Layer composites reinforced with fabrics – laminates are construction materials in which mechanical properties can be shaped by designing their microstructure appropriately. However, the multi-phase microstructure of laminates makes it difficult to calculate the strength of the laminate constructions, especially when the reinforcement is a fabric. The article presents a special calculation model for determining the strength of an exemplary construction element made of laminate reinforced with a roving fabric with a plain weave made of carbon fibers. The computational model reflected in a simplified way the laminate microstructure, i.e. the number and orientation of the reinforcement fabric layers and its weave, and enabled a simulation of the behavior of the construction element under load up to fracture, which occurred as a result of breaking the reinforcement and interlayer crack – delamination. The simulation results were compared with the results of experimental stand tests. A method of modifying the computational model for laminates reinforced with non-plain weave was also suggested.
Keywords
- Layered composite
- delamination
- woven fabric
- carbon fiber roving
- finite element method
Contemporary textile industry, apart from clothing and decorative textiles, also provides special textiles for technical use. Among them there are fabrics in which the warp and weft are rovings, i.e. bundles of parallelly arranged and untwisted glass, carbon or aramid fibers. Such fabrics, due to the high mechanical strength of the fibers, well-mastered and efficient weaving technology as well as consistency caused by the appropriate weave, are used as reinforcement in polymer layered structural composites – laminates. Reinforcement layers in laminates are impregnated with a matrix on the basis of polyester or epoxy resin and it is the type of fibers, fabric weave, number of layers and their mutual orientation and production technology (laminating) that determine whether the laminate will be an isotropic or anisotropic construction material [1]. Laminates reinforced with fabrics made of fibers of very high strength are characterized by high specific strength, i.e. the strength related to density and that is why they are more commonly used in the production of parts of planes, cars, wind turbines and many other constructions [2,3,4]. Figure 1 shows the roving fabrics most often used in laminates. In the unidirectional fabric in Figure 1(a), the warp made of durable rovings is interwoven with a weft made of thin polyester roving of low strength, which is only to ensure fabric consistency and keep warp rovings parallel during lamination. A single layer of such reinforcement after impregnation with a matrix is durable practically in one direction – along the warp. In the fabric in Figure 1(b), the warp and weft are durable rovings connected by plain weave (
That is why the main aim of the article is to present a new proposal for modelling laminate construction elements using the formalism of the finite element method, which allows the determination of limit operating loads due to delamination and simulation of crack initiation and development, and at the same time:
it takes into account the microstructure of the laminate, i.e. not only the number and orientation of layers but also the fabric weave of reinforcement,
it makes it possible to determine crack initiation sites in the laminate and simulate their development,
it uses the conditions and material constants which can be determined by means of a developed experimental and computational procedure based on easy-to-perform tests of specimens of a particular laminate, thanks to which it also takes into account the influence of laminate microstructure and laminating technology on its mechanical properties,
it does not excessively increase the labour consumption of developing the computational model or the time of performing computer simulation.
Modeling and analyzing the strength of fabric reinforced laminates is most often carried out using the finite element method (FEM) because, compared to analytical methods, it allows one to take into consideration in one's analyses many factors determining the mechanical properties of laminates.
In commercial computational programs using FEM, fabric-reinforced laminates are modeled at the macroscale level without reflecting their microstructure and, in particular, the weave of the reinforcement fabric (Figure 2(a)). The model is built of finite elements of shell or solid type and the laminate microstructure is most often taken into consideration indirectly, replacing the laminate with a homogeneous material. This modeling method, due to good numerical efficiency of the model, can be used in the static analyses of structural elements made of laminates [6,7]. However, such a modeling method can cause erroneous simulation results [7,8], and simulating delamination with its help requires the use of special computational methods (e.g. VCCT [9], cohesive zone method [10]), which significantly reduce the numerical efficiency of the model and require predicting the location of potential damage sites.
The strength of laminafte is determined by the strength of the fibers, matrix and fiber-matrix connection. In order to calculate it, FE models with different degrees of detail are used (Figure 2(b) and 2(c)). The reinforcement in the microscale model is modeled with finite elements of the solid type [11,12], which accurately reflect the cross-section and shape of individual fibers (Figure 2(b)). However, such models – due to a very large number of degrees of freedom, are not numerically effective. For this reason, microscale models are not used in engineering calculations of laminate construction elements. A reasonable compromise between macro and microscale models are mesoscale models (Figure 2(c)), which take into account – in a simplified way – the laminate microstructure resulting from the weave of the reinforcement fabric. That is why the mesoscale models are much more numerically effective compared to the microscale ones and, moreover, they do not require discretization of reinforcement rovings into single fibers (there is an assumption about roving homogenization [13]). Determining laminate strength then consists in checking the strength criteria for roving and matrix and for the connection between reinforcement and matrix [14,15]. Despite good numerical efficiency, the mesoscale models described in the literature are not used in engineering practice for modeling delamination in laminate construction elements. Just like microscale models, they require the use of special computational methods for modeling cracking, which worsen their numerical efficiency and require the prediction of the location of potential crack sites.
