Failure criteria for C/SiC composites under plane stress state

Abstract

The applicability and limitation of several quadratic strength theories were investigated with respect to 2D-C/SiC and 2.5D-C/SiC composites. A kind of damage-based failure criterion, referred to as D-criterion, is proposed for nonlinear ceramic composites. Meanwhile, the newly developed criterion is preliminarily validated under tension-shear combined loadings. The prediction shows that the failure envelope given by D-criterion is lower than that from Tsai-Hill and Hoffman criteria. This reveals that the damage-based criterion is reasonable for evaluation of damage-dominated failure strength.

Full text

THEORETICAL & APPLIED MECHANICS LETTERS 4, 021007 (2014)
Failure criteria for C/SiC composites under plane stress state
Chengpeng Yang,
a) Guiqiong Jiao,Hongbao Guo
Department of Engineering Mechanics, Northwestern Polytechnical University, Xi’an
710129, China
(Received 30 November 2013; revised 13 February 2014; accepted 20 February 2014)
Abstract The applicability and limitation of several quadratic strength theories
were investigated with respect to 2D-C/SiC and 2.5D-C/SiC composites. A kind
of damage-based failure criterion, referred to as D-criterion, is proposed for non-
linear ceramic composites. Meanwhile, the newly developed criterion is prelim-
inarily validated under tension-shear combined loadings. The prediction shows
that the failure envelope given by D-criterion is lower than that from Tsai–Hill
and Hoffman criteria. This reveals that the damage-based criterion is reasonable
for evaluation of damage-dominated failure strength.
c
⃝2014 The Chinese Society of Theoretical and Applied Mechanics. [doi:10.1063/2.1402107]
Keywords C/SiC composite, damage evolution, failure criterion
Few studies have been dedicated to the failure criterion for ceramic matrix composites due to
the material nonlinearity, anisotropy and inhomogeneity, as well as the complexity of the direc-
tional interior microstructural damages. In this study, the applicability and limitation of several
quadratic strength theories are investigated with respect to C/SiC composites. To overcome the
drawbacks, a new failure criterion based on damage evolution mechanisms is developed for C/SiC
composites and/or other nonlinear brittle matrix composites.
Quadratic failure theories such as Tsai–Hill criterion,1Hoffman criterion,2and Tsai–Wu ten-
sorial criterion3are widely accepted for orthotropic materials. However, these phenomenological
theories are obviously based on data-fitting approach, without any physical significance. In the
present analysis, we encounter highly anisotropic failure problems, and a Cartesian material ref-
erence system is considered under plane stress states. We focus on the failure assessment in two
fracture planes, one perpendicular to axis-1 (referred to as plane-1) and the other perpendicular to
axis-2 (referred to as plane-2).
The Tsai–Hill criterion was formulated by referring to distortional energy and is thus an in-
teractive criterion. If the longitudinal stress
σ
1and shear stress
τ
12 are imposed on plane-1, the
failure condition can be given by the following equation
(
σ
1/X)2+ (
τ
12/S12 )2=1,(1a)
where Xis the allowable value of
σ
1and S12 is the positive pure shear strength. Similarly, if one
impose the transverse stress
σ
2and shear stress
τ
21 on plane-2, the failure condition is
(
σ
2/Y)2+ (
τ
21/S21 )2=1,(1b)
a)Corresponding author. Email: yang@mail.nwpu.edu.cn.

