The determination of thermal residual stresses in
unidirectional and cross-ply titanium matrix composites using
an etch removal method
Gerald CR Watt1,2, Andrew D Crocombe1, Stephen L Ogin1, and Stephen Kyle-Henney2
1
Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey,
UK
2
Composites Group, TISICS Ltd, Farnborough, Hampshire, UK
Abstract
Recent work has shown that a simple rule-of-mixtures approach may be used to predict
the stress strain behaviour of a cross-ply metal matrix composite (MMC) laminate.
However, the low-strain behaviour was not predicted accurately, probably because
thermal residual stresses are obviously not included in such an approach. To increase the
understanding of the limitations of the rule-of-mixtures approach for predicting the stressstrain response, the residual strain-state of the fibre reinforcement has been determined
using an etching technique (henceforth referred to as the ‘total etch removal method’),
and results have been compared both with finite element modelling and with thermal
residual strain measurements derived from stress-strain curves. The results show that the
residual strain distribution in a cross-ply composite may be more complex than
previously thought, with the fibres in internal 00 plies having considerably higher thermal
residual strains than fibres in external plies. The results confirm that the rule-of-mixtures
approximation can be used, with some reservations with regard to the low strain
behaviour.
Keywords
Metal matrix composites (MMCs), Residual stress, Finite element analysis (FEA), Micromechanics.
Introduction
Recent interest from the space industry has meant that continuous fibre metal matrix
composites (MMCs) are now being considered for a wider variety of components than
simply loaded booms and struts. In particular, the merits of silicon carbide (SiC)
continuous fibre reinforced pressure vessels are currently being assessed [1, 2]. At
present, a wide range of monolithic tank constructions are in service within the
commercial satellite industry but, compatible alloys required for the long-term storage of
propellants do not have the desired specific strength and stiffness, and MMCs offer a
potential advantage over conventional materials.
To contain loading from internal pressurisation, MMC layups must be able to withstand
considerable biaxial stresses. However, it is well established that the transverse properties
of unidirectional MMCs are poor in comparison to their parent alloys [3] as a result of the
weak interface that exists between the fibre and matrix [4-6]. Consequently, where biaxiality is required, a layup of mixed directionality is necessary.
To date, a limited amount of research has been conducted on continuous fibre bi-axial
layups, and the role and contribution of thermal residual stresses to the mechanical
performance is generally only theoretical in nature [7, 9-10]. Ultimately, the design of
structures for pressure containment requires the behaviour of biaxial and multiaxial MMC
layups to be better understood and more experimental work is required to substantiate
FEA results. Previous work conducted on the tensile response of cross-ply TMCs by Watt
et al [12] has shown that a good prediction of cross-ply stress-strain response up to failure
could be obtained by averaging the stress strain data from the component 0° and 90°
laminates (i.e. a “rule-of-mixtures” approach). Figure 1 shows typical stress-strain curves
for a 00 ply, a 900 ply and a cross-ply laminate ([0/90]2s), together with the rule-ofmixtures prediction for the cross-ply laminate.
Figure 1. Prediction of the stress strain behaviour of a [0/90]2s TMC in tension from the
stress-strain behaviour of UD plies for longitudinal and transverse loading, using the ruleof-mixtures.
An inaccuracy associated with this prediction was the inability to account for the
premature onset of non-linearity in the [0/90]2s coupons, which was suggested to be a
consequence of the thermal residual stress state of the matrix in the laminate. In order to
investigate and quantify the residual stresses, a version of the etch removal method
(called here the “total etch removal method”) has been employed for the determination of
residual stresses in MMCs. Previous researchers have experimented with variations of the
etch removal method to release the induced compressive thermal strain in the fibres [1319]. For example, an etching agent was dropped onto a site perpendicular to the fibre
ends and where the matrix was removed, fibre extension was measured (see Fig 2) [1315].
Figure 2. Etch removal technique from Pickard and Miracle [14].
A modification of this technique was to use a ‘selective etch’ over a bundle of 20 to 30
fibres contained in a narrow slit of MMC by masking off either side of the strip before
exposure to the etchant [16-18]. The resulting fibre height change (illustrated in Fig 3)
could then be measured via travelling, or confocal, microscopy and axial strain calculated
based on measured extensions. In more recent years, a method of back calculating hoop
stresses with the Lamé equations for an axisymmetric cylinder model [19] has been tested
on existing selective matrix etch test data [16,17] and validated with FEA.
Figure 3. Etch removal technique with wire eroded slit and masked off areas from
Ramamurty et al [16].
