- Email: [email protected]
Cyclically thermomechanical plasticity analysis for a broken fiber in ductile matrix composites using shear lag model
Composites Science and Technology 62 (2002) 641–654 www.elsevier.com/locate/compscitech
Cyclically thermomechanical plasticity analysis for a broken fiber in ductile matrix composites using shear lag model Junqian Zhanga,*, Jian Wua, Shaolun Liub a
Department of Engineering Mechanics, Chongqing University, 400044 Chongqing, China b Beijing Institute of Aeronautical Materials, PO Box 81-23, 100095 Beijing, China Received 16 November 2001; accepted 17 December 2001
Abstract The local cyclic plasticity of the interface around a broken fiber in ductile matrix composites under the in-phase and out-of-phase thermomechanical fatigue (TMF) loads is analyzed by using the single-fiber shear-lag model. The elastic, perfectly-plastic shear stress–strain relation is used to model the thermomechanical behavior of the fiber/matrix interface. It is shown that the alternating plastic shearing of the interface takes place under an appropriate combination of mechanical stress and thermal load. In the stress versus temperature diagram the so-called cyclic plasticity zone is identified. A new parameter, i.e. the cyclic plasticity length, Ls, is found which is smaller than the yield length, Ls, caused by monotonic loading. The closed-form solutions for Ls, Ls, the fiber stress profiles and the cyclic plastic shear strain range, p , are obtained. Ls and p increase for both the in-phase and out-of-phase TMF conditions as the mechanical load and/or thermal load increase. The in-phase condition produces a higher plastic shear strain range than the out-of-phase condition does. The solutions obtained may be used for modeling fiber/matrix debonding caused by the fiber breakage under TMF fatigue loading. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fiber composites
1. Introduction The dominant damage and failure modes in continuous fibers reinforced metal matrix composites (MMCs) can change with the magnitude, type and path of thermomechanical loading. The fatigue life diagram of the maximum applied strain versus the cycles to failure was proposed in [1,2] for understanding fatigue of the MMCs. The diagram shows three regimes: regime 1, 2, 3 (high, intermediate and low applied strain), in which the dominate failure mode is the fiber breakage, fiber-bridged matrix cracking and matrix cracking, respectively. The modeling of fiber-bridged matrix cracking has been investigated extensively in the literature (see [3] and references therein). In this paper we will consider the fiber breakage that is the dominant failure mode observed in the MMCs loaded in high stress * Corresponding author at present address: Laboratory for Technical Mechanics, University of Paderborn, 33098 Paderborn, Germany. Fax: +49-5251-603483. E-mail address: [email protected] (J. Zhang).
(regime I). The fiber breaks cause stress concentration that stimulates either the local plasticity or debonding around the breaks even if the overall behaviors of the composite are still elastic. As pointed out by Beyerlein and Phoenix [4], plastic deformation of the matrix will occur instead of fiber-matrix debonding if the interfacial strength is much higher than the matrix yield stress in shear. In this case under thermomechanical fatigue (TMF) loads the matrix may experience an alternating plastic deformation, which leads to the fiber/matrix debonding. Understanding of the cyclic deformation around the fiber break is essential to model the debonding under fatigue, in turn, could help us to develop mechanistic-based life prediction methodologies for thermomechanical fatigue. Shear-lag models of varying degrees of complexity have been used to determine the stresses in a broken fiber with reasonable accuracy, although they made some simplified assumptions. The first shear-lag model in the literature was proposed by Cox [5] to determine the stresses in a fiber embedded in an elastic matrix. This model assumes that the interface between the fiber
0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00022-2
642
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Fig. 1. The composite unit cell for the single-fiber shear-lag model, (a) Cross-section, (b) Side view.
and the matrix is well bonded. Plasticity and debonding in multiple-fiber systems are modeled in the references [6,7] by adding the shear yield at the interface into Cox’s model. And the models result in a linear fiber stress recovery from the break. The so-called bi-linear model was proposed in [3,8] where the interface was characterized by two shear stress values associated with matrix yielding and frictional sliding, respectively. More recently, Landis and McMeeking [9] made a further modification by introducing both axial and shear stresses into an elastic/perfectly-plastic flow rule for the matrix. Finite element analyses were conducted in [9,10], suggesting the shear-lag model is adequate in predicting the fiber stress recovery although it makes several simplifying assumptions. The mentioned models considered the monotonic mechanical loading only. This paper concerns the local cyclic plasticity of the matrix around a broken fiber in MMCs under thermomechanical fatigue loading. The in-phase and out-of-phase TMF conditions are considered. In order to obtain the stress profiles and the cyclically thermoplastic shear strain of interface the analysis is done by using the shear-lag model and the closed-form solutions will be derived.
equation for a differential element of the broken fiber is @f ¼ 4 D @z
ð2Þ
where f is the average axial stress of the fiber and is the shear stress acting on the interface between the fiber and the matrix. The fiber is considered to be an elastic spring such that radial, hoop and shear stresses are neglected, and the thermoelastic constitutive law for the fiber gives @u f ðT T0 Þ f ¼ Ef ð3Þ @z where u is the axial displacement of the fiber; Ef and f are the Young’s modulus and the coefficient of thermal expansion of the fiber, respectively; T and T0 are temperatures of the current state and the stress-free state, respectively. The relationship between the interfacial shear stress, , and the fiber displacement, u, is obtained by substituting Eq. (3) into Eq. (2) 1 d2 u ¼ DEf 2 4 dz
ð4Þ
2. Basic equations for the shear-lag model Consider a unidirectional continuous fiber composite with a hexagonal array of cylindrical fibers (Fig. 1). It is assumed that a single-fiber break takes place at z=0. The distance between the broken fiber and the six immediate neighbor fibers, w, can be related to the fiber volume fraction, Vf [9], rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 D ð1Þ w¼ 2 3V f where D is the fiber diameter. The composite is subjected to a cyclically thermomechanical load. Equilibrium
The shear strain of the matrix surrounding the broken fiber is approximated by [9] ¼
uc u w
ð5Þ
where uc is the displacement of the outer boundary of the unit cell. The approximation made to the singlefiber shear-lag model is that the axial displacement at the outer boundary of the cell is the same as the displacement that would exist in an undamaged composite at the same applied stress [9]. This condition is given by
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
f uc ¼ "z ¼ þ f ðT T0 Þ z Ef
ð6Þ
where " denotes the nominal composite strain and f the remote fiber stress. Eqs. (5) and (6) imply assumptions: (1) the matrix does not carry the load released by the fiber breaking, and only transmits shear between the fibers; (2) the load released by the breaking is equally carried by all of the intact fibers. The assumption (2) can be released by using the multiple-fiber shear-lag models [4]. uc would be replaced by the displacement of the nearest neighboring fiber and follows a differential equation other than the Eq. (6). And these would result in a set of coupled differential equations governing the displacements of the fibers. The assumption (1) is most accurate for composites with the high fiber volume fraction and the high modulus ratio of fiber to matrix. Most polymer composites and some metal-matrix composites, such as Boron/Al and SiC/Ti, satisfy these conditions [8]. The remote fiber stress, f , is related to applied composite stress and temperature by using the rule of mixture as follows, f ¼
Ef ½ þ pðT T0 Þ; p ¼ E m ð1 Vf Þðm f Þ Ec
ð7Þ
where denotes the applied composite stress; Ec stands for the Young’s modulus of the composite. At the fiber break the axial stress in the fiber is zero, and far from the break the fiber strain must be equal to the applied composite strain. Mathematically these boundary conditions are @u f ðT T0 Þ ¼ 0 Ef ð8Þ @z z¼0 @u f ¼ þ f ðT T0 Þ @z z ! 1 E f
Eqs. (2)–(9) are common to any shear-lag analysis for a single-fiber break. In the succeeding section they will be used along with the cyclic behavior of the interface to describe the stress and deformation responses under the TMF loads with both in-phase and out-of-phase conditions.
