Metallic materials — Principles and designs for multiaxial fatigue testing

ISO/TR 12112:2018 discusses the general principles of multiaxial fatigue testing and the design recommendations for specific classes of multiaxial testing machines and test specimens.

Matériaux métalliques — Principes et conceptions associés aux essais de fatigue multiaxiale

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REPORT 12112
First edition
Metallic materials — Principles and
designs for multiaxial fatigue testing
Matériaux métalliques — Principes et conceptions associés aux essais
de fatigue multiaxiale
Reference number
ISO/TR 12112:2018(E)
ISO 2018

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ISO/TR 12112:2018(E)

© ISO 2018
All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may
be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting
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Phone: +41 22 749 01 11
Fax: +41 22 749 09 47
Published in Switzerland
ii © ISO 2018 – All rights reserved

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ISO/TR 12112:2018(E)

Contents Page
Foreword .v
Introduction .vi
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 General principles . 2
4.1 Methodology . 2
4.2 Historical development . 2
4.3 Specific multiaxial test methods . 6
4.3.1 Bending + torsion . . 6
4.3.2 Axial + torsion . 6
4.3.3 Axial + internal pressure . 6
4.3.4 Axial + internal + external pressure . 6
4.3.5 Axial + internal + external pressure + torsion . 6
4.3.6 Cruciform — LCF . 6
4.3.7 Cruciform — Crack growth . 7
4.4 Multiaxial fatigue analysis . 7
4.4.1 Computer aided design . 7
4.4.2 Fatigue life prediction . . 7
4.5 Multiaxial fatigue failure criteria . 8
5 Axial + torsion testing systems and specimen design. 9
5.1 Historical development . 9
5.2 Specimen design .11
5.2.1 Design considerations .11
5.2.2 Design recommendations .11
5.2.3 Comparison with ASTM E2207 .11
5.3 Machine design .12
5.3.1 Frame .12
5.3.2 Loadcells .12
5.3.3 Strain measurement.12
5.3.4 Control .12
5.3.5 Data acquisition .12
5.3.6 Software.12
6 Cruciform testing systems and specimen design .13
6.1 Historical development .13
6.2 Specimen design .13
6.3 Machine design .14
6.3.1 Frame .14
6.3.2 Loadcells .14
6.3.3 Strain measurement.15
6.3.4 Crack growth monitoring .15
6.3.5 Control .15
6.3.6 Data acquisition .15
6.3.7 Software.15
7 Axial + differential pressure systems and specimen design .16
7.1 Historical development .16
7.2 Specimen design .19
7.2.1 Design considerations .19
7.2.2 Design recommendations .19
7.2.3 Axial stress due to pressure .20
7.3 Machine design .20
7.3.1 Frame .20
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7.3.2 Pressure containment .20
7.3.3 Differential pressure .20
7.3.4 Force measurement .20
7.3.5 Pressure measurement .20
7.3.6 Strain measurement.20
7.3.7 Control .20
7.3.8 Data acquisition .20
7.3.9 Software.21
Annex A Historical analysis of specimen geometry .22
Bibliography .29
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ISO/TR 12112:2018(E)

ISO (the International Organization for Standardization) is a worldwide federation of national standards
bodies (ISO member bodies). The work of preparing International Standards is normally carried out
through ISO technical committees. Each member body interested in a subject for which a technical
committee has been established has the right to be represented on that committee. International
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ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of
electrotechnical standardization.
The procedures used to develop this document and those intended for its further maintenance are
described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the
different types of ISO documents should be noted. This document was drafted in accordance with the
editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).
Attention is drawn to the possibility that some of the elements of this document may be the subject of
patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of
any patent rights identified during the development of the document will be in the Introduction and/or
on the ISO list of patent declarations received (see www .iso .org/patents).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation on the voluntary nature of standards, the meaning of ISO specific terms and
expressions related to conformity assessment, as well as information about ISO's adherence to the
World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see the following
URL: www .iso .org/iso/foreword .html.
This document was prepared by Technical Committee ISO/TC 164, Mechanical testing of metals,
Subcommittee SC 4, Fatigue, fracture and toughness testing.
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ISO/TR 12112:2018(E)

