1. Introduction

2. Materials

3. Numerical Damage Accumulation Modeling in Composite Plate Reinforced by Knitted Fabric

3.1. Stress Distribution in the CMHR. Assumptions

We assume MF is isotropic, its elastic constants are obtained by averaging over volume. HCM plate is stretched in one direction. According to stochastic failure nature, load increase is leading to single broken MF sites accumulation in the plate. Each such single isolated breakage is designated as a defect with the size j=1. Failure is observed as the sequential rupture of yarn loops (MF) in the material due to increasing overstress on the MF located around the defect. In the model, more heavily overloaded by stretching stress MF orthogonal crossection, (along its length) is potential place for overloaded neighbor’s failure. In the works of other researcher [8,25,26,27] polymer matrix- textile-reinforced composite failure was modeled using different phenomenological approaches. Compared with them, this approach is structural. The structural approach can give new information about internal failure mechanisms in materials with sophisticated internal structure and create recipes for material properties improvement. The plate reinforced by single-in plane knitted fabric. Observing a single-layer plate, the plate is stretched by acting in-plane loads. The matrix material is epoxy resin with OSA, and the reinforcement is weft-knitted fabric impregnated with basalt fiber yarns. It should be mentioned that ideologically similar modeling can be used for HCM with other types of polymer matrix with other than OSA powder filler and synthetic fiber threads.

In the framework of our calculations, the composite matrix material (epoxy resin with OSA) is elasto-plastic. The addition of OSA is changing the matrix behavior to more brittle. Deformation till crack formation is decreasing. For highly brittle matrix calculation model must be changed, because matrix cracks formation will play initial role. Basalt fiber yarn impregnated by composite resin is observed as a “large” diameter fiber, macrofiber with elastic properties that are calculated according to the rule of mixture [27,34]. Such fiber is accepted as elastic until its rupture. The elastic properties of the matrix and reinforcement (MF) material are shown in the Table 1.

Table 1. Elastic properties of matrix with OSA.

Table 1. Elastic properties of matrix with OSA.

	Matrix with epoxy resin and 0% (weight) OSA	Macrofiber- basalt fiber thread impregnated by epoxy resin and 0% (weight) OSA
Density ρ (g/cm3)	1.19	1.74
Young’s modulus E (GPa)	3.15	63.4
Poisson’s ratio ν	0.35	0.3
	Matrix with epoxy resin and 10% (weight) OSA	Macrofiber- basalt fiber thread impregnated by epoxy resin and 10% (weight) OSA
Density ρ (g/cm3)	1.26	1.89
Young’s modulus E (GPa)	3.28	63.5
Poisson’s ratio ν	0.34	0.29
	Matrix with epoxy resin and 20% (weight) OSA	Reinforcement with basalt fiber thread in epoxy resin and 20% (weight) OSA
Density ρ (g/cm3)	1.34	2.07
Young’s modulus E (GPa)	3.62	64.1
Poisson‘s ratio ν	0.33	0.28

The geometric model for the HCM reinforced by the plain weft-knitted fabric is shown in Figure 2. This structure is loaded. The constant displacement is applied in horizontal direction, as is shown in Figure 3. Surface C is not moving in horizontal direction, surface B in vertical. The structure was numerically analyzed (using FEM) for prediction of the sequential yarn failure progress. The main parameters of the knitted fabric geometry are as follows: macrofiber diameter d = 0.55 mm; wale number W = 1.43 loop/cm, which is defined as the number of repeating loops per unit length along the course direction; and course number C = 2.5 loops/cm, which is the number of loops per unit length in the wale direction. Leaf and Glaskin’s geometrical model was also used in [6,15,19,24,25].

The geometric parameters of the investigated composite material plate were as follows: width 21 mm, height 20 mm, and thickness 2.43 mm (Figure 2 and Figure 3). Observed Epoxy resin is elasto - plastic material with tensile yield stress equal to 30 MPa. In the Figure 3a is possible to see the FEM mesh for whole HCM plate and meshed macrofibers which are forming the textile spatial structure inside the HCM plate.