3. Thermomechanical fatigue loads Both in-phase and out-of-phase thermomechanical fatigue (TMF) loads are considered (Fig. 2). The tensile stress applied to composites varies between min and max while the temperature changes between Tmin and Tmax. Consequently, the applied stress range and temperature range are defined by ¼ max min
ð10Þ
T ¼ Tmax Tmin
ð11Þ
For convenience the state ‘A’ is used to represent the state of TMF load with the maximum applied stress for both in-phase and out-of-phase conditions, and state ‘B’ the state of TMF load with minimum mechanical load, (Fig. 2). In what follows the quantities with superscripts ‘A’ and ‘B’ belong to the state ‘A’ and state ‘B’, respectively. Assuming that the fiber and the matrix far from the fiber break in the composite are thermoelastic, the remote fiber stress range is derived from the Eq. (7) for in-phase and out-of-phase TMF loads, respectively, B f ¼ A f f
¼ ð9Þ
643
8E f > > < E ð þ pTÞ in-phase c
> > : Ef ð pTÞ out-of-phase Ec
Fig. 2. A sketch of the thermomechanical fatigue loads, (a) In-phase condition, (b) Out-of-phase condition.
ð12Þ
644
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
As the thermomechanical loads increase the local yielding of matrix around the fiber break or the fiber/ matrix debonding will occur due to stress concentrations. If the interfacial strength is higher than the matrix yield stress in shear the plastic deformation of the matrix will occur first instead of fiber/matrix debonding [4]. This was observed in ductile-resin and some metal matrix composites with a strong interface, such as Boron/Al composite system [11]. For the composites with a weak interface the debonding and the frictional slip take place. Here we consider the strong interfaces. It is assumed that the local matrix yield is caused by the shear stress only, ignoring the effect of the radial, hoop and tensile stresses. The local plasticity is modeled by the elastic, perfectly-plastic shear stress–strain relation, i.e.
¼
Gm jj < s s ðTÞ yielded
jump at the moment that the slip displacement changes sign. This difference between the constitutive laws will cause differences in the solutions. Discussion will be given in some details in the last section. Analysis of the stresses and the deformation, involving the cyclic thermoplasticity, must be carefully carried out by applying thermomechanical fatigue load to the composites incrementally. 3.1. Initial loading (O!A) Let us consider the TMF loading process (O!A), for either the in-phase or out-of-phase condition. The local yield takes place within the length of Ls, which is to be determined by the continuity conditions between yield and elastic segments. By combining Eqs. (2)–(7) the governing differential equations for the state ‘A’ are obtained,
ð13Þ A A @ 2 uA 2 A 2 f D k u þk þ f T T0 z ¼ 0 @z2 Ef 2
where s and Gm are the yield shear stress and the shear modulus of the matrix, respectively; and denote the interfacial shear stress and shear strain, respectively. This interface constitutive law allows the interface to undergo the cyclic plasticity under thermomechanical fatigue loading (Fig. 3). The yield shear stress depends upon the temperature by s ¼ c1 þ c2 ½1 expðc3 TÞ
ð14Þ
where c1, c2 and c3 are material constants. The elastic unloading path of the elastic, perfectly-plastic model differentiates it from the frictional slip model where the shear stress changes sign immediately as long as the slip displacement changes sign regardless how small the slip displacement is. In other word, the stress–strain curve of the elastic, perfectly-plastic model is continuous everywhere whereas the shear stress of the frictional slip model has a
ð15Þ
for z>Ls @fA A ¼ 4 s for z
ð16Þ
ð17Þ
The boundary conditions Eqs. (8) and (9) become A @u Ef f T A T0 jz¼0 ¼ 0 ð18Þ @z A @uA jz ! 1 ¼ f þ f TA TA 0 @z Ef
ð19Þ
for the state ‘A’. The continuity conditions between yield and elastic segments lead to
Fig. 3. An illustration of constitutive behavior of the interface.
uA ðLs þ 0Þ ¼ uA ðLs 0Þ
ð20Þ
A ðLs þ 0Þ ¼ sA
ð21Þ
fA ðLs þ 0Þ ¼ fA ðLs 0Þ
ð22Þ
By solving the governing Eqs. (15) and (16) and by using the boundary conditions and the continuity conditions, one can derive the yield length, fiber break opening displacement, the axial displacement and the axial stress of the fiber, and the shear stress at the fiber/matrix interface for the state ‘A’,
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
A 1 Ls ¼ D fA k 4s A ¼ 2uA 0 ¼
ð23Þ
2 4DsA DsA A f þ Ef k2 4Ef sA
ð24Þ
8 > 2 A z2 > > s þ f T A T 0 z þ u A > 0 > Ef D > > > > >
< 4DsA z Ls A u ¼ exp k > k2 E f D > > > > > > A > A > f > þ f T T 0 z :þ Ef
0 4 z 4 Ls
Ls 4 z 4 1 ð25Þ
8 A z > < 4s D
fA ¼ 4 A z Ls A > : f s exp k k D
0 4 z 4 Ls Ls 4 z 4 1
645
unloading, and the shear stress at z=0 relaxes faster than that at any other positions. Thus, the shear stress of z=0 will change sign most first. Subsequently, there are two possible cases: the complete elastic unloading and the reverse shear plastic yielding within a length, depending on the TMF load range. If the TMF load range is not such large that the shear stress of z=0 does not reaches the reverse yield stress before state ‘B’, the TMF unloading is completely elastic. In this case the TMF unloading-reloading will be totally elastic, i.e. shakedown occurs. If the TMF load range is sufficiently large the shear stress in the matrix of z=0 will change sign and will reach the reverse yield stress at some moment before the thermomechanical load reaches state ‘B’. With continuous thermomechanical unloading the reverse plastic yield of the matrix will extend forward from the fiber break tip (z > 0). Up to the moment ‘‘B’’ the reverse plastic yield of the matrix has occurred within a certain length, say Ls. Let us consider the case that the reverse shear yielding takes place. Within the reverse yield length, Ls, which is still unknown, the equilibrium equation is
ð26Þ
A ¼
8 A < s
z Ls : sA exp k D
@fB B ¼ 4 s for z
0 4 z 4 Ls Ls 4 z 4 1
ð27Þ
where A stands for the fiber break opening displacement. With the aid of Eqs. (5), (6) and (25), the plastic shear strain of the matrix near the interface is obtained, 8 A k Ls z < sA k2 Ls z2 0 z Ls s A p ¼ Gm D D : 2Gm 0 z Ls ð28Þ The Eqs. (23)–(28) are valid only if Ls > 0, i.e. 4Ec A max þ p T A T0 > kEf s
ð30Þ
The stress and displacement of the fiber for z < Ls can be easily obtained by solving the Eq. (30). In the portion ðLs < z < 1Þ where the interface undergoes elastic deformation during the unloading from state ‘A’ to state ‘B’, the differential equation governing the incremental displacement of the fiber is derived by appropriately using the equations in Section 2, @2 uAB k2 uAB @z2 f þ k2 þ f T B T A z Ef
D2
¼ 0 for z > Ls ð29Þ
Otherwise the elastic shear lag solutions of Cox should be used. Formula (29) defines what thermomechanical loads can cause the local yielding. The mechanical load required for the local yielding decrease as the temperature increases. From Eq. (23) the local yielding (Ls > 0) indicates that the remote fiber stress, A f , is positive.
