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Yield and Plastic Flow by the Numbers

Dislocations, tensile tests, work hardening, yield criteria, proof tests, bending, thread rolling and residual stress, worked through one hydraulic cylinder

How metals yield and flow and how to use it: elastic constants, tensile tests, true stress and work hardening, hardness, von Mises and Tresca, proof tests, bending, rolling and residual stress.

10 chapters · About 100 minutes · Free to read · Intermediate mechanical and manufacturing engineering

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KEVOS® LibraryENGINEERINGYield andPlastic Flow bythe NumbersKEVOS® ORIGINAL BOOK10 CHAPTERS

About this book

Every metal part is designed around one number more than any other: its yield strength, the stress at which it stops springing back and starts to take a permanent set. Designers keep parts below it; fabricators deliberately go past it to bend, roll and form; test engineers prove parts against it; and failure investigators ask whether it was exceeded. Yet the physics of yielding, why it happens where it does and what happens after it, is often left to metallurgists.

This book explains yield and plastic flow from the crystal up and turns them into numbers that engineers use every day. It starts with the elastic constants and the dislocations that make metals yield at a tiny fraction of their theoretical strength, then reads a tensile test, converts it to true stress and strain, and describes work hardening. It relates hardness and grain size to strength, checks parts under combined stresses, finds the pressures at which a thick tube first yields and finally collapses, and then turns to deliberate plastic flow in bending, tube bending and thread rolling. It ends with thermal restraint, residual stress and annealing.

It is for mechanical and manufacturing engineers, designers, fabricators, and test and quality staff. It assumes school physics, algebra and a first course in stress analysis. Each chapter works the running case with real numbers, includes one plausible mistake and how it was found, links the KEVOS® tools that do the chapter’s calculations, and ends with an exercise and its worked answer.

The running case

Every chapter follows one product. This is an illustration. A hydraulic cylinder builder in Bayswater, Western Australia, makes the HC-63, a 63 mm bore cylinder for mobile mining equipment.

  • Duty: 21 MPa working pressure, proof tested at 31.5 MPa, 1.5 times working.
  • Barrel: cold-drawn, stress-relieved and honed steel tube, 63 mm bore and 76 mm outside diameter, with a mill certificate yield strength of 520 MPa.
  • Rod: 36 mm, from 0.45% carbon steel bar, with an M27 × 2 rolled thread at its end.
  • Fabricated parts: 10 mm grade 350 steel plate mounting brackets bent on a press brake, and 16 × 2 mm cold-drawn seamless steel tube for the hydraulic lines.
  • Test bench: a pressure test bench with an intensifier, an oil heater and a frame tied by 25 mm steel bars.

Strengths, test results, loads, pressures, prices and times are the case builder’s working figures or are calculated from stated assumptions. They do not describe KEVOS® work or any real business, product or supplier’s data.

How to use the book

Read Chapters 1 to 3 first: they set out the elastic and plastic behaviour that every later chapter uses. Chapter 4 relates hardness and grain size to strength. Chapters 5 and 6 deal with yield under combined stress and the behaviour of a pressurised tube up to collapse. Chapters 7 to 9 cover deliberate plastic flow in bending, tube bending and thread rolling, and Chapter 10 covers thermal restraint, residual stress and annealing.

Every calculation in the book was checked by running it, and the KEVOS® tools linked from each chapter were checked against the calculations they perform. The formulas are sound first estimates for ductile steels; mill certificates, tests and trial pieces are the final word on real material.

Source limits and professional responsibility

This book is educational. Its methods give first estimates of yield, collapse, springback, forming limits and thermal stress under the stated assumptions; they do not replace the design standards that apply to pressure equipment, structures or lifting gear, the material supplier’s data, or the judgement of a competent engineer. Hydraulic cylinders, pressure test rigs and presses store large amounts of energy and must be designed, tested and operated under a safe system of work as the model WHS laws require. Decisions about real parts and equipment remain with the competent people responsible for them.

Who this book is for

Mechanical and manufacturing engineers, designers, fabricators, test and quality staff who specify materials, read mill certificates, check parts against yield, or bend, roll and form metal.

What you will learn

  • Explain yield and plastic flow through the movement of dislocations, and why real metals yield far below their theoretical strength.
  • Evaluate a tensile test, convert it to true stress and strain, and describe work hardening with a flow curve.
  • Relate hardness and grain size to strength, and check a part under combined stresses by von Mises and Tresca.
  • Find the first-yield and collapse pressures of a thick tube, and judge a part after an overload.
  • Plan bends, tube bends and rolled threads within the material's ductility, and manage thermal restraint and residual stress.

