Interfaces between a fibre and its matrix

The interface between a fibre and its matrix represents an important element in the characterization and exploitation of composite materials. Both theoretical models and analyses of experimental data have been presented in the literature since modern composite were developed and many experiments have been performed. A large volume of results for a wide range of composite systems exists, but rather little comparison and potential consistency have been reached for fibres and/or for matrices. Recently a materials mechanics approach has been presented to describe the interface by three parameters, the interfacial energy [J/m2], the interfacial frictional shear stress [MPa] and the mismatch strain [-] between fibre and matrix. The model has been used for the different modes of fibre pull-out and fibre fragmentation. In this paper it is demonstrated that the governing equations for the experimental parameters (applied load, debond length and relative fibre/matrix displacement) are rather similar for these test modes. A simplified analysis allows the direct determination of the three interface parameters from two plots for the experimental data. The complete analysis is demonstrated for steel fibres in polyester matrix. The analysis of existing experimental literature data is demonstrated for steel fibres in epoxy matrix and for tungsten wires in copper matrix. These latter incomplete analyses show that some results can be obtained even if all three experimental parameters are not recorded.


Introduction
In fibrous composite materials the fibres and matrix interact to produce the resulting properties of the composites. This interaction takes several forms. The fibres and matrix separately contribute with their "own" properties (often in a fraction-weighted way), and these properties of the constituents are both the stiffness, the stress and the strain (elongation). The fibres and the matrix also interact with each other through the interface between them and contribute additional behavior and properties. Normally composite materials are composed of (one type of) fibres of cylindrical geometry with one long dimension and two small transverse dimensions, which normally form a roughly equiaxed cross section, that is often represented by a circular cross section. The representative model for the composite is thus a straight circular fibre in a block of matrix. The interface parameters are the stress and the mismatch strain. The equations also include materials parameters for the fibre and the matrix, as well as the geometry of the test specimens. The parameters are listed: Interface parameters Giic interface energy, J/m 2 τ s frictional shear stress between fibre and matrix, MPa Δe T thermo-elastic mismatch strain between fibre and matrix, [-] Material parameters E f fibre stiffness, GPa r fibre radius, mm α f fibre thermal expansion coefficient, K -1 E m matrix stiffness, GPa α m matrix thermal expansion coefficient, K -1 Experimental parameters P f load on the fibre, N σ c stress on the composite, MPa l d debond length along the interface, mm δ relative displacement between fibre and matrix, mm A c cross sectional area of matrix block / composite, mm 2 ΔT test temperature minus manufacturing temperature, o C In the original development of the model the equations were typically derived and presented as relations for the debond length as a function of relevant parameters, and for the relative displacement as a function of relevant parameters. In order to demonstrate the similarity for the three test cases, pull-out case PO-1 and PO-2, and the single fibre fragmentation case SFFT, the equations will be rewritten, to present (i) load (fibre load or composite stress) as a function of debond length and (ii) displacement/debond length ratio as a function of debond length. The first relation will give the interfacial shear stress, and the second relation will give the interfacial shear stress and the interface energy, and in combination the two relations will give the mismatch strain.
This rewriting of the original equations, the related graphical plots and the analysis of these plots will be illustrated in the following for the three test cases.

Common concepts and relations
Some of the concepts and related equations are identical for all three test cases, and will presented here, even if they originally were derived in slightly different form in the individual papers. These concepts are the general force balance and equality of strains at any cross section of the composite (because the composite has unidirectional and long fibres), and the thermal stresses and strains, as well as the fibre stresses and the matrix stresses in the uncracked composite (ahead of the debond crack tip) and in the debond region, and the fibre strain and the matrix strain in the debond region.  The force balance for the fibre and the matrix is, with sign convention as shown in figure 1: where σ is stress and V is volume fraction, with indices c for composite, f for fibre and m for matrix.
The three test cases are compared in figure 4 of [5], and the condition valid for each case is as follows: PO-1: condition σ m =0, and thus σ c -V f · σ f = 0 PO-2: condition σ c = 0, and thus V f · σ f + V m · σ m = 0 SFFT: condition σ f = 0, and thus σ c -V m · σ m = 0 The equality of strains for fibre, matrix and composite implies that e f = e m = e c . The thermo-elastic strains and stresses are caused by the different stiffness (E) and thermal expansion coefficient (α) of the fibres and the matrix , respectively, and by the fact that the composite (test specimen) is manufactured at (typically) a higher temperature than the test temperature. The thermal strains (which are stress-free) of each component are E E The relation between stress (load) on the fibre σ f and debond length l d is [4] [8]: (1) The load on the fibre P f ( = σ f · π r 2 ) is written as a function of l d : This indicates a linear plot of P f vs l d , with slope and cut-off on y-axis given as The relation between stress (load) on the fibre σ f , the debond length l d , and and the relative displacement between fibre and matrix δ is [4] [8]: The ratio δ/l d is rewritten as a function of l d , using the term ( This indicates a linear plot of δ/l d vs l d , with slope and cut-off on y-axis given as The two (linear) equations (2) and (3) allow the three interface parameters to be determined from two linear plots. Eq (2) gives the interface shear stress τ s from the slope SL1, and yields a combination of G iic and Δe T from the cut-off CO1. Eq (3) gives the interface shear stress τ s from the slope SL2, and the interface energy G iic from the cut-off CO2. Inserting this value for G iic into CO1 gives a value for