ð31Þ
Solving the differential equation leads to the incremental solutions with some undetermined constants for the unloading process (A!B). Then, the solutions for the stresses and the deformation at the state ‘B’ can be obtained by superposing the incremental solutions between A and B onto the solutions of the state ‘A’, i.e., uB ¼ uA þ uAB
ð32Þ
fB ¼ fA þ fAB
ð33Þ
B ¼ A þ AB
ð34Þ
3.2. Unloading (A!B) When the TMF load changes from state ‘A’ to state ‘B’ under either in-phase or out-of-phase condition, the yielded matrix around the fiber break starts the elastic
for the portion ðLs < z < 1Þ. The boundary conditions (8) and (9) become
646
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
B @u f T B T0 z¼0 ¼ 0 @z
Ef
@uB B jz ! 1 ¼ f þ f TB TB0 @z Ef
ð35Þ
ð36Þ
for the state ‘B’. The continuity conditions must be specified at the cross-sections z=Ls and z=Ls, i.e. uB ðLs þ 0Þ ¼ uB ðLs 0Þ
ð37Þ
B ðLs þ 0Þ ¼ BðLs 0Þ
ð38Þ
fB ðLs
þ 0Þ ¼
fB ðLs
0Þ
uB ðLs þ 0Þ ¼ uB ðLs 0Þ B
ðLs þ 0Þ ¼
sB
fB ðLs
fB ðLs
þ 0Þ ¼
ð39Þ
ð40Þ
ð41Þ
0Þ
ð42Þ
The solutions for state ‘B’ can be derived as follows:
f 1 Ls ¼ D 8 s k
uB0 ¼
uB ¼
2 2DsB DsA A D s f 2 f þ Ef k2 8Ef sA 16Ef s
ð43Þ
B B f þ f T T 0 z Ef
4DsA z Ls exp k þ Ef k2 D 8D s z L-s exp k Ef k2 D
>
> > > 8 s z L-s > > exp k > > k D > > > > > > > > > > 4sA z Ls > > : exp k þ Bf k D
0 < z < L-s L-s < z < Ls Ls < z < 1
ð46Þ 8 B s > > > > > > > > > z L-s > A > > þ 2 exp k s > s > D < B
¼ > z L-s > > 2 s exp k > > D > > > > > > > z Ls > A > : s exp k D
0 < z < L-s
L-s < z < Ls Ls < z < 1 ð47Þ
where s ¼
s ðTmax Þ þ s ðTmin Þ 2
ð48Þ
3.3. Cyclic plasticity zone ð44Þ
8 > 2 B > > s z2 þ f T B T0 z þ uB0 0 < z < L-s > > Ef D > > > > > > > 2sA z2 8D s z L-s > > > exp k > > Ef D Ef k2 D > > L-s < z < Ls
> > B > D f > A > f T T0 z þ u0 > > > Ef < > > > > > > > > > > > > > > > > > > > > > > > > > > > :
fB ¼
8 4 B > > > s z > D > > > > > > 8 s z L-s > A z > > > 4s D þ k exp k > D > > > > > < f
For the thermomechanical cycles after the first cycle, the analyses can be conducted by the method similar to that done for the first cycle. It can be shown that the solutions for all cycles are the same as those for the first cycle. The derivation details are not included here because there is no fundamental difficulty but the tedious mathematics. To be emphasized are two aspects. First, each cycle is divided into two increments by loading and unloading. Secondly, the composite length is divided into sub-segments according to yield length caused by the present loading increment as well as the yield lengths have been induced in the past loading histories. The cyclic shear plasticity of the interface will take place only if L-s > 0, i.e.
Ls < z < 1
ð45Þ
þ pT >
8Ec s in-phase TMF kEf
ð49Þ
pT >
8Ec s out-of-phase TMF kEf
ð50Þ
647
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Fig. 4. The cyclic thermoplasticity zone for the B/Al composite. Above the curves the TMF loads can produce the cyclic plastic shearing of matrix, below the curves the complete elastic unloading–reloading.
Otherwise there is no alternating plastic shear, and interface undergoes cyclic elastic deformation over the whole length after initial yielding of length Ls, i.e. shakedown. In T diagram the curve defined by L-s=0 separates the T plane into two area, one is called the cyclic plasticity zone, of which a TMF load produces the cyclic plastic shear deformation of interface. Fig. 4 illustrates such zones associated with four cases (two minimum temperatures along with each of the IP and OP conditions) for the B/Al composite. Each curve defines the boundary of the cyclic plasticity zone of that situation. The area above the curve is where the cyclic thermoplastic shearing of interface occurs. It can be seen that the cyclic plasticity zone of the IP condition is larger than that of the OP condition. And the zone gets wider as the minimum temperature is getting higher for both the IP and OP conditions. Under the TMF loads belonging to the cyclic plasticity zone the interface experiences an alternating shear yielding within the length L-s with the TMF cycles. The alternating plastic shear strain range can be derived from Eqs. (5), (6), (25) and (45)
The fiber stress range and the opening displacement range of the fiber break, , have the forms: f ¼ fA fB 8 z > 8 s > > < D ¼
> > > f 8 s exp k z L-s : k D
0 < z < Ls L-s < z < 1 ð52Þ
8D s D s f 2 B ¼ 2 uA u þ ¼ 0 0 Ef k2 8Ef s
ð53Þ
The cyclic plastic shear strain range may be used to predict the debonding caused by low cycle fatigue by incorporating Eq. (51) into the fatigue equations of Manson–Coffin type.
