Description of Mesoscale Static and Fatigue Analysis of 2D Woven Roving Plates with Convex Holes Subjected to Axial Tension

Abstract

The static and fatigue analysis of plates made of 2D woven roving composites with holes is conducted. The parametrization of convex holes is proposed. The experimental results of the specimens without holes and with different shapes of notches are discussed. The experiments and the appropriate procedures are carried out with the aid of ASTM codes. The fatigue behavior is considered with the use of the low cycle fatigue method. The analysis is supplemented by numerical finite element modeling. The present work is an extension of the results discussed in the literature. The damage of plates with holes subjected to tension always occurs at the tip of the holes, i.e., (x = a, b = 0), both for static and fatigue failure. The originality and the novelty of this approach are described by the failure’s dependence on two parameters: n and the ratio of the a/b ratio characterizing the hole geometry. The fuzzy approach is employed to reduce the amount of experimental data.

1. Introduction

The present state of the art in materials engineering is directly connected and determined with the application of composites in various areas of mechanical or building sciences (e.g., wind turbine blades, aerospace, and automotives). It relates to the use of the following:

-

Classical composites such as chopped fibers (nonwovens); long fibers grouped together and assembled into fabrics, called tows or yarns, constituting unidirectional laminates, (wovens, braids or knits) 2D woven fabrics, in which the yarns are divided into two components, i.e., the warp and the weft running in the cross direction to the warp, including 2.5D and 3D fabrics. The definitions and illustrations of these composites are presented by Gowayed [1]. Each type of fabric has its own advantages and disadvantages, and they are discussed in detail in ref. [2];

-

Non-classical composites such as nanostructural reinforcements (nanoplatelets, nanoribbons, various forms of graphene, and hexagonal nanostructures) [3];

-

Functionally graded materials (FGMs) [4];

-

Piezo electrics (PZTs) used as sensors or actuators—a detailed discussion of their material properties is provided in ref. [5].

Constructions of structures made of composite materials take the classical forms of beams, plates, shells, and 3D structures. Each physical problem is subjected to different boundary and loading conditions. As is shown in Figure 1, various physical problems can be formulated and solved.

Due to the variety of problems and the number of studies published in the literature, reviews of the literature should be limited to the most significant references in this area. Since the present work is devoted to the analysis of the stress concentrations of 2D composite structures, the discussion of the literature should be limited to the presentation of the most significant papers in the area of failure damage of 2D composites.

In general, for composites, the notch sensitivity and stress concentration around the hole problems can be divided into different categories:

The strength (static and fatigue) performance of laminated unidirectional constructions is discussed in refs. [6,7,8,9,10,11], which considers their failure morphology, depending on the fiber orientations, stacking sequences, and forms of their holes. The allowable loads for laminated composites are determined/defined by various parameters, such as the initial loads when the failure load occurs. They are dependent on design parameters such as the geometry (Figure 1), defect, load type, material types, lay-up of composites (configuration), etc. The work in [12] is specifically dedicated to developing an efficient load space of notched laminates taking into account various design variables (the allowable load space—ALS). In ref [13], special attention is focused on the influence of delaminations between unidirectional laminates on failure modes. Finite element fracture mechanics [14,15] is applied to determine both the critical static and fatigue loads of notched laminated plates subjected to tension.

Kim et al. [12] compared the static strength of notched laminated and 2D woven composite structures. Pandita et al. [16] analyzed the strain concentrations in woven fabric composites with circular and elliptical holes since 2D woven plain composites are treated as orthotropic structures. For 2D woven roving composites, the validity of different failure criteria is discussed in ref. [17]. Guo et al. [18] describe the results for centrally located waist, slit, octagon, square, hexagon, and rhombus holes. For 2D woven roving composites, an analysis was also carried out for plates made of a hybrid natural (juta) material [19,20]. Pandita et al. [21], Kaminski et al. [22], and Li Liongbiao [23] discussed the fatigue failure of 2D woven roving composites.