With the goal to optimize calculation time and diminish boundary conditions effect on overstress calculation, the same model was used with the elastic frame (two plates with similar width as for the structural model). Plates (frame) had the averaged, homogenized HCM elastic properties. Figure 4 shows the model with the frame. All loops and the matrix between them were meshed. Perfect bonding conditions were applied between matrix and the outer surface of each macrofiber (without delaminations and voids).

Figure 3. Boundary conditions for HCM volume without the frame.

Figure 3. Boundary conditions for HCM volume without the frame.

Figure 3. A. Mesh for HCM (Matrix and fiber) volume without the frame; a) meshed matrix and macrofibers; b) meshed macrofibers.

Figure 3. A. Mesh for HCM (Matrix and fiber) volume without the frame; a) meshed matrix and macrofibers; b) meshed macrofibers.

Side plane ABC remained planar during the whole loading process. Nodes on the butt-end surface C were fixed in the y direction, and points on surface B could not move in the x direction. Another butt-end surface A obtains displacement ∆_y. External stretching raised a non-equal stress distribution in the yarn loops (MF) that form the material reinforcement. Finite element analysis was carried out for this plate and results for the model with the frame are shown in Figure 5. In the Figure 5 is possible to see stresses in macrofibers (matrix is not shown on this picture).

Figure 4. Boundary conditions for HCM volume with the frame.

Figure 4. Boundary conditions for HCM volume with the frame.

Figure 4. A. Mesh for HCM (Matrix and macrofiber) volume with the frame.

Figure 4. A. Mesh for HCM (Matrix and macrofiber) volume with the frame.

Applied external loads led to stress formation in every orthogonal cross-section of every macrofiber in the material. Calculations were carried out assuming that failure propagation is governing by threads failure. The deformations till rupture of the HCM matrix is much greater than those of the reinforcing macrofibers. The finite element method gives us detailed picture of internal stresses and deformations in the material. Our approach is focused on macrofibers rupture, matrix is working as a media transforming stresses between MF loops. Was found more heavily loaded macrofibers cross-section (with highest value of stretching normal stress along MF local longitudinal axes). According to periodical structure of loops in the fabric such cross-sections will be many. Within every repeating loop (repeating element) such cross-section will be one. If one of them fails, we will have one broken macrofiber surrounded by virgin loops (MF) within matrix. Ruptured macrofiber increases the overload on the closest macrofibers - neighbors in adjacent reinforcing fabric loops. More heavily loaded cross-sections in neighboring loops are dependent on applied external load orientation to the plate axis xy. In every calculation they were recognized. Stress parameter which is used for characterization of the macrofiber overloading statement is averaged over crossection area stretching stress acting in the direction along the fiber in its current orthogonal cross-section. If we have two adjacent broken macrofibers plate was re-meshed and more heavily loaded cross-sections (locations of the cross-sections with highest value of the stress parameter) were found in the loops adjacent to the defect, consisting of two broken macrofibers. If we have three adjacent broken macrofibers, calculations show more heavily loaded cross-sections (position of the cross-sections) in the loops adjacent to the defect consisting of three broken macrofibers. In every such calculation, we are obtaining the stress parameter numerical values for maximally overloaded cross-sections. These calculations are possible to continue for defects with bigger size.

3.2. Calculations Using Model with a Frame

Curvature-based mesh was used for simulation, with maximal element SOLID 186 linear size equal to 0.1675 mm. Calculations were carried out using software ANSYS for all volume, and the results for normal stretching stresses (not averaged over cross-section area) in MF are shown in Figure 5. Stretching stress was calculated as:

$${\sigma }_{\nu }(\eta ,\xi )={\sigma }_x{l}^2+{\sigma }_y{m}^2+{\sigma }_z{n}^2+2{\tau }_{yz}mn+2{\tau }_{zx}nl+2{\tau }_{xy}lm,$$

(1)

where σ_ν is normal stress in the point of current orthogonal cross-section, σ_x is normal stress in the x-axis direction, σ_y is normal stress in the y-axis direction, σ_z is normal stress in the z-axis direction, τ_yz is shear stress in the yz plane, τ_zx is shear stress in the zx plane, τ_xy is shear stress in the xy plane, and l, m, n are direction cosines. Calculations were done combining ANSYS with MATLAB (for σ_ν calculation).

