Modification of the model for calculating the long SiC fibre-reinforced metal matrix composites transverse strength and study for interfacial influences

The transverse strength of long SiC fibre-reinforced metal matrix composites is superior to its longitudinal strength owing to interface factors. To accurately predict the transverse strength of long SiC fibre-reinforced metal matrix composites, a micro-mechanics representative volume element (RVE) model was developed; periodic boundary conditions were applied to the model to ensure its displacement and stress continuity. Taking into account the influence of the stress concentration coefficient of the matrix, the failure strength of long SiC fibre-reinforced metal matrix composites under transverse tensile loads and compressive loads were calculated, where the error between the calculation results of the model and the test results was found to be large. A novel calculation method based on the interfacial cohesion model is proposed herein, to improve the accuracy of the RVE model. It was found that the accuracy of the corrected model calculation has been improved through a comparison with the experimental values. The stress/strain relationship between interfaces of different strengths under tensile and compressive loads was analysed, the failure index of interface strength was extracted, and the relationship between the influence of interface strength on failure was determined.


Introduction
Long SiC fibre-reinforced titanium matrix composites have higher specific strength, stiffness, and modulus along the fibre direction than pure matrix materials. They can be used to prepare reinforced parts such as fan blades and low-pressure turbine shafts, which are widely used in high-performance aero-engines [1,2]. However, the poor performance of fibres in the transverse direction makes the transverse strength of the composite less than one-third of that in the longitudinal direction; fan blades, low-pressure turbine shafts, and other reinforced parts of aero-engines are subject to significant transverse loads during practical operation [3,4]. For the parts of aero-engines, large transverse loads often exceed the transverse strength limits of the material, resulting in structural failure [5] of the component. Therefore, studying the strength of long SiC fibre-reinforced titanium matrix composites under transverse loading is of great importance for the design and manufacture of reinforced components for aero-engine applications [6][7][8].
A prerequisite for the application of fibre-reinforced composites to the structural design of aero-engine reinforced components is the accurate prediction of their strength. The transverse tensile strength of long SiC fibre-reinforced titanium matrix composites has been studied extensively [9]. Akser and Choy [5] showed that temperature has a significant effect on the transverse tensile strength of SiC f /TC4 composites. Gao et al [10] investigated SSD characteristics and their mechanism of effect on tensile behaviour and fracture mechanisms for SiCp/Al/45% based on experimental analyses and numerical modelling. They established a representative Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. volume element (RVE) model considering SSD to understand crack initiation and propagation in uniaxial tension. Zahl et al [11] studied the effect of fibre arrangement on the transverse tensile strength of long SiC fibrereinforced titanium matrix composites. et alwed various analytical and computational micro-mechanical methods to evaluate the mechanical properties of particulate-reinforced metal matrix composites. The effects of particle size, shape, and orientation, along with the interface strength on the mechanical behaviour were presented. Zhou et al [12] studied the effects of high strain rates and different orientations on tensile behaviour and microcosmic evolution of Ti-6Al-4V sheets. The final damage of unidirectional composites under transverse loading has been demonstrated to be caused some reasons,they are interfacial cracking and matrix damage, mainly due to the stress concentration resulting from the addition of fibres to the matrix [6]; therefore, the calculation of the transverse strength of long SiC fibre-reinforced titanium matrix composites m into account the effect of stress concentration [13]. However, the predicted transverse strength results are still much larger than the experimental transverse strength [14,15]. On this basis, the model needs to be refined to fully account for the effects caused by the interface. At present, there is no method for determining the transverse strength of long SiC fibre-reinforced titanium matrix composites using a micro-mechanics RVE model based on the stress concentration factor and the influence of the interface.
In this study, the transverse strength of long SiC fibre-reinforced metal matrix composites was calculated, and strength tests under tensile load and compression loads were conducted. The micro-mechanics RVE model was established, considering the influence of the stress concentration coefficient of the matrix. Next, the failure strength of long SiC-fibre reinforced titanium matrix composites was calculated under transverse tensile loads and compression loads. After comparison with the test results, a calculation method based on the interfacial cohesion model was proposed to improve the accuracy of the calculation results. The failure index of interfacial strength was calculated, and the relationship between the influence of interfacial strength on failure was determined.