One of two finite element approaches is generally applied to the modelling of MMCs,
either micro-modelling or macro-modelling [20]. The most-widely used method for the
detailed modelling of residual stresses within a representative composite model is micromodelling, where a small section of the overall body is created and symmetry and
boundary conditions are used. Axisymmetric or concentric cylinder assemblage models
(CCMs) provide good approximations for unidirectional composites, but to model more
complex layups (such as the cross-ply laminates used in this work) a 3D unit
cell/representative volume element (RVE) is required. Uniaxial stress models (USMs)
and concentric cylinder models were investigated by Neu et al [7], who incorporated
viscoplastic matrix effects; they found the maximum difference between axial residual
stress in the models to be 15% at room temperature. However, it was pointed out by
Zhang et al [8] that the axial stress component is not a direct function of fibre spacing, so
CCMs and USMs should produce similar results.
The primary purpose of the current paper is to use a new variation of the etch removal
method to determine the residual thermal stresses in the axial fibres of a cross-ply
titanium matrix composite and to validate the use of the rule-of-mixtures approximation
as a summation of the behaviour of 0° and 90° plies tested separately. Measurements
from unidirectional composites are used to predict the lock-in temperature, enabling
comparisons with mechanical test results and finite element modelling predictions.
Experimental methods
Coupon fabrication
Fabrication of the unidirectional [0]8 and cross-ply [0/90]2s TMCs, and monolithic
coupons, for testing followed the standard TISICS procedure: Titanium foil (Ti-3Al2.5V) was cut to size on a guillotine and degreased with acetone. Sheets of aligned fibre
containing a fugitive binder were then sandwiched between the degreased foils until the
desired ply thickness was reached (eight plies in this case). The fibre was produced inhouse (via a chemical vapour deposition process) before being filament wound and
sprayed with a polymeric binder to maintain uniform fibre spacing; for all the coupons in
this work a 140 μm diameter SiC monofilament was used (SM3156), and the titanium foil
thickness was also 140 μm thick. The foil/fibre layups were contained between yttria
release agent coated steel packing plates and placed into a stainless steel can. The layups
fabricated were [0]8 for the unidirectional panels and [0/90]2s for the cross-ply panels.
The can was welded to contain the layup and then evacuated through a steel degas pipe.
A cycle of heating was used to remove the fugitive binder while evacuating the can at
moderate to high vacuum levels. Subsequently, the degas pipe was sealed, containing the
high internal vacuum (approximately 10-3 Pa). The can was then subject to a hot isostatic
press (HIP) cycle with a maximum temperature of around 950 ˚C and 100 MPa pressure.
After processing, the stainless steel can was mechanically removed and the yttria coated
packing plates were released from the consolidated panel. To produce monolithic
coupons of the titanium matrix alloy, ten sheets of 140 μm thick cold-rolled Ti-3Al-2.5V
foil were diffusion bonded into a flat panel using the same manufacturing methodology,
but without the addition of fibre.
Coupons conforming to ASTM D3552-12 were cut from their parent panels using wire
electro-discharge machining (EDM) and the edges were wet ground with a diamond
impregnated grinding wheel to remove the recast layer produced by the EDM. Monolithic
aluminium end tabs were then affixed to the tensile coupons with a structural adhesive
(3M 9323 B/A) to prevent damage from the knurled wedge grips when testing. The
compressive coupons were supplied with no tabs to fit the IITRI test fixture (as shown in
ASTM D3410). Consolidated thicknesses for the unidirectional and cross-ply coupons
were 1.897 mm ± 5 μm and 1.948 mm ± 7 μm, respectively.
Mechanical testing of coupons
An electro-mechanical Instron 4505 test frame (100 kN load cell) was used to load
parallel sided coupons 20 mm wide x 170 mm long (conforming to ASTM D3552
guidelines) to failure in tension at a cross-head displacement rate of 2 mm min-1 and
longitudinal strain gauges (TML BFLA-2-5) were used to measure the strain. Ambient
temperature mechanical test data was obtained for both unidirectional [0]8 and cross-ply
[0/90]2s TMCs. For the compression tests, parallel sided coupons measuring 20 mm wide
x 150 mm long were tested in an Illinois Institute of Technology Research Institute
(IITRI) style fixture by Airbus Defence and Space. The IITRI rig supported the coupon
from out of plane bending thus preventing premature buckling. The specimens were
gripped over 70 mm at each end, leaving a gauge length of 10 mm between the two
clamping wedges, strain gauges on both sides of the coupon gauge section were used.
The tensile and compressive coupon tests had the same nominal fibre volume fraction of
35% as the TMC specimens used in the etch removal technique.