021007-2 C. P. Yang, G. Q. Jiao, H. B. Guo Theor. Appl. Mech. Lett. 4, 021007 (2014)
where Yis the maximum value of
σ
2and S21 is the negative pure shear strength. It is consid-
ered that the Tsai–Hill criterion fails to account for the different strength behavior in tension and
compression of the anisotropic materials. In this aspect, Hoffman made some improvements by
incorporating linear terms into the fracture condition.
Under plane stress state, the Hoffman criterion for combined loading of longitudinal stress
and shear stress on plane-1 can be stated as
σ
2
1/(XtXc) + (Xc−Xt)/(XtXc)
σ
1+
τ
2
12/S2
12 =1,(2a)
where Xtand Xcare the longitudinal tensile and compressive strength, respectively. While for the
similar combined loadings on plane-2, the failure condition can be expressed as
σ
2
2/(YtYc) +
σ
2(Yc−Yt)/(YtYc) +
τ
2
21/S2
21 =1,(2b)
where Ytand Ycare the transverse tensile and compressive strength, respectively. Once the tensile
strength is equal to the compressive strength in each principal material direction (i.e., axis-1 and
axis-2), Eqs. (2a) and (2b) will revert to Eqs. (1a) and (1b), respectively.
The failure criterion of Tsai–Wu is conceived, different from Tsai–Hill and Hoffman, for an
entirely anisotropic material. Consequently, all of the quadratic interaction terms about normal
stresses and shear stress are included in the scalar formulation. When a state of plane stress is
applied to plane-1 or plane-2, the Tsai–Wu failure condition can be formulated as
Fi j
σ
i
σ
j+Fi
σ
i=1,i,j=1,2,6,(3)
where
σ
6is the in-plane shear stress (
τ
12 or
τ
21). The components of Fi j and Fiare related to
the strength properties of the material, Xt,Xc,Yt,Yc,S12,S21. If mechanical rupture initiates in
plane-1 by tension-shear or compression-shear combinations, the following condition can be used
for failure prediction
σ
2
1/(XtXc) +
σ
1(Xc−Xt)/(XtXc) +
τ
12(S21 −S12 )/(S12S21 ) +
τ
2
12/(S12 S21) = 1.(4a)
However, if the fracture originates in plane-2 due to similar combinations, the simplified Tsai–Wu
condition showed below can be suggested
σ
2
2/(YtYc) +
σ
2(Yc−Yt)/(YtYc) +
τ
21(S12 −S21 )/(S12S21 ) +
τ
2
21/(S12 S21) = 1.(4b)
Note that in the case of S12 =S21, Eqs. (4a) and (4b) will revert to Eqs.(2a) and (2b), respectively.
In order to validate the applicability of the noted strength theories described previously, an
examination is performed by comparing the theoretical predictions with experimental data listed
in literatures.
For 2D-C/SiC composites with weak fiber/matrix interface, the average measured data under
tension-shear combination loadings4are depicted in Fig. 1. The reported compression strength is
Xc=430.2 MPa. It can be seen from this figure that the distribution of the data is mainly in ellip-
tical shape. Therefore, the quadratic criteria may be appropriate for curve fitting. But obviously,

021007-3 Failure criteria for C/SiC composites under plane stress state doi:10.1063/2.1402107
the Tsai–Hill criterion overestimates the bearing capacity of the material to some extent, whereas
the failure envelope simulated by Tsai–Wu and/or Hoffman is relatively in better agreement with
the experimental data.
According to the test results, Guan et al.4suggested a new quadratic equation to describe
the strength character of 2D-C/SiC composites. The criterion takes into account the inequality
between tensile strength and compressive strength, and shear stress recovery due to crack closure
caused by compression. Furthermore, the failure condition given in Ref. 4can be transformed
into a more complicated equation for plane stress state as
A(
σ
2
1+
σ
2
2)−2A(
σ
1+
σ
2)
σ
0+A
σ
2
0+B
τ
2
12 =1.(5)
The coefficients A,B, and
σ
0can be determined by
A=4/(Xt+Xc)2,
σ
0= (Xt−Xc)/2,B= [4XtXc/(Xt+Xc)2](1/S2
12).(6)
Note that Eq. (5) is acceptable for brittle composites that present the same mechanical properties
along each principal material direction, i.e., Xt=Yt,Xc=Yc. The estimated result of Eq. (5) is
very close to that given by Hoffman and Tsai–Wu (as shown in Fig. 1).
For 2D-C/SiC composites with strong fiber/matrix interface, the reported tensile strength
is Xt=246.6 MPa, the compressive strength is Xc=389.6 MPa, and the shear strength is
S12 =128.5 MPa.5The rupture data of the material along with theoretical predictions under
tension-shear combined loadings5are depicted in Fig. 2. It is presented in this figure that both
Tsai–Hill and Hoffman criterion fail to give accurate results, because the mechanical fracture
depends on the damage initiation and propagation path. This means that the damage mechanics
of fibrous composites may be appropriate for characterizing these brittle failure cases.
Normal stress/MPa
Shear stress/MPa
Shear enhance
Tested data
Tsai-Hill
Tsai-Wu and Hoffman
Eq. (5)
D-criterion
0 50 100 150 200 250 300
150
120
90
60
30
0
Fig. 1. Predicted failure curves by different cri-
teria for 2D-C/SiC composites with weak inter-
face.
Normal stress/MPa
Shear stress/MPa
Tested data
Tsai-Hill
Hoffman
D-criterion
0 50 100 150 200 250 300
150
120
90
60
30
0
Fig. 2. Predicted failure envelops by different
criteria for 2D-C/SiC composites with strong in-
terface.
For 2.5D-C/SiC composites, experimental studies6suggested that the damage evolution paths
and fracture modes of this highly anisotropic material are diverse and elusory under complex stress
state. Therefore, the failure behavior is examined on two fracture planes, as shown in Fig. 3.
The tested results of the material under tension-shear and compression-shear combinations
are described in Fig. 4. Based on the distribution regularity of the experimental data represented
in Fig. 4(a), the Sun proposal7is accepted herein to account for the variation in shear strength due

