1 Introduction

Composites have been widely used in aerospace, machinery, shipbuilding and other fields [1]. Carbon fiber composites have good mechanical properties in various aspects, it is one of the most popular composite materials in our society today, and also need a more stringent working environment temperature compared to other composite materials [2], so excessive temperature will have a destructive effect on the material. Particle-reinforced metal materials are of isotropy, which is similar to metal materials, it has been mass-produced with different smelting technologies.

Materials will fail during working process, if a multiple cyclic stress far below the yield limit, it is defined as fatigue failure. The most failures are belong to fatigue failures in engineering program, and fatigue life prediction plays a significant role for reliability evaluation [3]. Failure occurrence is dangerous for both worker and product, it will lead to unnecessary economic losses in society. Therefore, it is important for the production to predict fatigue life of component more accurately [4,5,6]. The fatigue life prediction of traditional metal materials can be obtained by strength degradation model [7,8,9], damage accumulation model [10], energy method [11], S-N curve [12] and other methods [13, 14]. The fatigue life of materials will be affected by loading strength, load-stress ratio, and different load loading sequences [15,16,17]. In addition, the surface roughness and operating temperature of materials will also affect its fatigue life [18,19,20]. Many scholars have improved the model based on all these factors, and proposed many prediction methods considering material properties and working environment [21,22,23]. And some scholars analyzed the fatigue life by their numerical models [24,25,26,27]. Based on the research results of metal materials, scholars obtained fatigue life for metal matrix composites [28,29,30]. Metal matrix composites have better fatigue properties than metal alloys [31,32,33,34]. Fatigue or failure may be caused by these reasons, some people think that crack closure will be affected by the reinforcements of MMCs in crack propagation process [35, 36]. The fatigue life of MMCs is related to the increase in fatigue crack length and the reduction of the effective driving force for crack propagation caused by the deflection angle.

Path planning was originally part of preset algorithm of robot mobile behavior. According to some evaluation standard, an collision-free path can be found from the starting position to the target position in the obstacles environment [37]. The different distribution of these obstacles will affect the planned path in the environment. Path planning can be defined as an active behavior that allows a robot or mechanical device to find a path around an obstacle based on the environment. There is a variety of algorithms with different logic to be applied in path planning, such as Dijkstra algorithm, genetic algorithm, neural network algorithm [38], ant colony algorithm and so on. If the reinforcement particles of the material are distributed uniformly and disorderly, fatigue crack growth of particle-reinforced metal matrix composites will deflect or break through the reinforcement particles. Reinforcing particles hinder the growth of fatigue cracks, and the composite material has higher fatigue life than the metal in the macro. If the tip stress is large enough, the crack can go through the reinforcing particles. In an ideal condition, all the reinforcing particles can’t penetrate, and crack growth and propagation can be prevented to the greatest extent, and MMCs will have higher life. The cracks bypass all reinforcing particles, and the behavior is similar to robot path planning. It is possible to predict the crack path by path planning algorithm. The fatigue behavior of composite materials is quite different from traditional metals. To predict MMCs’ life more accurate, an improved fracture and fatigue crack growth (FCG) model is given by combining crack length and path planning algorithm.

2 Statement of the Problem

The fatigue life prediction of components is of great significance to product design and production safety. Composite materials have superior performance than traditional materials, which are replaced in some fields. The current prediction models are applied to obtain fatigue life of composite parts in a limited range. To predict the life of composite parts more accurately, the following issues should to be studied urgently:

  1. (1)

    As a new type of material, the studies for composite materials are far less mature than the traditional, and the test data of composite materials are also less than common metals. It is very difficult to collect the parameters of composites than alloys, such as the strength, Poisson's ratio, and section shrinkage. There are many factors to affect the performance of composites, such as the reinforcing matrix volume fraction, metal-based alloy properties and the material manufacturing process. In summary, it is difficult for composite materials to reproduce the experimental data.

  2. (2)

    There are different structures between composite materials and metals. Metal atoms are usually arranged in a certain shape, and some properties can be obtained. There are many forms of composite materials, and the ceramic particle reinforced metal matrix composites will be discussed. The composites are made of silicon carbide particles and alloys, and two materials can be connected by the interface. Due to the different microstructure, the internal stress distribution and fatigue crack growth of composite materials are totally different from the traditional metal.

