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Deformation behavior of model MMC scarf joints
Materials Science and Engineering A281 (2000) 113 – 125 www.elsevier.com/locate/msea
Deformation behavior of model MMC scarf joints D.D. Brink, J.C. Mailand 1, C.G. Levi *, F.A. Leckie Departments of Materials and Mechanical and En6ironmental Engineering, High Performance Composites Center, 1316D Engineering 11, Uni6ersity of California, Santa Barbara, CA 93106 -5050, USA Received 24 August 1999; received in revised form 26 October 1999
Abstract The mechanical response of metal interlayers constrained between two fiber-reinforced MMC (metal – matrix composites) sub-elements was investigated. The fibers were polycrystalline Al2O3, discontinuous at the joint, embedded in Al – 4.5Mg, that was continuous through the joint and constituted both the composite matrix and the interlayer material. The specimens were produced by pressurized melt infiltration and comprised interlayers oriented from 0 to 75° relative to the plane normal to the fiber axis. Analytical and finite element models were developed and compared with experimental results in order to elucidate the deformation behavior of these interlayers and to identify the key factors controlling joint performance. Two predominant regimes were identified in this manner. Joints in the intermediate angle range (45 9 15°) exhibit limit load behavior and develop large strains prior to failure, whereas joints at lower (5 15°) and higher (75°) angles show only modest plasticity. The models suggest that the latter group could, in principle, develop large strains and limit loads, but this behavior is precluded by the intervention of failure owing to debonding in the lower angle joints, and to composite fracture in the highest angle ones. The analysis further reveals that the constitutive behavior of the metal in the interlayer exhibits substantial hardening relative to that of the monolithic matrix processed in the same manner. The reasons for this behavior are discussed. The models set the stage for future work to elucidating failure criteria for interlayers in these joint configurations. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Structural behavior; Composite; Plastic; Computer simulation; Joining
1. Introduction The specific strength and stiffness of metal–matrix composites (MMCs) make them attractive materials for weight-sensitive components and structures. Full exploitation of these properties, however, is often hindered by inadequate understanding of the mechanisms that govern the performance of MMC joints and attachments to other elements of the structure. Joint design is typically based on the conservative assumption that interfaces normal to the fibers can carry no significant load. Hence, load transfer must be effected primarily by shear, leading to bulky joints and transitions that undermine the weight-saving potential of the MMC.
* Corresponding author. Tel. + 1-805-893-2381; fax: +1-805-8938486. 1 J.C. Mailand, previously graduate student researcher at UCSB, is now with CAMCO/Schlumberger Advanced Completions Group.
Previous work on Al-matrix/Al2O3 fiber composites has demonstrated that clean interfaces normal to the fiber ends have the capacity to support loads well in excess of the strength of the adjoining metal [1]. When coupled with model scarf configurations (Fig. 1) these interfaces have enabled joint strengths comparable to those of the composite [2]. Finite element analyses of butt joints (u= 0°) consisting of thin metal interlayers between two composite sub-elements [1] revealed the existence of two distinct regions within the interlayer, as illustrated in Fig. 1. The perimeter of the interlayer is characterized by localized inward shear over a distance L, with a buildup of hydrostatic stress away from the free surface as a consequence of the constraint imposed by the surrounding composite. Conversely, the stresses within the interlayer core are essentially uniform. An increase in the applied stress results in an increase in the slip length L and higher stresses in the core. The above scenario allowed the development of an analytical model to estimate the relevant stresses in the joint [1], a model subsequently extended to scarf joints
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with u ranging from 15 to 75° [2]. The interlayer metal in these analytical models was taken to be perfectly plastic, with a nominal yield strength so. For the nonzero angles it was assumed that the composite sub-elements are rigid and constrained from lateral displacements, so that the interlayer extension is confined to the axial direction. The resulting analytical model, coupled with experimental results, indicated that failure in these joints occurred at an approximately constant value ( 3so) of the stress normal to the interlayer-MMC interface [2]. The finding was consistent with the typical failure mode, which involves debonding at the fiber tips and thus suggests the existence of a threshold stress for the separation of the interface. Notwithstanding the apparent agreement, it was also evident that the model assumptions were not fully consistent with the large plastic strains measured experimentally for the intermediate angles (45915°). The analysis also implies that failure should initiate at the core, since that is the location of the highest normal stresses. Conversely, observations in the intermediate angle joints suggest failure initiation at the specimen edges [3] wherein the normal stresses can be lower but the plastic strains are often considerably higher. The present investigation was undertaken to address the above limitations of the current models. A full 3D finite element analysis of the scarf joint in Fig. 1 was developed and used to simulate the mechanical response of the interlayer as a function of u. In addition, two analytical approximations were developed to add confidence to the FEM results and to allow further insight into the physical phenomena. One of the analytical models retains the ‘rigid’ constraint of the previous models but incorporates interlayer elasticity, giving rise to an upper bound type solution. The second approxi-
Fig. 1. Schematic diagram of scarf joint geometry. Enlarged portion shows the locations of the core and slipping regions within the interlayer. The shear stress, t21, restricts inward motion of the interlayer near the free edge and causes large hydrostatic stresses to develop in the slipping region.
mation removes the constraint and allows essentially free lateral displacement of the composite sub-elements along the major axis of the interlayer plane, resulting in a lower bound solution. The manuscript is organized in the following manner. A brief review of the experimental results is presented first, to provide context. The models are then described and compared with each other. In applying these models to two different sets of experimental results it was found that the inherent constitutive behavior of the metal in the interlayer differs substantially from that of the unconstrained monolithic alloy. This problem and its potential sources are discussed next. It is shown how the ‘lower bound’ solution can be used to extract the constitutive behavior of the interlayer metal from one of the experiments. The models are then applied to the rest of the experiments using the modified constitutive law for the interlayer, and the predictions are compared with the experimental results. A final section addresses the stress and plastic strain distributions within the interlayer to assess the magnitude and implication of the edge effects.
2. Experiments Model joints with the configuration depicted in Fig. 1 and 05 u5 75° were fabricated using a melt infiltration process described elsewhere [1]. Briefly, Al alloys are infiltrated under pressures of 180 MPa into preforms of Al2O3 fibers containing prescribed discontinuities. Because matrix and interlayer are formed simultaneously, the interfaces at the fiber ends are free of gross defects and thus can achieve strengths superior to those produced by conventional joining methods. The constraint imposed by the specimen geometry allows the interfaces to be loaded to higher stresses than would be possible if the composite were surrounded by bulk metal [4]. Therefore, the technique allows the establishment of a baseline for the mechanical behavior of the interface against which the performance of more practical joints can be compared. The tensile behaviors of two sets of joint specimens used in this study are depicted in Fig. 2a and b. These sets were fabricated with the same matrix but two different commercial Al2O3 fibers, viz. Almax™ (Mitsui Mining Co., Japan) and Nextel 610™ (3M Corporation, Minneapolis, MN). The interlayer thickness, h, is in the range from 75 to 350 mm. The aspect ratio, B/h, exceeds 15 in all cases and thus limits the influence of edge effects. The strain plotted is that of the metal interlayer and was calculated from two gauges, as described elsewhere [1,2]. The accuracy of the strain measuring technique has been validated by means of strain mapping methods, as reported separately [5].
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by composite fracture rather than along the compositeinterlayer interface. Further discussion of the experimental results is provided in Section 4.4.
3. Deformation models The modeling approximations used to simulate the mechanical response of the scarf joints are summarized in Fig. 3. The finite element approach is closest to the experimental situation, where the ends of the composite subelements are firmly gripped by the testing frame but the material near the softer interlayer is less constrained, as depicted in Fig. 3a. Fig. 3b represents the fully constrained model, where lateral displacements are precluded along the length of the specimen. This is the approximation made in the upper bound solution. In the lower bound approximation the composite subelements are essentially free to slide along the interlayer plane, as depicted in Fig. 3c. Since the interlayer thickness h is in all cases much smaller than the interlayer breadth B, which is in turn smaller than the interlayer width W, edge effects are neglected in the analytical approximations. (The error associated with neglecting edge effects for an average aspect ratio, B/h =30, is estimated at B10% [2]. There are, however, some important ramifications of this assumption as discussed later.) Fig. 2. Mechanical response of the scarf joints manufactured with (a) Almax fibers and (b) Nextel 610 fibers. Interlayer orientations with u= 459 15° exhibit large strains to failure and reach an apparent limit load while the other joints fail prior to developing appreciable strains.
In both composite systems in Fig. 2 the butt joint (u = 0°) behavior is nearly linear to failure, consistent with the ‘rigid’ constraint approximation of earlier models. Conversely, joints with scarf angles between 30 and 60° exhibit substantial interlayer plasticity and a limit-load type behavior characterized by large increments of strain for modest changes in the applied stress. These experimental strains are incurred by ‘sliding’2 along the plane of the joint, which is at variance with the assumption of no lateral displacements in the ‘rigid’ model. The 15° joints exhibit mixed behavior, wherein the Almax specimen shows features reminiscent of the higher angle joints while the Nextel one is significantly stiffer and fails at strains similar to the butt joint. The 75° joints are considerably stiffer than most others and achieve substantially higher failure loads, with failure 2 To differentiate between the different forms of shear relevant to this problem ‘slipping’ will refer to the inward shear in the periphery of the interlayer whereas ‘sliding’ will be used for shear involving lateral displacement of the composite sub-elements.
Fig. 3. Schematic diagram representing the boundary conditions of the three models. The finite element model is depicted in (a) wherein the composite is fixed rigidly at the ends, the best correlation with experiment. Transverse motion is precluded in (b) and corresponds to the upper bound. Conversely, the composite subelements are free to slide along the interlayer in (c) and represent the boundary conditions of the lower bound.
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3.2. Upper bound approximation
Fig. 4. Schematic of a scarf joint interlayer depicting the relevant interfacial displacements. The upper bound is satisfied when uz =0, and the conditions for the lower bound require that d2 = 0. Displacements in the finite element models are such that uz ] 0 and d2 \0. Throughout this paper, strains are measured by the change in gage length from the initial size (A) to the deformed length (B).
Because of the finite sliding effect present in the joints for u \0° it is necessary to define an appropriate interlayer strain to be calculated in the model and compared with the experiments. This is done by considering the relative displacements of two points at the boundaries of the interlayer which are initially aligned in the axial direction, as illustrated in Fig. 4. For the case of the ‘rigid’ approximation uz =0 and the strain is purely axial. For the ‘sliding’ approximation d2 = 0 and the ‘gage length’ will rotate during deformation relative to the vertical axis. A drawback of this approach is that the increased effective gage length (A =h secu) results in an apparent stiffening of the interlayer.