4. Results pA
pB
p ¼ 8 2 > 2 s k L-s z < s k2 L-s z 0 < z < L-s ¼ D Gm D Gm > : 0 z > L-s ð51Þ
For illustration the Boron/Al continuous fiber composite is examined. The material system and the TMF loading paths are given in accordance with the TMF tests that were carried out for B/Al in Reference [12]. It is assumed that there are no manufacturing initial stresses by assuming stress-free temperature T0=20 C
648
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
and the yield shear stress of the interface at room temperature s0 ¼185 MPa. The properties of the constituents are given below Ef ¼ 400 GPa; f ¼ 6:3 mE= C; D ¼ 0:14 mm; Vf ¼ 0:48; Em ¼ 70:2 GPa; m ¼ 0:33; m ¼ 23:9 mE= C; c1 ¼ 191 MPa; c2 ¼ 3:68 MPa; c3 ¼ 0:011 ð1= CÞ: The constants c1, c2 and c3 are obtained by fitting the Eq. (14) to the yield stress data ranging from temperature T=20 C up to 350 C. The effects of the thermomechanical phase conditions as well as applied stress and temperature levels on the stress profiles, cyclic plastic shear strain range of the interface are illustrated. For the purpose each figure includes four curves, two of which belong to each of in-phase (IP) and out-of-phase (OP) thermomechanical loads. And the two curves of the same phase condition are associated with either a constant mechanical stress along with two temperature levels or an unchanged temperature along with two stress levels. In all figures the data belonging to the IP are represented by the open symbols while the data associated with the OP by the solid symbols. 4.1. The effects of mechanical load Figs. 5 and 6 show the effects of the mechanical loads on the interface shear stress profiles and the fiber stress distributions, respectively, along the z-axis at the state
of maximum mechanical load (state ‘A’). The four curves in each figure are associated with the same thermal cycle (250 C() 350 C). The plots of the stresses versus z-coordinate include two regions, the yield portion and the fiber stress recovery region. The yield length, Ls, increases with the applied mechanical stress for both IP and OP conditions. For the two mechanical load levels the interface yield length, Ls, associated with the IP is considerably longer than that for the OP. There are two reasons. First, IP condition makes the remote fiber tensile stress bigger, which can be seen in Fig. 6, consequently causes a longer yield length Ls from Eq. (22). Secondly, under IP condition the maximum mechanical stress is accompanied by the maximum temperature, in turn the smaller yield shear stress, which causes the yield length longer further through Eq. (22). Figs. 7 and 8 illustrate the interface shear stress profiles and the fiber stress distributions, respectively, along the z-axis at the state of the minimum mechanical load (state ‘B’). Again, the four curves in each figure are associated with the same thermal cycle (250 C () 350 C). The plots of the shear or fiber stress against distance from the break consist of three regions. When the mechanical stress is released from state ‘A’ to state ‘B’, the reverse yielding of the interface takes place within the portion, 04z4L-s, where the interface shear stress is constant but changes sign, comparing with the state ‘A’. The second region,L-s 4z4Ls, is where the interface experiences the initial yielding followed by cyclic elastic shearing, and the shear stress varies with
Fig. 5. Interfacial shear stress distributions versus distance from the break for state ‘A’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
z-coordinate dramatically. The fiber stress is compressive in these two regions, reaching a maximum value considerably high in the second region. The third one is recovery region where the fiber stress
649
approaches the far-field stress with increasing distance from the break. In Fig. 9 included are the four plots of the plastic shear strain range of interface, p , against z-coordinate
Fig. 6. Fiber stress distributions versus distance from the break for state ‘A’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
Fig. 7. Interfacial shear stress distributions versus distance from the break for state ‘B’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
650
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
under the same thermal cycle (250 C () 350 C) plus the different cyclic mechanical loads. It can be seen that the interface undergoes an alternating plastic shear in the region, 04z4L-s for both IP and OP thermo-
mechanical fatigue loads. However, the cyclic thermoplastic length, L-s, and the plastic strain range, p , of the IP thermomechanical fatigue condition are larger than those of the OP thermomechanical fatigue condi-
Fig. 8. Fiber stress distributions versus distance from the break for state ‘B’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
Fig. 9. Cyclic plastic shear strain range distributions versus distance from the break: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
tion. Moreover, L-s and Dp increase with an increasing the maximum applied stress. 4.2. The effects of thermal load In order to see the effects of thermal loads, in Figs. 10– 14 the stress and plastic strain range profiles are plotted
651
by using different temperature loads the but keeping mechanical fatigue load unchanged max ¼ 12 . It can 0 s be seen that the influence of the thermal load is similar with the effect of the mechanical load. The initial yield length, Ls, the cyclic thermoplastic length, L-s, and the plastic strain range, p increase for both IP and OP thermomechanical fatigue conditions as the maximum
Fig. 10. Interfacial shear stress distributions versus distance from the break for state ‘A’: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
Fig. 11. Fiber stress distributions versus distance from the break for state ‘A’: two thermal load levels under each of the IP and OP conditions, but
652
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Fig. 12. Interfacial shear stress distributions versus distance from the break for state ‘B’: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
Fig. 13. Fiber stress distributions versus distance from the break for state ‘B’: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
653
Fig. 14. Cyclic plastic shear strain range distributions versus distance from the break: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
temperature increases. The IP condition produces a higher plastic shear strain range than the OP condition does.
5. Concluding remarks Modeling the thermomechanical fatigue life of the fiber composite materials with ductile matrix requires an understanding of how stresses are distributed for the each individual failure mode. The fiber breakage is the main failure mode in metal matrix composites under high stress thermomechanical fatigue. A model to predict the local cyclic plasticity of the matrix around a broken fiber in ductile matrix composites under the inphase and out-of-phase thermomechanical fatigue (TMF) loads has been developed based on the singlefiber shear-lag analysis. In T plane the so-called cyclic plasticity zone was identified within which the TMF loads will cause the cyclic plasticity of the matrix. A new parameter, L-s, i.e. cyclic plasticity length, was discovered within which the matrix undergoes alternating plastic deformation. L-s is smaller that the static yield length, Ls, caused by the first half cycle. The closed-form expressions for L-s, Ls and the plastic strain range, p , was obtained. The cyclic plasticity length, L-s, and the plastic strain range, p increase for both the in-phase and out-of-phase TMF conditions as the mechanical load and/or thermal load
increase. The in-phase condition produces a higher plastic shear strain range than the out-of-phase condition does. The solutions obtained may be used for modeling fiber/matrix debonding caused by the fiber breakage under TMF fatigue loading. The generalization is needed and also possible to multiple-fiber breakage under TMF loads in order to use the solutions to predict the TMF life. Now let us discuss the differences between the elastic, perfectly-plastic model and the frictional slip model. In the first half cycle (loading) the distribution of the shear stress from the plasticity model is continuous along the whole length whereas the shear stress from the frictional slip model is discontinuous across the debond front between the debonded and bonded parts. In the plasticity model there is a critical TMF load range. The cyclic plasticity, which dissipates energy, can occur only if the TMF load range is higher than the critical value, otherwise there is no cyclic plasticity at all, i.e. shakedown. In contrast, the frictional slip always dissipates the energy regardless how small the TMF load range is. Most importantly, cyclic plasticity length, L-s, does not change as the TMF cycles increase whereas the slip length increases monotonically with cycles. The both models have their own material regimes wherein the models are valid. The plasticity model is likely suitable for the material systems with a strong interface while the slip model applies to the materials with a weak interface.
654
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Acknowledgements The authors gratefully acknowledge the financial supports by the National Science Foundation of China under grant number 19972076. References [1] Talreja R. A conceptual framework for interpretation of MMC fatigue. Materials Science and Engineering 1995;A200:21–8. [2] Majumdar BS, Newaz GM. Constituent damage mechanisms in metal matrix composites under fatigue loading, and their effects on fatigue life. Materials Science and Engineering 1995;A200:114–29. [3] Evans AG, Zok FW, McMeeking RM. Fatigue of ceramic matrix composites. Acta Metall Mater 1995;43:859–75. [4] Beyerlein IJ, Phoenix SL. Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic influence superposition. J Mech Phys Solids 1996; 44:1997–2039. [5] Cox HL. The elasticity and strength of paper and other fibrous materials. Br J Appl Phys 1952;3:72–9. [6] Hedgepeth JM, van Dyke P. Local stress concentrations in imperfect filamentary composite materials. J Composite Materials 1967;1:294–310.
[7] Henstenburg RB, Phoenix SL. Interfacial shear strength studies using the single-filament-composite test. Part II: a probability model and Monte Carlo simulation. Polymer Composites 1989; 10:389–408. [8] Beyerlein IJ, Phoenix SL. Stress profiles and energy release rates around fiber breaks in a lamina with propagating zones of matrix yielding and debonding. Composites Science and Technology 1997;57:869–85. [9] Landis CM, McMeeking RM. A shear-lag model for a broken fiber embedded in a composite with a ductile matrix. Composites Science and Technology 1999;59:447–57. [10] Tripathi D, Chen F, Jones FR. The effect of matrix plasticity on the stress fields in a single-filament composite and the value of interfacial shear strength obtained from the fragmentation test. Proc Roy Soc Lond A 1996;452:621–53. [11] Goree JG, Dharani LR, Jones WF. Crack growth and fracture of continuous fiber metal matrix composites: analysis and experiments. In: Johnson WJ, editor. Metal matrix composites: testing, analysis, and failure modes (ASTM STP 1032). Philadelphia: American Society for Testing and Materials, 1989. p. 251–69. [12] Liu SL, He YH. Thermo-mechanical fatigue behavior of a unidirectional B/Al metal matrix composite. In: Zhang Y, editor. Extended abstract of 13th International Conference on Composite Materials, Beijing China, 2001. p. 326.