The aim of the present work is to present a comparison of experimental and theoretical (numerical) studies for plates (specimens) made of 2D woven roving composites. The analysis deals with constructions of centrally located holes subjected to tensile loading. Since the static experimental studies demonstrate an evident material nonlinear (σ-ε) characteristic, the fatigue numerical (finite element) description is carried out with the use of the low fatigue cycle (LFC) method. The experimental (static and fatigue) results illustrate, essentially, the occurrence of the random (probabilistic) nature of both static and fatigue cracks. The classical approach to such problems is based on statistical (Weibull) distributions. However, in such a description, a lot of experimental data is required. Therefore, we propose to use, herein, the limit data (lower and higher bounds) to characterize the LFC behavior of structures.

In the present paper, we intend to discuss the similarities and differences between the static and fatigue behavior of plates made of 2D woven roving composites having central holes and subjected to uniaxial tension. The analysis is conducted with the application and comparison of experimental results with finite element computations.

The novelties of the paper are as follows:

-

It is demonstrated that the LFC behavior of plates made of plain weave 2D composites shows the similarities to elastic–plastic deformations.

-

The description of static and fatigue damages of plate with a convex hole (plain weave 2D composites) can be parametrized with the use of three values characterizing the area of the hole and the two lengths of the superellipses constituting the hole.

2. Method of the Experimental Analysis for Plates Made of Plain Weave

The category of fiber architecture is that formed by 2D, 2.5-D or 3-D weaving, braiding, or knitting the fiber bundles or “tows” to create interlocking fibers that often have orientations slightly or fully in an orientation orthogonal to the primary structural plane. This approach is taken for a variety of reasons, including the ability to have structural, thermal, or electrical properties in the third or “out-of-plane” dimension. Another often-cited reason for using these architectures is that the “unwetted” or dry fiber preforms (fibers before any matrix is added) are easier to handle, lower in cost, and conform to highly curved shapes more readily than the highly aligned, continuous fiber form.

Three different types of 2-D woven roving composites are distinguished—Figure 2.

Woven fiber-reinforce plastics are becoming increasingly important because they have a lot of advantages over laminates made from individual layers of unidirectional material.

To determine the methods of the analysis, it is necessary to characterize the set of parameters required in the evaluation of results. Let us note that the present investigations deal with the analysis of structures subjected to tensile loads so that the studies are limited to description of the stress–strain relations for such loading and boundary conditions both for experimental and theoretical considerations.

2.1. Static Behavior

Tensile testing is used to measure the force required to break a polymer composite specimen and the extent to which the specimen stretches or elongates to that breaking point. Tensile tests produce a stress–strain diagram, which is used to determine tensile modulus. The specimen is a constant rectangular cross-section. The tabs are bonded to the ends to prevent gripping damage. As it is plotted in Figure 3, the holes are drilled at the center of the specimen.

To model theoretically mechanical properties of 2-D woven roving composites, two possible approaches can be used—see, e.g., L. Wang et al. [24]:

• Two-level modeling, where at the first level, the fiber bundle (tow) is represented in the microscale, and at the second level, the representative volume element (RVE) is illustrated and modeled in the mesoscale.
• One-level modeling (mesoscale), where the mesh of composites (RVE) contains three parts: resin pocket, warp tows, and fill tows—each of the parts is represented by 3-D (hexahedron) finite elements.

The homogenization can be also carried out for the unit cell presented in Figure 4—see refs. [25,26]. They are dependent on the form of the cross-sections that are demonstrated in Figure 4.

The geometry of the cross-sections of 2-D woven fabric has a crucial influence on the derivation of mechanical properties. The homogenized mechanical properties can be determined with the use of experimental tension tests or by the numerical finite element analysis.

2.2. Fatigue Behavior of Woven Roving 2D Composites

Nowadays, the fatigue of structures can be approached using the following concepts—Figure 5:

• Low cycle fatigue (LCF).
• High (Mega) cycle fatigue (HCF) from A. Whoeler—both infinite and finite cyclic life can be analyzed, where the small strain increment results in large stress increment.