3.1. Finite element model Full three-dimensional models of the scarf joints were developed using the ABAQUS™ general finite element code and implemented on a Silicon Graphics Origin 200 workstation. The mesh consisted of 2816, 20-node solid, brick elements arranged in an eight by eight grid at the composite ends and refined to an eight by 32 grid within the interlayer. The boundary conditions were set to reflect experimental conditions and specimen dimensions. Namely, a uniform displacement in the axial direction was applied along the top surface of the models, while both the top and bottom surfaces were constrained against transverse motion (Fig. 3a). The loading direction is along the y-axis in the figure and is coincident with the fiber orientation within the composite. The strain reported in the results is calculated as discussed for the analytical models in Fig. 4, except that uz ]0 and d2 \ 0.
This analytical approximation represents an extension of the model originally detailed in [2], which provides a description of the interlayer stress distribution as a function of angle and applied load. The previous model assumed perfect plasticity in the interlayer while the present analysis incorporates elastic contributions. (Because of the absence of strain hardening the model is not a true ‘upper bound’. However, the hardening effects are expected to be modest owing to the small strains involved in this case.) The previous model also considered the contribution of the slipping regions depicted in Fig. 1, which results in a more compliant response. These effects are neglected in the present approximation. In addition to the assumption of rigid attachments (uz = 0), the composite itself is considered rigid and therefore the metal interlayer is essentially constrained from shrinking laterally as it is stretched in the axial direction, whereupon oxx = ozz = 0. These conditions differ from the plane strain (o33 = 0) assumption in [2] and result in a less compliant response, but the net effect is consistent with the upper bound nature of the solution. Since the overall displacement is limited to the y-direction, the orientation of the principal stresses and hence the stress-strain response should not change with the angle of the interlayer. Under this assumption, the upper bound model is most conveniently demonstrated with the butt joint geometry because the boundary conditions can be applied without additional strain transformations. Therefore, o11 = o33 = 0. The general expression for the elastic strains in the butt joint geometry is: oii =
1 (s − sjj − 6skk ); E ii
i" j" k
(1)
wherein n= 0.34 and is the Poisson’s ratio for the interlayer metal. From o11 = o33 = 0 it follows that: s11 = s33 = o22 =
6s22 1− 6
s22 26 2 1− E 1− 6
(2)
(3)
The latter equation is valid until yield is reached. For the given boundary conditions the von Mises yield criterion reduces to s22 − s11 = so, whereupon yield should occur at an applied load of: sa = s22 = so
1− 6 1− 26
(4)
Above this stress the interlayer is modeled as elasticperfectly plastic. The increments of plastic strain follow the relation do pij = lSij where l is the plastic multiplier and Sij are the deviatoric stress components. For s11 = s22 Sij is given by:
D.D. Brink et al. / Materials Science and Engineering A281 (2000) 113–125
1 S11 =S33 =s11 − (2s11 +s22) 3
(5a)
1 S22 =s22 − (2s11 +s22) 3
(5b)
The plastic strain increments are then given by: l do p11 = do p33 = (s11 −s22) 3
(6a)
2l (s −s11) 3 22
(6b)
do p22 =
The total strain is the sum of the elastic and plastic strains, doii =do eii +do pii
(7)
with do11 = 0 and do22 \0. From Eq. (1) and Eq. (2) the incremental elastic strains are:
1 26 do = ds22 − ds11 E E
(8b)
do11 =
1−26 l ds22 + (s11 −s22) =0 E 3
(9a)
do22 =
1−26 2l ds22 + (s22 −s11) E 3
(9b)
Solving for the plastic multiplier in Eq. (9a), substituting l into Eq. (9b), and integrating gives:
n
3(1−26) 2 s22 − sO E 3
(10)
This equation describes the upper bound response after yield. Because the composite stiffness is finite, the interlayer is not completely constrained from contracting laterally, whereupon the strains o11 and o33 are non-zero, albeit small. This will tend to lower the in-plane interlayer stresses and increase the longitudinal strain predicted by the model for any given stress. To a first approximation, the difference between the transverse strains in the composite and interlayer can be expressed as:
sa 1+ tan2 u
(12a)
t23 =
sa tan u 1+ tan2 u
(12b)
The relevant yield criterion under plane strain3 is:
Substituting Eqs. (6a) and (6b) and Eqs. (8a) and (8b) into Eq. (7) and invoking the yield criterion for perfect plasticity, which implies ds11 =ds22:
In contrast to the previous model, the lower bound approximation assumes essentially unconstrained interlayer shearing in the plane of the joint so that d2 =0 in Fig. 4. This allows the interlayer to attain high levels of plastic strain, wherein work hardening can be significant. To account for this, the model incorporates the actual constitutive behavior of the interlayer metal in the plastic regime, with the appropriate transformations into equivalent stresses and strains. The analysis requires full interlayer plasticity and assumes plane strain along the width of the joint. Neglecting the effects of the slipping regions (Fig. 1), it can be shown from simple equilibrium considerations that:
(8a)
e 22
o22 =
3.3. Lower bound approximation
s22 =
1−6 6 ds11 − ds22 do e11 = E E
117
Em m o c/m o =6effo m 11 = 6m − 6c 22 Ec 22
(11)
In this equation neff is an effective Poisson’s ratio relating the longitudinal and constrained transverse strains in the interlayer (neff/n :0.76). The effect of the reduced constraint on the upper bound model may then be approximated by using neff in lieu of the nominal n for the metal in Eq. (3) and Eq. (10).
3 (s − s11)2 + 3t23 = s 2 4 22
(13)
Where the left hand side of the equation is the equivalent stress and s is the strain-dependent yield stress, to differentiate it from the yield stress in perfect plasticity so, (s [ so at the onset of yield). Letting s11 = as22 and substituting Eqs. (12a) and (12b) into Eq. (13) gives: sa 2(1+tan2 u) = s 3(1 −a)2 + 12 tan2 u
(14)
The relevant lower bound is obtained by maximizing this equation, i.e. when a= 1: sa 1+tan2 u = s 3 tan2 u
(15)
which is the relationship between the stress applied to the scarf joint, sa, and the concomitant effective stress in the interlayer, s . Transforming the equivalent strain in a manner consistent with previous results is more involved. From the condition of plane strain and the normality rule, the strain normal to the interlayer can be expressed as: o22 =
3l s22 − s11 =0 4 s
(16)
where o22 equals zero since s22 = s11 for the highest lower bound (a=1). The effective plastic strain, o , is 3
The yield criterion in Eq. (13) is different than that expressed for the upper bound appromiation due to the difference in boundary conditions.
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then reduced to the shear component along the width, given by the relation: o =
g23
3
=
2
d3
3 h
(17)
wherein d3 is the displacement along the plane of the joint, illustrated in Fig. 4. From this diagram it is also inferred that the strain measured by the gage is ointerlayer = (B− A)/A, where A is the initial gage length and B is the length in the displaced state. One can readily show that: A =h secu
(18a)
B =[A 2 + 4d 23 + 4Ad3 sin u]1/2
(18b)
The resulting expression for the gage strain is then: ointerlayer = [3o 2 cos2 u + 3 sin (2u)o +1]1/2 −1
(19)
Thus, if the constitutive behavior of the metal in the interlayer, s (o ), is known, one can use Eq. (15) and Eq. (19) to generate the response of the interlayer, sa(ointerlayer), in a joint of arbitrary scarf angle u.
4. Results and discussion
4.1. Finite element 6ersus analytical approximations A comparison between the stresses calculated with the FEM and lower bound models for different levels of strain (0.01 5o 5 0.05) and scarf angles (30 5 u5 60°) is given in Fig. 5. The applied load is normalized by the yield stress (so) so that the result is independent of the input constitutive law, which is the same for both the analytical and FEM solutions. The dependence of
the absolute stress values on the interlayer constitutive law will be addressed in Section 4.2. It is evident that the correlation between the FEM and lower bound models is excellent for the 45 and 60° joints, and quite reasonable for the 30° joint. This indicates that the behavior of the interlayers in this range of scarf angles is more adequately represented by the ‘sliding’ approximation and thus that the rigid constraint assumed in previous models is largely absent. (As will be shown later, the opposite is true for the butt joint and low angle interlayers.) The agreement for u= 45° is not surprising as the conditions for sliding along the plane of the interlayer are most favorable at this angle. The maximum deviation ( 6%) occurred for the 30° joint at the lower plastic strains (1%). More significant, however, is the fact that the ‘lower bound’ result is higher than that predicted from the full 3D FEM calculation. The primary source of this discrepancy is the omission of the edge effects in the lower bound analysis. One can estimate from the FEM results that the slipping region at the edge depicted in Fig. 1 comprises 25% of the interlayer area for the 30° joint (cf. Fig. 9). The average axial stress in this area is lower than that in the core by 20%, but the lower bound analysis assumes that the whole interlayer is supporting a uniform stress, i.e. that corresponding to the core. This effect reduces the lower bound values for the 30° joint by 5%, bringing a majority of them slightly below the line representing perfect agreement, as expected for a lower bound calculation. The development of the slipping regions in the 45 and 60° joints result in a similar stress reduction. Notwithstanding this correction, however, the normalized applied load for the 30° joint at low strains (1%) still remains above the values predicted by FEM. This is ascribed to the effects of a significant elastic strain component normal to the interlayer in the FEM calculation (d2 \ 0 in Fig. 4) that is neglected in the lower bound analysis. Calculated for the 30° joint with the FEM model, the inclusion of the additional displacement increases the interlayer strain by a factor of 1.2, equivalent to an effective decrease in applied load of about 4% at the same nominal strain. The effect decreases rapidly with increasing strain in the 30° joint, and is not significant in the region of concern for the 45 and 60° joints.
4.2. Interlayer constituti6e beha6ior
Fig. 5. Correlation between results from the FEM and lower bound analytical models. Normalized applied stress levels are presented for interlayer strains between 1 and 5%. Results for the lower bound lie above those of the FEM primarily due to omission of edge effects in the analytical model.
