110 minutes · Free to read · Intermediate materials and quality engineering

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KEVOS® LibraryENGINEERINGMicrostructuresby the NumbersKEVOS® ORIGINAL BOOK11 CHAPTERS

About this book

A metallographic report turns a polished, etched slice of metal into numbers: a pearlite fraction, a grain size number, a nodule count, a porosity percentage, an inclusion rating, a hardness. Those numbers decide whether a casting is accepted, whether a heat treatment worked and whether a failure was the material’s fault. Yet each one comes from a flat section through a three-dimensional structure, from a handful of microscope fields chosen by a person, and from counting rules that change the answer if they change.

This book teaches the arithmetic that lies behind those numbers, which is mostly stereology and counting statistics, so that a reader can ask for the right measurement, judge how precise it is and act on it sensibly. It starts with what a section shows and how to sample a part. It then works through phase fractions by point counting, grain size by counts and intercepts, and the correction of particle sizes seen in sections. It applies these to graphite shape in ductile iron, to porosity, to non-metallic inclusions and to microhardness traverses, and ends with the uncertainty of the whole chain and the decisions made from it.

It is for mechanical, materials and quality engineers, laboratory technicians and supervisors, and the buyers who order castings and forgings to a microstructure requirement. It assumes algebra, logarithms, a little probability and the vocabulary of steels and cast irons. Specimen preparation and etching are described only where they change the numbers. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, and ends with an exercise and its worked answer.

The running case

Every chapter follows one fictional laboratory. This is an illustration. Plant P makes centrifugal pumps and runs a small metallography laboratory with one technician, a metallurgical microscope with a camera and image analysis software, a Brinell tester and a microhardness tester. The laboratory checks two purchased parts.

  • The pump casing, a ductile (spheroidal graphite) iron casting ordered to grade EN-GJS-500-7, with a 20 mm wall and a 32 mm sealing flange. The plant’s drawing adds microstructure requirements: nodularity at least 80%, at least 100 nodules per mm², pearlite no more than 40% of the matrix, porosity on the sealing face below a stated limit, and a hardness of 170 to 230 HBW.
  • The pump shaft, a quenched and tempered 42CrMo4 (AS 1444 grade 4140) steel bar, 40 mm in diameter, with an induction-hardened bearing journal. The drawing requires a prior austenite grain size of ASTM 5 or finer, a stated inclusion cleanliness, and an effective hardened depth on the journal.

The story begins when a batch of casings leaks through the sealing flange on the plant’s pressure test, and a shaft from a new supplier fails its journal hardness check. The laboratory is asked whether the material is at fault.

The plant, its suppliers and every measurement and result are invented. None describes KEVOS® work, a real company or a real supplier’s product. The drawing requirements are illustrative choices, not values taken from any standard.

How to use the book

Read Chapters 1 to 3 in order: they set out what a section shows, how to sample and the point-counting method that later chapters build on. Chapters 4 and 5 cover grain size. Chapters 6 to 9 deal with particles, pores and inclusions, where the sectioning effect matters most. Chapter 10 covers hardness traverses, and Chapter 11 the uncertainty of microstructure numbers and the decisions made from them.

KEVOS® tools are linked where they perform a step of the method, with a note of exactly what each one calculates and where its model stops. Where a tool solves a related problem rather than the chapter’s own, the text says so.

Source limits and professional responsibility

The relations used here are the established ones of quantitative metallography and stereology: Delesse’s principle, the planimetric and intercept grain size relations, Wicksell’s corpuscle relations and extreme-value statistics. Each chapter states the assumptions they need, chiefly that the structure is sampled without bias and, for some relations, that particles are spheres. Test standards are named by number and scope only; their procedures, charts and acceptance values govern real testing and are not reproduced.

Metallographic results feed decisions about pressure-containing castings, rotating shafts and failure investigations, some of them safety-critical. Real acceptance, failure analysis and specification work are matters for competent, accredited laboratories and qualified engineers working to the applicable standards and contracts. This book is educational. It explains how the numbers are made and how far to trust them; it is not a test procedure, a specification or a substitute for accredited testing.

Who this book is for

Mechanical, materials and quality engineers, laboratory technicians and supervisors, and buyers of castings and forgings who order, read or act on metallographic reports.

What you will learn

  • Explain what a plane section can and cannot show about a three-dimensional structure, and choose where to cut and how many fields to measure.
  • Estimate a phase fraction by point counting and state its standard error and the number of fields needed.
  • Measure grain size by the planimetric and intercept methods, convert it to an ASTM grain size number and judge field-to-field spread.
  • Correct particle and pore sizes seen in sections for the sectioning effect, and relate nodule count per area to number per volume.
  • Use porosity, inclusion and hardness measurements with their uncertainties to accept or reject material, and compare two laboratories fairly.