  3. (3)

    The fatigue failure mechanism of composite materials is different from traditional materials. Traditional prediction models are usually used to calculate fatigue life of the composites. Material performance parameters are replaced with composites, and rough estimates of fatigue life can be obtained. The crack fracture length can be expressed as follows:

$$a_{c} = \frac{1}{\pi }\left( {\frac{{K_{IC} }}{F\sigma }} \right)^{2},$$
(1)

where \(a_{c}\) is the crack length when the material breaks, KIC is fracture toughness, F is the shape factor, σ is the stress.

If it is necessary to calculate the crack fracture length of SiC/Al, the fracture toughness of SiC/Al can be replaced with Al. The force change is caused by the SiC reinforcing particles of the material, and the influence of force change is ignored to obtain the results. This method can be applied to deal with engineering problems within a certain range, and it is different to find the failure mechanism of the composite material. According to the failure process of composite materials, a new model is established to improve the accuracy of life prediction.

3 Crack Propagation Path Model

3.1 Dijkstra Algorithm

Dijkstra algorithm is used to simulate the path planning of random paths. This algorithm can be used to find the single-source shortest path of weighted directed graphs by breadth-first search. After many repetitions, the last shortest path will be found from the end point to the start.

The algorithm will be given to find the shortest path from the starting point A to the ending point F, and the schematic diagram of local shortest path can be shown in Figure 1. There are two arrays U and V, U indicates points that have been backtracked, V indicates points that have not been backtracked, and the number in parentheses is the shortest distance from point F. The process can be expressed as follows:

figure a
Figure 1
figure 1

Schematic diagram of local shortest path

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According to step 2, the shortest distance between point C and F is 13, and the shortest path at this moment is (C, E, F). After an iteration, point D is added to the U array as the shortest path point. The shortest distance between point C and point F is 12, and the shortest path is changed to (C, D, F). After continuous iterations, the shortest path between two points can be obtained.

3.2 Random Barriers

SiC/Al composite material is made of aluminum alloy as the metal matrix, and SiC particles are evenly distributed in the metal matrix as a reinforcement. SiC particles are randomly distributed within a certain range on a microscopic scale, and their shapes are mostly random convex quadrilaterals. According to the composite characteristics, a model is established to simulate the distribution of random SiC particles.

If the crack propagation path extends to the position of the reinforcing base in the composite materials, the crack propagation path will deflect, and it will be lengthened. The rules of composite materials are different from the metals. It is easy to find the propagation path of cracks in composites, and the law of crack propagation is similar to robot path planning. The crack bypasses the reinforcement from the beginning to end, cracking rule is obtained by simulating the crack propagation path, and the relationship between the crack and the fatigue life will be obtained.

The reinforcement base is an irregular quadrangle and randomly distributed in the metal matrix. Supposing each quadrilateral is located as a cell, square interval is used in the path planning algorithm. If four random points are taken in a cell to represent the endpoints of the polygon, maybe the random quadrilateral formed with a "concave polygon". The "concave polygon" structure is prone to stress concentration at the shortest diagonal, and the reinforcing base particles often exist in the shape of "convex polygon" in practical engineering. In the modeling process, the strengthening basis is treated as a random quadrilateral, and the four endpoints fall into four square sub-regions that bisect the cell. Each sub-region is a grid, and points are randomly taken in each grid to form a random quadrilateral representing the position and shape of the reinforcing base. In composite materials, the reinforcing matrix is uniformly distributed in the metal matrix on a macroscopic scale. Based on microscopic observations, the spacing between particles is randomly distributed within a certain order of magnitude. The points are randomly taken from four adjacent grids with side length of 1.5 as the endpoint of the random quadrilateral, and its area size matches the actual particle size of 3–5 mm. During the modeling process, a random number is added to the cell distance. The coordinates of each endpoint for the random quadrilateral should to be stored in matrix A for subsequent calculations.

There is very little contact or even superposition between the reinforcing groups in Figure 2, and it can be considered as the phenomenon of reinforcing group polymerization in composite materials. The endpoint coordinates can be seen in Table 1.