Even though the agreement between the analytical (lower bound) and FEM predictions provides confidence in the models, initial comparison with the experiment met with less success. FEM calculations incorporating the interlayer constitutive behavior measured from the monolithic alloy consistently underestimated the stress at a given level of strain for joint angles in which the interlayer experienced significant
D.D. Brink et al. / Materials Science and Engineering A281 (2000) 113–125
s =
3 tan u sa 1+ tan2 u
119
(20)
And the effective strain, o , is similarly related to the measured strain in the interlayer by: 1
(21) {[(1+ointerlayer)2sec2u− 1]1/2 − tan u}
3 These equations apply only when the lower bound assumptions are satisfied; i.e. when sliding along the interlayer is dominant and the plastic strain is substantially larger than the elastic strain. The overall constitutive behavior was fitted to a Ramberg–Osgood model [6]: o =
Fig. 6. Plot comparing the constitutive law measured from the monolithic alloy to that calculated for the constrained interlayer. Using the lower bound approximations, the interlayer behavior was deconvoluted from the response of the 45° Almax specimen (shown in the figure) for strains in the plastic regime. This was coupled to the elastic response of the metal using a Ramberg–Osgood fit (s0 =162 MPa; o0 = 0.23%; a = 3/7; n= 8).
plasticity. After extensive analysis of the experimental conditions and testing the models with different mesh refinements and boundary conditions, it was concluded that the most likely source of the disparity was the constitutive behavior of the interlayer metal that was used in the calculations. In essence, the comparison between models and experiments suggests that the stresses in the interlayer must be substantially higher than those exhibited by the monolithic alloy at the same equivalent strain. This hypothesis was examined using a dual strategy. First, a constitutive law for the metal was extracted from the measured stress-strain behavior of one of the experiments. The result was then used in the FEM model to simulate the response of the other specimens. As this approach produced good agreement with the experiments, the second element of the strategy focused on probing independently whether the interlayer does exhibit the additional hardening suggested by the modified constitutive law, and what the possible reasons for this behavior might be. The derivation of the modified constitutive law is presented in this section and the verification and analysis of the hardening in Section 4.3. Owing to the geometry and dimensions of the metal interlayer one cannot deduce its constitutive behavior directly from the experimental stress-strain response. However, if the conditions of the lower bound analysis are reasonably satisfied, one may deconvolute the constitutive law from the stress-strain response using Eq. (15) and Eq. (19) and reversing the procedure outlined at the end of Section 3.3. The equivalent stress in the plastic regime is calculated from a simple rearrangement of Eq. (15):
s s o = +a so oo so
n
(22)
wherein so and oo are the nominal yield stress and strain, and a and n are fitting parameters. The elastic response of the model was set to match that of the monolithic alloy (70 GPa), while at higher strains the model was fitted to the data generated with Eq. (20) and Eq. (21). The procedure described above was applied to the stress-strain curve measured for the 45° Almax specimen. The lower bound conditions are expected to fit most closely the 45° interlayer, as noted above4. The selection of the Almax instead of the Nextel specimen was arbitrary. (The 45° Nextel specimen was part of the model validation that included the rest of the specimens from both sets, as discussed in Section 4.4.) The resulting constitutive law for the metal in the interlayer is shown in Fig. 6 and compared with the response measured independently for the monolithic alloy. This exercise indicates that the 0.2% offset interlayer yield stress is 162 MPa. In comparison, the yield strength of the unconstrained monolithic alloy was 879 3 MPa.
4.3. E6idence and origin of interlayer hardening Independent experimental confirmation of the elevation in yield stress deduced above is not trivial, as a monolithic body of metal that shares the characteristics of the interlayer cannot be easily generated. However, it is feasible to probe the hardness of the interlayer by microindentation and compare it with values obtained from the specimens used to evaluate the stress-strain behavior of the monolithic alloy in Fig. 6. A higher interlayer hardness would qualitatively confirm the elevation in yield stress. It is also possible to estimate the elevation in strength if a correlation between hardness and yield strength is available. 4 A similar constitutive response for the interlayer is obtained using the 30 or 60° data, supporting the validity of the analysis. This will be apparent in the verification of the model against the complete set of experimental data.
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Vickers microindentation tests using 10 g loads were performed on two pristine butt joint specimens and the two 75° specimens which failed by composite fracture at the grips after relatively small plastic strains. The results were very similar as one 0 and one 75° sample had a VHN of 72 and the remaining two had a value of 79. In contrast, the measured hardness for a monolithic alloys specimen with a yield strength of 87 MPa was 65. (A set of Rockwell H measurements on the monolithic specimen produced a value of 91, equivalent to 56 VHN.) Although the scatter is significant, the measurements support the view that the metal in the interlayer has a higher yield strength than that of the monolithic material cast under similar conditions. These results are compared to literature values for commercial Al–Mg alloys in the appendix to assess whether the estimated increase in interlayer yield stress is consistent with the microhardness measurements. A more elusive issue is the identification of the operating mechanism(s) responsible for the interlayer hardening. Microstructural analysis revealed an average grain size of 1 mm in the cast monolithic alloy used to measure the constitutive law in Fig. 6, and 10 mm in the interlayers of the joints. Significant grain refinement may be anticipated from the physical dimensions of the interlayer and the numerous sources of heterogeneous nucleation available during the infiltration through a porous ceramic and into a space surrounded by fiber tips with diameters on the order of the metal grain size. While the level of grain refinement is substantial, however, the magnitude of the concomitant Hall–Petch effect (with a coefficient of 0.068 [7]) is only about 20 MPa.
Fig. 7. Mechanical response of scarf joint interlayers generated from the finite element models. Limit-load behavior is apparent for u ] 15° and corresponds to ‘sliding’ in the interlayer. The butt joint reaches a limit load at higher stresses due to the convergence of the slipping regions. Superimposed are the average failure stress of the experimental specimens for 0 5u5 15° (wherein the response is sensitive to strain) and the average failure strain for u] 30° (wherein the response is stress sensitive).
A second source of hardening is associated with the thermal expansion mismatch between the metal in the interlayer and the surrounding combination of composite and porous ceramic mold. It is well known that such effect is responsible for dislocation generation and in-situ hardening of the matrix in discontinuously reinforced Al composites when cooling from the processing temperature [8,9]. An accurate assessment of this contribution is hindered by the geometric complexity of the problem. (The joint specimen is embedded in a porous mold, which in turn is surrounded by a large mass of metal and the steel die. Moreover, there are thermal gradients, and pressures higher than the yield stress applied during solidification and part of the cooling process.) A simple analysis which considers the interlayer as constrained solely by the composite and only in the plane of the joint gives estimated plastic strains of 1%. This amount of strain could produce an increase in the yield strength as high as 40 MPa, based on the constitutive law for the monolithic specimen in Fig. 6. Added to the grain size effect, one might anticipate an elevation of the yield stress of 60 MPa, which is somewhat lower than the value estimated from the lower bound analysis but of the right order. The issue, however, requires further investigation.
4.4. Comparison of predicted and experimental constituti6e beha6ior Fig. 7 summarizes the predicted stress-strain behavior of the interlayer in scarf joints with 0° 5 u5 75°, calculated with the FEM model using the modified constitutive law for the metal. The trend in predicted interlayer behavior with increasing angle is remarkably consistent with the experimental results. The differences in form between the lower angles (0 and 15°) and the intermediate angles (30–60°) are quite evident. The low angle joints achieve only modest plasticity even at loads past the failure strengths of the experimental specimens (300–320 MPa), whereas the higher angle joints develop large plastic strains below the experimental failure loads. It should be noted, however, that large interlayer strains consistent with limit load behavior would be predicted for the 15° joint if failure by debonding were not to intervene and the applied stress could reach 600 MPa. Although the 0° curve does not behave in the same manner within the range investigated ( 5 650 MPa), it can be shown that a butt joint would reach a limit load when the slipping regions in Fig. 1 overlap [1]. For the present specimens the requisite stresses would be on the order of 1600 MPa, and are clearly unattainable because interfacial failure intervenes at much lower loads. The predicted response of the 75° joint is similar to the 15° in that it exhibits limit load behavior at much higher values of stress than the 30–60° joints. As in the case of the lower angle joints,
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the loading is interrupted by fracture well below the limit, although in this case it is the composite that fails and not the interlayer-composite interface. The limit load is thus predicted to decrease rapidly as u deviates from 0°, reaching a minimum at 45° and then rising again as the angle increases above 45°. The elastic response of the interlayers also varies with scarf angle, as evident in Fig. 7. The elastic stiffening produced by the interlayer constraint in the butt joint, Eq. (3), is relaxed as u increases. There are, however, more significant geometric effects that arise from the manner in which the interlayer strain is calculated, as noted before (cf. Fig. 4). If d2 d3, the strain would increase with angle for a fixed gage length A and sliding displacement d3, as inferred from Eq. (18b). Conversely, for a fixed displacement and interlayer thickness (h) the gage length would increase with increasing u, reducing the apparent strain Eq. (18a). Results of the FEM analysis indicate that the latter effect is dominant for the higher angles. Within the elastic and near yield regime the calculated magnitude of d3 is approximately constant for a given load, and essentially independent of angle. As d3 A, the increased gage length effect leads to an artificially lower strain and thus an apparent stiffening. This is manifested most evidently in the response of the 75° joint, which exhibits a higher modulus than the 0° joint. Comparisons between the experimental response of the interlayers with the predictions from the different models are given in Fig. 8a – f for each of the scarf angles investigated. Both the lower bound analytical approximation and the FEM model utilize the constitutive law derived in Section 4.2, while the upper bound model uses an elastic-perfectly plastic response with the same yield stress (162 MPa). Since the upper bound approximation is closer to the FEM results for the 0 and 15° joints, the lower bound solution is not included for these cases. For the same reasons, only the lower bound approximation is included for the higher angle specimens, u] 30°. Consider first the lower angle joints in Fig. 8a and b. For the 0° case there is good agreement between the two experimental results, and excellent correlation with both the upper bound and FEM models, especially in the case of the Almax specimen. At higher levels of applied stress edge effects become more important, causing the finite element model to be more compliant than the upper bound solution. The 15° specimens exhibit a higher degree of experimental variability than the butt joints. The Nextel 610 specimen is quite consistent with the calculations, but the interlayer in the Almax composite is substantially more compliant. While the details have not been ascertained, it is likely that the behavior of the Almax joint reflects simply a larger than normal variability in the experimental conditions during specimen manufacture, and possibly dur-
121
ing testing. The good experimental agreement with the upper bound model suggests that the early response of the 0 and 15° joints is dominated by elasticity. The stresses needed to activate significant interlayer sliding and plasticity are above the failure point of the actual specimens and thus the deviation from the ‘rigid’ approximation is not too critical in the context of these experiments. However, the results are insightful in predicting superior joint performance if interfacial failure were postponed to higher loads, e.g. by strengthening the interlayer with particles [10]. The experimental responses of the higher angle joints are compared with the FEM and lower bound models in Fig. 8c through f. In general, there is good agreement between the lower bound and FEM models and the experimental measurements. The discrepancy between the Almax and Nextel responses for a given angle is reasonable, as the apparent limit loads for each pair of specimens are within 10% for 305 u5 60°. The correlation for the 75° joints is not as clear. Failure of both Almax and Nextel specimens occurred within the grips and not by separation at the composite-interlayer interface, as in all other angles. However, the shape of the Nextel 610 curve in conjunction with the FEM result suggests that the interlayer was at the onset of the limit load regime when composite failure intervened. This indicates that the mechanical behavior of 75° joints is likely to be controlled by sliding and interlayer plasticity, as is the case for 305 u5 60°. The correlation between the finite element and experiment for the geometries investigated indicates that the FEM model accurately represents joint behavior in both the elastically dominated regime at low angles and the plastically dominated regime at u] 30°. The finite element work can thus be used with confidence to investigate the interlayer stress and strain distributions, and how these stresses influence joint failure. For instance, possible changes in the limiting failure mechanism between the elasticity-controlled response at low angles and the plasticity-controlled behavior at higher angles can be studied more rigorously. These issues will be explored further in a later publication dealing with the failure criteria for these joints [5].