Fine mechanics theory
Fine mechanics takes into account the inhomogeneity of composites, analysing them on a very small scale. The actual inhomogeneous material is replaced with an equivalent homogeneous medium; therefore, when studying their mechanical properties, only RVEs are used to represent the overall model. The RVE model can be expressed in many ways, such as a single cell or multiple cells [16]. Long SiC fibre-reinforced titanium matrix composites with uniformly aligned unidirectional fibres are studied, whose RVEs have exactly the same mechanical properties, fibre contents, and arrangements. The arrangement of the fibres determines the way in which the RVE model is built [17]. The RVE model with quadrilateral arrangement is used to calculate the transverse tensile stress concentration coefficient of long SiC fibre-reinforced titanium matrix composites, as shown in figure 1.
Since composites often comprise periodic arrangements of RVE cells, when they are subjected to external loading, they exhibit similar stress and strain fields to RVEs. Therefore, the stress and strain fields of an RVE are used to represent the fine-scale stress and strain fields of the composite body.
The RVE model is mapped to the plane and represented geometrically in two dimensions, as shown in Stress continuity requires that the stresses on the corresponding faces of the RVE cells are reversed and of equal value, i.e. there is a continuous stress field between the RVE cells. The stress continuity condition will hold when the displacement continuity condition is imposed [18].

Failure guidelines
The maximum stress criterion is based on the principle of comparing each stress component of a material with its corresponding permissible value of strength, and is characterised by the simple independence of the stress components and the lack of interaction between them. The maximum stress criterion is used in threedimensional failure damage problems by comparing each stress component of the material with its permissible maximum stress (in compression, tension, or shear). If one of these stress components is greater than the permissible maximum, the material starts to fail at the corresponding integration point and the material degrades [19,20]. For the three-dimensional failure problem of the structural damage of composite materials, the corresponding maximum stress criterion can be expressed using the following equation: )represents the ratio of each stress component to its respective failure strength, with the subscripts C and T denoting compression and tension, respectively. The relationship between stress and failure can be expressed by the following equation:

Tests
The basic mechanical properties of long SiC fibre-reinforced titanium matrix composites (SiC f /TC4) were subjected to tensile and compression tests using the test pieces shown in figure 3. The long SiC fibres were manufactured at the Institute of Metal Research, Chinese Academy of Sciences. The SiC layer was deposited on a tungsten monofilament by the chemical vapour deposition process. A carbon coating with a thickness of 2 μm was also produced to prevent a reaction between the SiC layer and titanium alloy. The total diameter of the SiC fibres was about 100 μm, and the average tensile strength was about 3600 MPa. The Ti matrix composites were fabricated through the matrix-coated fibre route to achieve homogeneous fibre distribution. The matrix with a nominal composition of Ti-6Al-4V was deposited on long SiC fibres using magnetron sputtering [21]. The dimensions of the test piece were 173 mm × 24.5 mm × 5.4 mm. The material parameters for the fibres and the matrix are shown in table 1. The tensile test was performed by affixing four strain gauges back-to-back at the centre of the specimens. During the test, the sample was clamped in the middle, as shown in figure 4. The specimens were loaded continuously at a rate of 2 mm/min until damage occurred. The damage load, damage strain, and damage mode of the specimens were recorded, and the modulus was measured in the longitudinal strain interval of 1000 to 3000. The tensile test piece is shown in figure 5.
In the compression test, two strain gauges were pasted back-to-back at the centre of the sample, as shown in figure 6. The samples were then tightened with a 3 Nm torque 3 times, and each end of the clamp was tightened diagonally to ensure that the clamping force on the surface of the sample was uniform. The assembled clamp was placed in the centre of the platform of the testing machine. The specimens were continuously loaded to failure at a loading rate of 1.3 mm/min. The load-strain data of the specimen were continuously collected, and the failure load and failure mode were recorded. The compression test is shown in figure 7.