Total etch removal method (TERM)
Two sets of TMC coupons of nominal dimensions 20 mm wide x 100 mm long were
removed from their parent panels using EDM. The first specimens analysed were
unidirectional [08] TMC with TISICS SM3156 fibre in a Ti-3Al-2.5V matrix. The second
set of specimens were cross-ply [0/90]2s TMC with the same nominal volume fraction
and fibre/matrix combination. A surface grinder with a diamond grinding wheel was used
to wet grind the edges of the coupons, thereby removing the recast layer left from EDM.
Once all coupons had been ground parallel, dimensional measurements were taken on a
Nikon V12 profile projector (shadowgraph) to determine the long axis fibre length.
Volume fraction measurements of the composites were made using optical microscopy
images of polished mounted specimens (Figures 4 & 5) taken from the parent panels and
the average volume fibre fraction was found to be 35%.
Figure 4. A micrograph of a mounted and polished uniaxial TMC specimen viewed on a
plane perpendicular to the fibre axes.
Figure 5. A micrograph of a mounted and polished cross-ply TMC specimen, viewed on
a plane perpendicular to the 00 fibres (note: blemishing is due to the etchant).
Prior to etching the specimens, measurements were performed with the shadowgraph in
reflected light mode with a 100x objective lens to measure the edge-to-edge distance
along the coupon parallel to the 00 fibres. Connected to the traversing table was a
Metronic 200 digital read-out which output X and Y coordinates to a precision of 1 µm.
The shadowgraph was used again once the fibres had been liberated from the matrix with
a strong etch consisting of a 5% HF / 23% HNO3 solution. The solution created for
etching was not, however, of sufficient strength to cause damage to the fibres but acted to
dissolve the matrix completely. Samples of approximately one hundred full-length fibres
from each coupon were randomly selected, and these were placed individually into a
milled alignment block and the fibre length was measured. The purpose of the block was
to provide an index point and to support and align the fibres during measurement under
the shadowgraph, thus removing inaccuracies resulting from possible fibre curvature.
Mechanical test and TERM results
Derivation of residual stresses from the unidirectional composite
mechanical test data
Results from the unidirectional TMC coupon testing are shown in Figure 6, where the
solid lines represent tensile behaviour and the dashed lines denote compressive stressstrain behaviour. A clear change of gradient or ‘knee’ is evident in both sets of curves and
this represents the point at which macroscopic yield of the matrix occurs. Once the matrix
shows plastic behaviour, the instantaneous modulus of the composite is dominated by the
presence of the fibres, which exhibit linear elastic behaviour up to failure.
Figure 6. Compressive and tensile behaviour of a unidirectional SM3156 TMC.
The difference in measured strain between the tensile and compressive knees in the
stress-strain curves is known to be a consequence of the residual stress state imparted
upon cool-down from processing [21, 22]; the matrix reaches a stress state sufficient to
cause macroscopic yielding and a noticeable change in instantaneous modulus occurs. It
should be noted that the required stress to yield the ‘as-HIPed’ condition monolithic
matrix alloy (Ti-3Al-2.5V) remains virtually identical in both tension and compression
loading at about 0.5% strain (as shown by Figure 7). However, premature failure of the
strain gauges used for strain measurement meant that strain-to-failure of the coupons was
not recorded.
Figure 7. Compressive and tensile behaviour of ‘as-HIPed’ monolithic Ti-3Al-2.5V
coupons. Dashed lines show the compressive specimen, solid lines are the tensile
specimen.
Due to the CTE mismatch between the fibres and the matrix in the unidirectional MMC,
the fibres are under a residual state of compression and the matrix is in a state of residual
tension. When compression testing of the composite begins, the state of residual tension
in the matrix must be fully reversed before an axial compressive stress can be induced in
the matrix, and yield reached. Consequently, matrix yield and change of instantaneous
modulus in the unidirectional TMC, occur at a higher applied strain than for that of the
monolithic matrix alloy (Fig 7) in compression (εcmy = - 0.75%). Similarly, the applied
tensile strain required to cause yield in a TMC is lower than that of the monolithic alloy
as the matrix is already in a state of residual tension (εcmy= 0.35%).