Structural components in industry are frequently subject to some form of multiaxial stressing. Fatigue
cracks generally initiate from surface defects or discontinuities and are thus primarily influenced by
the surface biaxial stress system. This can vary from equibiaxial, where surface principal stresses are
equal in magnitude and sign (present under conditions of pressurization, rotation and thermal loading)
to pure shear where surface stresses are equal in magnitude, opposite in sign (as in shafts and shear
The majority of fatigue test data gathered worldwide have been and will continue to be under uniaxial
conditions for reasons of simplicity and cost. A secondary goal of multiaxial testing is therefore to
develop behavioural models which relate failure under specified multiaxial conditions to established
uniaxial cases.
This document utilizes data gathered from the past 80 years spanning most multiaxial fatigue research.
It can be of interest to new researchers in the field and form a basis for full International Standards as
the need arises.
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Metallic materials — Principles and designs for multiaxial
fatigue testing
1 Scope
This document discusses the general principles of multiaxial fatigue testing and the design
recommendations for specific classes of multiaxial testing machines and test specimens.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminological databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https: //www .iso .org/obp
— IEC Electropedia: available at http: //www .electropedia .org/
biaxial strain ratio
ratio of the surface principal strains, smaller/larger
biaxial stress ratio
ratio of the surface principal stresses, smaller/larger
principal strains
ε > ε > ε
1 2 3
principal direct strains at a point in a multiaxial strain field
principal stresses
σ > σ > σ
1 2 3
principal direct stresses at a point in a multiaxial strain field
Poisson’s ratio
negative ratio of transverse to longitudinal strain under uniaxial tensile stressing
specimen diameter
diameter of a cylindrical tubular specimen
Note 1 to entry: The symbols d , d and d are used to express outside, inside and mean diameters, respectively.
0 i m
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parallel length
parallel length of a cylindrical tubular specimen
fillet radius
fillet radius of a cylindrical tubular specimen
directional suffix
suffix identifying a direction in a cylindrical tubular specimen
Note 1 to entry: The suffixes z, r and θ are used to express axial, radial and circumferential directions,
strain component suffix
suffix identifying a strain component
Note 1 to entry: The suffixes e, p and t are used for elastic, plastic and total strain components, respectively.
internal pressure
internal pressure within a cylindrical tubular specimen
4 General principles
4.1 Methodology
Multi-axial fatigue testing sets out to simulate the dynamic stress-strain conditions at key locations on
components, on test specimens of constant geometry for a given test series, and to determine the cyclic
stress-strain history, crack initiation and propagation behaviour, fatigue life and failure mode.
Dependent on the level of geometric constraint in the real component, it can be more useful to
test specimens under stress or strain control, e.g. a test specimen representative of a relatively
unconstrained gas turbine blade can be tested in stress control whereas it can be more relevant to
utilize strain control for a test specimen simulating part of a steam turbine disc subject to thermal
straining during start-up.
Further, where stress amplitudes are sufficient to take test specimen materials well into the region
of cyclic plasticity (LCF), it can be preferable to employ strain control in order to better control cyclic
amplitude during the test and failure at end of test.
4.2 Historical development
Multiaxial fatigue has been addressed since the 1930s. Initially, testing machine and specimen designs
were created to address specific biaxial stress conditions, e.g. torsion, bending + torsion, cantilever
bending, anticlastic bending and plate pressurization. However, a criticism of much of the early research
was that specimen design had to change in order to change the biaxial stress or strain ratio, leading to
uncertainty in the interpretation of results.
The benefit of being able to test a single specimen design over a wide range of biaxiality led to the
choice of two generic specimen types, tubular and cruciform, together with associated multi-axis
testing machine designs.
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Table 1 summarizes the attributes of the different test methods applicable to tubular and plate
Biaxiality is shown in terms of the range of surface strains with ε held constant. Only cruciforms
and systems employing axial force plus internal and external pressure are capable of applying fully
reversed fatigue cycles over the full range of biaxiality (−1 ≤ ϕ ≤ +1) to test specimens.
Buckling is a key concern in the design of effective LCF specimens.
A reasonable gauge area of essentially constant strain is beneficial.
Ideally, strain should be constant through the thickness.
If all the applied forces are carried by the gauge area, then all stresses and strains can be determined;
otherwise, only total (not plastic) strains can be measured.
The ability to visually observe the specimen is useful especially for surface crack monitoring.
Some designs are suitable for high temperature and thermo-mechanical fatigue (TMF) testing.
Systems involving torsion cause the principal axes to rotate up to 45°.
System cost can be scaled by the number of actuators, and therefore closed servo-loops, in the design.