Bathias [27] noticed that in metals, the fatigue damage is strongly related to the cyclic plasticity that is to say the dislocation mobility and slip systems. Due to the environmental effect and the plane stress states, the initiation of fatigue damage is often localized near the surface of metals. In polymer matrix composite materials, the fatigue damage is not related to plasticity. Considering only polymer-matrix composites reinforced by long fibres, it is acknowledged that the first damage that appears under loading is matrix cracks before the fracture of the fibres. These are micro cracks with an initial thickness of one layer, and their presence constitutes the initiation of damage. Propagation will develop next, by a multiplication of cracks building to a critical density and resulting in the development of delamination until the eventual fracture of the fibers, should arise.

At the mesoscopic level, the fatigue damage propagation in metal is a single crack perpendicular to the tension loading. This fatigue crack tip is surrounded by a plastic zone. In composites materials, the fatigue damage is multidirectional and the damage zone, much larger than the plastic zone, is related to the complex morphology of the fracture. For 2D woven roving composites the results are presented in Section 3.1.

In the later part of the paper, the LCF problems are investigated.

Low cycle fatigue (LCF) from Coffin [28] and Manson [29]—only finite fatigue life is possible and can be considered using the LCF criteria in such a case, a small strain increment corresponds to a large stress increment—Figure 6; according to this approach, the fatigue life of a member subjected to fully reversed constant amplitude strain-controlled loading is given by an equation of the following form:

$$\epsilon ={\epsilon }_a+{\epsilon }_p=\frac{{\sigma }_f^{\prime }}{E}{\left(N_f\right)}^b+{\epsilon }_f^{\prime }{(N_f)}^c$$

(1)

where ε—total strain amplitude, ${\sigma }_f^{\prime }$—fatigue strength coefficient, b—fatigue strength exponent, ${\epsilon }_f^{\prime }$—fatigue ductility coefficient, c—fatigue ductility exponent, E—modulus of elasticity, N_f—reversals to failure; the determination of coefficients is presented schematically in Figure 6 and Figure 7.

The LCF curve is defined by five coefficients. Manson proposed the method of universal slope, where the equation for the method can be given as follows:

$$\mathsf{\Delta }\epsilon /2_{total}=1.75\left(UTS/E\right)N_f^{-0.12}+D^{0.6}{N}_f^{-0.6}$$

(2)

where UTS denotes the ultimate tensile strength, D is the ductility, i.e., the percentage reduction in of the area.

Since the static stress–strain curves demonstrate the evident nonlinear behavior (see Section 4) the fatigue analysis is carried out with the use of the LCF method, where the static load carrying capacity is not exceeded. The form of the tensile load is illustrated in Figure 8, where P_m = 0.8 P_static and ΔP = 0.1 P_m.

2.3. Definition of Convex Holes

A lot of planar convex curves can be expressed in the form of the super-ellipse defined in the following way:

$${\left(\frac{x}{a}\right)}^n+{\left(\frac{y}{b}\right)}^n=1$$

(3)

where a, b, and n are positive rational numbers. A supercircle will obviously correspond to setting a = b. For each value of n—the supercircular exponent—we obtain a different convex curve. The shapes of the supercircles for different values of n are shown in Figure 9. Evidently, n = 2 corresponds to a circle, whereas n= 1 corresponds to a triangle with its sides rotated by an angle of π/4. The case $n\to \infty$ also corresponds to a square with its sides parallel to the axes. Finally, the curve characterizing the convex hole (the supercircle) can be represented in the form of the Fourier series.

The above relation exploits the symmetry with respect to the OX and OY axes.

Applying the form of the curve (3), one-quarter of the surface closed by the superelipse can be written in the following form:

$$A_C=ab\frac{{\mathrm{\Gamma }}^2(1+1/n)}{\mathrm{\Gamma }(1+0.5/n)}$$

(4)

where Γ(…) denotes the Euler gamma function.