The mechanical test results enable a simple derivation of the residual strain in the fibres
for the unloaded unidirectional composite to be made. Assuming that matrix yield is a
consequence of axial stress, rather than von Mises stress, the strain that would need to be
applied to the unidirectional (UD) composite to achieve zero matrix stress is|𝜀_𝑡𝑚𝑦 | −
|𝜀_𝑐𝑚𝑦 |/2. Here, εcmy refers to the strain required for composite compressive matrix
yield, and εtmy denotes the strain for yield of the matrix in tension. At zero matrix stress,
all the load is carried by the fibres, giving the fibre stress, which is compressive, as
𝜎𝑓 = [
|𝜀𝑡𝑚𝑦 |−|𝜀𝑐𝑚𝑦 | 𝐸𝑐
2
]𝑉
𝑓
(1)
where Vf is the volume fraction of fibres and Ec is the composite modulus. Unloading the
|𝜀𝑡𝑚𝑦 |−|𝜀𝑐𝑚𝑦 |
composite from the compressive strain of
to zero strain, reduces the
2
compressive strain in the fibre such that for the unloaded composite, the fibre residual
stress, σresf, is
𝜎𝑟𝑒𝑠𝑓 = [
|𝜀𝑡𝑚𝑦 |−|𝜀𝑐𝑚𝑦 |
2
𝐸
] [𝑉𝑐 − 𝐸𝑓 ]
𝑓
(2)
where Ef is the fibre modulus. The corresponding residual stress in the matrix, which is
tensile, is given by
𝜎𝑟𝑒𝑠𝑚 = [
|𝜎𝑐𝑚𝑦 |−|𝜎𝑡𝑚𝑦 |
2
][
𝐸𝑚
𝐸𝑐
]
(3)
where, σcmy is the stress to macroscopic matrix yield in compression loading (onset of
non-linearity) and σtmy is the onset of non-linearity in the tensile regime; Em is the matrix
Young’s modulus. It should be noted that equation (3) is identical to the expression
derived by Newaz and Majumdar [23]. From equation (2), the fibre residual strain, εresf,
for the unloaded UD MMC, is then
𝜀𝑟𝑒𝑠𝑓 = [
|𝜀𝑡𝑚𝑦 |−|𝜀𝑐𝑚𝑦 |
2
] [𝑉
𝐸𝑐
𝑓 𝐸𝑓
− 1]
(4)
Tension and compression test data for the fibre volume fraction of Vf = 0.35 produced a
composite modulus, Ec = 213 ± 5 GPa. A value for the Young’s modulus of the SM3156
monofilament is more problematic. A similar fibre (SM2156) has a modulus of about 385
GPa [24] but the SM2156 has more free carbon in the SiC region and would be expected
to exhibit a lower modulus than SM3156 which maintains a stoichiometric balance.
Applying a rule-of-mixtures approach to the UD tensile data obtained in this work
suggests a SM3156 modulus of about Ef = 400 GPa, which does not seem unreasonable.
Using these values, equation (4) provides a residual axial thermal strain in the fibres, as a
consequence of thermal stresses, of εresf = - 0.104%. This value will be compared with the
results of the etch removal experiments in the next section. The calculated strain would
result in a residual axial fibre stress of 416 MPa and a matrix stress of 224 MPa to
maintain equilibrium.
Derivation of residual stresses from the cross-ply composite
mechanical and physical property data
Figure 8. Compressive and tensile behaviour of a cross-ply SM3156 TMC coupons.
Typical results from the cross-ply TMC coupon testing are shown in Figure 8, where the
solid lines represent tensile behaviour and the dashed lines denote compressive stressstrain behaviour. A change of gradient or ‘knee’ is evident in both sets of curves;
however, unlike the unidirectional stress strain curves in which the knee is primarily a
result of plastic deformation in the matrix [25], the tensile knees in the cross-ply TMCs
are partly the consequence of matrix plasticity but also of additional mechanisms (such as
fibre/matrix debonding). The fibre debonding in the transverse plies causes a reduction in
the instantaneous modulus in tension from a very small strain. Replication work
conducted by Johnson et al [26] has confirmed that debonding occurs in the transverse
plies of similar [0/90]2s cross-ply TMCs at stresses above the initial knee (0.08%) in the
tensile stress strain response. Other researchers, such as Xia et al [27], have used
micromodelling techniques to investigate the propagation of debonding in cross-ply
TMCs suggesting that an induced strain of approximately 0.06% is sufficient to initiate
debonding of the transverse plies. Consequently, the knees in the stress-strain curves
cannot here be employed to determine the thermal residual strains and a different
approach must be used to predict the residual fibre strain in the cross-ply composites.
Using mechanical property data from the unidirectional and transverse laminates, along
with known thermal expansion coefficients, the residual fibre strain can be determined,
again using a simple one-dimensional model, by considering the force-balance parallel to
the 00 fibres. Firstly, the strain in the laminate (εrx,0) parallel to the 00 fibres is calculated
from the force balance for the axial and transverse plies, taking into account strain
compatibility i.e.
𝜀𝑟𝑥,0 =
𝐸90 ∆𝑇(𝛼90 −𝛼0 )
𝐸0 +𝐸90
(5)
Where E0 is the axial ply modulus, E90 is the transverse ply modulus, ΔT refers to the
change in temperature, α0 defines the axial ply CTE and α90 defines the transverse CTE.
