A model for longitudinal tensile strength prediction of low braiding angle three-dimensional and four-directional composites

2.1 Basic hypothesis

In this paper, the model to predict the longitudinal tensile strength of 3D four-directional composites with low braiding angle is based on the following hypotheses: (1) the strength characters of the whole composite depend on the strength characters of inner single cells; (2) unidirectional composite fiber bundles in inner single cells always keep straight state; (3) when the composite carries longitudinal tensile load, the unidirectional composite fiber bundles are the main force-bearing component and the contribution of the matrix between fiber bundles to tensile load can be ignored; (4) the fiber fracture in the composite follows a certain probability distribution; (5) before the damage of a unidirectional composite fiber bundle, linear elasticity is satisfied by both the composite and the unidirectional composite fiber bundles.

2.2 Elastic constant prediction of inner single cells

Suppose the length, width and height of an inner single cell of a 3D four-directional braided composite are a, b and h, respectively, as shown in Figure 1. Their relationships with the inner braiding angle γ and horizontal orientation angle β are as follows:

Figure 1:

Schematic diagram of a unidirectional composite fiber bundle in an inner single cell.

The fiber volume fraction in a unidirectional composite fiber bundle, namely yarn filling factor, denoted by κ, can be obtained by theoretical calculation or section measurement, and is usually between 0.5 and 0.8. Suppose five independent elastic constants of a fiber are E_f1, E_f2, G_f12, G_f23, and ν_f12, respectively, and three independent elastic constants of matrix are E_m, G_m and ν_m, so the five independent elastic constants of a unidirectional composite fiber bundle can be calculated by the following expressions [19]:

Accordingly, the stiffness matrix of a unidirectional composite fiber bundle [C_b] can be determined.

An inner single cell contains unidirectional composite fiber bundles with four interleaving directions, and the directions of the bundles are different from the main direction of the 3D four-directional braided composite. Therefore, the coordination of the unidirectional composite fiber bundles needs to be transformed and unified to the coordination of the whole 3D four-directional braided composite. The directional angles of the unidirectional composite fiber bundles in the inner single cell are (γ, -β), (γ, β), (γ, π-β) and (γ, π+β), respectively. The stiffness matrices in the whole coordinate system of the unidirectional composite fiber bundles with four directions, denoted by [Cb,1′],[Cb,2′],[Cb,3′] and [Cb,4′], respectively, can be obtained by coordinate system transformations [20].

The volume fraction of unidirectional composite fiber bundles, V_b, is defined as the ratio of the volume of the bundles in a single cell to the volume of the whole single cell, and its numerical value equals the ratio of the fiber volume fraction V_f to the yarn filling factor κ, that is

According to the average stiffness method, the stiffness of a single cell in the composite is equal to the volume-weighted average of the stiffness of each component, that is

where [C_m] is the stiffness matrix of the matrix. Thus, nine independent elastic constants of an inner single cell of a 3D four-directional braided composite can be obtained.

2.3 Longitudinal tensile strength prediction of unidirectional composite fiber bundles

Owing to the strength dispersion of single fibers, some weak fibers may break first when the composite withstands a finite tensile load, and then a certain number of random crack cores germinate in the composite [18].

The density of random crack cores ρ is defined as the number of random crack cores contained in the fiber per unit length in the composite. When the average fiber tensile stress is σ_f, the density of random crack cores is

where F(σ_f, δ) is a strength distribution function of single fibers and δ₀ is the ineffective length (the length of stress-recovery zones of broken fibers).

The ineffective length δ₀ can be calculated with the Kelly-Tyson shear lag model [21], that is

where d is the diameter of fibers and τ_face is the shear strength of the fiber-matrix interface.

The number of random crack cores m is the total number of cracks produced in the composite, and its numerical value equals the product of the density of random crack cores ρ and the total fiber length of the composite Φ, that is,

where Int[] is the function of getting integers.

As the tensile load increases, a random crack core tends to propagate around. Furthermore, the influenced length of the random crack core increases with the number of broken fibers. According to the prefect evolvement process of random crack cores [18], when the number of broken fibers in a random crack core is i, the corresponding influenced length can be approximately expressed as

When i=1, δ₁=δ₀.

When the further propagation probability of a random crack core is great enough, it indicates that the random crack core begins to unstably propagate, and the size of the random crack core at this time could be considered as the critical size. According to the random crack core theory [18], the critical size of random crack cores N_cri needs to meet the following expressions:

and

where K_i=max(K_i,j), i≤j≤n_con, and K_i,j is the stress concentration factor of the fiber whose serial number is j when i fibers have broken in the random crack core [22]; n_con is the biggest serial number of fibers enduring concentrative stresses around the random crack core when i fibers have broken.

Random crack cores distribute discretionarily in a composite, and the failure probability of a unidirectional composite fiber bundle under certain stress equals the probability that a random crack core evolving to critical size exists. Consequently, when the fiber stress is σ_f, the failure probability of a unidirectional composite fiber bundle can be expressed as

where P_Ncri(σ_f) is the probability of a random crack core evolving to critical size, and can be obtained by the probability algorithm of the prefect evolvement process of random crack cores [18].