Figure 2
figure 2

Random distribution of reinforcing base in matrix

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Table 1 Obstacle endpoint matrix
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3.3 Propagation Path Model

The midpoint of the line connecting the two adjacent quadrilateral endpoints is taken as the passable point, which can be given based on the position information of the random quadrilaterals. All possible paths for crack growth can be chosen from the line connecting these passable points. The coordinates of the passable points can be calculated from the data in matrix A, and the coordinates of each passable point are stored in matrix B.

After determining the starting point and the ending of the crack, the shortest crack is selected by the Dijkstra algorithm. Considering the ideal situation of the modeling, the stress intensity factors at the crack tip are not enough to break through the reinforcing base, and the path between the passing points must bypass the quadrilateral region. The connecting line of the passable points will be considered as possible path, the possible path can not intersect any random quadrilateral edge. The rapid exclusion test and straddle experiment can be used to select the connecting lines. The validation can be given as follows.

A passable path and one side of the quadrilateral can be taken as an example, the line segment P1P2 is a passable path, and Q1Q2 is an edge of a quadrilateral. Suppose a rectangle with P1P2 as the diagonal, Q1Q2 is a rectangle made diagonally. The two line segments will not intersect as if the two rectangles are not intersect. If the two rectangles do not intersect, they will not pass the fast rejection experiment. If they fail to pass the fast rejection experiment, the two line segments will inevitably disjoin. If rapid rejection experiment is passed, a straddle experiment can be performed.

If a line segment P1P2 intersects with a line segment Q1Q2, then P1P2 is distributed at both ends of Q1Q2, that is straddle experiment. The logical relationship can be shown in Figure 3. It should be satisfied as follows:

$$\left( {\left( {{\text{P}}_{1} - {\text{Q}}_{1} } \right) \times \left( {{\text{Q}}_{2} - {\text{Q}}_{1} } \right)} \right) * \left( {\left( {{\text{Q}}_{2} - {\text{Q}}_{1} } \right) \times \left( {{\text{P}}_{2} - {\text{Q}}_{1} } \right)} \right) > 0,$$
(2)

where "×" is the symbol of vector products, and "*" is the symbol of quantity products.

Figure 3
figure 3

Diagram of intersection determination

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The connection relationship between the passable points is stored in matrix C, which can be expressed as 1 and the unconnected is recorded as 0 for subsequent program calls, the connected matrix partial value are shown in Table 2. The Dijkstra algorithm will traverse all the paths between the set start and end points and find the shortest path through backtracking. The calculation results can be seen in Figure 4.

Table 2 Connected matrix partial value
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Figure 4
figure 4

Shortest path of crack propagation in composites

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According to Figure 4, the crack needs to “bypass” the reinforcing base, and the crack will deflect without changing the overall direction. If the total length of the crack lengthens, the fatigue life of the composite material will be increased.

4 Fatigue Life Model

4.1 Residual Stress Model

The preparation temperature of the SiC / Al composite material is between 680° and 780°, and the material is cooled to normal temperature after the preparation. It will cause the change of material volume. Since the composite material is made of a mixture of metal and non-metal, the two materials have different expansion coefficients based on temperature, and residual stress will be generated during the cooling process. As shown in Figure 5, during the working process of the test piece, the residual stress will inevitably affect the stress ratio of the test piece.

Figure 5
figure 5

The change of stress ratio related to residual stress

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The residual stress \(\sigma_{\alpha }\) can be expressed as follows [37]:

$$\sigma_{\alpha } = \Delta \alpha \Delta t\frac{{K_{m} K_{i} }}{{K_{m} + K_{i} }},$$
(3)

where \(\Delta \alpha\) is the difference of expansion coefficients for the two materials, \(\Delta t\) is the difference between the preparation temperature and loading temperature, which fluctuates around 650°. \(K\) is the elastic modulus, the subscripts m and i are the matrix and the reinforcement, respectively.

The particle reinforced particles are added to the metal, and the stress field distribution can be shown in the composite in Figure 6. L is the length of the residual stress field. The existence of residual stress will affect the stress ratio of the load on the test piece.

Figure 6
figure 6

The contour field picture

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According to the residual stress, the stress ratio R can be shown as follows:

$$R = \frac{{\sigma_{\min } + \sigma_{\alpha } }}{{\sigma_{\max } + \sigma_{\alpha } }},$$
(4)

where \(\sigma_{\min }\) is minimum stress, \(\sigma_{\max }\) is maximum stress.