The residual tensile fibre strains, εresf, for the 00 fibres within the axial plies of the crossply laminate is then found to be:
𝜀𝑟𝑒𝑠𝑓 =
𝐸0 𝜀𝑟𝑥,0 +𝐸𝑚 (1−𝑉𝑓 )∆𝑇(𝛼𝑚 −𝛼𝑓 )
𝐸𝑓 𝑉𝑓 +𝐸𝑚 (1−𝑉𝑓 )
(6)
Here, Em is the monolithic Ti-3Al-2.5V modulus, Ef denotes the fibre modulus, αf and αm
are the fibre and matrix CTEs, respectively. Equation (6) predicts a residual 00 fibre strain
in the cross-ply laminate of εresf = - 0.17%. It should be noted that this approach assumes
that the strains are the same for the innermost and outermost axial (00) plies.
Derivation of residual stresses from the TERM of unidirectional
composites
Figure 9. Gaussian distribution of released fibre strains from the unidirectional coupons.
Experimental results for the residual strains measured from three UD coupons (using the
TERM method) are shown in Figure 9, with approximately 100 fibres analysed
individually in each case; the data points for each set of results has been joined together
for clarity. The strain was calculated from a comparison of the free fibre lengths with the
coupon lengths. Averaged over the three coupons, the fibre residual strain was found to
be εresf = - 0.13% ± 0.02 (see Table 1). This is 25% higher than the strain found from the
analysis of the mechanical test data in Section 3.1 (i.e. - 0.104%). A fibre residual strain
of - 0.13% corresponds to an average [compressive] residual axial stress of about 520
MPa in the fibres, and 280 MPa in the matrix.
Table 1. Results of the total etch removal method on unidirectional TMC.
UD Coupon 1
UD Coupon 2
UD Coupon 3
Average fibre extension (μm)
137 ± 17
125 ± 13
129 ± 23
Longitudinal fibre strain (%)
0.14 ± 0.02
0.13 ± 0.01
0.13 ± 0.02
Longitudinal fibre stress (MPa)
546 ± 68
511 ± 52
526 ± 96
Longitudinal matrix stress (MPa)
294
275
283
Derivation of residual stresses from the TERM method applied to
cross-ply TMC coupons
Figure 10. Gaussian distribution of released fibre strain from the cross-ply coupons.
The total etch removal method was also applied to cross-ply coupons. Again, samples of
one hundred axial fibres were selected from each of the three coupons for measurement
under the shadowgraph. As the residual stress state in the cross-ply will be different from
the UD specimen, a different fibre residual strain was expected. Figure 10 shows the
residual strains for the three specimens and it is clear that there is a bi-modal distribution.
Average results for the three coupons are shown in Table 2 where the average residual
strain is about - 0.32 ± 0.08%. However, Figure 10 shows there are peak average strains
at strains of about - 0.23 ± 0.03% and - 0.37 ± 0.04%. The presence of the two peaks
warranted further investigation.
Table 2. Results of the total etch removal method on cross-ply TMC coupons.
CP Coupon 1
CP Coupon 2
CP Coupon 3
Average fibre extension (μm)
319 ± 100
339 ± 43
355 ± 72
Longitudinal fibre strain (%)
0.320 ± 0.100
0.340 ± 0.044
0.295 ± 0.072
Longitudinal fibre stress (MPa)
1280 ± 400
1360 ± 176
1180 ± 288
A parallel-sided coupon of cross-ply composite (approximately 50 mm long by 20 mm
wide), from the same parent panel as the previous coupons, was prepared. Two parallel
slits were cut with a length of 24 mm into the panel, equidistant from the centre, with a
Struers Accutom and a 0.6 mm thick diamond slitting blade. The coupon was
subsequently encapsulated in paraffin wax to act as an etch resist, and once cooled, the
wax coating over the central TMC strip was removed. The matrix was removed (as in the
slit etch removal method) from the TMC strip with the same etchant mixture as used in
the TERM experiments. The whole coupon was submerged in the etchant and although
the matrix was removed more rapidly on the outer surfaces, after prolonged exposure
(approximately 16 hours), no remaining matrix was visible between the plies. The ends of
the etched fibres were then viewed using a confocal microscope (Zeiss LSM 800). The
microscopy established that the fibres of the inner 0° ply were on average 24 μm longer
than the fibres of the outer ply over the 24 mm length i.e. the fibres of the inner ply were
approximately 0.1% longer than the fibres of the outer ply; Figures 11 and 12 show
overlaid images of the two sets of fibres.