Cyclically thermomechanical plasticity analysis for a broken fiber in ductile matrix composites using shear lag model Junqian Zhanga,*, Jian Wua, Shaolun Liub a
Department of Engineering Mechanics, Chongqing University, 400044 Chongqing, China b Beijing Institute of Aeronautical Materials, PO Box 81-23, 100095 Beijing, China Received 16 November 2001; accepted 17 December 2001
Abstract The local cyclic plasticity of the interface around a broken fiber in ductile matrix composites under the in-phase and out-of-phase thermomechanical fatigue (TMF) loads is analyzed by using the single-fiber shear-lag model. The elastic, perfectly-plastic shear stress–strain relation is used to model the thermomechanical behavior of the fiber/matrix interface. It is shown that the alternating plastic shearing of the interface takes place under an appropriate combination of mechanical stress and thermal load. In the stress versus temperature diagram the so-called cyclic plasticity zone is identified. A new parameter, i.e. the cyclic plasticity length, Ls, is found which is smaller than the yield length, Ls, caused by monotonic loading. The closed-form solutions for Ls, Ls, the fiber stress profiles and the cyclic plastic shear strain range, p , are obtained. Ls and p increase for both the in-phase and out-of-phase TMF conditions as the mechanical load and/or thermal load increase. The in-phase condition produces a higher plastic shear strain range than the out-of-phase condition does. The solutions obtained may be used for modeling fiber/matrix debonding caused by the fiber breakage under TMF fatigue loading. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Fiber composites
1. Introduction The dominant damage and failure modes in continuous fibers reinforced metal matrix composites (MMCs) can change with the magnitude, type and path of thermomechanical loading. The fatigue life diagram of the maximum applied strain versus the cycles to failure was proposed in [1,2] for understanding fatigue of the MMCs. The diagram shows three regimes: regime 1, 2, 3 (high, intermediate and low applied strain), in which the dominate failure mode is the fiber breakage, fiber-bridged matrix cracking and matrix cracking, respectively. The modeling of fiber-bridged matrix cracking has been investigated extensively in the literature (see [3] and references therein). In this paper we will consider the fiber breakage that is the dominant failure mode observed in the MMCs loaded in high stress * Corresponding author at present address: Laboratory for Technical Mechanics, University of Paderborn, 33098 Paderborn, Germany. Fax: +49-5251-603483. E-mail address: [email protected] (J. Zhang).
(regime I). The fiber breaks cause stress concentration that stimulates either the local plasticity or debonding around the breaks even if the overall behaviors of the composite are still elastic. As pointed out by Beyerlein and Phoenix [4], plastic deformation of the matrix will occur instead of fiber-matrix debonding if the interfacial strength is much higher than the matrix yield stress in shear. In this case under thermomechanical fatigue (TMF) loads the matrix may experience an alternating plastic deformation, which leads to the fiber/matrix debonding. Understanding of the cyclic deformation around the fiber break is essential to model the debonding under fatigue, in turn, could help us to develop mechanistic-based life prediction methodologies for thermomechanical fatigue. Shear-lag models of varying degrees of complexity have been used to determine the stresses in a broken fiber with reasonable accuracy, although they made some simplified assumptions. The first shear-lag model in the literature was proposed by Cox [5] to determine the stresses in a fiber embedded in an elastic matrix. This model assumes that the interface between the fiber
0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00022-2
642
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Fig. 1. The composite unit cell for the single-fiber shear-lag model, (a) Cross-section, (b) Side view.
and the matrix is well bonded. Plasticity and debonding in multiple-fiber systems are modeled in the references [6,7] by adding the shear yield at the interface into Cox’s model. And the models result in a linear fiber stress recovery from the break. The so-called bi-linear model was proposed in [3,8] where the interface was characterized by two shear stress values associated with matrix yielding and frictional sliding, respectively. More recently, Landis and McMeeking [9] made a further modification by introducing both axial and shear stresses into an elastic/perfectly-plastic flow rule for the matrix. Finite element analyses were conducted in [9,10], suggesting the shear-lag model is adequate in predicting the fiber stress recovery although it makes several simplifying assumptions. The mentioned models considered the monotonic mechanical loading only. This paper concerns the local cyclic plasticity of the matrix around a broken fiber in MMCs under thermomechanical fatigue loading. The in-phase and out-of-phase TMF conditions are considered. In order to obtain the stress profiles and the cyclically thermoplastic shear strain of interface the analysis is done by using the shear-lag model and the closed-form solutions will be derived.
equation for a differential element of the broken fiber is @f ¼ 4 D @z
ð2Þ
where f is the average axial stress of the fiber and is the shear stress acting on the interface between the fiber and the matrix. The fiber is considered to be an elastic spring such that radial, hoop and shear stresses are neglected, and the thermoelastic constitutive law for the fiber gives @u f ðT T0 Þ f ¼ Ef ð3Þ @z where u is the axial displacement of the fiber; Ef and f are the Young’s modulus and the coefficient of thermal expansion of the fiber, respectively; T and T0 are temperatures of the current state and the stress-free state, respectively. The relationship between the interfacial shear stress, , and the fiber displacement, u, is obtained by substituting Eq. (3) into Eq. (2) 1 d2 u ¼ DEf 2 4 dz
ð4Þ
2. Basic equations for the shear-lag model Consider a unidirectional continuous fiber composite with a hexagonal array of cylindrical fibers (Fig. 1). It is assumed that a single-fiber break takes place at z=0. The distance between the broken fiber and the six immediate neighbor fibers, w, can be related to the fiber volume fraction, Vf [9], rffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 1 D ð1Þ w¼ 2 3V f where D is the fiber diameter. The composite is subjected to a cyclically thermomechanical load. Equilibrium
The shear strain of the matrix surrounding the broken fiber is approximated by [9] ¼
uc u w
ð5Þ
where uc is the displacement of the outer boundary of the unit cell. The approximation made to the singlefiber shear-lag model is that the axial displacement at the outer boundary of the cell is the same as the displacement that would exist in an undamaged composite at the same applied stress [9]. This condition is given by
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
f uc ¼ "z ¼ þ f ðT T0 Þ z Ef
ð6Þ
where " denotes the nominal composite strain and f the remote fiber stress. Eqs. (5) and (6) imply assumptions: (1) the matrix does not carry the load released by the fiber breaking, and only transmits shear between the fibers; (2) the load released by the breaking is equally carried by all of the intact fibers. The assumption (2) can be released by using the multiple-fiber shear-lag models [4]. uc would be replaced by the displacement of the nearest neighboring fiber and follows a differential equation other than the Eq. (6). And these would result in a set of coupled differential equations governing the displacements of the fibers. The assumption (1) is most accurate for composites with the high fiber volume fraction and the high modulus ratio of fiber to matrix. Most polymer composites and some metal-matrix composites, such as Boron/Al and SiC/Ti, satisfy these conditions [8]. The remote fiber stress, f , is related to applied composite stress and temperature by using the rule of mixture as follows, f ¼
Ef ½ þ pðT T0 Þ; p ¼ E m ð1 Vf Þðm f Þ Ec
ð7Þ
where denotes the applied composite stress; Ec stands for the Young’s modulus of the composite. At the fiber break the axial stress in the fiber is zero, and far from the break the fiber strain must be equal to the applied composite strain. Mathematically these boundary conditions are @u f ðT T0 Þ ¼ 0 Ef ð8Þ @z z¼0 @u f ¼ þ f ðT T0 Þ @z z ! 1 E f
Eqs. (2)–(9) are common to any shear-lag analysis for a single-fiber break. In the succeeding section they will be used along with the cyclic behavior of the interface to describe the stress and deformation responses under the TMF loads with both in-phase and out-of-phase conditions.