Pull-out case PO-2
This pull-out case has been used in some tests and implies a clamping device to support the matrix block at the fibre end of the specimen. The practical geometry has often been a drop of matrix (polymer) on the fibre, which is pulled out of this drop. The macroscopic specimen geometry of this droplet test is not well defined. The present model (which implies a circular cylindrical geometry) was presented [5] with a detailed analysis and comments to the assumptions and concepts used to obtain the model. Here the results are presented and rewritten into a form, which is comparable to the other test cases. The load condition is σ c = 0, and the force balance is thus It should be noted that the second term with σ c /E c is not present because the condition is σ c = 0. The down-stream stresses are This indicates a linear plot of P f vs l d , with slope and cut-off on y-axis given as It should be noted that eq (5) is similar to eq (2) of case PO-1, with the scaling factor V m E m /E c , such that loads (stresses) on the fibre relate by P The relation between stress (load) on the fibre σ f , the debond length l d , and the relative displacement between fibre and matrix δ is ([5] eq 11): The ratio δ/l d is rewritten as a function of l d , using the term This indicates a linear plot of δ/l d vs l d , with slope and cut-off on y-axis given as It should be noted that eq (6) is identical to eq (3) of case PO-1.
The two (linear) equations (5) and (6) allow the three interface parameters to be determined from two linear plots. Eq (5) gives the interface shear stress τ s from the slope SL1, and yields a combination of G iic and Δe T from the cut-off CO1. Eq (6) gives the interface shear stress τ s from the slope SL2, and It should be noted that although eq (3) for case PO-1 and eq (6) for PO-2, respectively, are identical, the relations to find Δe T are not identical, but (partly) scaled by the factor V m E m /E c .

Single fibre fragmentation case SFFT
This test case represents a different test geometry and specimen geometry, as well as a different progress of the experiment. In the pull-out cases, one specimen gives one set of data. In the SFFT experiment, the (single) fibre will break several times (multiple fracture), and each fibre break gives two fibre ends which retract into the hole of the matrix, giving two sets of data for load, debond length and relative fibre/matrix displacement. This event is repeated at the next fibre break, and this continues until the composite fails. The (potentially) many fibre breaks cause concern over the possible interaction of the debond lengths from nearby fibre breaks. No experiments exist to illustrate the importance of this (possible) course of events.
The model was presented [6] with a detailed analysis and comments to the assumptions and concepts used to obtain the model. Here the results are presented and rewritten into a form, which is comparable to the other test cases. The load condition is σ f = 0, and the force balance is thus It should be noted, in contrast to the pull-out cases, that for the SFFT case (only) the composite stress (load) can be recorded experimentally, and thus eq (8) must be written in terms of σ c .This indicates a linear plot of σ c vs l d , with slope and cut-off on y-axis given as The relation between stress on the composite σ c , the debond length l d , and the relative displacement between fibre and matrix δ is ([6] eq 27): The ratio δ/l d is rewritten as a function of l d , using the term (σ c /V m E m -Δe T ) from eq (7): This indicates a linear plot of δ/l d vs l d , with slope and cut-off on y-axis given as It should be noted that eq (9) is identical to eq (3) of case PO-1, and to eq (6) of case PO-2. The two (linear) equations (8) and (9) allow the three interface parameters to be determined from two linear plots. Eq (8) gives the interface shear stress τ s from the slope SL1, and yields a combination of G iic and Δe T from the cut-off CO1. Eq (9) gives the interface shear stress τ s from the slope SL2, and the interface energy G iic from the cut-off CO2. Inserting this value for G iic into CO1 gives a value for Δe T : It should be noted that although eq (3) for case PO-1 and eq (6) for PO-2 and eq (9) for SFFT are identical, the relations to find Δe T are not identical, but (partly) scaled by the factor V m E m .

Experimental test and analysis
The three test cases have similarities and some marked differences. The two pull-out cases give one set of experimental data (fibre load P f , debond length l d , and relative fibre/matrix displacement δ) for each specimen. In the SFFT case, one specimen (potentially) gives several (double) sets of data. The (potentially) many fibre breaks cause concern over the possible interaction of the debond lengths from nearby fibre breaks. No experiments exist to illustrate the importance of this (possible) course of events.
There are two equations for each test case, both giving a linear plot, where the slope and the cut-off values form the basis for calculating the (three) interface parameters. The experimental data and their analysis are summarized here.