Figure 11. Confocal microscopy images of the TERM exposed axial plies (LHS: side on,
RHS: end on); the subsurface axial ply is shaded red, while the surface ply is shaded blue.
Figure 12. A confocal micrograph with false colour showing the difference in extension
(24 μm) between the top ply (shaded blue) and subsurface axial ply (shaded red).
Consequently, the bimodal distribution of fibre lengths seen in Figure 10 is the
consequence of different residual thermal strains in the axial plies, with fibres in the inner
0° plies having a residual compressive strain approximately 0.1% higher than fibres in the
outer plies; consequently, when the matrix is removed, the fibres in the inner plies are
longer. It is possible that the bimodal distribution of axial fibres strains within the crossply laminate is principally due to the outer plies having a free surface for which relatively
unhindered movement can occur in HIP processing (e.g. by creep deformation during
cooling), whilst the inner plies have a greater constraint and therefore experience greater
residual strains.
Taken overall, the average thermal residual strain for the 00 fibres of the cross-ply
composite (εresf ≈ - 0.32%) is significantly higher than in the unidirectional composite
(εresf ≈ - 0.13%). This higher residual compressive strain in the axial fibres of the crossply coupons is reflected in the axial failure strains for the cross-ply and unidirectional
composites. The axial strain to failure of both types of TMC is dominated by the
behaviour of the axial fibres; consequently, the strain-to-failure of the cross-ply coupons
was approximately 1.1%, whereas the UD coupons failed at a strain of about 0.9%, as
shown in Figure 1. The higher strain-to-failure of the cross-ply coupons (by 0.2%)
reflects the higher residual compressive strain of the 0° fibres in the cross-ply coupons
compared to the UD coupons (also about 0.2%). The total etch removal method was also
applied to cross-ply coupons. Again, samples of one hundred axial fibres were selected
from each of the three coupons for measurement under the shadowgraph. As the residual
stress state in the cross-ply will be different from the UD specimen, a different fibre
residual strain was expected. Figure 10 shows the residual strains for the three specimens
and it is clear that there is a bi-modal distribution. Average results for the three coupons
are shown in Table 2 where the average residual strain is about - 0.32 ± 0.08%. However,
Figure 10 shows peaks in the average strain at about and there are peak average strains at
strains of about - 0.23 ± 0.03% and - 0.37 ± 0.04%. The presence of the two peaks
warranted further investigation.
Prediction of stress-strain curves using FE modelling
Introduction
To investigate the build-up of residual stresses, and the subsequent stress strain response
of the unidirectional and cross-ply composites in tension and compression, simple threedimensional representative volume element (RVE) models were created. The modelling
had several purposes. First, if the unidirectional test data could be replicated closely, it
would increase confidence in the material properties used in the FE models. Secondly, if
the radial residual fibre stresses in the cross-ply model were found to be removed by
tensile interfacial stresses at low applied strains, then this would help to explain the initial
non-linearity in the tensile stress-strain response of the cross-ply coupons; consequently,
the disparity between the simple rule-of-mixtures approach to predicting the stress-strain
behaviour and the actual response of the cross-ply laminates could be understood.
The consequence of the fabrication technique and processing was that the composite fibre
architecture approximated well to a simple repeating array (see Figures 4 and 5) and this
was represented in the FE model with a rectangular RVE model as a reasonable
simplification of the material structure. Dimensions for the RVE models were obtained
by measuring the average fibre spacing in micrographs of the composite. The RVE
models used to represent the composites consisted of 1/4 of a unidirectional cell and 1/16
of a cross-ply cell (Fig 13 and 14 respectively).
Figure 13. Caption: 3D RVE for the unidirectional TMC.
Figure 14. 3D RVE for the cross-ply TMC.
Constituent material properties for the models were available from prior in-house testing
and literature; Table 3 shows room temperature properties, and Table 4 shows the yield
strength and Young’s modulus of the matrix as a function of temperature.
Table 3. Physical properties of fibre and matrix constituents. (Strength and modulus data
from [12] and CTE data from [28])
Material
Yield
UTS (MPa)
strength
Young’s modulus
Post-yield
Poisson’s
Avg. CTE
(GPa)
modulus
ratio
(m/m C-1)
(MPa)
(GPa)
Ti-3Al-2.5V
520
675
106
0.5
0.39
9.9 x10-6
SM3156
N/A
3800
400
N/A
0.17
4.5 x 10-6
Table 4. Ti-3Al-2.5V matrix properties at temperature ([28]).