Assuming the constant value of the area A_C, one can find that the area is the function of three values, a, b, and n. Therefore, for the identical value of the n parameter, there is an infinite number of the a and b parameters satisfying Equation (4)—see Figure 9.

3. Comparison of Experimental and Numerical Results

3.1. Static Strain–Stress Relations

Figure 4a,b depict the photographs of 2D woven composites having different thicknesses of bundles corresponding to packing density of fibers.

Using the experimental results plotted in Figure 10, the average Young’s and Kirchhoff’s moduli are written in

Table 1. Static mechanical properties of plain 2D glass.

	Young’s Modulus in the Warp (Longitudinal) Direction Linear Part of the Curves Plotted in Figure 10 in [GPa]	Young’s Modulus in the Weft (Transverse) Direction [GPa]	Kirchhoff’s Modulus Linear Part of the Curves Plotted in Figure 10 in [GPa]
Experiment	13.142	13.004	9.621
Finite element modeling	12.958	12.930	9.143

—see ref. [30]. The numerical (Finite Element) homogenization is also carried out for the unit cell presented in Figure 4—see [30]. The agreement between experimental and numerical results is very good. In addition, let us note that the values of the Young’s moduli for warp and weft directions are almost the same.

Figure 11 presents the form of boundary conditions for a quarter of the specimens used in the FE analysis both for the static and fatigue loads.

The failure modes of rectangular specimens are presented in Figure 12. The damage occurs in the perpendicular direction to the tension load. Since the thickness of the aramid specimen is very low (Figure 4 and Figure 12a), its mechanical behavior (σ-ε curve) is almost linear, and the failure is characterized by a brittle fracture. For glass woven roving, the deformation (Figure 10) is nonlinear and can be described by elastic–plastic behaviour. The failure is not localized along one line (compare Figure 12b). It is observed that failure forms is associated with delaminations illustrated by white zones.

It is mainly caused by the non-uniform thickness distributions of glass specimens. It has a significant influence on the load distributions along specimens at the particular points, as it is plotted in Figure 13. The broader discussion of those problems is presented by Muc, Kędziora [30]. For woven roving composites, the scatter of results is usually observed.

Usually, as discussed in ref. [22] for 2-D woven composites (fabric formed by interlacing the longitudinal yarns (warp) and the transverse yarns (weft)), such as plain, twill, or satin, four types of damage mechanisms occur under static and fatigue loadings: intra-yarn cracks in yarns oriented transversely to the loading direction, inter-yarn decohesion between longitudinal and transverse yarns, fiber failure in longitudinal yarns, and yarn failures. The failure forms plotted in photographs in Figure 14 are identical for fatigue problems.

3.2. Stress Concentration around Convex Holes

Figure 14 demonstrates the forms of the final damage of the specimens. As it may be seen, the failures occur always in the perpendicular direction to the tensile loads, similarly as for isotropic materials. However, the failure mode is associated with delaminations and the inelastic deformations of the matrix—white zones in Figure 14.

The 2-D solid elements (plane stress elements NKTP = l) in the NISA finite element package are included in the present analysis. The element may be shaped as a 4 to 12 node quadrilateral or as a 3- or 6 node triangle, depending on the selected approach [31]. Each node has two degrees of freedom (two displacements u_x, u_y), and the state of stress is characterized by three components: σ_x, σ_y, and σ_xy. In addition, the nonuniform thickness distributions at each of the nodes can be introduced.

The influence of the meshing on the accuracy of computations is illustrated in Figure 15.

Using the planar finite elements, it is possible to numerically evaluate the stress distributions around the holes. Figure 16 illustrates the plots of the static stress concentrations around the convex holes. The maximal stress concentrations occurs at the tip of the notch. The stresses are related to the value P/(Wt).