Pull-out case PO-1
The plot of P f vs l d gives the interface shear stress from the slope SL1 as 1  It should be noted that the equations governing the calculations from the experimental analyses, are rather similar and in some situations identical.

Model characteristics
The model is basically one and the same, and is applied to slightly different loading cases. This leads to the rather high degree of similarity for the various equations for load vs debond length and for displacement/debond length vs debond length, and to the scaling for the pull-out cases PO-1 and PO-2 as indicated above. It should be noted that the load vs debond length equation is individual for the three test cases, while the delta/debond length vs debond length equation is identical for all three test cases and is independent of load. This is expected because the delta and the debond length are both connected to the geometry of the debond process. This process is essentially a shear crack (mode II crack) of the interface, which is characterized by the interface energy (chemical nature) and the interfacial shear stress (interface topography).
This combined situation gives two linear equations, which are the basis for the simple and "smart" linear plots, leading to establishing the three characteristic interface parameters, as described above in the analyses.

Case V f = 0
The equations have all been written with the term V m E m /E c as a practical term, entering the equations where required. This term is normally very close 1, because normally one rather thin fibre is embedded in a fairly large block of matrix, which makes the fibre volume fraction very low, and thus the matrix volume fraction V m very close to 1 and E c very close to E m . This fact can be used to simplify the equations by setting V m E m /E c = 1. This situation is not always present as the experimental examples below illustrates.

Traditional experiments
Many interface characterizations and related pull-out experiments have been reported in the literature, e.g. [9] [10] [11] for the end-gripped test geometry PO-1, and [12] [13] for the droplet test geometry PO-2. The tests have normally been performed by using several specimens with different embedded lengths of fibre. The curve for pull-out force (stress) vs fibre displacement is described in detail in [5] and figure 2. In most cases an apparent interfacial shear stress has been calculated from the maximum pull-out force P max and the embedded fibre length l emb .  It has often been noted in the individual publications in the literature, that this apparent shear stress was dependent on the embedded length, as discussed in [5] and illustrated in figure 3. This dependency is caused by the fact that the (maximum) fibre load is linear with but not directly proportional to embedded length. For the sake of clarity the eqs (2) and (5) At the maximum fibre load P max the embedded fibre has debonded completely, as sketched in figure  2, and starts to pull out of the matrix block (against frictional sliding). Therefore, at l d = l emb the fibre load is P f = P max . From this equation (10) The constant is controlled by G iic and Δe T (and material parameters), and only if this constant happens to be zero, will the apparent shear stress give the true interfacial shear stress τ s . For all other situations the apparent shear stress will depend on the embedded length, with increasing or decreasing values dependent on the sign of the constant, as discussed in [5] and figure 3. The equation (10) suggests a direct way of making a correct analysis of simple pull-out experiments, based on various embedded fibre lengths and recorded maximum force from the pull-out curve, (figure 2): by plotting the P max vs l emb and using a linear fitting equation, the slope will give the true interfacial shear stress τ s , and the cut-off will give a constant including a combination of G iic and Δe T , but it will not allow a determination of these two parameters individually. Examples will be given below to illustrate this incomplete analysis of pull-out experiments from the literature.

Practical examples
The model and its method of analysis will be illustrated by three examples. The one example is a complete experiment for steel fibres in polyester matrix, where all three experimental parameters load, debond length and relative displacement are recorded. The other two examples are incomplete and are from the literature, one is steel fibres in epoxy and one is tungsten wires in copper matrix. The relevant properties for the materials systems are collected in Table 1.

Steel fibres in polyester matrix (complete analysis)
The pull-out process is case PO-1 with gripping of the matrix block. These experimental data from Prabhakaran et al [8] have been analyzed earlier, using a slightly more complex procedure [8], where the specimens and the experimental test fixture is described in detail. From the video recording of the debond process a total of 30 pictures were used to measure the debond length l d at the corresponding fibre load P f . From the (continuous) recordings of the relative fibre/matrix displacement, (30) values for δ were selected at these fibre loads. These (30) simultaneous sets of data are the basis for establishing the plot of P f vs l d in figure 4 and the plot of δ/l d vs l d in figure 5. The experimental and material parameters are listed in Table 1. It should be noted that with the fairly thick fibre of radius 0.155 mm the fibre volume fraction of 0.003 clearly is not close to zero, and thus the parameter V m E m /E c is about 0.825 and must be included in the calculations. The straight lines in the plots show rather good fit (R 2 is ca. 0.98). From the slopes and cut-offs of the straight lines the three interface parameters are calculated using the equations for case PO-1 from section 7. the temperature difference between manufacturing temperature and test temperature. With reasonable assumptions as listed in Table 1, the estimated value is: ΔeT = (12 -70) · 10 -6 · (20 -50) = 0.0017.
A comparison with the results of the previous analysis [8] shows that the shear stress is very closely the same, while the interface energy and the mismatch strain deviate significantly.