3. Thermomechanical fatigue loads Both in-phase and out-of-phase thermomechanical fatigue (TMF) loads are considered (Fig. 2). The tensile stress applied to composites varies between min and max while the temperature changes between Tmin and Tmax. Consequently, the applied stress range and temperature range are defined by ¼ max min
ð10Þ
T ¼ Tmax Tmin
ð11Þ
For convenience the state ‘A’ is used to represent the state of TMF load with the maximum applied stress for both in-phase and out-of-phase conditions, and state ‘B’ the state of TMF load with minimum mechanical load, (Fig. 2). In what follows the quantities with superscripts ‘A’ and ‘B’ belong to the state ‘A’ and state ‘B’, respectively. Assuming that the fiber and the matrix far from the fiber break in the composite are thermoelastic, the remote fiber stress range is derived from the Eq. (7) for in-phase and out-of-phase TMF loads, respectively, B f ¼ A f f
¼ ð9Þ
643
8E f > > < E ð þ pTÞ in-phase c
> > : Ef ð pTÞ out-of-phase Ec
Fig. 2. A sketch of the thermomechanical fatigue loads, (a) In-phase condition, (b) Out-of-phase condition.
ð12Þ
644
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
As the thermomechanical loads increase the local yielding of matrix around the fiber break or the fiber/ matrix debonding will occur due to stress concentrations. If the interfacial strength is higher than the matrix yield stress in shear the plastic deformation of the matrix will occur first instead of fiber/matrix debonding [4]. This was observed in ductile-resin and some metal matrix composites with a strong interface, such as Boron/Al composite system [11]. For the composites with a weak interface the debonding and the frictional slip take place. Here we consider the strong interfaces. It is assumed that the local matrix yield is caused by the shear stress only, ignoring the effect of the radial, hoop and tensile stresses. The local plasticity is modeled by the elastic, perfectly-plastic shear stress–strain relation, i.e.
¼
Gm jj < s s ðTÞ yielded
jump at the moment that the slip displacement changes sign. This difference between the constitutive laws will cause differences in the solutions. Discussion will be given in some details in the last section. Analysis of the stresses and the deformation, involving the cyclic thermoplasticity, must be carefully carried out by applying thermomechanical fatigue load to the composites incrementally. 3.1. Initial loading (O!A) Let us consider the TMF loading process (O!A), for either the in-phase or out-of-phase condition. The local yield takes place within the length of Ls, which is to be determined by the continuity conditions between yield and elastic segments. By combining Eqs. (2)–(7) the governing differential equations for the state ‘A’ are obtained,
ð13Þ A A @ 2 uA 2 A 2 f D k u þk þ f T T0 z ¼ 0 @z2 Ef 2
where s and Gm are the yield shear stress and the shear modulus of the matrix, respectively; and denote the interfacial shear stress and shear strain, respectively. This interface constitutive law allows the interface to undergo the cyclic plasticity under thermomechanical fatigue loading (Fig. 3). The yield shear stress depends upon the temperature by s ¼ c1 þ c2 ½1 expðc3 TÞ
ð14Þ
where c1, c2 and c3 are material constants. The elastic unloading path of the elastic, perfectly-plastic model differentiates it from the frictional slip model where the shear stress changes sign immediately as long as the slip displacement changes sign regardless how small the slip displacement is. In other word, the stress–strain curve of the elastic, perfectly-plastic model is continuous everywhere whereas the shear stress of the frictional slip model has a
ð15Þ
for z>Ls @fA A ¼ 4 s for z
ð16Þ
ð17Þ
The boundary conditions Eqs. (8) and (9) become A @u Ef f T A T0 jz¼0 ¼ 0 ð18Þ @z A @uA jz ! 1 ¼ f þ f TA TA 0 @z Ef
ð19Þ
for the state ‘A’. The continuity conditions between yield and elastic segments lead to
Fig. 3. An illustration of constitutive behavior of the interface.
uA ðLs þ 0Þ ¼ uA ðLs 0Þ
ð20Þ
A ðLs þ 0Þ ¼ sA
ð21Þ
fA ðLs þ 0Þ ¼ fA ðLs 0Þ
ð22Þ
By solving the governing Eqs. (15) and (16) and by using the boundary conditions and the continuity conditions, one can derive the yield length, fiber break opening displacement, the axial displacement and the axial stress of the fiber, and the shear stress at the fiber/matrix interface for the state ‘A’,
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
A 1 Ls ¼ D fA k 4s A ¼ 2uA 0 ¼
ð23Þ
2 4DsA DsA A f þ Ef k2 4Ef sA
ð24Þ
8 > 2 A z2 > > s þ f T A T 0 z þ u A > 0 > Ef D > > > > >
< 4DsA z Ls A u ¼ exp k > k2 E f D > > > > > > A > A > f > þ f T T 0 z :þ Ef
0 4 z 4 Ls
Ls 4 z 4 1 ð25Þ
8 A z > < 4s D
fA ¼ 4 A z Ls A > : f s exp k k D
0 4 z 4 Ls Ls 4 z 4 1
645
unloading, and the shear stress at z=0 relaxes faster than that at any other positions. Thus, the shear stress of z=0 will change sign most first. Subsequently, there are two possible cases: the complete elastic unloading and the reverse shear plastic yielding within a length, depending on the TMF load range. If the TMF load range is not such large that the shear stress of z=0 does not reaches the reverse yield stress before state ‘B’, the TMF unloading is completely elastic. In this case the TMF unloading-reloading will be totally elastic, i.e. shakedown occurs. If the TMF load range is sufficiently large the shear stress in the matrix of z=0 will change sign and will reach the reverse yield stress at some moment before the thermomechanical load reaches state ‘B’. With continuous thermomechanical unloading the reverse plastic yield of the matrix will extend forward from the fiber break tip (z > 0). Up to the moment ‘‘B’’ the reverse plastic yield of the matrix has occurred within a certain length, say Ls. Let us consider the case that the reverse shear yielding takes place. Within the reverse yield length, Ls, which is still unknown, the equilibrium equation is
ð26Þ
A ¼
8 A < s
z Ls : sA exp k D
@fB B ¼ 4 s for z
0 4 z 4 Ls Ls 4 z 4 1
ð27Þ
where A stands for the fiber break opening displacement. With the aid of Eqs. (5), (6) and (25), the plastic shear strain of the matrix near the interface is obtained, 8 A k Ls z < sA k2 Ls z2 0 z Ls s A p ¼ Gm D D : 2Gm 0 z Ls ð28Þ The Eqs. (23)–(28) are valid only if Ls > 0, i.e. 4Ec A max þ p T A T0 > kEf s
ð30Þ
The stress and displacement of the fiber for z < Ls can be easily obtained by solving the Eq. (30). In the portion ðLs < z < 1Þ where the interface undergoes elastic deformation during the unloading from state ‘A’ to state ‘B’, the differential equation governing the incremental displacement of the fiber is derived by appropriately using the equations in Section 2, @2 uAB k2 uAB @z2 f þ k2 þ f T B T A z Ef
D2
¼ 0 for z > Ls ð29Þ
Otherwise the elastic shear lag solutions of Cox should be used. Formula (29) defines what thermomechanical loads can cause the local yielding. The mechanical load required for the local yielding decrease as the temperature increases. From Eq. (23) the local yielding (Ls > 0) indicates that the remote fiber stress, A f , is positive.