4.5. Interlayer stress distributions The relative magnitudes of the core and slipping regions are illustrated in Fig. 9, which shows stress profiles through the mid-plane of the specimen bisecting the long axis of the interlayer, at the interface with the composite sub-elements. The normal and hydrostatic stresses in this figure were calculated with the FEM models at an interlayer strain of 4%, corresponding to applied loads of 345, 300, and 346 MPa for the 30, 45, and 60° joints, respectively. Although the load for the 30° joint is above its experimental failure stress, the
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Fig. 8. The experimental stress-strain behavior of the 0 and 15° specimens are compared to the FEM and upper bound analytical results in (a) and (b) using neff =0.26. The experimental behavior of the 30 – 75° joints are shown in (c) through (f) and compared to the FEM and lower bound analytical models.
stress distributions are well developed at this level of strain and more clearly show the relevant features. Comparison of the normal stresses (s22 in the coordinates of Fig. 4) in Fig. 9a reveals a decrease in overall magnitude with angle. This is readily explained by a simple force balance in the axial direction wherein the shear stress along the sliding direction, t23, is shown to carry an increasing amount of the applied load as the
scarf angle increases [2]. This holds even though the applied load in the 45° joint is lower than that of the 60° at the chosen level of interlayer strain. A decrease in normal stress near the free edge is common to all angles and corresponds to the slipping region in Fig. 1, along which large hydrostatic stresses build up with distance from the free surface. As assumed in the present models and in agreement with previous work
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[1,2], the stresses within the core are essentially independent of position. (The slight stress concentrations at the intersection of the core and slipping regions may have implications in the failure process, but are not particularly significant for the purposes of the present analysis.) The slipping regions comprise 25% of the interlayer area for the joints depicted, reducing the average stress in the joint below that of the core level and leading to the discrepancy between the lower bound analytical and finite element models discussed in Section 4.1. The interfacial pressure stresses for the joints are shown Fig. 9b, where negative values represent tension. The build-ups of stress in the slipping regions are evident and correspond to the rise in the normal stress in Fig. 9a. The stress decreases with increasing angle owing to the higher interlayer shear component, as discussed in [2]. For the particular example in this figure the average magnitude of the pressure stress in the slipping regions is about 15 – 20% below the core value. As the load increases the magnitude of the slipping regions increases with a consequent divergence of the core and average stresses. For example, the slipping regions would comprise 40% of the interlayer area in the 60° joint at 385 MPa, and the average stress would be 88% of that in the core. (This is hypothetical as the assumed stress is above the experimental failure loads.) The result indicates that the analytical models become increasingly inaccurate as the load increases due to the omission of edge effects. However, the moderate deviations calculated within the relevant stress range of the present joints (B 5%) indicate that the upper and lower bound analytical models remain a good approximation of the interlayer behavior in these experiments.
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occurs in the composite rather than at the interface. Analytical (upper and lower bound) and finite element models were developed to describe the response of the joints in an attempt to understand the development of the interlayer stress state and the concomitant influence on performance. Comparison with experiment indicates that the constitutive behavior of the monolithic alloy differs from that of the interlayer, an effect originating from the variation in processing conditions between the monolithic material and the constrained interlayer. The interlayer constitutive law was identified using the models, and resulted in good experimental agreement for the upper bound model at low angles, the lower bound model at higher angles, and FEM for all angles. The correlation suggests that within the range of experimentally accessible loads, joint behavior for 0 5 u5 15° is dictated by elastic effects within the interlayer while that of higher angle joints is dominated by plasticity. Limit loads are expected for all angles above the stresses at which interfacial failure is initiated, with a minimum established at u= 45°. Also, the close correlation between FEM and experiment sug-
5. Conclusions Model scarf joints consisting of thin metal interlayers inclined at angles of 0 5u 575° between continuous fiber reinforced MMC’s were manufactured and tested in tension. The high level of constraint on the interlayers causes the development of large hydrostatic stresses within them, and allows the joints to support loads much larger than the UTS of the monolithic interlayer alloy. The mechanical response of the interlayers changes dramatically with scarf angle, however. The low angle joints (05u5 15°) are nearly elastic until failure ensues at the composite-monolith interface. In contrast, interlayers with higher angles (30 5 u5 60°) exhibit much larger strains and appear to reach a limit load prior to failure. Although the highest loads are reached for u =75°, these interlayers develop less plastic strain than when u =45 9 15°. Moreover, failure
Fig. 9. Interfacial distributions of the (a) normal stress and (b) pressure stress for 30 ]u ]60° at an interlayer strain of 4%. The extent of the slipping regions is apparent for these conditions, and results in a reduction in load carrying capacity of 5%.
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mens. The Nextel-610™ fibers were kindly provided by Dr Colin McCullough of the 3M Corporation.
Appendix A
Fig. 10. The relationship between hardness and yield strength for various Al alloys is explored by comparing as-cast (this study) alloys with annealed and cold worked alloys from the literature. Correlations for the three conditions are established in (a) and appear to be insensitive to alloy content. Trends for specific alloys are shown in (b) and have slopes ranging between 0.15 and 0.18, agreeing well with the proposed relation for the as-cast monolithic and interlayer alloy.
gests that the stress distributions obtained from the model are accurate and can be used to explore the failure mechanisms which limit joint performance. The development of appropriate failure criteria for the scarf joints will be the subject of a subsequent publication [5].
Acknowledgements This investigation was supported by the Mechanics and Materials Program of the National Science Foundation under grant CMS-9634927. The authors are grateful to Dr M.Y. He and Dr H. Kim for their assistance in implementing the ABAQUS FEM code, to Dr Chad Landis for useful discussions pertaining to the development of the upper bound model, and to Dr J.Y. Yang for his assistance in the preparation of the speci-
Correlations between hardness and yield strength for aluminum alloys are rather scarce in the literature. An early analysis by Tabor [11] suggests that the ratio of the Meyer hardness to the yield strength for heavily work-hardened materials is constant and has a value of 2.8 for Al alloys. One can then show for typical indentation diameters that the ratio of the Brinell hardness (HB, kg mm − 2) to the yield strength (so, MPa) should be 0.27. This correlation is compared in Fig. 10a with experimental values for several 5XXX and 5XX Al–Mg alloys5 in the annealed or as-cast condition, as well as after various degrees of strain hardening [12]. Included in this figure are values for three Al–Mg alloys with different amounts of Mg (1, 4.5 and 7%) pressure cast for this study under conditions similar to those used in fabricating the joint specimens. (These hardness values were determined as Vickers and then converted to Brinell using a standard correlation [13] to facilitate comparison with the values for the commercial alloys.) A linear regression analysis shows that there is a reasonable correlation for each group of data (pressure cast, annealed/as-cast and strain hardened). The slopes of the lines decrease from the pressure cast specimens to the cold worked materials, approaching in the latter case the value of 0.27 suggested by Tabor’s analysis (Fig. 10a). However, it is evident that Tabor’s correlation is not useful for the present exercise as neither the monolithic alloy nor the interlayers fit the ‘heavily work hardened’ description. A more elaborate relationship proposed by Chang et al. [14] was not found to be any more satisfactory. A more useful correlation, albeit totally empirical, is suggested when the commercial alloy values are plotted by individual alloy, as depicted in Fig. 10b. Data for the different 5XXX series alloys could be fitted to a simple linear relationship with reasonable correlation coefficients (\ 0.96) and slopes within a relatively narrow range (0.15–0.18). Plotted in the same figure, the values corresponding to the monolithic specimen (HB 549 4 and so 87 MPa) and the interlayer (HB 6693 and so 162 MPa) define a band with a slope close to that of the commercial alloys (0.16). Albeit semi-quantitative, this comparison indicates that the magnitude of the yield stress elevation estimated for the interlayer is consistent with the microhardness measurements.
5 For each Mg content the alloy selected was one which had the minimum amounts of other alloying elements.
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References [1] A. Burr, J.Y. Yang, C.G. Levi, F.A. Leckie, Acta Metall. Mater. 43 (5) (1995) 3361–3373. [2] D.D. Brink, C.G. Levi, A.C.F. Cocks, F.A. Leckie, Acta Mater. 45 (7) (1997) 2765–2775. [3] D.D. Brink, J.C. Mailand, C.G. Levi, F.A. Leckie, The Third Pacific Rim International Conference on Advanced Materials and Processing (PRICM3), The Minerals, Metals and Materials Society, 1998, pp. 441–446. [4] R. Hill, The Mathematical Theory of Plasticity, Oxford University Press, London, 1950, p. 228. [5] D.D. Brink, C.G. Levi, F.A. Leckie, in press. [6] W. Ramberg, W.R. Osgood, Technical Notes National Advisory Committee for Aeronautics No. 902, NACA, 1943, Washington, D.C.
.
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[7] T.H Courtney, The Mechanical Behavior of Materials, McGrawHill, NY, 1990, p. 171. [8] T.W. Clyne, P.J. Withers, An Introduction to Metal Matrix Composites, Cambridge University Press, Great Britain, 1993, p. 84. [9] S. Suresh, A. Mortensen, A. Needleman, Fundamentals of Metal Matrix Composites, Butterworth – Heinemann, Boston, MA, 1993, pp. 124 – 126. [10] D.D. Brink, S.A. Waltner, J.Y. Yang, C.G. Levi, F.A. Leckie, Mater. Sci. Eng. (2000) in press. [11] D. Tabor, J. Inst. Metals 79 (1951) 1 – 18. [12] Metals Handbook, Technical Eds., S.R. Lampman, T.B. Zorc, 10th ed., ASM International, 2 (1990) pp. 89-100. [13] 1989 Annual Book of ASTM Standards, Managing Ed., R.A. Storer, Volume 03.01, ASTM, Easton, MD, 1989, pp. 338. [14] S.C. Chang, M.T. Jahn, C.M. Wan, J.Y.M. Lee, T.K. Hsu, J. Mater. Sci. 11 (1976) 623 – 630.