Temperature (°C)
Yield strength (MPa)
Young’s modulus (GPa)
22
520
106
150
434
96
350
269
83
400
255
82
450
244
77
500
225
74
550
201
50
600
153
40
Finite-element model description
Symmetry boundary conditions were applied to three orthogonal faces of the 3D RVE
models and nodal coupling equations were applied to the three opposing faces so that all
nodes on a face were only allowed to move perpendicular to their parent face and by the
same amount. This setup forced the faces of the RVE to remain planar, thus simulating
the effect of the RVE being at the centre of a much larger region of composite. In order to
simulate the effect of thermal residual stresses on the stress-strain curves, a first
approximation of the lock-in temperature was derived from the TERM measurements on
the unidirectionally reinforced TMCs.
As discussed above, due to differences in the CTE between the matrix and the fibre, upon
liberation of the fibre from the matrix, the fibre will relax (elongate) as residual stresses
are released. At the lock-in temperature, it is known that the coupon is longer than at
room temperature, due to the positive coefficient of thermal expansion of the composite
(c). The length of the coupon at the lock-in temperature (LcLI) can therefore be written in
terms of the coupon length at room temperature (LcRT) as
𝐿𝑐𝐿𝐼 = 𝐿𝑐𝑅𝑇 (1 + 𝛼𝑐 ∆𝑇)
(7)
Here, ΔT is the increase in temperature from ambient to the lock-in temperature and αc is
the coefficient of thermal expansion of the coupon (αc = 6.6 x 10-6 K-1).
As the coefficient of thermal expansion of the fibre is known (αf = 4.5 x 10-6 K-1), the
length of the free fibre at room temperature (LfRT) can be expressed as
𝐿𝑓𝑅𝑇 = 𝐿𝑐𝐿𝐼 (1 − 𝛼𝑓 ∆𝑇)
(8)
Substituting equation (7) into (8) gives
𝐿𝑓𝑅𝑇 = (𝐿𝑐𝑅𝑇 (1 + 𝛼𝑐 ∆𝑇))(1 − 𝛼𝑓 ∆𝑇)
(9)
Solving equation (9) for ΔT, using fibre measurements obtained from the UD TERM
experiments, the lock-in temperature was found to be 667 °C above ambient. This lock-in
temperature is in agreement with other authors who have used temperatures between 500700 °C as the lock-in temperature [3, 5, 9, 17-19]. The lower end of this temperature
range is thought to correspond to the temperature at which stress-relief through creep
stops being significant in a titanium alloy [29]. Cool-down from HIPing typically occurs
over a period of hours and would facilitate such creep conditions.
The stress strain response of the representative volume element models was obtained by
applying a displacement condition to a node coupled face in-plane with a fibre end. The
displacement control was deactivated until the thermal drop had been implemented in the
first time-step, thus simulating the build-up of thermal residual stress. After the
introduction of residual stresses to the model, the displacement control was set to increase
incrementally over an arbitrary period of time (10 seconds). Lastly, the applied stress was
calculated by dividing the axial reaction force by the cross-sectional area of the RVE. The
applied stress was then plotted against the total strain to give the stress-strain response.
Predictions of stress-strain behaviour
Figure 15. Axial residual strain in the fibre and matrix upon cooldown.
The FE model for the UD TMC predicted a residual axial fibre strain of - 0.132% (Fig
15), which is in excellent agreement with the TERM value of - 0.132%, and in reasonable
agreement with the value derived from the tensile and compressive stress/strain curves of
- 0.104 % (Table 5).
Table 5. Averaged residual strains and deduced stresses.
TERM approach
Stress-strain
Finite Element Analysis Predictions
curve derived
Panel
Fibre Strain
Fibre Strain
Fibre Strain
Fibre Stress
Matrix Stress
(%)
(%)
(%)
(MPa)
(MPa)
[0]8
- 0.132
- 0.104
-0.132
-589
320
[0,90]2s
- 0.318
N/A
-0.194
-820
243
Figure 16. A comparison of tensile behaviour of the uniaxial 3D RVE and experimental
test data.
Figure 17. A comparison of compressive behaviour of the uniaxial 3D RVE and
experimental test data.
Figure 16 shows a comparison of the experimentally obtained tensile stress-strain curves
against a curve predicted by the FE modelling for the UD TMC. Although there is
reasonable agreement, the FE model predicts a departure from linearity approximately
250 MPa lower than found experimentally; this is also found for the compression stressstrain curves (Figure 17). Reductions of the same magnitude predicted for the
macroscopic yield for both tension and compression loading suggest that this discrepancy
is not due to the use of an incorrect ‘lock-in’ temperature, but rather an indication that the
effective matrix yield strength in the composite is significantly higher than anticipated
from monolithic experimental data. When component stresses upon cool-down to room
temperature are investigated in the FE model, the von-Mises stress state in the UD model
is seen to exceed the 520 MPa yield strength (as taken from Figure 7) of as-HIPed Ti-32.5 in localised areas (Fig 18). Hence, it appears likely that local constraint in the matrix
between the fibres is delaying the onset of yield.