ð31Þ
Solving the differential equation leads to the incremental solutions with some undetermined constants for the unloading process (A!B). Then, the solutions for the stresses and the deformation at the state ‘B’ can be obtained by superposing the incremental solutions between A and B onto the solutions of the state ‘A’, i.e., uB ¼ uA þ uAB
ð32Þ
fB ¼ fA þ fAB
ð33Þ
B ¼ A þ AB
ð34Þ
3.2. Unloading (A!B) When the TMF load changes from state ‘A’ to state ‘B’ under either in-phase or out-of-phase condition, the yielded matrix around the fiber break starts the elastic
for the portion ðLs < z < 1Þ. The boundary conditions (8) and (9) become
646
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
B @u f T B T0 z¼0 ¼ 0 @z
Ef
@uB B jz ! 1 ¼ f þ f TB TB0 @z Ef
ð35Þ
ð36Þ
for the state ‘B’. The continuity conditions must be specified at the cross-sections z=Ls and z=Ls, i.e. uB ðLs þ 0Þ ¼ uB ðLs 0Þ
ð37Þ
B ðLs þ 0Þ ¼ BðLs 0Þ
ð38Þ
fB ðLs
þ 0Þ ¼
fB ðLs
0Þ
uB ðLs þ 0Þ ¼ uB ðLs 0Þ B
ðLs þ 0Þ ¼
sB
fB ðLs
fB ðLs
þ 0Þ ¼
ð39Þ
ð40Þ
ð41Þ
0Þ
ð42Þ
The solutions for state ‘B’ can be derived as follows:
f 1 Ls ¼ D 8 s k
uB0 ¼
uB ¼
2 2DsB DsA A D s f 2 f þ Ef k2 8Ef sA 16Ef s
ð43Þ
B B f þ f T T 0 z Ef
4DsA z Ls exp k þ Ef k2 D 8D s z L-s exp k Ef k2 D
>
> > > 8 s z L-s > > exp k > > k D > > > > > > > > > > 4sA z Ls > > : exp k þ Bf k D
0 < z < L-s L-s < z < Ls Ls < z < 1
ð46Þ 8 B s > > > > > > > > > z L-s > A > > þ 2 exp k s > s > D < B
¼ > z L-s > > 2 s exp k > > D > > > > > > > z Ls > A > : s exp k D
0 < z < L-s
L-s < z < Ls Ls < z < 1 ð47Þ
where s ¼
s ðTmax Þ þ s ðTmin Þ 2
ð48Þ
3.3. Cyclic plasticity zone ð44Þ
8 > 2 B > > s z2 þ f T B T0 z þ uB0 0 < z < L-s > > Ef D > > > > > > > 2sA z2 8D s z L-s > > > exp k > > Ef D Ef k2 D > > L-s < z < Ls
> > B > D f > A > f T T0 z þ u0 > > > Ef < > > > > > > > > > > > > > > > > > > > > > > > > > > > :
fB ¼
8 4 B > > > s z > D > > > > > > 8 s z L-s > A z > > > 4s D þ k exp k > D > > > > > < f
For the thermomechanical cycles after the first cycle, the analyses can be conducted by the method similar to that done for the first cycle. It can be shown that the solutions for all cycles are the same as those for the first cycle. The derivation details are not included here because there is no fundamental difficulty but the tedious mathematics. To be emphasized are two aspects. First, each cycle is divided into two increments by loading and unloading. Secondly, the composite length is divided into sub-segments according to yield length caused by the present loading increment as well as the yield lengths have been induced in the past loading histories. The cyclic shear plasticity of the interface will take place only if L-s > 0, i.e.
Ls < z < 1
ð45Þ
þ pT >
8Ec s in-phase TMF kEf
ð49Þ
pT >
8Ec s out-of-phase TMF kEf
ð50Þ
647
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Fig. 4. The cyclic thermoplasticity zone for the B/Al composite. Above the curves the TMF loads can produce the cyclic plastic shearing of matrix, below the curves the complete elastic unloading–reloading.
Otherwise there is no alternating plastic shear, and interface undergoes cyclic elastic deformation over the whole length after initial yielding of length Ls, i.e. shakedown. In T diagram the curve defined by L-s=0 separates the T plane into two area, one is called the cyclic plasticity zone, of which a TMF load produces the cyclic plastic shear deformation of interface. Fig. 4 illustrates such zones associated with four cases (two minimum temperatures along with each of the IP and OP conditions) for the B/Al composite. Each curve defines the boundary of the cyclic plasticity zone of that situation. The area above the curve is where the cyclic thermoplastic shearing of interface occurs. It can be seen that the cyclic plasticity zone of the IP condition is larger than that of the OP condition. And the zone gets wider as the minimum temperature is getting higher for both the IP and OP conditions. Under the TMF loads belonging to the cyclic plasticity zone the interface experiences an alternating shear yielding within the length L-s with the TMF cycles. The alternating plastic shear strain range can be derived from Eqs. (5), (6), (25) and (45)
The fiber stress range and the opening displacement range of the fiber break, , have the forms: f ¼ fA fB 8 z > 8 s > > < D ¼
> > > f 8 s exp k z L-s : k D
0 < z < Ls L-s < z < 1 ð52Þ
8D s D s f 2 B ¼ 2 uA u þ ¼ 0 0 Ef k2 8Ef s
ð53Þ
The cyclic plastic shear strain range may be used to predict the debonding caused by low cycle fatigue by incorporating Eq. (51) into the fatigue equations of Manson–Coffin type.
4. Results pA
pB
p ¼ 8 2 > 2 s k L-s z < s k2 L-s z 0 < z < L-s ¼ D Gm D Gm > : 0 z > L-s ð51Þ
For illustration the Boron/Al continuous fiber composite is examined. The material system and the TMF loading paths are given in accordance with the TMF tests that were carried out for B/Al in Reference [12]. It is assumed that there are no manufacturing initial stresses by assuming stress-free temperature T0=20 C
648
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
and the yield shear stress of the interface at room temperature s0 ¼185 MPa. The properties of the constituents are given below Ef ¼ 400 GPa; f ¼ 6:3 mE= C; D ¼ 0:14 mm; Vf ¼ 0:48; Em ¼ 70:2 GPa; m ¼ 0:33; m ¼ 23:9 mE= C; c1 ¼ 191 MPa; c2 ¼ 3:68 MPa; c3 ¼ 0:011 ð1= CÞ: The constants c1, c2 and c3 are obtained by fitting the Eq. (14) to the yield stress data ranging from temperature T=20 C up to 350 C. The effects of the thermomechanical phase conditions as well as applied stress and temperature levels on the stress profiles, cyclic plastic shear strain range of the interface are illustrated. For the purpose each figure includes four curves, two of which belong to each of in-phase (IP) and out-of-phase (OP) thermomechanical loads. And the two curves of the same phase condition are associated with either a constant mechanical stress along with two temperature levels or an unchanged temperature along with two stress levels. In all figures the data belonging to the IP are represented by the open symbols while the data associated with the OP by the solid symbols. 4.1. The effects of mechanical load Figs. 5 and 6 show the effects of the mechanical loads on the interface shear stress profiles and the fiber stress distributions, respectively, along the z-axis at the state
of maximum mechanical load (state ‘A’). The four curves in each figure are associated with the same thermal cycle (250 C() 350 C). The plots of the stresses versus z-coordinate include two regions, the yield portion and the fiber stress recovery region. The yield length, Ls, increases with the applied mechanical stress for both IP and OP conditions. For the two mechanical load levels the interface yield length, Ls, associated with the IP is considerably longer than that for the OP. There are two reasons. First, IP condition makes the remote fiber tensile stress bigger, which can be seen in Fig. 6, consequently causes a longer yield length Ls from Eq. (22). Secondly, under IP condition the maximum mechanical stress is accompanied by the maximum temperature, in turn the smaller yield shear stress, which causes the yield length longer further through Eq. (22). Figs. 7 and 8 illustrate the interface shear stress profiles and the fiber stress distributions, respectively, along the z-axis at the state of the minimum mechanical load (state ‘B’). Again, the four curves in each figure are associated with the same thermal cycle (250 C () 350 C). The plots of the shear or fiber stress against distance from the break consist of three regions. When the mechanical stress is released from state ‘A’ to state ‘B’, the reverse yielding of the interface takes place within the portion, 04z4L-s, where the interface shear stress is constant but changes sign, comparing with the state ‘A’. The second region,L-s 4z4Ls, is where the interface experiences the initial yielding followed by cyclic elastic shearing, and the shear stress varies with
Fig. 5. Interfacial shear stress distributions versus distance from the break for state ‘A’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
z-coordinate dramatically. The fiber stress is compressive in these two regions, reaching a maximum value considerably high in the second region. The third one is recovery region where the fiber stress
649
approaches the far-field stress with increasing distance from the break. In Fig. 9 included are the four plots of the plastic shear strain range of interface, p , against z-coordinate
Fig. 6. Fiber stress distributions versus distance from the break for state ‘A’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
Fig. 7. Interfacial shear stress distributions versus distance from the break for state ‘B’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
650
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
under the same thermal cycle (250 C () 350 C) plus the different cyclic mechanical loads. It can be seen that the interface undergoes an alternating plastic shear in the region, 04z4L-s for both IP and OP thermo-
mechanical fatigue loads. However, the cyclic thermoplastic length, L-s, and the plastic strain range, p , of the IP thermomechanical fatigue condition are larger than those of the OP thermomechanical fatigue condi-
Fig. 8. Fiber stress distributions versus distance from the break for state ‘B’: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
Fig. 9. Cyclic plastic shear strain range distributions versus distance from the break: two mechanical load levels under each of the IP and OP conditions, but the same thermal cycle (250 C () 350 C).