Deformation behavior of model MMC scarf joints D.D. Brink, J.C. Mailand 1, C.G. Levi *, F.A. Leckie Departments of Materials and Mechanical and En6ironmental Engineering, High Performance Composites Center, 1316D Engineering 11, Uni6ersity of California, Santa Barbara, CA 93106 -5050, USA Received 24 August 1999; received in revised form 26 October 1999
Abstract The mechanical response of metal interlayers constrained between two fiber-reinforced MMC (metal – matrix composites) sub-elements was investigated. The fibers were polycrystalline Al2O3, discontinuous at the joint, embedded in Al – 4.5Mg, that was continuous through the joint and constituted both the composite matrix and the interlayer material. The specimens were produced by pressurized melt infiltration and comprised interlayers oriented from 0 to 75° relative to the plane normal to the fiber axis. Analytical and finite element models were developed and compared with experimental results in order to elucidate the deformation behavior of these interlayers and to identify the key factors controlling joint performance. Two predominant regimes were identified in this manner. Joints in the intermediate angle range (45 9 15°) exhibit limit load behavior and develop large strains prior to failure, whereas joints at lower (5 15°) and higher (75°) angles show only modest plasticity. The models suggest that the latter group could, in principle, develop large strains and limit loads, but this behavior is precluded by the intervention of failure owing to debonding in the lower angle joints, and to composite fracture in the highest angle ones. The analysis further reveals that the constitutive behavior of the metal in the interlayer exhibits substantial hardening relative to that of the monolithic matrix processed in the same manner. The reasons for this behavior are discussed. The models set the stage for future work to elucidating failure criteria for interlayers in these joint configurations. © 2000 Elsevier Science S.A. All rights reserved. Keywords: Structural behavior; Composite; Plastic; Computer simulation; Joining
1. Introduction The specific strength and stiffness of metal–matrix composites (MMCs) make them attractive materials for weight-sensitive components and structures. Full exploitation of these properties, however, is often hindered by inadequate understanding of the mechanisms that govern the performance of MMC joints and attachments to other elements of the structure. Joint design is typically based on the conservative assumption that interfaces normal to the fibers can carry no significant load. Hence, load transfer must be effected primarily by shear, leading to bulky joints and transitions that undermine the weight-saving potential of the MMC.
* Corresponding author. Tel. + 1-805-893-2381; fax: +1-805-8938486. 1 J.C. Mailand, previously graduate student researcher at UCSB, is now with CAMCO/Schlumberger Advanced Completions Group.
Previous work on Al-matrix/Al2O3 fiber composites has demonstrated that clean interfaces normal to the fiber ends have the capacity to support loads well in excess of the strength of the adjoining metal [1]. When coupled with model scarf configurations (Fig. 1) these interfaces have enabled joint strengths comparable to those of the composite [2]. Finite element analyses of butt joints (u= 0°) consisting of thin metal interlayers between two composite sub-elements [1] revealed the existence of two distinct regions within the interlayer, as illustrated in Fig. 1. The perimeter of the interlayer is characterized by localized inward shear over a distance L, with a buildup of hydrostatic stress away from the free surface as a consequence of the constraint imposed by the surrounding composite. Conversely, the stresses within the interlayer core are essentially uniform. An increase in the applied stress results in an increase in the slip length L and higher stresses in the core. The above scenario allowed the development of an analytical model to estimate the relevant stresses in the joint [1], a model subsequently extended to scarf joints
0921-5093/00/$ - see front matter © 2000 Elsevier Science S.A. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 9 9 ) 0 0 7 3 1 - 5
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with u ranging from 15 to 75° [2]. The interlayer metal in these analytical models was taken to be perfectly plastic, with a nominal yield strength so. For the nonzero angles it was assumed that the composite sub-elements are rigid and constrained from lateral displacements, so that the interlayer extension is confined to the axial direction. The resulting analytical model, coupled with experimental results, indicated that failure in these joints occurred at an approximately constant value ( 3so) of the stress normal to the interlayer-MMC interface [2]. The finding was consistent with the typical failure mode, which involves debonding at the fiber tips and thus suggests the existence of a threshold stress for the separation of the interface. Notwithstanding the apparent agreement, it was also evident that the model assumptions were not fully consistent with the large plastic strains measured experimentally for the intermediate angles (45915°). The analysis also implies that failure should initiate at the core, since that is the location of the highest normal stresses. Conversely, observations in the intermediate angle joints suggest failure initiation at the specimen edges [3] wherein the normal stresses can be lower but the plastic strains are often considerably higher. The present investigation was undertaken to address the above limitations of the current models. A full 3D finite element analysis of the scarf joint in Fig. 1 was developed and used to simulate the mechanical response of the interlayer as a function of u. In addition, two analytical approximations were developed to add confidence to the FEM results and to allow further insight into the physical phenomena. One of the analytical models retains the ‘rigid’ constraint of the previous models but incorporates interlayer elasticity, giving rise to an upper bound type solution. The second approxi-
Fig. 1. Schematic diagram of scarf joint geometry. Enlarged portion shows the locations of the core and slipping regions within the interlayer. The shear stress, t21, restricts inward motion of the interlayer near the free edge and causes large hydrostatic stresses to develop in the slipping region.
mation removes the constraint and allows essentially free lateral displacement of the composite sub-elements along the major axis of the interlayer plane, resulting in a lower bound solution. The manuscript is organized in the following manner. A brief review of the experimental results is presented first, to provide context. The models are then described and compared with each other. In applying these models to two different sets of experimental results it was found that the inherent constitutive behavior of the metal in the interlayer differs substantially from that of the unconstrained monolithic alloy. This problem and its potential sources are discussed next. It is shown how the ‘lower bound’ solution can be used to extract the constitutive behavior of the interlayer metal from one of the experiments. The models are then applied to the rest of the experiments using the modified constitutive law for the interlayer, and the predictions are compared with the experimental results. A final section addresses the stress and plastic strain distributions within the interlayer to assess the magnitude and implication of the edge effects.
2. Experiments Model joints with the configuration depicted in Fig. 1 and 05 u5 75° were fabricated using a melt infiltration process described elsewhere [1]. Briefly, Al alloys are infiltrated under pressures of 180 MPa into preforms of Al2O3 fibers containing prescribed discontinuities. Because matrix and interlayer are formed simultaneously, the interfaces at the fiber ends are free of gross defects and thus can achieve strengths superior to those produced by conventional joining methods. The constraint imposed by the specimen geometry allows the interfaces to be loaded to higher stresses than would be possible if the composite were surrounded by bulk metal [4]. Therefore, the technique allows the establishment of a baseline for the mechanical behavior of the interface against which the performance of more practical joints can be compared. The tensile behaviors of two sets of joint specimens used in this study are depicted in Fig. 2a and b. These sets were fabricated with the same matrix but two different commercial Al2O3 fibers, viz. Almax™ (Mitsui Mining Co., Japan) and Nextel 610™ (3M Corporation, Minneapolis, MN). The interlayer thickness, h, is in the range from 75 to 350 mm. The aspect ratio, B/h, exceeds 15 in all cases and thus limits the influence of edge effects. The strain plotted is that of the metal interlayer and was calculated from two gauges, as described elsewhere [1,2]. The accuracy of the strain measuring technique has been validated by means of strain mapping methods, as reported separately [5].
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by composite fracture rather than along the compositeinterlayer interface. Further discussion of the experimental results is provided in Section 4.4.
3. Deformation models The modeling approximations used to simulate the mechanical response of the scarf joints are summarized in Fig. 3. The finite element approach is closest to the experimental situation, where the ends of the composite subelements are firmly gripped by the testing frame but the material near the softer interlayer is less constrained, as depicted in Fig. 3a. Fig. 3b represents the fully constrained model, where lateral displacements are precluded along the length of the specimen. This is the approximation made in the upper bound solution. In the lower bound approximation the composite subelements are essentially free to slide along the interlayer plane, as depicted in Fig. 3c. Since the interlayer thickness h is in all cases much smaller than the interlayer breadth B, which is in turn smaller than the interlayer width W, edge effects are neglected in the analytical approximations. (The error associated with neglecting edge effects for an average aspect ratio, B/h =30, is estimated at B10% [2]. There are, however, some important ramifications of this assumption as discussed later.) Fig. 2. Mechanical response of the scarf joints manufactured with (a) Almax fibers and (b) Nextel 610 fibers. Interlayer orientations with u= 459 15° exhibit large strains to failure and reach an apparent limit load while the other joints fail prior to developing appreciable strains.
In both composite systems in Fig. 2 the butt joint (u = 0°) behavior is nearly linear to failure, consistent with the ‘rigid’ constraint approximation of earlier models. Conversely, joints with scarf angles between 30 and 60° exhibit substantial interlayer plasticity and a limit-load type behavior characterized by large increments of strain for modest changes in the applied stress. These experimental strains are incurred by ‘sliding’2 along the plane of the joint, which is at variance with the assumption of no lateral displacements in the ‘rigid’ model. The 15° joints exhibit mixed behavior, wherein the Almax specimen shows features reminiscent of the higher angle joints while the Nextel one is significantly stiffer and fails at strains similar to the butt joint. The 75° joints are considerably stiffer than most others and achieve substantially higher failure loads, with failure 2 To differentiate between the different forms of shear relevant to this problem ‘slipping’ will refer to the inward shear in the periphery of the interlayer whereas ‘sliding’ will be used for shear involving lateral displacement of the composite sub-elements.
Fig. 3. Schematic diagram representing the boundary conditions of the three models. The finite element model is depicted in (a) wherein the composite is fixed rigidly at the ends, the best correlation with experiment. Transverse motion is precluded in (b) and corresponds to the upper bound. Conversely, the composite subelements are free to slide along the interlayer in (c) and represent the boundary conditions of the lower bound.
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3.2. Upper bound approximation
Fig. 4. Schematic of a scarf joint interlayer depicting the relevant interfacial displacements. The upper bound is satisfied when uz =0, and the conditions for the lower bound require that d2 = 0. Displacements in the finite element models are such that uz ] 0 and d2 \0. Throughout this paper, strains are measured by the change in gage length from the initial size (A) to the deformed length (B).