Inside this book

Chapters

  1. 01What a section showsHow a flat polished section samples a three-dimensional structure, which quantities it shows without bias, which it distorts, and why one worst-looking field misleads.
  2. 02Sampling and preparing sectionsWhere to cut a casting or bar, which orientation to section, how to place microscope fields without bias, and which preparation faults change the numbers.
  3. 03Phase fractions by point countingEstimating the volume fraction of a phase by counting grid points, its standard error from field-to-field scatter, the number of fields needed, and image analysis thresholds.
  4. 04Grain size by counts and interceptsMeasuring grain size by counting grains in a known area or boundaries crossed by test lines, converting both to the ASTM grain size number, and the precision of each.
  5. 05Grain size acceptance and spreadTurning several field readings into one grain size result, averaging correctly, judging field-to-field spread, recognising duplex structures and writing a defensible acceptance decision.
  6. 06Particle sizes from sectionsWhy sections of particles look smaller and over-represent large ones, how to get particle size and number per volume from sections of spheres, and why nodule counts depend on a size cutoff.
  7. 07Graphite shape and nodularityDescribing particle shape with circularity and roundness, measuring nodularity in ductile iron, how section thickness affects it, and checking casting hardness with valid Brinell tests.
  8. 08Porosity in sectionsMeasuring casting porosity by area fraction and by density, telling shrinkage from gas pores, why sections cannot show whether pores connect, and tracing porosity to feeding.
  9. 09Inclusions and the largest oneRating non-metallic inclusions on the right section, why their amount matters less than their largest size, estimating the largest inclusion by extreme values, and inclusions in tensile tests.
  10. 10Hardness traverses and microhardnessMeasuring a hardened case with a microhardness traverse, keeping indentations valid, finding the effective hardened depth, and how diagonal measurement uncertainty limits the result.
  11. 11From microstructure to decisionThe uncertainty of a microstructure result from sample to report, comparing laboratories and batches fairly, avoiding pseudo-replication, decision rules near limits, and writing specifications.

Apply This in Practice

Use these KEVOS® tools to try the method from this reading with your own numbers.

References and source notes

The manuscript and worked examples are original KEVOS® writing. References inform particular concepts; their authors do not endorse this book. Web references were consulted on 31 October 2026.

  • K. R. Mecke and D. Stoyan (eds), Lecture Notes in Physics 554, Springer, Berlin (ISBN 3 540 67750 X). Statistical Physics and Spatial Statistics: The Art of Analyzing and Modeling Spatial Structures and Pattern Formation (2000). Supplied source, in copyright. Read only as a subject map: its contents and preface name stereology, the Wicksell corpuscle problem, porous media and spatial statistics as shared ground between physics and statistics. No text, equations, figures, examples or exercises are reproduced or adapted; the metallographic treatment, running case and numbers here are original.
  • S. D. Wicksell, Biometrika, vol. 17, pp. 84–99. The corpuscle problem. A mathematical study of a biometric problem (1925). The original statement of the problem of recovering sphere sizes from their plane sections, used in Chapters 6 and 8.
  • E. E. Underwood, Addison-Wesley, Reading, Massachusetts. Quantitative Stereology (1970). Standard text for the basic stereological relations used in Chapters 1, 3, 4 and 6: point, line and area fractions, and the relation between number per area and number per volume.
  • J. C. Russ and R. T. DeHoff, Kluwer Academic/Plenum, New York. Practical Stereology, second edition (2000). Background for sampling design, systematic field selection and the precision of counting methods in Chapters 2 and 3.
  • A. Baddeley and E. B. Vedel Jensen, Chapman & Hall/CRC, Boca Raton. Stereology for Statisticians (2005). Background for the unbiased sampling principles in Chapter 2: random position, systematic sampling and the need to sample orientation for anisotropic structures.
  • E. O. Hall, Proceedings of the Physical Society, Section B, vol. 64, pp. 747–753. The deformation and ageing of mild steel: III Discussion of results (1951). One of the two original papers on the grain size dependence of yield strength, named in Chapter 11. No constants are taken from it.
  • Y. Murakami, Elsevier, Oxford. Metal Fatigue: Effects of Small Defects and Nonmetallic Inclusions (2002). Source of the idea, used in Chapter 9, of characterising an inclusion or defect by the square root of its projected area and estimating the largest one in a volume by extreme-value statistics. No data or formulae for fatigue strength are reproduced.
  • E. J. Gumbel, Columbia University Press, New York. Statistics of Extremes (1958). The Gumbel distribution and return period used for the largest inclusion in Chapter 9.
  • Joint Committee for Guides in Metrology (BIPM). Evaluation of measurement data: Guide to the expression of uncertainty in measurement (JCGM 100:2008) (2008). The method of combining standard uncertainties used for hardness and image measurements in Chapters 10 and 11.
  • ASTM International. ASTM E112, Standard Test Methods for Determining Average Grain Size (current edition). Named by number and scope in Chapters 4 and 5. The relations between grain size number, grains per area and mean intercept used here are those also implemented in the KEVOS® grain size tools; the standard's procedures and tables are not reproduced.
  • ASTM International. ASTM E562 (point counting), ASTM E1382 (grain size by image analysis) and ASTM E1181 (duplex grain sizes) (current editions). Named by number and scope only in Chapters 3 to 5. Nothing is quoted.
  • International Organization for Standardization. ISO 945-1 (classification of graphite in cast irons) and ISO 1083 (spheroidal graphite cast irons) (current editions). Named by number and scope only in Chapter 7. The nodularity rules and grade requirements of the running case are illustrative, not values from these standards.
  • ASTM International; International Organization for Standardization. ASTM E45 and ISO 4967 (non-metallic inclusion content of steel) (current editions). Named by number and scope only in Chapter 9 as the chart-comparison methods of rating inclusions. Nothing is quoted.
  • International Organization for Standardization; ASTM International. ISO 6506-1 (Brinell) and ISO 6507-1 (Vickers) hardness test methods, and ASTM E384 (microindentation hardness) (current editions). Named by number and scope only in Chapters 7 and 10. The validity rules used are those entered in the KEVOS® hardness tools, chosen by the user.

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