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4 © ISO 2018 – All rights reserved
Table 1 — Multiaxial test methods for tubular and plate specimens
Biaxial Range of Single Immune Invariant Min. Moni- Speci- Crack High tem- TMF Rota- No. of
specimen surface geome- to buck- σ and ε ε-gra- toring men ob- growth perature studies tion of actuators
schemat- principal try ling on gauge dient biaxial σ servation studies capability principal propor-
ics and strains area through and ε stresses tional to
modes of thickness cost
√ √ √ √ √ 1
+ torsion
√ √ √ √ √ √ √ √ √ 2
+ torsion
√ √ √ √ √ √ 2
+ P
+ P
+ con- √ √ √ √ √ 2
+ P
+ P
√ √ √ √ √ 4
+ P
+ torsion

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Table 1 (continued)
Biaxial Range of Single Immune Invariant Min. Moni- Speci- Crack High tem- TMF Rota- No. of
specimen surface geome- to buck- σ and ε ε-gra- toring men ob- growth perature studies tion of actuators
schemat- principal try ling on gauge dient biaxial σ servation studies capability principal propor-
ics and strains area through and ε stresses tional to
modes of thickness cost
√ √ √ √ 1
ver bend
√ √ √ √ √ 1
tic bend
pressuri- √ √ 1
√ √ √ √ √ 4
form LCF
√ √ √ √ √ √ √ √ 4