Because of the finite sliding effect present in the joints for u \0° it is necessary to define an appropriate interlayer strain to be calculated in the model and compared with the experiments. This is done by considering the relative displacements of two points at the boundaries of the interlayer which are initially aligned in the axial direction, as illustrated in Fig. 4. For the case of the ‘rigid’ approximation uz =0 and the strain is purely axial. For the ‘sliding’ approximation d2 = 0 and the ‘gage length’ will rotate during deformation relative to the vertical axis. A drawback of this approach is that the increased effective gage length (A =h secu) results in an apparent stiffening of the interlayer.
3.1. Finite element model Full three-dimensional models of the scarf joints were developed using the ABAQUS™ general finite element code and implemented on a Silicon Graphics Origin 200 workstation. The mesh consisted of 2816, 20-node solid, brick elements arranged in an eight by eight grid at the composite ends and refined to an eight by 32 grid within the interlayer. The boundary conditions were set to reflect experimental conditions and specimen dimensions. Namely, a uniform displacement in the axial direction was applied along the top surface of the models, while both the top and bottom surfaces were constrained against transverse motion (Fig. 3a). The loading direction is along the y-axis in the figure and is coincident with the fiber orientation within the composite. The strain reported in the results is calculated as discussed for the analytical models in Fig. 4, except that uz ]0 and d2 \ 0.
This analytical approximation represents an extension of the model originally detailed in [2], which provides a description of the interlayer stress distribution as a function of angle and applied load. The previous model assumed perfect plasticity in the interlayer while the present analysis incorporates elastic contributions. (Because of the absence of strain hardening the model is not a true ‘upper bound’. However, the hardening effects are expected to be modest owing to the small strains involved in this case.) The previous model also considered the contribution of the slipping regions depicted in Fig. 1, which results in a more compliant response. These effects are neglected in the present approximation. In addition to the assumption of rigid attachments (uz = 0), the composite itself is considered rigid and therefore the metal interlayer is essentially constrained from shrinking laterally as it is stretched in the axial direction, whereupon oxx = ozz = 0. These conditions differ from the plane strain (o33 = 0) assumption in [2] and result in a less compliant response, but the net effect is consistent with the upper bound nature of the solution. Since the overall displacement is limited to the y-direction, the orientation of the principal stresses and hence the stress-strain response should not change with the angle of the interlayer. Under this assumption, the upper bound model is most conveniently demonstrated with the butt joint geometry because the boundary conditions can be applied without additional strain transformations. Therefore, o11 = o33 = 0. The general expression for the elastic strains in the butt joint geometry is: oii =
1 (s − sjj − 6skk ); E ii
i" j" k
(1)
wherein n= 0.34 and is the Poisson’s ratio for the interlayer metal. From o11 = o33 = 0 it follows that: s11 = s33 = o22 =
6s22 1− 6
s22 26 2 1− E 1− 6
(2)
(3)
The latter equation is valid until yield is reached. For the given boundary conditions the von Mises yield criterion reduces to s22 − s11 = so, whereupon yield should occur at an applied load of: sa = s22 = so
1− 6 1− 26
(4)
Above this stress the interlayer is modeled as elasticperfectly plastic. The increments of plastic strain follow the relation do pij = lSij where l is the plastic multiplier and Sij are the deviatoric stress components. For s11 = s22 Sij is given by:
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1 S11 =S33 =s11 − (2s11 +s22) 3
(5a)
1 S22 =s22 − (2s11 +s22) 3
(5b)
The plastic strain increments are then given by: l do p11 = do p33 = (s11 −s22) 3
(6a)
2l (s −s11) 3 22
(6b)
do p22 =
The total strain is the sum of the elastic and plastic strains, doii =do eii +do pii
(7)
with do11 = 0 and do22 \0. From Eq. (1) and Eq. (2) the incremental elastic strains are:
1 26 do = ds22 − ds11 E E
(8b)
do11 =
1−26 l ds22 + (s11 −s22) =0 E 3
(9a)
do22 =
1−26 2l ds22 + (s22 −s11) E 3
(9b)
Solving for the plastic multiplier in Eq. (9a), substituting l into Eq. (9b), and integrating gives:
n
3(1−26) 2 s22 − sO E 3
(10)
This equation describes the upper bound response after yield. Because the composite stiffness is finite, the interlayer is not completely constrained from contracting laterally, whereupon the strains o11 and o33 are non-zero, albeit small. This will tend to lower the in-plane interlayer stresses and increase the longitudinal strain predicted by the model for any given stress. To a first approximation, the difference between the transverse strains in the composite and interlayer can be expressed as:
sa 1+ tan2 u
(12a)
t23 =
sa tan u 1+ tan2 u
(12b)
The relevant yield criterion under plane strain3 is:
Substituting Eqs. (6a) and (6b) and Eqs. (8a) and (8b) into Eq. (7) and invoking the yield criterion for perfect plasticity, which implies ds11 =ds22:
In contrast to the previous model, the lower bound approximation assumes essentially unconstrained interlayer shearing in the plane of the joint so that d2 =0 in Fig. 4. This allows the interlayer to attain high levels of plastic strain, wherein work hardening can be significant. To account for this, the model incorporates the actual constitutive behavior of the interlayer metal in the plastic regime, with the appropriate transformations into equivalent stresses and strains. The analysis requires full interlayer plasticity and assumes plane strain along the width of the joint. Neglecting the effects of the slipping regions (Fig. 1), it can be shown from simple equilibrium considerations that:
(8a)
e 22
o22 =
3.3. Lower bound approximation
s22 =
1−6 6 ds11 − ds22 do e11 = E E
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Em m o c/m o =6effo m 11 = 6m − 6c 22 Ec 22
(11)
In this equation neff is an effective Poisson’s ratio relating the longitudinal and constrained transverse strains in the interlayer (neff/n :0.76). The effect of the reduced constraint on the upper bound model may then be approximated by using neff in lieu of the nominal n for the metal in Eq. (3) and Eq. (10).
3 (s − s11)2 + 3t23 = s 2 4 22
(13)
Where the left hand side of the equation is the equivalent stress and s is the strain-dependent yield stress, to differentiate it from the yield stress in perfect plasticity so, (s [ so at the onset of yield). Letting s11 = as22 and substituting Eqs. (12a) and (12b) into Eq. (13) gives: sa 2(1+tan2 u) = s 3(1 −a)2 + 12 tan2 u
(14)
The relevant lower bound is obtained by maximizing this equation, i.e. when a= 1: sa 1+tan2 u = s 3 tan2 u
(15)
which is the relationship between the stress applied to the scarf joint, sa, and the concomitant effective stress in the interlayer, s . Transforming the equivalent strain in a manner consistent with previous results is more involved. From the condition of plane strain and the normality rule, the strain normal to the interlayer can be expressed as: o22 =
3l s22 − s11 =0 4 s
(16)
where o22 equals zero since s22 = s11 for the highest lower bound (a=1). The effective plastic strain, o , is 3
The yield criterion in Eq. (13) is different than that expressed for the upper bound appromiation due to the difference in boundary conditions.
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then reduced to the shear component along the width, given by the relation: o =
g23
3
=
2
d3
3 h
(17)
wherein d3 is the displacement along the plane of the joint, illustrated in Fig. 4. From this diagram it is also inferred that the strain measured by the gage is ointerlayer = (B− A)/A, where A is the initial gage length and B is the length in the displaced state. One can readily show that: A =h secu
(18a)
B =[A 2 + 4d 23 + 4Ad3 sin u]1/2
(18b)
The resulting expression for the gage strain is then: ointerlayer = [3o 2 cos2 u + 3 sin (2u)o +1]1/2 −1
(19)
Thus, if the constitutive behavior of the metal in the interlayer, s (o ), is known, one can use Eq. (15) and Eq. (19) to generate the response of the interlayer, sa(ointerlayer), in a joint of arbitrary scarf angle u.
4. Results and discussion
4.1. Finite element 6ersus analytical approximations A comparison between the stresses calculated with the FEM and lower bound models for different levels of strain (0.01 5o 5 0.05) and scarf angles (30 5 u5 60°) is given in Fig. 5. The applied load is normalized by the yield stress (so) so that the result is independent of the input constitutive law, which is the same for both the analytical and FEM solutions. The dependence of
the absolute stress values on the interlayer constitutive law will be addressed in Section 4.2. It is evident that the correlation between the FEM and lower bound models is excellent for the 45 and 60° joints, and quite reasonable for the 30° joint. This indicates that the behavior of the interlayers in this range of scarf angles is more adequately represented by the ‘sliding’ approximation and thus that the rigid constraint assumed in previous models is largely absent. (As will be shown later, the opposite is true for the butt joint and low angle interlayers.) The agreement for u= 45° is not surprising as the conditions for sliding along the plane of the interlayer are most favorable at this angle. The maximum deviation ( 6%) occurred for the 30° joint at the lower plastic strains (1%). More significant, however, is the fact that the ‘lower bound’ result is higher than that predicted from the full 3D FEM calculation. The primary source of this discrepancy is the omission of the edge effects in the lower bound analysis. One can estimate from the FEM results that the slipping region at the edge depicted in Fig. 1 comprises 25% of the interlayer area for the 30° joint (cf. Fig. 9). The average axial stress in this area is lower than that in the core by 20%, but the lower bound analysis assumes that the whole interlayer is supporting a uniform stress, i.e. that corresponding to the core. This effect reduces the lower bound values for the 30° joint by 5%, bringing a majority of them slightly below the line representing perfect agreement, as expected for a lower bound calculation. The development of the slipping regions in the 45 and 60° joints result in a similar stress reduction. Notwithstanding this correction, however, the normalized applied load for the 30° joint at low strains (1%) still remains above the values predicted by FEM. This is ascribed to the effects of a significant elastic strain component normal to the interlayer in the FEM calculation (d2 \ 0 in Fig. 4) that is neglected in the lower bound analysis. Calculated for the 30° joint with the FEM model, the inclusion of the additional displacement increases the interlayer strain by a factor of 1.2, equivalent to an effective decrease in applied load of about 4% at the same nominal strain. The effect decreases rapidly with increasing strain in the 30° joint, and is not significant in the region of concern for the 45 and 60° joints.
4.2. Interlayer constituti6e beha6ior
Fig. 5. Correlation between results from the FEM and lower bound analytical models. Normalized applied stress levels are presented for interlayer strains between 1 and 5%. Results for the lower bound lie above those of the FEM primarily due to omission of edge effects in the analytical model.