Transverse strength calculation model
Due to the transverse isotropy of the SiC f /TC4 composite, an RVE model is established, as shown in figure 8. The model consists of a TC4 matrix and SiC f fibres with a fibre volume fraction of 35%; the fibre diameter is 100 μm, and the side length of the RVE model is 0.14976 mm. C3D8R solid cells are selected for both fibres and the matrix, and the RVE model is meshed periodically.
Fixed constraints are applied to the left side of the RVE model, and transverse tensile and transverse compressive loads are applied to the right side.
According to the physical interpretation of the stress concentration factor, the location where damage occurs under external loading is where the stress concentration occurs. Therefore, the stresses should be averaged along

Clamp bolts
Guide rode Test piece Figure 6. Compressive test fixture.
the direction normal to the outer damage surface [22]. As shown in figure 9, the external load is applied along the x 2 direction. Under transverse tensile loading, the damage surface is generally perpendicular to the direction of the external load x , 2 as shown in figure 9(a). Under transverse compression loading, the damage surface forms an angle f with the external load direction x , 2 as shown in figure 9(b). When unidirectional composites are subjected to uniaxial transverse loads, the stress concentration factor of the matrix is defined as the ratio of the line average stress to the body average stress [23]; then

Model modification
In the process of manufacturing and during high-temperature application, interfacial reactions of SiC f /TC4 occurred between the fibres and the matrix to varying degrees, forming interfacial layers due to the chemical activity of Ti. When the stresses and displacements of the fibres and the matrix are continuous at their common boundary, the corresponding interface is called the ideal interface [24]. If either of the boundary conditions for stress or displacement continuity cannot be met, the interface is called non-ideal. In composites, the interface plays an important role in transmitting loads between the matrix and the fibres; therefore, it heavily affects their mechanical properties. The presence of interfaces will improve the accuracy of mechanical property predictions for composites. When the interface between the fibres and the matrix is cracked, the transfer of load between the two is directly affected, which reduces the overall strength of the composite, with the transverse tensile strength being most affected by the cracking of the interface. Then, the transverse tensile strength considering the effect of interface cracking is calculated. In the RVE model, the interface element is inserted, and the interface is selected based on the bilinearity cohesion model as the calculation method for the COH3D8 cohesion element [25]; the interface parameters are shown in table 2. At the beginning of loading, the normal stress of the interface increases linearly until it reaches the normal strength , max s at which the maximum stress criterion is selected and the interface unit experiences damage, the stiffness begins to decline, the damage index increases from 0, and the stress is redistributed. As the damage increases, the interface displacement increases and the normal stress of the damaged element gradually decreases. The maximum displacement criterion is chosen, i.e. when the normal stress of the damaged unit decreases to 0, the interface displacement reaches the maximum failure displacement , n d the interface stiffness decays to 0, the damage index increases to 1, and the interface loses its load-carrying capacity completely. A transverse tensile load of 0 is applied to the RVE model and the load is gradually increased until the interface between the fibres and the substrate cracks, as shown figure 10.   figure 11. When the transverse tensile and compressive load reached 1020 MPa, stress and displacement were apparent along the transverse external loading direction. As the Young's modulus of fibres is much greater than that of the matrix, the deformation of the fibres under transverse loading is less than the deformation of the matrix. The matrix is the main load-bearing component under transverse loading, and the transverse loadbearing capacity of metal matrix composites is mainly dependent on the matrix. Stress concentrations occur in the matrix at the edge of the hole when the strength of the matrix is weaker than the strength of its constituent material. Under transverse tensile loads and transverse compression loads, j is the angle between the shortest external normal on the damage surface and the external load direction. The average value of the stress components in the external load direction of the matrix along the shortest external normal is ,