Stress distributions in every macrofiber internal cross-section’s point were obtained, then locations of N orthogonal cross-sections for each macrofiber along its spatial geometrical line were selected and numbered. In every orthogonal cross-section, stress parameter - the average stretching stress value (across the cross-section’s area) orthogonal to the cross-section’s plane was calculated as follows:

$$<{\mathsf{\sigma }}_{\mathsf{\nu }}>=\frac{1}{S}{\iint }_0^S{\sigma }_{\nu }\left(\eta ,\xi \right)dS ,$$

(2)

where <σ_ν> is average normal stress in the cross-section, ${\sigma }_{\nu }(\eta ,\xi )$ is local normal stress in different points of the fiber cross-section, η and ξ are coordinates in crossections plane, S is the area of the macrofiber cross-section.

Figure 5. Local stretching stress (in MPa) orthogonal to each macrofiber orthogonal crossection in macrofibers inside the HCM plate. Simulation results.

Figure 5. Local stretching stress (in MPa) orthogonal to each macrofiber orthogonal crossection in macrofibers inside the HCM plate. Simulation results.

Figure 6. Stress parameter <σ_ν > values (averaged over MF cross-section’s area normal stretching stress) in numbered cross-sections of 2^nd, 3^rd, 4^th, and 5^th MF in virgin material without broken MF: (a) numbered cross-sections location on the loops; (b) stress parameter values in numbered cross-sections along selected MF. Calculation was made for Matrix-Epoxy resin without OSA.

Figure 6. Stress parameter <σ_ν > values (averaged over MF cross-section’s area normal stretching stress) in numbered cross-sections of 2^nd, 3^rd, 4^th, and 5^th MF in virgin material without broken MF: (a) numbered cross-sections location on the loops; (b) stress parameter values in numbered cross-sections along selected MF. Calculation was made for Matrix-Epoxy resin without OSA.

In virgin material, without broken macrofibers, according to the internal thread’s architecture, local overstresses can be found in different macrofibers loop’s places are shown in Figure 5 (scale is in MPa). Averaged over current cross-section areas local stretching stress <σν > – stress parameter (SP) values in cross-sections along 2nd, 3rd, 4th, and 5th MF are shown in Figure 6. Looking on Figure 6a, it could be concluded - there is one more heavily stretched cross-section. In yarns 2, 3 and 4 (Figure 6), this is cross-section 79 (numbering is the same for every thread). Stress parameter values calculated along macrofibers are shown in Figure 6b. All other cross-sections are loaded with smaller overstress (have smaller value of the stress parameter). Crossections 28 in all threads have the same stress parameter as in 79. Small difference in calculated values is because vicinity of the plate borders. About cross-sections 28 and 79 in the fives thread. In virgin material with many threads (big size plate) stress parameter must be the same as in threads 2-4. Deviations are coming from vicinity of the plate’s border in FEM calculations. Calculations shown, when we use representative volume having only one loop stress deviations in peak stress parameter values was bigger (with the same mesh).

Suppose the first thread breaks. According to our model, it will happen in one of the more heavily loaded cross-sections. In our situation, it will be cross-section 79 in the MF 3. It is worth to mentioning that the choice of the first broken cross-section is rather arbitrary, and it could be referred to the broken cross-section as the cross-section in a randomly chosen representative part of the material. Cross-section 79 in the MF 3 is stretched with the maximal normal stress with the SP value 1672.2 MPa. In the situation with pure epoxy resin matrix, the value is 1672.2 MPa, and in the situation with epoxy resin with 20% OSA, the value is 1679.0 MPa.