4.2 Initial Crack Model

The initial size of crack initiation should be determined and related to crack initiation life calculation. Crack initiation life Ni can be described as follows [39]:

$$\begin{aligned} N_{i} &= \left[ {\frac{2}{{\varepsilon_{f} }}\left( {\frac{{\Delta \sigma_{egv}^{2} }}{EK}} \right)^{{\frac{1}{1 + n}}} - \frac{{\Delta \varepsilon_{c} }}{{\varepsilon_{f} }}} \right]^{ - 2} \\ & = \frac{1}{4}\left\{ {\left. {\left( {\frac{1}{{EK\varepsilon_{f}^{1 + n} }}} \right)^{{\frac{1}{1 + n}}} \left[ {\Delta \sigma_{egv}^{{2}{^{{\frac{2}{1 + n}}} }} - \frac{{\Delta \varepsilon_{c} }}{{2\varepsilon_{f} }}\left( {EK\varepsilon_{f}^{1 + n} } \right)^{{\frac{1}{1 + n}}} } \right]} \right\}^{ - 2} } \right. \\ & = \frac{1}{4}\left( {\frac{1}{{E\sigma_{f} \varepsilon_{f} }}} \right)^{{\frac{2}{1 + n}}} \left[ {\Delta \sigma_{egv}^{{\frac{2}{1 + n}}} - \frac{{\Delta \varepsilon_{c} }}{{2\varepsilon_{f} }}\left( {E\sigma_{f} \varepsilon_{f} } \right)^{{\frac{1}{1 + n}}} } \right]^{ - 2} , \end{aligned}$$
(5)

where the fracture fatigue strength \(\sigma_{f}\) and the fracture fatigue strain \(\varepsilon_{f}\) are related to the reduction of area \(\psi\), \(\Delta \sigma_{egv}\) is equivalent stress, \(\Delta \varepsilon_{c}\) is the threshold of strain range, E is elastic modulus, n is strain hardening exponent.

Equivalent stress \(\Delta \sigma_{egv}\) can be given as follows:

$$\Delta \sigma_{egv} = \sqrt {\frac{1}{{2\left( {1 - R} \right)}}} K_{t} \Delta \sigma ,$$
(6)

where \(K_{t}\) is the stress concentration factor and \(K_{t} = 1\), Δσ is stress amplitude.

The threshold of strain range \(\Delta \varepsilon_{c}\) can be obtained as follows:

$$\Delta \varepsilon_{c} = \frac{{2\tau_{ - 1} }}{E} - \frac{{\varepsilon_{f} }}{{10^{3.5} }},$$
(7)

where \(\tau_{ - 1}\) is torsional fatigue limit.

The fracture fatigue strain \(\varepsilon_{f}\) can be expressed as follows:

$$\varepsilon_{f} = - In\left( {1 - \psi } \right),$$
(8)

where ψ is reduction of area.

The fracture fatigue strength \(\sigma_{f}\) can be calculated as follows:

$$\sigma_{f} = \left( {1 + In\frac{1}{1 - \psi }} \right)\sigma_{b} ,$$
(9)

where \(\sigma_{b}\) is tensile strength.

The initial crack size a0 can be given as follows:

$$a_{0} = \frac{{\left( {{{K_{IC} } \mathord{\left/ {\vphantom {{K_{IC} } {\sigma_{b} }}} \right. \kern-0pt} {\sigma_{b} }}} \right)^{2} }}{\pi }.$$
(10)

Fracture toughness KIC can be given as follows:

$$K_{IC} = 0.032E\sqrt {\pi n} .$$
(11)

4.3 Crack Growth Model

If a component is subjected to cyclic loads, fatigue cracks propagation will occur until the component begins to fatigue or fail. The relationship between crack length and propagation life can be constructed according to the Paris formula [40]:

$$\frac{\text{d}a}{{\text{d}N_{f} }} = C\left( {\Delta K} \right)^{m} ,$$
(12)

where a is the crack length, Nf is the number of stress cycles, C and m are material coefficients, ΔK is stress intensity factor range,

$$\Delta K = F\sqrt {\pi a} \left( {\sigma_{\max } - \sigma_{\min } } \right).$$
(13)

To obtain the crack propagation life prediction more accurate, the Paris formula can be improved considering the stress ratio as follows [41]:

$$\frac{\text{d}a}{{\text{d}N_{f} }} = \frac{{C\left( {\Delta K} \right)^{m} }}{{\left( {1 - R} \right)K_{c} - \Delta K}},$$
(14)

\(K_{c}\) is stress intensity factor threshold.