Modified model calculation results
A fixed constraint was applied to the left side of the RVE model containing the interface layer, and a transverse tensile load of 1020 MPa was uniformly applied to the right side; the calculated results are shown in figure 12.
Under the 1020 MPa transverse tensile load, the interface cracks along the load direction, but not in the vertical load direction. The RVE model undergoes transverse tensile deformation; because the Poisson's ratio of the matrix is nearly twice that of the fibres, the matrix is more likely to produce greater deformation in the vertical transverse external load direction than the fibres under transverse tensile loading, and there is compressive stress along the vertical transverse tensile load direction at the interface between the fibres and the matrix. Figure 13 shows the evolution of transverse tensile damage at the interface. The blue elements in the diagram represent the area where no damage has occurred; the elements in other colours represent areas where damage has occurred and the stiffness has degraded but the elements have not failed; the elements that have disappeared represent the area where damage has occurred and the stiffness has degraded to 0, alongside complete failure. When the load is increased to 204 MPa, the interface is slightly deformed, but no interface cracking occurs; when the load is increased to 235 MPa, the interface reaches the damage initiation criterion on both sides along the direction of external load, damage begins to appear, and the interface cracks; as the external load increases, the interface damage area expands from the initial damage to both sides, and the interface cracks. As the external load increases, the interface damage area expands from the initial damage to both sides and the interface cracks open along the circumferential direction; when the load increases to 1020 MPa, the element stiffness on both  sides of the interface degrades to 0 and the load carrying capacity is lost. Further, the interface cracks gradually expand from 0°to near 70°. The transverse tensile interfacial cracking angle of the SiC f /TC4 composite was obtained experimentally to be 70°, as shown in figure 14(a). The modified non-ideal interface RVE model calculated an interfacial cracking angle of 70.3°for the SiC f /TC4 composite, as shown in figure 14(b), with an error of 0.4%.
Interfacial cracking affects the load transfer from the matrix to the fibres and has a significant effect on the transverse tensile strength of the composite. Table 5 shows a comparison of stress concentration coefficient and strength values when interface cracking is not considered and when it is considered. The transverse tensile stress concentration coefficient of the composite material after considering interface cracking is 1.72 and the transverse tensile strength is 551.24 MPa, which is 6.51% more accurate than the transverse strength of the composite material predicted without considering interface cracking. It is proven that the modified model considers the influence of the interface on the transverse tensile strength of composite materials, and can predict the strength of composite materials more accurately.

Interface influencing factors 4.3.1. Transverse tensile load
The strength limits corresponding to different interface strengths were studied under transverse tensile loading, and finite-element calculations were carried out by taking the normal, 0.2-times, and tenfold strengths. The relationship between stress-strain and interface strength was obtained as shown in figure 15.
In the graph, the horizontal axis represents the strain and the vertical axis represents the stress. From the stress-strain relationship, it can be seen that the transverse tensile strength of the composite increases slowly at the weak interface and reaches a maximum value at the stress of 96 MPa, subsequently decreasing slowly. At normal interface strengths, the transverse tensile strength of the composite increases more obviously and reaches a maximum at the stress of 460 MPa, subsequently decreasing at a relatively fast rate. At the tenfold interface strength, the transverse tensile strength of the composite increases rapidly and reaches a maximum value. When the interface strength increases to a certain value, the influence of the interface strength on the strength of the composite is not so obvious, and the tensile strength of the matrix plays the main role. It can also be observed visually that irrespective of how much the interfacial strength increases, the resulting transverse tensile strength of the composite is always lower than the tensile strength of the matrix at 980 MPa, which fully illustrates that the strength limit of the composite is determined by its weakest component.    Under the action of transverse tensile loading, the RVE model of the normal interface strength and tenfold interface strength is calculated using the finite-element method. In combination with the failure criterion, the damage evolution is calculated as shown in figure 16.
The horizontal axis in figure 16 represents the strain and the vertical axis represents the failure index. From the curves, it can be seen that the damage rate of the interface sharply increases with increasing strain, and the corresponding curve is very steep. In the case of a normal interface strength, the transverse tensile progressive damage process is as follows: the interface first appears to be damaged at a very small strain; as the loads increase continuously, the degree of damage to the interface further expands. When the interface strain is around 0.5 × 10 −4 , the damage to the interface is close to saturation. The matrix is beginning to yield, and the plastic deformation of the matrix also continues to increase. The degree of damage at the interface rapidly reaches saturation and then basically remains constant, while the plastic deformation of the substrate rises slowly with increases in the load.
As shown in figure 16, the failure index-strain curve for the tenfold strength shifts to the right and the failure index-strain curve for the matrix shifts to the left. The gap between the two decreases, and the transverse tensile progressive damage process is as follows: the substrate first exhibits plastic deformation at a very small strain. The interface also begins to exhibit damage with increases in the load, and the degree of damage of the matrix and the interface increase simultaneously. When the strain reaches 1 × 10 −4 , the damage of the interface reaches saturation and no longer changes with the changes in the load, while the degree of plastic deformation of the interface slowly increases with increases in the load after the strain reaches 2.5 × 10 −4 .