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4.3 Specific multiaxial test methods
4.3.1 Bending + torsion
This was the first technique used to apply combined stresses in high cycle fatigue (HCF) at room
temperature. Oscillating vertical forces were applied to a horizontally clamped cylindrical specimen
which could be rotated by up to 90° in the horizontal plane so as to introduce bending, bending
+ torsion, or torsion in the waisted centre section. Specimens were either solid or hollow. A number of
these electro-mechanical testing machines were built between 1930 and 1950 to investigate fatigue of
aero-engine steels, especially for crankshaft applications.
4.3.2 Axial + torsion
This popular technique employs a single tubular specimen design with a gauge length over which stress
and strain are substantially invariant and access for strain measurement and crack monitoring. The
principal stress and strain directions progressively rotate through 45° as the test moves from uniaxial
to torsion. Elevated temperature testing and thermo-mechanical fatigue (TMF) are achievable with
relevant accessories and control software. Despite a limited range of strain biaxiality (–ν ≥ ϕ ≥ −1),
this approach is widespread and standard testing machines with dual servo-hydraulic actuators are
available from commercial manufacturers.
4.3.3 Axial + internal pressure
This approach permits a single tubular specimen design with essentially invariant stress and strain
over the gauge length. Crack studies are difficult as maximum stress occurs at the bore, so cracks can
only be visible after penetration of the wall shortly prior to failure. In addition, cyclic plasticity results
in strain ratchetting as external radial compression cannot be applied to fully reverse the stress —
strain cycle. Hence this approach is essentially restricted to elastic HCF studies. The testing machine
typically utilizes a dual actuator servo-hydraulic design.
4.3.4 Axial + internal + external pressure
This design enables fully reversed cycling without ratchetting because radial compression can be
applied. Axial and circumferential stresses and strains are measurable, enabling LCF hysteresis loops
on both surface axes, which makes the approach suitable for fundamental behavioural studies. Because
a pressure vessel is located around the specimen, visual observations are difficult. Also elevated
temperature testing above about 200 °C requires gas pressurization which presents safety issues. By
employing variable internal pressure and fixed external pressure, a design with just 2 servo-hydraulic
actuators is achievable.
4.3.5 Axial + internal + external pressure + torsion
The addition of torsion introduces rotation of principal stress or strain axes which allows, in principle,
material anisotropy and the effects of the different symmetries (in the axial and circumferential
directions) to be investigated. The mechanical design is complex with 4 servo-hydraulic actuators, but
has been successfully achieved.
NOTE Multiaxial testing machines featuring axial force and differential pressure are typically used for
academic research or specific R&D applications and are usually designed and manufactured to order.
4.3.6 Cruciform — LCF
Four orthogonal loading arms apply biaxial strain to a central circular gauge area on the specimen. This
area is usually spherically recessed on both sides in order to resist buckling and ensure that cracks
initiate near the centre. In consequence, the gauge area does not support all the applied forces, i.e.
some of the force is shunted around the outside. As a result, stresses and plastic strains are not readily
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determinable. However, visual observation of developing fatigue cracks is straightforward and elevated
temperature testing, including TMF, is readily achievable.
4.3.7 Cruciform — Crack growth
The four orthogonal arms are slotted to minimize grip constraint. A central square, constant thickness,
gauge area typically features a central hole stress raiser to initiate fatigue cracks. There is a large
region of essentially constant biaxial strain ideal for crack initiation and propagation studies. Elevated
temperature testing, including TMF, is achievable. Maximum compressive strains are limited to avoid
buckling in the gauge area and arms.
NOTE Cruciform designs provide the opportunity for testing single geometry plate specimens with dual
symmetry over the range of surface biaxiality. Testing systems employ 4 servo-hydraulic actuators within an
annular frame and are typically specified according to application and manufactured to order.
4.4 Multiaxial fatigue analysis
4.4.1 Computer aided design
In the design of structural components subject to multiaxial fatigue, it is common to use finite element
analysis (FEA) to determine stresses and strains. For elastic behaviour, such analyses are useful to
predict stress concentrations and local yield in order to evolve specimen designs.
4.4.2 Fatigue life prediction
Yield criteria such as Tresca (maximum shear), Von Mises or octahedral shear strain, coupled with the
Palmgren-Miner linear damage hypothesis, are frequently employed to predict “multiaxial fatigue life”.
However, research evidence does not necessarily support this approach.
Multiaxial LCF fatigue studies on specimens capable of being tested over the full biaxial range
showed that Tresca and Von Mises did not correlate all the fatigue life data, especially over the range
between uniaxial and torsion, i.e. (0 ≥ ψ ≥ −1) and (−ν ≥ ϕ ≥ −1).
For example, in Figure 1, Mohr’s strain circles drawn with principal strain (ε ) constant and Poisson’s
ratio = 0,5, show that the maximum shear strain (γ ) is the lowest in the uniaxial stress (ϕ = −ν) case.
However, ranking these biaxial fatigue cases from most to least damaging, the order was equibiaxial
strain (ϕ = +1), plane strain (ϕ = 0), uniaxial stress (ϕ = −ν) and pure shear (ϕ = −1).
Current consensus indicates that a critical shear plane analysis including, as a modifier, the direct
stress or strain acting normal to that plane, offers the best approach to correlating multiaxial fatigue
behaviour across the complete range of applied biaxial surface stresses or strains.
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a)  Pure shear (ϕ = −1) b)  Uniaxial stress (ϕ = −ν)
c)  Equibiaxial strain (ϕ = +1) d)  Plane strain (ϕ = 0)
Figure 1 — Mohr’s strain circles, ε constant, for Poisson’s Ratio (ν) = 0,5
4.5 Multiaxial fatigue failure criteria
The definition of fatigue failure criteria can have a significant effect on attempts to correlate theoretical
analysis with experimental results.
Axial force + torsion (without pressurization) results in the gauge area seeing all the applied stresses.

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