Inside this book

Chapters

  1. 01Elastic, then plasticHow a metal stretches elastically, why it yields at a tiny fraction of its theoretical strength, and how dislocations carry plastic flow and make metals harden as they deform.
  2. 02Reading a tensile testWhat a tensile test measures and how to evaluate one: yield by offset or yield point, tensile strength, elongation, reduction of area, modulus and true fracture values, and when a result is invalid.
  3. 03True stress, true strain and work hardeningWhy engineering stress misleads after yield, how true stress and strain and the flow curve describe work hardening, when necking starts, and what cold drawing does to the HC-63's barrel.
  4. 04Hardness and grain sizeWhat a hardness test measures and how far it predicts strength, how grain size raises yield strength by the Hall–Petch relation, and why a coarse-grained heat passed a hardness check.
  5. 05Yield under combined stressWhy shear, not pressure, makes metals yield, how the Tresca and von Mises criteria compare, and how to check the HC-63 barrel's bore without dropping the radial stress.
  6. 06First yield, collapse and proof testsHow a thick tube goes from first yield at its bore to plastic collapse, what a proof test proves, and how to judge a barrel after a test bench overpressure.
  7. 07Bending plate past yieldWhat happens through the thickness of a bend, the tightest radius a plate takes without cracking, how much it springs back, and why a punch change and a nesting change cracked a batch of brackets.
  8. 08Bending tubeHow a tube wall thins on the outside and thickens on the inside of a bend, how to check strain and wall against limits, and why a check with the nominal wall passed tubes that came out too thin.
  9. 09Rolling threads: forming instead of cuttingHow thread rolling forms a thread by plastic flow, why the blank is sized near the pitch diameter, how rolls and starts are chosen, and what rolling does for the thread's strength.
  10. 10Restraint, residual stress and annealingHow restrained thermal strain makes stress, why welds always leave yield-level residual stress, how a heated bar can buckle, and how stress relief and annealing undo plastic flow.

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.

  • H. Kleinert, World Scientific, Singapore. Gauge Fields in Condensed Matter, Volume II: Stresses and Defects (Differential Geometry, Crystal Melting) (1989). In copyright. Used only as a map for the physics behind Chapters 1, 3 and 10: crystals as ordered lattices, displacement and strain, dislocations as the line defects that carry plastic flow, their interactions, and the extension of elasticity to defects. No text, equations, figures or problems are reproduced, and all cases and calculations are KEVOS® originals.
  • D. Hull and D. J. Bacon, Butterworth-Heinemann. Introduction to Dislocations, fifth edition (2011). Background for the theoretical shear strength, dislocation movement, the Orowan relation and work hardening by dislocation interaction in Chapters 1 and 3.
  • G. E. Dieter, McGraw-Hill. Mechanical Metallurgy, SI metric edition (1988). Background for the tensile test, true stress and strain, the flow curve, necking, yield criteria, strain ageing and the Bauschinger effect in Chapters 2 to 5.
  • R. Hill, Clarendon Press, Oxford. The Mathematical Theory of Plasticity (1950). Background for the elastic-plastic thick-walled tube under internal pressure, its first-yield and collapse pressures and its plastic zone, in Chapter 6.
  • N. J. Petch, Journal of the Iron and Steel Institute, volume 174, pages 25 to 28. The cleavage strength of polycrystals (1953). One of the two original sources of the grain size relation used in Chapter 4, with E. O. Hall's 1951 paper in the Proceedings of the Physical Society.
  • International Organization for Standardization. ISO 6892-1 Metallic materials — Tensile testing — Part 1: Method of test at room temperature (2019). Named by number and scope only in Chapter 2. The specimen size and acceptance minimums in the case are the case builder's purchase specification, not values quoted from the standard.
  • International Organization for Standardization. ISO 6506-1 Metallic materials — Brinell hardness test — Part 1: Test method (2014). Named by number and scope only in Chapter 4, as the method behind the Brinell validity rules that the KEVOS® tool applies.
  • ASTM International. ASTM E112 Standard Test Methods for Determining Average Grain Size (2013). Named by number and scope only in Chapter 4, as the definition of the grain size number that the KEVOS® tool calculates.
  • Safe Work Australia. Model Code of Practice: Managing the risks of plant in the workplace (2022). Named in Chapters 1 and 6 for the duty to manage the risks of plant, including pressure equipment, test rigs and presses, under the model WHS laws.

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