Figure 7. Location of the broken cross-section on third MF.

Figure 7. Location of the broken cross-section on third MF.

Further in calculations, the failed yarn’s cross-section is simulated by a thin, soft, cylindrical insertion into the yarn body (small red part in the Figure 7), with an elastic modulus close to zero. This insertion, which has the same diameter as the yarn, models the location of rupture in the thread. An investigation of debonding between matrix and MF, as well as the internal delamination and fraying of the yarn (MF), is not included in the present research.

We placed the soft insertion in cross-section 79 of the yarn 3. Perfect bonding took place between the separate MF parts and the MF and matrix in the finite element model. Failure in the composite material led to the formation of overstresses on the cross-sections of the nearest neighbors in the vicinity of the ruptured place. Figure 8 shows stress parameter distribution in the plate with one rupture.

Figure 8. Cross-sections with maximal normal stresses (stress parameter <σ_ν >) in material with one broken macrofiber. Cross-section 79 in the yarn 3 is broken. Epoxy resin matrix without OSA.

Figure 8. Cross-sections with maximal normal stresses (stress parameter <σ_ν >) in material with one broken macrofiber. Cross-section 79 in the yarn 3 is broken. Epoxy resin matrix without OSA.

Rupture is in the cross-section 79 in the MF number 3. More overloaded are crossections number 79 in the MF number 4 and 2. Stress parameter value in these cross-sections are equal to 1740.0 MPa (in MF 2 we obtained slightly smaller) for the case with epoxy matrix (without OSA) and 1748.0 MPa for the case with epoxy matrix having 20% OSA. It can be also observed that the stress value in the yarn 3 dropped nearly to 0 (in reality it is 0) when we added the soft inclusion. If cross-section 79 in the yarn 4, will fail, there would be two adjacent broken threads in the material, and these two breakages would form a defect consisting of two adjacent broken macrofibers (numbers 3 and 4). Threads forward (and back along the crack’s line)) to the two ruptured macro-fibers will be more heavily overloaded.

Then, three adjacent broken MF, then four adjacent broken macrofibers were introduced in the plate (broken cross-sections were replaced by soft inclusions) and overloads on adjacent to broken macrofibers were calculated. (see Figure 9).

Figure 9. Broken cross-sections replaced by soft inclusions.

Figure 9. Broken cross-sections replaced by soft inclusions.

The deformation of the CM with epoxy matrix is shown in Figure 10. In Figure 10b broken ends of MF are forming empty ”channels” in the matrix because of matrix elastoplastic properties.

Figure 10. FEM simulations (a) with and (b) without soft inclusions.

Figure 10. FEM simulations (a) with and (b) without soft inclusions.

Maximal overstress in overstressed cross-sections in the vicinity of j broken fibers is shown in the Table 2. For four broken fibers plate border is significantly affect the result. Normal stress values are shown for calculations done using models with and without the frame, as well as for CM with epoxy matrix and epoxy matrix resin with 20% OSA. Looking on the data in the table, we see that overstress values are slightly dependent on the addition of OSA to the Epoxy matrix. Higher dependence is possible to expect with very high OSA particles concentrations. Model geometry (with and without the frame) affects the failure scenario such way - without the frame, the model has slightly higher normal stress values comparing to the condition with the frame. The enlargement of observed broken threads (and number of loops) leads to a rapid increase in the numerical task size and calculation time. In order to address this situation, a linear approximation of the overstress dependence on the size of broken yarns clusters was accepted, as shown in Figure 11.

Figure 11. Maximal tensile stress parameter <σ_ν > on the nearest yarns-neighbors on the border of the cluster consisting of i broken yarns (MF).

Figure 11. Maximal tensile stress parameter <σ_ν > on the nearest yarns-neighbors on the border of the cluster consisting of i broken yarns (MF).