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
tion. Moreover, L-s and Dp increase with an increasing the maximum applied stress. 4.2. The effects of thermal load In order to see the effects of thermal loads, in Figs. 10– 14 the stress and plastic strain range profiles are plotted
651
by using different temperature loads the but keeping mechanical fatigue load unchanged max ¼ 12 . It can 0 s be seen that the influence of the thermal load is similar with the effect of the mechanical load. The initial yield length, Ls, the cyclic thermoplastic length, L-s, and the plastic strain range, p increase for both IP and OP thermomechanical fatigue conditions as the maximum
Fig. 10. Interfacial shear stress distributions versus distance from the break for state ‘A’: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
Fig. 11. Fiber stress distributions versus distance from the break for state ‘A’: two thermal load levels under each of the IP and OP conditions, but
652
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Fig. 12. Interfacial shear stress distributions versus distance from the break for state ‘B’: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
Fig. 13. Fiber stress distributions versus distance from the break for state ‘B’: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
653
Fig. 14. Cyclic plastic shear strain range distributions versus distance from the break: two thermal load levels under each of the IP and OP conditions, but the same stress cycle (0 MPa () 12s0 MPa).
temperature increases. The IP condition produces a higher plastic shear strain range than the OP condition does.
5. Concluding remarks Modeling the thermomechanical fatigue life of the fiber composite materials with ductile matrix requires an understanding of how stresses are distributed for the each individual failure mode. The fiber breakage is the main failure mode in metal matrix composites under high stress thermomechanical fatigue. A model to predict the local cyclic plasticity of the matrix around a broken fiber in ductile matrix composites under the inphase and out-of-phase thermomechanical fatigue (TMF) loads has been developed based on the singlefiber shear-lag analysis. In T plane the so-called cyclic plasticity zone was identified within which the TMF loads will cause the cyclic plasticity of the matrix. A new parameter, L-s, i.e. cyclic plasticity length, was discovered within which the matrix undergoes alternating plastic deformation. L-s is smaller that the static yield length, Ls, caused by the first half cycle. The closed-form expressions for L-s, Ls and the plastic strain range, p , was obtained. The cyclic plasticity length, L-s, and the plastic strain range, p increase for both the in-phase and out-of-phase TMF conditions as the mechanical load and/or thermal load
increase. The in-phase condition produces a higher plastic shear strain range than the out-of-phase condition does. The solutions obtained may be used for modeling fiber/matrix debonding caused by the fiber breakage under TMF fatigue loading. The generalization is needed and also possible to multiple-fiber breakage under TMF loads in order to use the solutions to predict the TMF life. Now let us discuss the differences between the elastic, perfectly-plastic model and the frictional slip model. In the first half cycle (loading) the distribution of the shear stress from the plasticity model is continuous along the whole length whereas the shear stress from the frictional slip model is discontinuous across the debond front between the debonded and bonded parts. In the plasticity model there is a critical TMF load range. The cyclic plasticity, which dissipates energy, can occur only if the TMF load range is higher than the critical value, otherwise there is no cyclic plasticity at all, i.e. shakedown. In contrast, the frictional slip always dissipates the energy regardless how small the TMF load range is. Most importantly, cyclic plasticity length, L-s, does not change as the TMF cycles increase whereas the slip length increases monotonically with cycles. The both models have their own material regimes wherein the models are valid. The plasticity model is likely suitable for the material systems with a strong interface while the slip model applies to the materials with a weak interface.
654
J. Zhang et al. / Composites Science and Technology 62 (2002) 641–654
Acknowledgements The authors gratefully acknowledge the financial supports by the National Science Foundation of China under grant number 19972076. References [1] Talreja R. A conceptual framework for interpretation of MMC fatigue. Materials Science and Engineering 1995;A200:21–8. [2] Majumdar BS, Newaz GM. Constituent damage mechanisms in metal matrix composites under fatigue loading, and their effects on fatigue life. Materials Science and Engineering 1995;A200:114–29. [3] Evans AG, Zok FW, McMeeking RM. Fatigue of ceramic matrix composites. Acta Metall Mater 1995;43:859–75. [4] Beyerlein IJ, Phoenix SL. Stress concentrations around multiple fiber breaks in an elastic matrix with local yielding or debonding using quadratic influence superposition. J Mech Phys Solids 1996; 44:1997–2039. [5] Cox HL. The elasticity and strength of paper and other fibrous materials. Br J Appl Phys 1952;3:72–9. [6] Hedgepeth JM, van Dyke P. Local stress concentrations in imperfect filamentary composite materials. J Composite Materials 1967;1:294–310.
[7] Henstenburg RB, Phoenix SL. Interfacial shear strength studies using the single-filament-composite test. Part II: a probability model and Monte Carlo simulation. Polymer Composites 1989; 10:389–408. [8] Beyerlein IJ, Phoenix SL. Stress profiles and energy release rates around fiber breaks in a lamina with propagating zones of matrix yielding and debonding. Composites Science and Technology 1997;57:869–85. [9] Landis CM, McMeeking RM. A shear-lag model for a broken fiber embedded in a composite with a ductile matrix. Composites Science and Technology 1999;59:447–57. [10] Tripathi D, Chen F, Jones FR. The effect of matrix plasticity on the stress fields in a single-filament composite and the value of interfacial shear strength obtained from the fragmentation test. Proc Roy Soc Lond A 1996;452:621–53. [11] Goree JG, Dharani LR, Jones WF. Crack growth and fracture of continuous fiber metal matrix composites: analysis and experiments. In: Johnson WJ, editor. Metal matrix composites: testing, analysis, and failure modes (ASTM STP 1032). Philadelphia: American Society for Testing and Materials, 1989. p. 251–69. [12] Liu SL, He YH. Thermo-mechanical fatigue behavior of a unidirectional B/Al metal matrix composite. In: Zhang Y, editor. Extended abstract of 13th International Conference on Composite Materials, Beijing China, 2001. p. 326.