Figure 18. Von Mises stress contours resulting from cool-down in the unidirectional 3D
RVE.
The FE modelling of the cross-ply (CP) composite predicts a residual axial fibre strain of
- 0.194%; this is higher than the UD residual strain prediction (- 0.132%) as expected,
and in reasonable agreement with the one-dimensional analytical model prediction of 0.17%. Although the predicted residual strain is in reasonable agreement with the
measured inner 00 ply fibre residual strain of the [0/90]2s TMC (about - 0.23%), it is
significantly lower than the averaged residual strain for all of the 00 fibres i.e. - 0.32%.
Further work on the differences in the residual strains between the inner and outer plies of
the [0/90]2s TMC will be required to understand this discrepancy.
The FE analysis of the cross-ply 3D-RVE model was also used to predict the tensile and
compressive stress/strain behaviour and this is compared with the rule-of mixtures
approach and the experimental data for both tension and compression loading in Figures
19 and 20.
Figure 19. A graph showing the cross-ply tensile test data, the cross-ply 3D RVE
response and the weighted ROM prediction.
Figure 20. A graph showing the cross-ply compressive test data, the cross-ply 3D RVE
response and the weighted ROM prediction.
Despite reasonable agreement between the CP models, similar discrepancies are noted to
the UD RVE models, with the knees occurring at stresses approximately 250 MPa lower
than the experimental data. The weighted ROM approach is adequate for an initial
approximation of the cross-ply behaviour, but fails to accurately replicate the low strain
behaviour and the post-knee tangent modulus of the experimentally obtained curves. It is
apparent that the role of thermal residual stresses and damage accumulation play a
significant role in the cross-ply stress-strain behaviour, and this is where the limitations
of the weighted rule-of-mixtures approach are realised. The thermal residual stresses
induced in the matrix due to the mismatch in CTE act to apply a compressive radial stress
to the fibres in the axial and transverse directions. For increasing load, debonding is
possible for the transverse fibres only once the radial compressive stresses have been
overcome, and this debonding may be responsible for the initial knee in the stress-strain
curve. That a similar debonding does not occur for the axial fibres (since the transverse
stresses do not oppose the thermal residual stresses for the axial fibres) is supported by
very limited fibre pull-out of the 00 fibres on the fracture surfaces of the cross-ply TMC.
Concluding remarks
In this work, the total etch removal method (TERM) has been used to measure the
thermal residual strains (and hence stresses) for the 00 fibres in unidirectional and crossply titanium matrix composites. These results were then compared to estimates of
residual strain obtained from mechanical test data and finite element analysis. For
unidirectional composites, excellent agreement was found between the measured residual
strain using TERM and finite-element predictions; the fibre strain derived from analysis
of the stress-strain curves in tension and compression was 30% lower. For the cross-ply
([0/90]2s) laminates, the thermal residual strain in the 00 fibres was predicted using the FE
analysis to be significantly lower than the average found for all of the 00 fibres in the
laminate. However, a bi-modal distribution of residual strains was found, with the inner
00 plies experiencing a higher compressive strain in the manufactured composite than the
outer 00 plies.
In addition, the results suggest that stress-strain curves for 00 plies and 900 plies can be
combined using the rule-of-mixtures to give a reasonable approximation of the behaviour
cross-ply ([0/90]2s) TMC, even though the approach obviously ignores: (i) the complexity
of constrained yield in the matrix; (ii) the possibility of fibre/matrix debonding in the 900
; and (iii) the differences in the thermal residual strains between the inner and outer plies
of the laminate which have been revealed using the TERM approach.
Acknowledgements
The authors of this paper would like to gratefully acknowledge the financial support of
InnovateUK in part funding the NGMP programme which enabled initial testing. The
authors would also like to thank Terry McCaul of Airbus Defence and Space for
coordinating compression testing, and Jerry Lord of NPL for his assistance in obtaining
mechanical property data for the TMCs. The lead author, Gerald Watt, would also like to
thank the EPSRC for their continued support of his EngD work under the MiNMAT
Industrial Doctorate Centre at the University of Surrey.
Declaration of conflicting interests
The authors declare that there is no potential conflict of interest with respect to the
research, authorship, and/or publication of this article.
Funding statement
This work was supported by the Engineering and Physical Sciences Research Council
(EPSRC) [EP/GO37388/1].
Supplementary Materials
Due to the commercially sensitive nature of the research materials supporting this
publication, not all of the data can be made publicly available. Please contact the author
for further details.
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