As it may be noticed in Figure 16, the stress concentrations increase with the growth of the ratio a/b (see Equation (3)). The identical behavior occurs for the higher values of the n parameter (i.e., n = 6), keeping the same values of the area A_c. The similar effects are obtained for plates with holes made of woven roving composites, isotropic materials, or laminated composites, as discussed by Pandita et al. [16]. In general, various stress criteria can be employed in the analysis. We are searching for the maximal values of the following:

-

The circumferential stresses.

-

The Hill criterion.

-

The Huber-Mises-Hencky criterion.

-

The Tsai-Wu criterion

-

The Hashin 3D or 2D criterion.

The detailed description of the above criteria is presented, e.g., in refs. [17,32]. Let us note that the Hill, Tsai–Wu, and Hashin criteria are used for the description of structures where both positive and negative values of stresses occur what is not observed in the analysed problem.

Since in the present analysis and modeling of 2D bidirectional woven roving plain composite materials can be treated as isotropic materials (see Table 1—the mesoscale analysis), the strain concentration factors can be approximated by the results valid for isotropy. The ratio between the local stresses around the hole and the nominal stresses is characterized by the stress concentration factor k_t1, i.e.,

$${\sigma }_{tip}=k_{t1}{\sigma }_{\infty }$$

(5)

where the nominal stress ${\sigma }_{\infty }=P/(Wt)$, and the local stress ${\sigma }_{tip}$ corresponds to the value of the σ_y stress component at x = a and y = 0 (the edge of the hole). The theoretical stress concentration factor is a function only of geometry and the type of loading and can be determined either analytically (using the theory of elasticity), through finite element analysis, or through experiments.

For isotropic materials, the stress concentration factor of an elliptical notch can be described in the following way:

$$k_{t1}=\left(1+2\frac{b}{a}\right)$$

(6)

Considering the values of the coefficients 1 + 2b/a, one can notice that the horizontal ellipses correspond to the high concentration effects, whereas the vertical position of ellipses corresponds to the low concentration effects. The comparisons of the stress concentration factors (4) are demonstrated in

Table 2. The comparison of the stress concentration effects on the ellipsoidal notches—the initiation of cracks.

Stress Concentration Factor ${\mathit{k}}_{\mathit{t}1}$	Theoretical	Numerical (FE) Analysis	Percentage Error $({\mathit{k}}_{\mathit{t}1}^{\mathit{n}\mathit{u}\mathit{m}\mathit{e}\mathit{r}}-{\mathit{k}}_{\mathit{t}1}^{\mathit{t}\mathit{h}\mathit{e}\mathit{o}\mathit{r}\mathit{e}\mathit{t}})/{\mathit{k}}_{\mathit{t}1}^{\mathit{t}\mathit{h}\mathit{e}\mathit{o}\mathit{r}\mathit{e}\mathit{t}}$
b/a = 2.812	6.624	6.943	10.24
b/a = 1.000	3.000	3.211	12.51
b/a = 0.336	1.672	1.745	13.71

.

The accuracy of the FE approximations is a function of the accuracy of meshing, which, in the present case, is quite good. Pandita [16] obtained the unsymmetric behavior of failure modes. It is difficult to verify such results since they depend on too many factors.

Let us note that for the parameter n greater than 6 the area A_C, Equation (2) can be approximated by the value ab(1–0.5/n). Therefore, the growth of the ratio a/b and the parameter n corresponds to the increase of the stress concentration factors and/or the HMH factor (Table 2 and Figure 17).

Figure 18 demonstrates the non-homogeneous strain distributions around elliptical holes. The results are obtained with the use of the digital image correlation (DIC) method. They resemble the plots presented by Pandita et al. [16].

3.3. Fatigue Behavior

Let us note that static stress–strain curves (Figure 10) have the identical form as the curve describing the low cycle fatigue in Figure 6. Therefore, it dictates the necessity of the use of the LCF criterion in the analysis of this method in the analysis of fatigue behavior for the structures considered.