Even though the agreement between the analytical (lower bound) and FEM predictions provides confidence in the models, initial comparison with the experiment met with less success. FEM calculations incorporating the interlayer constitutive behavior measured from the monolithic alloy consistently underestimated the stress at a given level of strain for joint angles in which the interlayer experienced significant
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s =
3 tan u sa 1+ tan2 u
119
(20)
And the effective strain, o , is similarly related to the measured strain in the interlayer by: 1
(21) {[(1+ointerlayer)2sec2u− 1]1/2 − tan u}
3 These equations apply only when the lower bound assumptions are satisfied; i.e. when sliding along the interlayer is dominant and the plastic strain is substantially larger than the elastic strain. The overall constitutive behavior was fitted to a Ramberg–Osgood model [6]: o =
Fig. 6. Plot comparing the constitutive law measured from the monolithic alloy to that calculated for the constrained interlayer. Using the lower bound approximations, the interlayer behavior was deconvoluted from the response of the 45° Almax specimen (shown in the figure) for strains in the plastic regime. This was coupled to the elastic response of the metal using a Ramberg–Osgood fit (s0 =162 MPa; o0 = 0.23%; a = 3/7; n= 8).
plasticity. After extensive analysis of the experimental conditions and testing the models with different mesh refinements and boundary conditions, it was concluded that the most likely source of the disparity was the constitutive behavior of the interlayer metal that was used in the calculations. In essence, the comparison between models and experiments suggests that the stresses in the interlayer must be substantially higher than those exhibited by the monolithic alloy at the same equivalent strain. This hypothesis was examined using a dual strategy. First, a constitutive law for the metal was extracted from the measured stress-strain behavior of one of the experiments. The result was then used in the FEM model to simulate the response of the other specimens. As this approach produced good agreement with the experiments, the second element of the strategy focused on probing independently whether the interlayer does exhibit the additional hardening suggested by the modified constitutive law, and what the possible reasons for this behavior might be. The derivation of the modified constitutive law is presented in this section and the verification and analysis of the hardening in Section 4.3. Owing to the geometry and dimensions of the metal interlayer one cannot deduce its constitutive behavior directly from the experimental stress-strain response. However, if the conditions of the lower bound analysis are reasonably satisfied, one may deconvolute the constitutive law from the stress-strain response using Eq. (15) and Eq. (19) and reversing the procedure outlined at the end of Section 3.3. The equivalent stress in the plastic regime is calculated from a simple rearrangement of Eq. (15):
s s o = +a so oo so
n
(22)
wherein so and oo are the nominal yield stress and strain, and a and n are fitting parameters. The elastic response of the model was set to match that of the monolithic alloy (70 GPa), while at higher strains the model was fitted to the data generated with Eq. (20) and Eq. (21). The procedure described above was applied to the stress-strain curve measured for the 45° Almax specimen. The lower bound conditions are expected to fit most closely the 45° interlayer, as noted above4. The selection of the Almax instead of the Nextel specimen was arbitrary. (The 45° Nextel specimen was part of the model validation that included the rest of the specimens from both sets, as discussed in Section 4.4.) The resulting constitutive law for the metal in the interlayer is shown in Fig. 6 and compared with the response measured independently for the monolithic alloy. This exercise indicates that the 0.2% offset interlayer yield stress is 162 MPa. In comparison, the yield strength of the unconstrained monolithic alloy was 879 3 MPa.
4.3. E6idence and origin of interlayer hardening Independent experimental confirmation of the elevation in yield stress deduced above is not trivial, as a monolithic body of metal that shares the characteristics of the interlayer cannot be easily generated. However, it is feasible to probe the hardness of the interlayer by microindentation and compare it with values obtained from the specimens used to evaluate the stress-strain behavior of the monolithic alloy in Fig. 6. A higher interlayer hardness would qualitatively confirm the elevation in yield stress. It is also possible to estimate the elevation in strength if a correlation between hardness and yield strength is available. 4 A similar constitutive response for the interlayer is obtained using the 30 or 60° data, supporting the validity of the analysis. This will be apparent in the verification of the model against the complete set of experimental data.
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Vickers microindentation tests using 10 g loads were performed on two pristine butt joint specimens and the two 75° specimens which failed by composite fracture at the grips after relatively small plastic strains. The results were very similar as one 0 and one 75° sample had a VHN of 72 and the remaining two had a value of 79. In contrast, the measured hardness for a monolithic alloys specimen with a yield strength of 87 MPa was 65. (A set of Rockwell H measurements on the monolithic specimen produced a value of 91, equivalent to 56 VHN.) Although the scatter is significant, the measurements support the view that the metal in the interlayer has a higher yield strength than that of the monolithic material cast under similar conditions. These results are compared to literature values for commercial Al–Mg alloys in the appendix to assess whether the estimated increase in interlayer yield stress is consistent with the microhardness measurements. A more elusive issue is the identification of the operating mechanism(s) responsible for the interlayer hardening. Microstructural analysis revealed an average grain size of 1 mm in the cast monolithic alloy used to measure the constitutive law in Fig. 6, and 10 mm in the interlayers of the joints. Significant grain refinement may be anticipated from the physical dimensions of the interlayer and the numerous sources of heterogeneous nucleation available during the infiltration through a porous ceramic and into a space surrounded by fiber tips with diameters on the order of the metal grain size. While the level of grain refinement is substantial, however, the magnitude of the concomitant Hall–Petch effect (with a coefficient of 0.068 [7]) is only about 20 MPa.
Fig. 7. Mechanical response of scarf joint interlayers generated from the finite element models. Limit-load behavior is apparent for u ] 15° and corresponds to ‘sliding’ in the interlayer. The butt joint reaches a limit load at higher stresses due to the convergence of the slipping regions. Superimposed are the average failure stress of the experimental specimens for 0 5u5 15° (wherein the response is sensitive to strain) and the average failure strain for u] 30° (wherein the response is stress sensitive).
A second source of hardening is associated with the thermal expansion mismatch between the metal in the interlayer and the surrounding combination of composite and porous ceramic mold. It is well known that such effect is responsible for dislocation generation and in-situ hardening of the matrix in discontinuously reinforced Al composites when cooling from the processing temperature [8,9]. An accurate assessment of this contribution is hindered by the geometric complexity of the problem. (The joint specimen is embedded in a porous mold, which in turn is surrounded by a large mass of metal and the steel die. Moreover, there are thermal gradients, and pressures higher than the yield stress applied during solidification and part of the cooling process.) A simple analysis which considers the interlayer as constrained solely by the composite and only in the plane of the joint gives estimated plastic strains of 1%. This amount of strain could produce an increase in the yield strength as high as 40 MPa, based on the constitutive law for the monolithic specimen in Fig. 6. Added to the grain size effect, one might anticipate an elevation of the yield stress of 60 MPa, which is somewhat lower than the value estimated from the lower bound analysis but of the right order. The issue, however, requires further investigation.
4.4. Comparison of predicted and experimental constituti6e beha6ior Fig. 7 summarizes the predicted stress-strain behavior of the interlayer in scarf joints with 0° 5 u5 75°, calculated with the FEM model using the modified constitutive law for the metal. The trend in predicted interlayer behavior with increasing angle is remarkably consistent with the experimental results. The differences in form between the lower angles (0 and 15°) and the intermediate angles (30–60°) are quite evident. The low angle joints achieve only modest plasticity even at loads past the failure strengths of the experimental specimens (300–320 MPa), whereas the higher angle joints develop large plastic strains below the experimental failure loads. It should be noted, however, that large interlayer strains consistent with limit load behavior would be predicted for the 15° joint if failure by debonding were not to intervene and the applied stress could reach 600 MPa. Although the 0° curve does not behave in the same manner within the range investigated ( 5 650 MPa), it can be shown that a butt joint would reach a limit load when the slipping regions in Fig. 1 overlap [1]. For the present specimens the requisite stresses would be on the order of 1600 MPa, and are clearly unattainable because interfacial failure intervenes at much lower loads. The predicted response of the 75° joint is similar to the 15° in that it exhibits limit load behavior at much higher values of stress than the 30–60° joints. As in the case of the lower angle joints,
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the loading is interrupted by fracture well below the limit, although in this case it is the composite that fails and not the interlayer-composite interface. The limit load is thus predicted to decrease rapidly as u deviates from 0°, reaching a minimum at 45° and then rising again as the angle increases above 45°. The elastic response of the interlayers also varies with scarf angle, as evident in Fig. 7. The elastic stiffening produced by the interlayer constraint in the butt joint, Eq. (3), is relaxed as u increases. There are, however, more significant geometric effects that arise from the manner in which the interlayer strain is calculated, as noted before (cf. Fig. 4). If d2 d3, the strain would increase with angle for a fixed gage length A and sliding displacement d3, as inferred from Eq. (18b). Conversely, for a fixed displacement and interlayer thickness (h) the gage length would increase with increasing u, reducing the apparent strain Eq. (18a). Results of the FEM analysis indicate that the latter effect is dominant for the higher angles. Within the elastic and near yield regime the calculated magnitude of d3 is approximately constant for a given load, and essentially independent of angle. As d3 A, the increased gage length effect leads to an artificially lower strain and thus an apparent stiffening. This is manifested most evidently in the response of the 75° joint, which exhibits a higher modulus than the 0° joint. Comparisons between the experimental response of the interlayers with the predictions from the different models are given in Fig. 8a – f for each of the scarf angles investigated. Both the lower bound analytical approximation and the FEM model utilize the constitutive law derived in Section 4.2, while the upper bound model uses an elastic-perfectly plastic response with the same yield stress (162 MPa). Since the upper bound approximation is closer to the FEM results for the 0 and 15° joints, the lower bound solution is not included for these cases. For the same reasons, only the lower bound approximation is included for the higher angle specimens, u] 30°. Consider first the lower angle joints in Fig. 8a and b. For the 0° case there is good agreement between the two experimental results, and excellent correlation with both the upper bound and FEM models, especially in the case of the Almax specimen. At higher levels of applied stress edge effects become more important, causing the finite element model to be more compliant than the upper bound solution. The 15° specimens exhibit a higher degree of experimental variability than the butt joints. The Nextel 610 specimen is quite consistent with the calculations, but the interlayer in the Almax composite is substantially more compliant. While the details have not been ascertained, it is likely that the behavior of the Almax joint reflects simply a larger than normal variability in the experimental conditions during specimen manufacture, and possibly dur-
121
ing testing. The good experimental agreement with the upper bound model suggests that the early response of the 0 and 15° joints is dominated by elasticity. The stresses needed to activate significant interlayer sliding and plasticity are above the failure point of the actual specimens and thus the deviation from the ‘rigid’ approximation is not too critical in the context of these experiments. However, the results are insightful in predicting superior joint performance if interfacial failure were postponed to higher loads, e.g. by strengthening the interlayer with particles [10]. The experimental responses of the higher angle joints are compared with the FEM and lower bound models in Fig. 8c through f. In general, there is good agreement between the lower bound and FEM models and the experimental measurements. The discrepancy between the Almax and Nextel responses for a given angle is reasonable, as the apparent limit loads for each pair of specimens are within 10% for 305 u5 60°. The correlation for the 75° joints is not as clear. Failure of both Almax and Nextel specimens occurred within the grips and not by separation at the composite-interlayer interface, as in all other angles. However, the shape of the Nextel 610 curve in conjunction with the FEM result suggests that the interlayer was at the onset of the limit load regime when composite failure intervened. This indicates that the mechanical behavior of 75° joints is likely to be controlled by sliding and interlayer plasticity, as is the case for 305 u5 60°. The correlation between the finite element and experiment for the geometries investigated indicates that the FEM model accurately represents joint behavior in both the elastically dominated regime at low angles and the plastically dominated regime at u] 30°. The finite element work can thus be used with confidence to investigate the interlayer stress and strain distributions, and how these stresses influence joint failure. For instance, possible changes in the limiting failure mechanism between the elasticity-controlled response at low angles and the plasticity-controlled behavior at higher angles can be studied more rigorously. These issues will be explored further in a later publication dealing with the failure criteria for these joints [5].