021007-4 C. P. Yang, G. Q. Jiao, H. B. Guo Theor. Appl. Mech. Lett. 4, 021007 (2014)
X25 1 mm 14.47 SEI X30 500 µm 12.47 SEI
(a) (b)
Fig. 3. Fracture plane of 2.5D-C/SiC material: (a) plane-1, (b) plane-2.
to
σ
1. If tensile
σ
1is applied to the material, the shear strength is expected to decrease, such as
(
σ
1Xt)2+ [
τ
12/(S12 −
η
1
σ
1)]2=1,(7a)
where
η
1is an experimentally determined constant. Contrarily, if compressive
σ
1is imposed on
the material, the shear strength will increase, that is
(
σ
1/Xc)2+ [
τ
12/(S12 −
η
2
σ
1)]2=1,(7b)
where
η
2is regarded as an internal material friction parameter. For the target material,
η
1=0.2,
η
2=0.15. The predicted curve of Eq. (7) is plotted in Fig. 4(a), and it is in better agreement
with experimental data, as compared with either the curve given by Tsai–Hill or that derived
from Hoffman criteria. The failure assessment on plane-2 is described in Fig. 4(b). It is worth
mentioned that the failure of the material is insensitive to a change of sign of shear stress. Thus,
although the linear term of shear stress is added to Tsai–Wu criterion, the prediction has little
difference with that of Hoffman.
-320 -240 -160 -80 0 80 160 -150 -100 -50 050 100 150 200
Experimental data
Hoffman
Tsai-Hill
Sun
Experimental data
Hoffman
Tsai-Wu
Tsai-Hill
120
90
60
30
120
90
60
30
τ12/MPa
σ1/MPa σ2/MPa
τ21/MPa
(a) plane-1 (b) plane-2
Fig. 4. Failure strength of 2.5D-C/SiC composite under combined loadings.
In general, the phenomenological strength theories are appropriate for orthotropic materi-
als with brittle fracture behavior. To nonlinear ceramic composites, such as C/SiC materials,
the Hoffman and Tsai–Wu criteria can also provide good approximation of mechanical failure
strength under certain loading conditions. However, since the classical criteria are not based on
the interior damage of materials, the influence of damage mechanisms on final failure can not be

021007-5 Failure criteria for C/SiC composites under plane stress state doi:10.1063/2.1402107
accounted for. In the following analysis, a kind of damage based criterion will be proposed.
In order to describe the highly anisotropic and the direction dependent damage of nonlinear
brittle matrix composites, a significant and special damage matrix is introduced herein, i.e.
Di j =




D11 D12 D16
D21 D22 D26
D61 D62 D66




.(8)
In the above equation, Di j (i=1,2, j=1,2,6) denotes damage in principal material direction i
caused by
σ
j; while D6j(j=1,2,6) represents shear damage induced by
σ
j. In the matrix, the
six off-diagonal terms are used to characterize the damage coupling effects.
For simplicity, the damage coupling terms are all assumed to be zero in this study. The damage
behavior is subsequently characterized by the residual damage components D11,D22, and D66. It
is considered that the failure of a ceramic matrix composite is driven by its interior damage. When
mechanical rupture occurs, these damage components should satisfy a mathematical relationship.
Luo et al.8had proposed a condition in which the principal maximum damage value reaches
1.0. This is similar to the maximum stress/strain criterion. However, it is also reasonable to
follow the conception of classical quadratic theories, and suggest a quadratic damage function.
Consequently, we introduce a reference damage variable, Deff, which is quantified in mathematical
perspective by a quadratic equation given as D2
11 +D2
22 +D2
66 =D2
eff. Generally, Deff varies with
loading conditions. Once mechanical failure occurs under a certain stress state, Deff will achieve
one of its critical values. This means that one collapse load corresponds to one reference value.
In order to unify the diverse failure conditions, we propose the following damage function
I11D2
11 +I22D2
22 +I66D2
66 =1.(9)
In the above equation, I11,I22, and I66 are simply set as invariable, and they can be determined by
some simple mechanical tests, such as uniaxial tension, compression and in-plane shear.
If one imposes a uniaxial tensile/compressive stress oriented along axis-1 or axis-2 on a spec-
imen, then the failure conditions are
I11 ˜
D2
11 =1,I22 ˜
D2
22 =1,(10)
where ˜
D11 and ˜
D22 are the maximum value of D11 and D22, respectively. Similarly, by imposing
pure in-plane shear stress on a sample, we obtain
I66 ˜
D2
66 =1,(11)
where ˜
D66 is the maximum shear damage. The solution of Eqs. (9)–(11) gives the damage-based
failure criterion as
(D11/˜
D11)2+(D22 /˜
D22)2+(D66 /˜
D66)2=1.(12)
Note that the basic conception of this criterion is that the final failure of composites is domi-