4. Model of Damage Accumulation in the CMHR

4.2. Damage Accumulation

Modeling damage accumulation in stretched CM reinforced by weft-knitted fabric assumes, that failure is a complex stochastic process that starts with scattered, isolated breaks accumulation in MF sections, redistributing overstresses on adjacent to broken virgin MF, rupture of overstressed neighbors and formation of broken MF clusters. Clusters are growing inside the bundles; each bundle is orthogonal to the applied stretching direction. This process starts. At beginning, if we have only small size clusters is necessary to increase applied force value for obtaining bigger clusters. Clusters are growing. Process will transform to ultimate catastrophic biggest cluster growth (when we have such) with very small increase of applied load leading to next cluster formation. Growth happened because the overstress distributed on the closest unbroken neighbors will practically immediately initiate a segment break in the adjacent MF. In the framework of this model, we neglected the interaction of clusters between two adjacent bundles.

We can introduce one random variable $\mathit{J}\left(\sigma \right)$ - size of the cluster, with the probability function $P\left\{\mathit{J}\left(\sigma \right)\ge \left.i\right\}\right.$. P{J≥ i} is the probability to find a cluster consisting of at least i adjacent (in one bundle) broken MF when the externally applied tensile stress reaches σ and stress parameter value on MF σ₀. It is obvious that

P(j=1) =$W\left({\sigma }_0\right)$.

The probability of find a cluster consisting of at least two adjacent broken MF is P(j≥2) =P(j=1) (1-(1-W (${\sigma }_1\left){)}^2\right).$

Here we want to show, that cluster consisting of at least two adjacent broken MF is forming by failure of one of the nearest neighbors to rupture place on the broken MF. Two neighbors are similarly overloaded on the border of the cluster consisting of one broken MF (see Figure 8). Suppose broken is crossection 79 belonging to the yarn 2, two neighbors (crossections 79 in 1^st yarn and 79 in 3^th yarn become overloaded). Stress parameter value in this crossections become equal to ${\sigma }_1$.

The probability of finding a cluster consisting of at least three adjacent broken MF is

P(j≥3) =P (j ≥ 2) (1-(1-W (${\sigma }_2\left){)}^2\right)$, because two neighbors are similarly overloaded on the border of the cluster consisting of two broken MF. (in Figure 6 a, cross-sections 79 belonging to yarns 1 and 4, if broken crossections are number 79 in yarns 2 and 3). The probability of finding a cluster consisting of at least four adjacent broken yarns is

Figure 12. a). chain of bundles. b Repeating yarn element in the geometric model of chain of bundles.

Figure 12. a). chain of bundles. b Repeating yarn element in the geometric model of chain of bundles.

P(j≥4) =P(j≥3) (1-(1-W (${\sigma }_3\left){)}^2\right)$.

The probability of finding a cluster consisting of at least five adjacent broken yarns is equal to

P(j≥5) =P(j≥4) (1-(1-W(σ₄))²), because two neighbors are similarly overloaded on the border of the cluster consisting of four broken yarns. Continuing this procedure, we can calculate the probability of finding clusters consisting of at least i adjacent broken MF under applied external stress σ:

P(j≥i) =P(j≥i-1) (1-(1-W(σ_i-1))²).

Figure 13. Stress parameter <σν > values distribution along the overloaded MF element in the chain of bundles model. Between crossections A and B MF is loaded by stress values.

Figure 13. Stress parameter <σν > values distribution along the overloaded MF element in the chain of bundles model. Between crossections A and B MF is loaded by stress values.

σ₀ $\in [$σ-20%σ; σ+20%σ]. This part starts from crossection A to crossection B (see Figure 12).

5. Numerical Example

Figure 14. Probability of finding clusters consisting of at least i adjacent broken yarns in stretched material.

Figure 14. Probability of finding clusters consisting of at least i adjacent broken yarns in stretched material.

6. Conclusions

Abbreviations:

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