The longitudinal tensile strength of a unidirectional composite fiber bundle X_bt is expressed as

where σf¯ is the average stress of the fibers when P_b(σ_f) is around 0.5, and σm¯ is the average stress of the matrix corresponding to the fiber stress σf¯ in a unidirectional composite fiber bundle under equal strain.

2.4 Damage analysis of unidirectional composite fiber bundles

An inner single cell of a 3D four-directional braided composite will be deformed under the longitudinal average tensile stress σ. As shown in Figure 1, the length a′, width b′ and height h′ of the inner single cell after deformation are given, respectively, as

where E_in1, ν_in12 and ν_in13 are the elastic constants of the inner single cell.

After the deformation, the inner braiding angle γ′ is expressed as

With the deformation of the inner single cells, unidirectional composite fiber bundles are also deformed. The axial strain ε_b1 and shear deformation α_b of the unidirectional composite fiber bundles are given, respectively, by

The average tensile stress σ_b and average shear stress τ_b of the unidirectional composite fiber bundles are given, respectively, by

Substituting Equations (18)–(23) in the above equations, the relation of the average tensile stress σ_b and average shear stress τ_b of the unidirectional composite fiber bundles with the longitudinal average tensile stress σ of the composite can be obtained.

The shear strength S_b of a unidirectional composite fiber bundle can be calculated by the following empirical formula:

where S_m is the shear strength of the matrix.

When a 3D four-directional braided composite withstands a longitudinal tensile load, the unidirectional composite fiber bundles in inner single cells bear mainly axial tensile stress and shear stress. According to the Tsai-Hill criterion, the damage criterion of a unidirectional composite fiber bundle bearing axial tensile stress and shear stress can be described as

With the combined effect of tensile stress and shear stress, the damage of unidirectional composite fiber bundles can be classified into stretching-dominated and shear-dominated damages. If the damage is stretching-dominated damage, it means the occurrence of a large number of fiber breakages. At this time, the unidirectional composite fiber bundles almost lose their bearing capability, so the whole composite can be considered to be failed. If the damage of unidirectional composite fiber bundles is shear-dominated damage, it indicates the occurrence of fiber-matrix interface debonding without the occurrence of a large number of fiber breakages. At this time, the unidirectional composite fiber bundles still have strong bearing capability, so the composite cannot be considered to be failed.

After the fiber-matrix interface debonding occurs, the shear modulus and transverse elastic modulus of unidirectional composite fiber bundles will decrease obviously, resulting in the stiffness degradation of the 3D four-directional braided composite. At this time, the unidirectional composite fiber bundles and the 3D four-directional braided composite do not satisfy the linearly elastic hypothesis, so Equations (24) and (25) are no longer applicable. In this paper, it is supposed that after the fiber-matrix interface debonding of unidirectional composite fiber bundles occurs, the longitudinal tensile load of the 3D four-directional braided composite equals the sum of the component forces, which are derived from the axial tension of each fiber bundle in the longitudinal direction of the composite. That is,

where A_b is the cross-sectional area of a fiber bundle, and Ab=14Vfκabcosγ, so

Therefore, after the fiber-matrix interface debonding, the condition for the second damage of unidirectional composite fiber bundles (i.e. fiber breakage) is

where Xbt′ is the tensile strength of the unidirectional composite fiber bundles after the interface debonding and can be calculated by the algorithm in Section 2.3. Owing to the interface debonding, the interfacial shear strength can be substituted by the slip resistance.

According to the critical condition (30) for the second damage of unidirectional composite fiber bundles, the average tensile stress of a 3D four-directional braided composite, which is corresponding to the second damage of the unidirectional composite fiber bundles, is expressed by

2.5 A longitudinal tensile strength model of low braiding angle 3D four-directional braided composites

Although different damage modes may appear during the stretching process of a low braiding angle 3D four-directional braided composite, such as fiber breakage, fiber-matrix interface debonding and so on, fiber-matrix interface debonding does not mean the failure of the composite. The final damage of a low braiding angle 3D four-directional braided composite is usually characterized by a large number of fiber breakages.

Suppose that the average longitudinal tensile stress of a composite is σ_t1, which corresponds to the first damage of unidirectional composite fiber bundles, and the average longitudinal tensile stress of the composite is σ_t2, which corresponds to the second damage of the bundles after fiber-matrix interface debonding. If σ_t1≥σ_t2, the first damage of the unidirectional composite fiber bundles is the fiber breakage, so the longitudinal tensile strength of the composite is σ_c=σ_t1; if σ_t1<σ_t2, the first damage of the unidirectional composite fiber bundles is the fiber-matrix interface debonding, and there are a large number of fiber breakages when the second damage occurs, so the longitudinal tensile strength of the composite is σ_c=σ_t2.

Therefore, the longitudinal tensile strength of a low braiding angle 3D four-directional braided composite is