021007-6 C. P. Yang, G. Q. Jiao, H. B. Guo Theor. Appl. Mech. Lett. 4, 021007 (2014)
nated by the interior damage degree. Equation (12) is a macroscopic proposal (referred to as
D-criterion), which is prepared for advanced ceramic composites with nonlinear mechanical be-
havior. The shortcoming of the theory lies in the difficulty of anisotropic damage evolution analy-
sis. Meanwhile, modifications are also essential to incorporate the damage coupling terms. Before
applications, the damage evolutionary equations of target materials need specific investigation.
The unloading modulus is suggested for damage definition, which gives
D11 =1−E1u/E1,D22 =1−E2u /E2,D66 =1−G6u/G6,(13)
where E1and E1u are initial modulus and unloading modulus in axis-1, E2and E2u are initial
modulus and unloading modulus in axis-2, and G6and G6u are initial shear modulus and shear
unloading modulus.
To 2D-C/SiC composites examined previously, one can derive D11 =D22 for structural sym-
metry. According to Ref. 9, by fitting of experimental data, D11 (or D22 )can be expressed as
D11 =3.6×10−3
σ
1−5.7×10−6
σ
2
1,06
σ
16246.6 MPa (14a)
for the 2D-C/SiC composite with strong interface. While for the material with weak interface,
one can get the following relations
D11 =


0,
σ
1<10 MPa,
0.59 −0.73/[1+exp(0.018
σ
1−1.6)],10 MPa 6
σ
16269.8 MPa
(14b)
according to Ref. 10. The relationship between shear damage and shear stress is linear, so that
one has
D66/˜
D66 =
τ
12/S12 .(15)
By substituting the maximum value of
σ
1into Eq. (14), ˜
D11 can be obtained as 0.541 for the
material with strong interface and 0.563 for the material with weak interface, respectively. But
the value of ˜
D66 is not needed here because one can use the stress instead of the damage para-
meter (see Eq. (15) and Table 1), though it can be determined in the same way.
The D-criterion is preliminarily validated with respect to the off-axis loading of the 2D-
C/SiC composite with weak interface. The reported off-axis strengths are 256.7 MPa (
θ
=15◦),
203.8 MPa (
θ
=30◦), and 195.6 MPa (
θ
=45◦), respectively.10 Note that
θ
is the off-axis angle
and the stress components (
σ
1,
σ
2,
τ
12)can be obtained by resolution of the off-axis strength. The
predictions of D-criterion, Tsai–Hill criterion and Hoffman criterion are summarized in Table 1.
Further comparison reveals that the newly proposed Eq. (12) is applicable even though it excludes
the damage coupling terms.
Further applications of D-criterion are conducted for 2D-C/SiC composites under tension-
shear combined loadings, and the predictions are exhibited in Figs. 1and 2. It can be seen from the
figures that the failure envelopes given by D-criterion are lower than that derived from Tsai–Hill
and Hoffman criteria. To 2D-C/SiC with strong interface, the prediction of D-criterion is closer

021007-7 Failure criteria for C/SiC composites under plane stress state doi:10.1063/2.1402107
Table 1. Evaluation results of different failure criteria.
Criterion Theoretical function10 Deviation
15◦30◦45◦
Tsai–Hill (
σ
1/269.8)2+ (
σ
2/269.8)2−
σ
1
σ
2/269.82+
(
τ
12/125.7)2−1–0.004 –0.258 –0.263
Hoffman [
σ
2
1+
σ
2
2+ (430.2−269.8)(
σ
1+
σ
2)]/(269.8×430.2)−
σ
1
σ
2/(269.8×430.2)+(
τ
12/125.7)2−1
–0.077 –0.069 –0.042
D-criterion
{{0.59 −0.73/[1+exp(0.018
σ
1−1.6)]}2+
{0.59 −0.73/[1+exp(0.018
σ
2−1.6)]}2}/0.5632+
(
τ
12/120)2−1,(
σ
1>10 MPa,
σ
2>10 MPa)
+0.197 +0.070 +0.013
to the experimental results for the stochastic damage initiation and brittle fracture. However, the
new criterion is not used to the 2.5D-C/SiC composite, because the damage evolution equations
are not available in the literature.6
The present paper has to be considered as a preliminary investigation. The applicability and
limitations of the classical phenomenological strength theories to C/SiC composites are investi-
gated. A kind of damage-based criterion named D-criterion is proposed for advanced ceramic
composites. The preliminary validation is performed under off-axis tension and tension-shear
combined loadings. The prediction shows that the failure envelope given by D-criterion is gov-
erned by damage evolution equations, and more complex failure envelops can be described besides
quadratic curves. Future work will concentrate on improving the D-criterion in order to account
for the influence of damage coupling on final failure of brittle matrix composites.
This work was supported by the Basic Research Funds of Northwestern Polytechnical University
(JC20110219) and the National Natural Science Foundation of China (11102160).
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