4.5. Interlayer stress distributions The relative magnitudes of the core and slipping regions are illustrated in Fig. 9, which shows stress profiles through the mid-plane of the specimen bisecting the long axis of the interlayer, at the interface with the composite sub-elements. The normal and hydrostatic stresses in this figure were calculated with the FEM models at an interlayer strain of 4%, corresponding to applied loads of 345, 300, and 346 MPa for the 30, 45, and 60° joints, respectively. Although the load for the 30° joint is above its experimental failure stress, the
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Fig. 8. The experimental stress-strain behavior of the 0 and 15° specimens are compared to the FEM and upper bound analytical results in (a) and (b) using neff =0.26. The experimental behavior of the 30 – 75° joints are shown in (c) through (f) and compared to the FEM and lower bound analytical models.
stress distributions are well developed at this level of strain and more clearly show the relevant features. Comparison of the normal stresses (s22 in the coordinates of Fig. 4) in Fig. 9a reveals a decrease in overall magnitude with angle. This is readily explained by a simple force balance in the axial direction wherein the shear stress along the sliding direction, t23, is shown to carry an increasing amount of the applied load as the
scarf angle increases [2]. This holds even though the applied load in the 45° joint is lower than that of the 60° at the chosen level of interlayer strain. A decrease in normal stress near the free edge is common to all angles and corresponds to the slipping region in Fig. 1, along which large hydrostatic stresses build up with distance from the free surface. As assumed in the present models and in agreement with previous work
D.D. Brink et al. / Materials Science and Engineering A281 (2000) 113–125
[1,2], the stresses within the core are essentially independent of position. (The slight stress concentrations at the intersection of the core and slipping regions may have implications in the failure process, but are not particularly significant for the purposes of the present analysis.) The slipping regions comprise 25% of the interlayer area for the joints depicted, reducing the average stress in the joint below that of the core level and leading to the discrepancy between the lower bound analytical and finite element models discussed in Section 4.1. The interfacial pressure stresses for the joints are shown Fig. 9b, where negative values represent tension. The build-ups of stress in the slipping regions are evident and correspond to the rise in the normal stress in Fig. 9a. The stress decreases with increasing angle owing to the higher interlayer shear component, as discussed in [2]. For the particular example in this figure the average magnitude of the pressure stress in the slipping regions is about 15 – 20% below the core value. As the load increases the magnitude of the slipping regions increases with a consequent divergence of the core and average stresses. For example, the slipping regions would comprise 40% of the interlayer area in the 60° joint at 385 MPa, and the average stress would be 88% of that in the core. (This is hypothetical as the assumed stress is above the experimental failure loads.) The result indicates that the analytical models become increasingly inaccurate as the load increases due to the omission of edge effects. However, the moderate deviations calculated within the relevant stress range of the present joints (B 5%) indicate that the upper and lower bound analytical models remain a good approximation of the interlayer behavior in these experiments.
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occurs in the composite rather than at the interface. Analytical (upper and lower bound) and finite element models were developed to describe the response of the joints in an attempt to understand the development of the interlayer stress state and the concomitant influence on performance. Comparison with experiment indicates that the constitutive behavior of the monolithic alloy differs from that of the interlayer, an effect originating from the variation in processing conditions between the monolithic material and the constrained interlayer. The interlayer constitutive law was identified using the models, and resulted in good experimental agreement for the upper bound model at low angles, the lower bound model at higher angles, and FEM for all angles. The correlation suggests that within the range of experimentally accessible loads, joint behavior for 0 5 u5 15° is dictated by elastic effects within the interlayer while that of higher angle joints is dominated by plasticity. Limit loads are expected for all angles above the stresses at which interfacial failure is initiated, with a minimum established at u= 45°. Also, the close correlation between FEM and experiment sug-
5. Conclusions Model scarf joints consisting of thin metal interlayers inclined at angles of 0 5u 575° between continuous fiber reinforced MMC’s were manufactured and tested in tension. The high level of constraint on the interlayers causes the development of large hydrostatic stresses within them, and allows the joints to support loads much larger than the UTS of the monolithic interlayer alloy. The mechanical response of the interlayers changes dramatically with scarf angle, however. The low angle joints (05u5 15°) are nearly elastic until failure ensues at the composite-monolith interface. In contrast, interlayers with higher angles (30 5 u5 60°) exhibit much larger strains and appear to reach a limit load prior to failure. Although the highest loads are reached for u =75°, these interlayers develop less plastic strain than when u =45 9 15°. Moreover, failure
Fig. 9. Interfacial distributions of the (a) normal stress and (b) pressure stress for 30 ]u ]60° at an interlayer strain of 4%. The extent of the slipping regions is apparent for these conditions, and results in a reduction in load carrying capacity of 5%.
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mens. The Nextel-610™ fibers were kindly provided by Dr Colin McCullough of the 3M Corporation.
Appendix A
Fig. 10. The relationship between hardness and yield strength for various Al alloys is explored by comparing as-cast (this study) alloys with annealed and cold worked alloys from the literature. Correlations for the three conditions are established in (a) and appear to be insensitive to alloy content. Trends for specific alloys are shown in (b) and have slopes ranging between 0.15 and 0.18, agreeing well with the proposed relation for the as-cast monolithic and interlayer alloy.
gests that the stress distributions obtained from the model are accurate and can be used to explore the failure mechanisms which limit joint performance. The development of appropriate failure criteria for the scarf joints will be the subject of a subsequent publication [5].
Acknowledgements This investigation was supported by the Mechanics and Materials Program of the National Science Foundation under grant CMS-9634927. The authors are grateful to Dr M.Y. He and Dr H. Kim for their assistance in implementing the ABAQUS FEM code, to Dr Chad Landis for useful discussions pertaining to the development of the upper bound model, and to Dr J.Y. Yang for his assistance in the preparation of the speci-
Correlations between hardness and yield strength for aluminum alloys are rather scarce in the literature. An early analysis by Tabor [11] suggests that the ratio of the Meyer hardness to the yield strength for heavily work-hardened materials is constant and has a value of 2.8 for Al alloys. One can then show for typical indentation diameters that the ratio of the Brinell hardness (HB, kg mm − 2) to the yield strength (so, MPa) should be 0.27. This correlation is compared in Fig. 10a with experimental values for several 5XXX and 5XX Al–Mg alloys5 in the annealed or as-cast condition, as well as after various degrees of strain hardening [12]. Included in this figure are values for three Al–Mg alloys with different amounts of Mg (1, 4.5 and 7%) pressure cast for this study under conditions similar to those used in fabricating the joint specimens. (These hardness values were determined as Vickers and then converted to Brinell using a standard correlation [13] to facilitate comparison with the values for the commercial alloys.) A linear regression analysis shows that there is a reasonable correlation for each group of data (pressure cast, annealed/as-cast and strain hardened). The slopes of the lines decrease from the pressure cast specimens to the cold worked materials, approaching in the latter case the value of 0.27 suggested by Tabor’s analysis (Fig. 10a). However, it is evident that Tabor’s correlation is not useful for the present exercise as neither the monolithic alloy nor the interlayers fit the ‘heavily work hardened’ description. A more elaborate relationship proposed by Chang et al. [14] was not found to be any more satisfactory. A more useful correlation, albeit totally empirical, is suggested when the commercial alloy values are plotted by individual alloy, as depicted in Fig. 10b. Data for the different 5XXX series alloys could be fitted to a simple linear relationship with reasonable correlation coefficients (\ 0.96) and slopes within a relatively narrow range (0.15–0.18). Plotted in the same figure, the values corresponding to the monolithic specimen (HB 549 4 and so 87 MPa) and the interlayer (HB 6693 and so 162 MPa) define a band with a slope close to that of the commercial alloys (0.16). Albeit semi-quantitative, this comparison indicates that the magnitude of the yield stress elevation estimated for the interlayer is consistent with the microhardness measurements.
5 For each Mg content the alloy selected was one which had the minimum amounts of other alloying elements.
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[7] T.H Courtney, The Mechanical Behavior of Materials, McGrawHill, NY, 1990, p. 171. [8] T.W. Clyne, P.J. Withers, An Introduction to Metal Matrix Composites, Cambridge University Press, Great Britain, 1993, p. 84. [9] S. Suresh, A. Mortensen, A. Needleman, Fundamentals of Metal Matrix Composites, Butterworth – Heinemann, Boston, MA, 1993, pp. 124 – 126. [10] D.D. Brink, S.A. Waltner, J.Y. Yang, C.G. Levi, F.A. Leckie, Mater. Sci. Eng. (2000) in press. [11] D. Tabor, J. Inst. Metals 79 (1951) 1 – 18. [12] Metals Handbook, Technical Eds., S.R. Lampman, T.B. Zorc, 10th ed., ASM International, 2 (1990) pp. 89-100. [13] 1989 Annual Book of ASTM Standards, Managing Ed., R.A. Storer, Volume 03.01, ASTM, Easton, MD, 1989, pp. 338. [14] S.C. Chang, M.T. Jahn, C.M. Wan, J.Y.M. Lee, T.K. Hsu, J. Mater. Sci. 11 (1976) 623 – 630.