The results presented in Table 1 show that the analyzed glass 2D plain weave can be treated as quasi-isoptropic materials. It allows to use the Coffin–Manson description (Figure 5 and Figure 7) for isotropic materials with the material constants described by Equations (1) and (2).

For notched plates made of woven roving composites and subjected to tension he typical fatigue final form is illustrated in Figure 19. It is analogous to fatigue behaviour of isotropic (metallic) structures.

The number of cycles corresponding to the final fatigue failure is presented in

Table 3. The critical number of cycles.

Specimen	1	2	3	4
The length [mm]	125.0	125.0	125.05	125.04
The average thickness [mm]	2.48	2.39	2.43	2.47
Number of cycles Nf	11,012	10,945	15,004	14,617
Average number of cycles	12,894.5

. They show the scatter of results caused by the nonuniform distributions of thickness for woven roving composites that have also a great influence on distributions of stresses along the specimen length—see Figure 12.

The scatter of the critical number of cycles should be represented by the statistical or fuzzy set. Using fuzzy set analysis, the experimental fatigue degradation analysis (the decrease of the secant Young modulus) is illustrated in the form drawn in Figure 20. The results are represented in the form of straight lines being the approximation (linearization) of the experimental data, where E(n) denotes the values of Young’s modulus for the prescribed values of the number of cycles n, and n = 0 corresponds to the static behavior. N_f is the number of cycles characterizing the fatigue failure—see Table 3.

In our opinion, the results presented in Figure 20 are much more convenient than the classical statistical analysis since it requires less experimental data.

3.4. Finite Element Analysis of Fatigue Problems

For low cycle fatigue problems, the numerical modeling is carried out in two steps:

• Finite element modeling of structures with convex holes.
• Derivation of final number of cycles using the Coffin–Manson relation (5).

The analysis is conducted with the use of the numerical package NISA II Endure version 17 [31]. The accuracy of the FE computations are shown in Figure 21.

The lowest value occurs at the tip of the circular notch (grey color—see Figure 22), and it corresponds to the experimental value—Table 2. The crack initiation is always characterized by the lowest value of the number of cycles. The maximal value of N_f exists in the direction of the remote tension—the axis y.

For different shapes of central notches, the identical numerical analysis can be repeated. The results are illustrated in Figure 23. The dimensions of the holes are identical to those discussed for static loads. As it may be seen, the distributions are almost identical to the curves plotted in Figure 11 describing the stress concentration effects for static loads.

Similarly, as for the stress concentration factors, the increase of the parameters a/b and n results in an increase in the critical number of cycles. Identical effects are observed in Refs. [33,34].

4. Concluding Remarks

The influence of notches in 2D woven fabric composites is investigated at the mesoscale based on the ultimate stress and on fatigue loads. The stress/strain concentration under static and fatigue loads in 2D woven fabric composites with holes is influenced by the tensile loading direction and by the hole geometry and its dimension relative to the unit cell of the plain woven fabrics. For different shape parameters, the strain concentration is located at the tip of holes.

For the analyzed problem, the similarity between the static stress–strain curves and the low fatigue cycle (LCS) is observed. Therefore, the fatigue load behaviour was described by the low fatigue cycle method that is described herein both in the experimental and theoretical way.

The computational finite element methods are also implemented in the paper. The comparison of experimental and numerical results is very good. The agreement is proved for the homogenization of the average mechanical properties and in determination of failure loads, both static and fatigue.

Both for static and fatigue damages, the failure occurs at the tips (x = a, y = 0) in the perpendicular direction to tensile loads.

The presented results show the necessity of the use of at least two parameters (convex hole) to compare the static and fatigue behavior of plates subjected to tension. The growth of the parameters a/b and n leads to the increase of the critical values characterizing the static and the LCS fatigue damages. The increase can be approximated with a very good accuracy by parabolic or straight lines.

The plain 2D woven roving composites have nonlinear behaviour and it has a significant influence on the final failure of static and fatigue damages. It results in the necessity of the use of statistical or fuzzy approaches/methods.