Steel fibres in epoxy matrix (incomplete analysis)
The pull-out process is case PO-2 with a clamp at fibre end of the specimen (droplet test). The experiments follow the traditional pull-out procedure with measurement of the maximum force for various embedded fibre lengths. The data originate from Gorbatkina [14] and were presented by Zhandarov and Mäder [15] in their figure 4b. The (7) sets of data for maximum force and embedded fibre length are analyzed according to the procedure described above for traditional, incomplete data and plotted as P max vs l emb in figure 6. The experimental and material parameters are listed in Table 1. It should be noted that with the moderately thick fibre of radius 0.075 mm the fibre volume fraction of 0.0009 clearly is not close to zero, and thus the parameter V m E m /E c is about 0.944 and must be included in the calculations. The straight line in figure 6 shows rather good fit (R 2 is ca. 0.94). From the slope for the straight line, the interface shear stress is calculated using the equation for P max vs l emb in the full form eq (5) for case PO-2 from section 7.2. The results are shown here: 15.533 N τ s 18.8 MPa As emphasized above the cut-off CO1 only gives a combination of G iic and Δe T . If the mismatch strain is estimated from its definition and the data in Table 1, the value is ΔeT = (12 -50) · 10 -6 ·(20 -120) = 0.0038, and with this value the interface energy is ca 250 J/m 2 .

Tungsten wires in copper matrix (incomplete analysis)
The study of the interface in composites was (probably) initiated in the early 1960´s, and Tyson [16] performed some of the first pull-out tests for tungsten wires in blocks of copper matrix. One of these test series, which is presented by Kelly [17] figure 5.18, is performed at 600 o C, and includes 8 sets of data for embedded fibre length and maximum force for the pull-out experiment. The test case is PO-1. The 8 sets of data are analyzed according to the procedure described above for traditional, incomplete data and plotted as P max vs l emb in figure 7. The experimental and material parameters are listed in Table 1. It should be noted that with the moderately thick fibre of radius 0.25 mm the fibre volume fraction of 0.007 clearly is not close to zero, and thus the parameter V m E m /E c is about 0.973 and must be included in the calculations. The straight line in figure 7 shows rather good fit (R 2 is ca. 0.94). From the slope for the straight line the interface shear stress is calculated using the equation for P max vs l emb in the full form eq (2) for case PO-1 from section 7.1. The results are shown here: SL1 47.863 N/mm CO1 28.326 N τ s 29.6 MPa As emphasized above the cut-off CO1 only gives a combination of G iic and Δe T . If the mismatch strain is estimated from its definition and the data in Table 1, the value is ΔeT = (4.3 -16.5) · 10 -6 · (600 -800) = 0.0024, and with this value the interface energy is ca 190 J/m 2 . It must be emphasized that this rough calculation is based on an effective manufacturing and stress-free state for the copper matrix at about 800 o C.

Summary
The characterization of the interface between fibre and matrix in composites has been discussed and three interface parameters have been defined: the interface energy G iic , the frictional interfacial shear stress τ s and mismatch strain between fibre and matrix Δe T . The (new) mechanical model for the interface has been presented, and it has been based on the (three) test cases: pull-out of a fibre from a block of matrix, (i) supported by gripping at the matrix end of the specimen, called case PO-1, (ii) supported by clamping at the fibre end of the specimen, called PO-2, and (iii) fibre fragmentation in the single fibre composite, called SFFT. The model is shown to give rather similar but not identical results for the three cases. This facilitates and simplifies the equations for the test cases, which are presented in the same or comparable format.
Two governing equations are derived for each test case, one equation relating the fibre load or composite stress to debond length, and the other equation relating the relative displacement/debond length ratio to the debond length. The first equation is very similar for the three cases, and the second equation is identical for the three cases. This latter derivation is a new observation.
These two equations allow two simple linear plots and the direct determination of the three interface parameters from the slope and cut-off values of these plots. This is possible (only) when the three experimental parameters are recorded: fibre load, debond length and relative fibre/mtatrix displacement. This is illustrated for the material system of a steel fibre in polyester matrix.
Many experiments exist in the literature, which (only) measure the maximum force for pull-out of fibres with different embedded lengths. The mechanical model and the equation for fibre load vs debond length can be used to establish the true interfacial shear stress, and (only) a combination of interface energy and mismatch strain. This is illustrated for the material system of a steel fibre in epoxy matrix, in test case PO-2, and for the material system of a tungsten wire in copper matrix, in test case PO-1.
It is believed that the results for the interface parameters obtained here are correct, but few or no reference values exist to compare and validate the numerical results.