Transverse compression load
The strength limits corresponding to different interface strengths were investigated under transverse compressive loading, and at the normal interface strength, 0.2-time, and tenfold strengths. The relationship between stress-strain and interface strength was obtained as shown in figure 17.
The stress-strain relationship shows that the transverse compressive strength of the composite does not always increase as the interfacial strength increases. After the interfacial strength increases to a certain value, the effect of the interfacial strength on the strength of the composite is not so obvious, and the compressive strength of the matrix plays the main role at this point. The stiffness of the material increases with the strength of the interface, and the non-linearity of the material first appears at the weak interface, due to the reduction in stiffness of the material caused by interface damage. At the same time, the increased strength of the interface can lead to a certain brittleness of the composite material. In practice, greater strengths are not optimal, and an appropriate interface strength should be selected according to the actual needs.
For interfaces with different strengths, the RVE model has different damage evolutions when subjected to loads, as shown in figure 18.
As shown in figure 18, the process of progressive damage by transverse compression at normal interface strength interfaces is as follows: the interface is first damaged at very small strains. As the load increases, the damage to the interface rises sharply, and the degree of damage to the interface increases rapidly. When the interface damage reaches a certain level, the matrix appears to yield; with increases in the load, the plastic deformation of the matrix also increases. The degree of damage to the interface rapidly reaches saturation and then remains largely constant at this value, while the plastic deformation of the matrix rises slowly with increasing loads.
As shown in figure 18, the tenfold strength's damage trends for the interface and the matrix are very similar, the transverse compression progressive damage process is as follows: the matrix first exhibits plastic deformation at a very small strain, and the interface also exhibits damage with increases in the load. The matrix and interface damage degrees increase with increases in the loads; when the strain reaches 2 × 10 −4 , the plastic deformation degree of the matrix reaches saturation and no longer varies with the load, while the damage at the interface increases slowly with increases in the load after the strain reaches 3 × 10 −4 .

Conclusion
(1) For the fine-mechanics RVE model, the stress concentration coefficient under transverse tensile loads was calculated to be 1.58 and the transverse tensile strength was 602.99 MPa; the stress concentration coefficient under transverse compression loads is 1.11 and the transverse compression strength is 991 MPa. The RVE model was modified to consider the effect of interface cracking on the transverse tensile strength of the composite material, and the stress concentration coefficient after interface cracking was calculated to be 1.72; the transverse tensile strength was 551.24 MPa. The prediction accuracy of the modified model was 24.17%, with an error of 6.51% from the experimental value.  (2) The interface between the fibre and the matrix reaches the bonding strength limit first, resulting in damage. The longitudinal tensile strength of the composite increases with increases in the interfacial strength, but the effect becomes smaller after a certain extent. At this time, the tensile strength of the matrix plays a major role. When the interface is weak, the damage percentage of the interface quickly reaches about 1, and then basically remains unchanged, while the plastic deformation of the matrix increases slowly with increases in the load. When the interface is strong, the gap between the two will be reduced. Based on the cohesive zone model, the interface strength is adjusted to calculate the failure index under transverse tensile and compressive loads, and the influence of interface strength on the failure mode is given. When the strength of the matrix is fixed, the bonding strength of the interface has a significant influence on the transverse tensile strength of the composite. Irrespective of how much the interfacial strength increases, the transverse tensile strength of the composites is always lower than that of the matrix, at 980 MPa.