The method of modeling the matrix impregnated fabric (lamina) presented below assumes that the fiber rovings are reflected with finite elements of beam type and the matrix with elements of shell type. Figure 3 shows a model of the so-called unit cell, i.e. the smallest repetitive element of the microstructure (RUC – repetitive unit cell) extracted from a fabric with plain weave. RUC consists of two pairs of rovings interwoven with each other and a fragment of the matrix. In order to simplify the RUC model, it was assumed that dimension Δz is small compared to dimensions Δx and Δy. Then RUC is defined by four nodes lying in one plane at the roving interweaving sites – N_{1}, N_{2}, N_{3}, N_{4}. Beam elements R modeling matrix-impregnated roving sections (weft and warp) are characterized by two cross-sectional parameters i.e. the area of cross section of roving
The possibility to modify the mesoscale model of unit cell makes it useful in modeling laminates reinforced with fabric of a non-plain weave. For example, in the half basket weave, which is a derivative of plain weave, there are non-interwoven roving sections which are arranged parallelly on the surface of the fabric. An example of such a fabric (
The usefulness of the mesoscale model (Figure 3) for assessing the strength of construction elements due to delamination was tested on the example of a laminate reinforcement beam for side doors of a passenger car (Figure 5). It is a construction element which protects passengers and absorbs part of the side impact energy by being deformed elastically and then destroyed. The beam was made of eight layers of roving fabric (3k CF200 style450), stacked on top of each other, with a plain weave of carbon fibers whose warp was parallel to the axis of the beam. The reinforcement layers during lamination were impregnated with a chemically hardenable polymer matrix based on epoxy resin Biresin® CR120. The door beam is an example of a construction element of an irregular and complex shape, because its outline is an irregular curve and the shape is made of flat and curved surfaces. The strength of the door beams is checked in a standard way using the three-point bending test (Figure 6). The construction of the measuring stand and the test procedure are described in detail e.g. in GMW16418 standard [16]. As a result of the test,
The construction of the beam mesoscale model began with the preparation of the geometric shell model, which was then discretized by means of four-node shell finite elements. The discretization concerned the external surface of the modeled beam. Then the nodes of the shell model were parallelly shifted by the thickness of the layer, thus building up the nodes of all other reinforcement layers. In the meshes of nets created in this way in the layers, beam elements R and shell elements S were generated, forming RUC in accordance with the diagram in Figure 7. In the next stage, beam elements P were inserted between appropriate nodes of adjacent layers, with matrix material constants and substitute cross-sectional parameters
The matrix material constants necessary for building the model were read from the technical card [17]. Due to the lack of data on the fabric, a roving tensile test was carried out, on the basis of which a substitute Young's modulus of roving (
Values of parameters of finite elements creating RUC model
1D Beam – P | – | ||
1D Beam – R | – | ||
2D Shell – S | – | – |
The cross-sectional area
In order to assess the resistance of the beam to delamination, criteria for the strength of the finite elements forming RUC (Figure 3) had to be defined. It was assumed that during simulation of delamination (initiation and propagation of crack between layers) the stiffness (cross-sectional parameters) of finite elements creating connections between layers (
The criteria were defined resulting from three possible basic modes of laminate fracture as a result of initiation of crack between layers, i.e. tension mode, sliding shear mode and scissoring shear mode [18]. Beam element P, in which the threshold value of tensile or shear force was exceeded, was searched and then, depending on the existing fracture mode, removed in one step (tension mode) or its stiffness was reduced similarly to beam elements R (sliding and scissoring shear mode). The method of determining the threshold values of tensile and shear forces for beam elements P for three fracture modes is presented in work [18]. The task was to carry out simulation of experimental tests of delamination of pre-delaminated specimens of laminate in the conditions of the following tests: DCB (double cantilever beam), ENF (end notched flexure) and MECT (modified edge crack torsion). Comparing the experimental curves and the ones determined during the simulation of experiments, the threshold values of forces in finite elements P modeling connections between layers corresponding to the moment of crack initiation were read. Yet, for the DCB test it was the tensile force
Such a procedure also required determining the threshold values of the shear force
In addition, an assumption was made that the destruction of the bending door beam can also take place as a result of a fracture of rovings forming the reinforcement fabric. That is why a special procedure was developed to simulate the crack of matrix-impregnated roving, which was implemented in the model by gradual reduction of stiffness (of cross-sectional parameters
On the basis of many numerical experiments, it was established that the procedure of reducing the stiffness of finite elements R modeling the roving sections in which force
In the second phase, after exceeding the threshold value of force
Constants in formulas (1) and (2) as well as ways of reducing the moment of inertia of section
Simulation of the three-point bending test of the laminate door beam, in which the proposed mesoscale FE model and the procedures for initiating and propagating interlayer cracks were used in practice in ANSYS program using internal APDL command language. APDL language was also used to develop a procedure for automated generation of a mesoscale door beam model.