According to Eq. (14), crack propagation life Nf can be expressed as follows:

$$N_{f} = \int_{{a_{0} }}^{{a_{c} }} {\frac{{\left[ {\left( {1 - r} \right)K_{c} - \Delta K} \right]\text{d}a}}{{C\left( {F\Delta \sigma \sqrt {\pi a} } \right)^{m} }}} ,$$
(15)

where \(a_{c}\) is the crack length in the material being broken.

Eq. (15) can be rewritten as follows:

$$\begin{aligned} N_{f} & = \frac{{\left( {1 - r} \right)K_{c} }}{{C\left( {F\Delta \sigma \sqrt \pi } \right)^{m} \left( {1 - \frac{m}{2}} \right)}}\left( {a_{c}^{{1 - \frac{2}{m}}} - a_{0}^{{1 - \frac{2}{m}}} } \right) \\ & \quad - \frac{2}{{C\left( {F\Delta \sigma \sqrt \pi } \right)^{m - 1} \left( {3 - m} \right)}}\left( {a_{c}^{{\frac{3 - m}{2}}} - a_{0}^{{\frac{3 - m}{2}}} } \right). \end{aligned}$$
(16)

A curve-fitted equation was given by Ref. [42] as follows:

$$\frac{{{\text{d}}a}}{{{\text{d}}n}} = \frac{{C\left( {1 - f} \right)^{n} \Delta K^{n} \left( {1 - \frac{{\Delta K_{th} }}{\Delta K}} \right)^{p} }}{{\left( {1 - R} \right)^{n} \left( {1 - \frac{\Delta K}{{\left( {1 - R} \right)K_{c} }}} \right)^{q} }},$$
(17)

where C, n, p and q are empirical constants, f is a crack-opening function empirical. The total life of the material is the sum of the initiation and extended life,

$$N = N_{i} + N_{f} .$$
(18)

When the crack path bends, the path direction deviates from the ideal crack direction, and the corresponding relationship between the crack length and the fatigue growth life also changes accordingly. The following equation can be modified as follows:

$$\begin{aligned} \frac{\text{d}a}{{\text{d}n}} & = \alpha_{1} \left( {\frac{{\text{d}a_{1} }}{\text{d}n}} \right) + \alpha_{2} \left( {\frac{{\text{d}a_{2} }}{\text{d}n}} \right) + \ldots + \alpha_{I} \left( {\frac{{\text{d}a_{I} }}{\text{d}n}} \right) \\ & = \sum\limits_{i = 1}^{I} {\alpha_{i} \left( {\frac{{\text{d}a_{i} }}{\text{d}n}} \right)} , \end{aligned}$$
(19)
$$\alpha = \frac{d\cos \theta + s}{{d + s}},$$
(20)

where \(\theta\) is the deflection angle, d is the distance to extend in the deflection direction, and s is the extension distance in the direction when the crack does not deflect. Crack path deflection can be seen in Figure 7.

Figure 7
figure 7

Crack path deflection diagram

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If a deflection coefficient \(\alpha_{i}\) are gotten in each possible path, all the deflection coefficient can be introduced into Eq. (20), the final crack propagation rate can be obtained.

5 Simulation Analyses

A model is proposed to predict fatigue life for metal matrix particle reinforced composites. To obtain the accuracy of the model, the test data in Ref. [43] is used for verification.

Composite material was produced and combined with a new type of aluminum alloy and SiC particles in Table 3 [43]. Table 4 shows the basic properties of the metal matrix. Material parameters C and m are equal to 7.8×10−4 and 0.09 respectively. Taking the material parameters into the life calculation model and giving different values to the fatigue stress, the fatigue life curve of Al alloy can be calculated, and it is as shown in Figure 8.