On the basis of the simulation results exported by Ansys program, the force-deflection curve
The assessment of the beam resistance to delamination was then verified in the experimental three-point bending test on a special stand (Figure 6), which met the requirements of the standard [16]. The door beam was screwed to the sliding supports which were connected to the fixed base of the stand by means of coil springs reflecting the stiffness of the car door frame. During the test, vertical displacement
As a result of the conducted test, the curve of the force
Table 2 summarizes the errors of force and deflection resulting from the comparison of calculated values with experimentally measured values. On their basis, relative errors of force and deflection were determined, relating absolute error values to measured values. The largest error of force was 8% and of deflection 4%, which – given the complexity of the modeled delamination phenomenon – indicates very good compliance of the simulation and experiment results. As an additional measure of the quality of the fit of the model to the results of experimental tests, one can propose
Errors of deflection and bending force for mesoscale model
NL_{1} and NL_{1(m)} | 0.87 | 4.24 | 156.12 | 8.42 |
NL_{2} and NL_{2(m)} | 0.93 | 2.59 | 234.59 | 7.87 |
MAX and MAX_{(m)} | 1.51 | 3.11 | 284.21 | 7.64 |
Another fact confirming the usefulness of the proposed mesoscale FE model of a construction element made of fabric-reinforced laminate is the comparison of the sites of formation of subsequent cracks during the bending test. As can be seen in Figure 10, during the simulation and the experiment the cracks formed at the same sites.
The proposed method of assessing strength with reference to delamination and stiffness of construction elements made of laminates reinforced with technical fabric, which uses the RUC concept proposed in the article and the mesoscale level of laminate microstructure observation, enables:
simplified reflecting of the laminate microstructure, in particular of the weave of the reinforcement fabric and the number and orientation of layers;
Experimental tests of laminate specimens and construction elements indicate that their mechanical properties, including delamination resistance, depend on the weave of the reinforcement fabric. The comparison of the results of delamination simulation with the results of experimental tests of an exemplary laminate construction element – a door beam – confirmed that the method of modelling the laminate microstructure described in the article combined with experimental tests of specimens ensures good consistency of the results in terms of deflections and of crack initiation sites as well as limit values of loads that cause them.
experimental determination of delamination strength criteria for a laminate with a specific microstructure from specific components produced by a specific lamination technology;
Although the necessity to carry out this procedure limits the use of the numerical values in the proposed criteria of laminate resistance to delamination to specific cases, it ensures good consistency of simulation and experiment results. The proposed experimental and computational procedure for determining these criteria can be easily carried out for other laminates and other laminating technology with only basic laboratory equipment.
simulation of delamination without the need to predict the location of crack formation and propagation;
Methods which are often used during the cumulation of cracking, such as VCCT or cohesive zone method, require predicting the potential crack initiation sites, which in engineering calculations can be difficult and sometimes even impossible. Removing this obstacle significantly broadens the scope of application of the method of modelling laminate construction elements described in the article.
assessment of strength to delamination of undamaged laminate construction elements as well as determination of permissible loads for partially damaged elements (with small cracks), which can be further exploited under lower load.
The experimental verification of the mesoscale FE model of a real construction element made of laminate with an irregular and complex shape carried out on the example of a three-point bending test of a passenger car door beam showed full usefulness of the proposed model in engineering calculations. It is worth adding that the authors’ experience shows that thanks to the automation of generating mesoscale FE model, e.g. while using the APDL language of the ANSYS program, the time of developing a laminate construction element model is comparable to the time of developing a model of this element made of sheet metal.
Errors of deflection and bending force for mesoscale model
NL_{1} and NL_{1(m)} | 0.87 | 4.24 | 156.12 | 8.42 |
NL_{2} and NL_{2(m)} | 0.93 | 2.59 | 234.59 | 7.87 |
MAX and MAX_{(m)} | 1.51 | 3.11 | 284.21 | 7.64 |
Values of parameters of finite elements creating RUC model
1D Beam – P | – | ||
1D Beam – R | – | ||
2D Shell – S | – | – |
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