Table 3 Composition of the Aluminum alloy (mass: %)
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Table 4 Material properties of SiC/Al
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Figure 8
figure 8

Comparison of experience data and fitting curve in Al alloy

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The S-N curve of Al-20Si metal is fitted by Eq. (19) as shown in Figure 8. It can be seen that most of the test value are close to the fitted curve. When the load is 140 MPa, there are some errors between the test result and curve. Ignoring experimental error factors, the load does not reach the fatigue expansion threshold, and leads to the actual life being longer than estimated life.

To calculate the fatigue life of composite materials, the stress ratio and path growth correction should be considered. As shown in Table 4, the coefficient of thermal expansion for Al alloy is 24.2×10−6. The expansion coefficient of SiC particles is 3.4×10−6. If the expansion coefficient of metal and nonmetal is brought into Eq. (3), the residual stress is −1.003×10−3. Suppose the stress amplitude is 233 MPa, the original stress ratio is r = 0.1, and the modified stress ratio is r = 0.7881. The path correction value calculated by the path planning algorithm is 0.6650. Substituting the above values into Eqs. (16) and (18), the life curve of the composite can be obtained, and the comparison with the test results is shown in Figure 9. Figure 9 shows the relationship between the calculated life curve of the composite material and the test results. The curve can be used to predict the life trend of experimental materials well. The tensile strength, elongation and other parameters of the metal matrix will change to varying degrees after adding the reinforcing base. Different from the regular fatigue life prediction method, crack deflection and residual stress are considered, and the change caused by the strengthening base is ignored in material properties. It can be seen from Figure 9, when the specimen is in a low stress environment, the new model data are not consistent with the experimental data very well, and the model is effective to obtain fatigue life in a certain extent.

Figure 9
figure 9

Comparison of experience data and fitting curve in SiC/Al composites

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According to Figures 8 and 9, when the stress level is 160 MPa, the error is the smallest in the predicted and the test life. When the stress level is higher than 160 MPa, the estimated life is higher than the test life, and the error increases with the increase of stress. Under very high stress levels, cracks will more easily penetrate the reinforced base without deflection. This process cannot be reflected by the life prediction model, and the test life will be larger than the calculation result which is not considered in penetrated condition. When the stress level is lower than 160 MPa, the predicted data is smaller than the test data. The presence of the reinforcement hinders the slip of the metal crystals, and will greatly increase the fatigue initiation life under small loads.

Based on this model, the path planning model can be used to analyze and discuss the composite fatigue life. The comparison of life increment for the two materials under different stresses is shown in Table 5.

Table 5 Life increment of composite materials
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It is easy to find that the increase of fatigue life of the composites reinforced by non-metallic particles conforms to a certain rule. The life prediction model of fatigue algorithm combined with path planning algorithm can reproduce this rule to a certain extent. The increase of fatigue life for composite materials decreases sharply in the region with large stress. The large fatigue stress destroys the non-metallic particles, and the path planning algorithm partially can used to simulate the failure.

6 Conclusions

Fatigue propagation characteristic of composite materials is studied, and a new model of the crack propagation path for composite materials is established according to the path planning algorithm. The differences between crack propagation length of composite and metal materials are estimated. The modified Paris model is used to estimate the fatigue life, and some conclusions can be drawn as follows:

  1. (1)

    The crack trajectory simulated by the path planning model is closer to the microscopic crack trajectory, and the fatigue life is more accurately calculated using the estimated crack length. During regular fatigue failure of composite materials, the crack path will expand around the reinforcement base. The Dijkstra algorithm can effectively avoid the simulation of cracks passing through the strengthening base, and the local path optimization of the Dijkstra algorithm can be used to simulate the characteristics of crack "steps". This model can be extended to predict crack growth length for other similarly-structured composite materials.

  2. (2)

    Considering the residual stress caused by temperature changing, the fatigue crack propagation model is improved. The residual thermal stress in the material is considered to improve the fatigue crack growth model.

  3. (3)

    Based on the propagation of microscopic cracks, a new model of the life prediction for composite materials is presented. Combined with the crack length estimated by the path planning model, the fatigue crack propagation life is calculated. The experimental data show that the new model has some defects, and some works can be done in future.