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Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing
Materials Today: Proceedings xxx (xxxx) xxx
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing O.A. Panina ⇑, A.S. Nemov, A.Yu. Zobacheva, I.A. Kobykhno, O.V. Tolochko, V.K. Yadykin Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya 29, St. Petersburg 195251, Russia
a r t i c l e
i n f o
Article history: Received 11 December 2019 Accepted 6 January 2020 Available online xxxx Keywords: Composite Defects Mechanical characteristics Carbon fibers Thermoplastic Nonlinear simulation
a b s t r a c t The paper is dedicated to the simulation of mechanical behavior of a unidirectional thermoplastic-based composite material reinforced with carbon fibers. The influence of microstructure imperfections on mechanical characteristics of the composite material has been studied, and numerical parameters that ensure a good correlation between nonlinear two-scale simulation of progressive damage and experimental data have been identified. Results of the study revealed limited importance of accounting for such imperfections of composite microstructure as deviation in fiber laying direction and fiber curvature. The two-scale simulation approach based on submodeling technique was proved to describe the nonlinear behavior of the thermoplastic-based carbon fiber material with a reasonable accuracy. Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the Materials Science: Composites, Alloys and Materials Chemistry.
1. Introduction Development of new reliable and cost-effective structures in mechanical engineering requires the use of materials with high physical-mechanical, technological and operational properties: high strength, heat resistance, corrosion resistance, resistance to crack propagation, low density. Today, the majority of natural and artificial materials do not meet the requirements of industry. Therefore, there is a need in use of composite materials which are characterized by a combination of unique properties: lightness, strength, sufficiently high rigidity, non-magnetic, radio transparency, etc. For a specific product it is often necessary to create a unique composite material that meets certain requirements by selection of appropriate components, choice of reinforcing pattern and of volumetric ratio of the reinforcing component, and by implementing an appropriate manufacturing process. Composite materials usually consist of reinforcing elements and a binder. One of the most promising reinforcing fillers is carbon fibers. Carbon fibers are characterized by high tensile strength, low weight, and low coefficient of thermal expansion, which makes carbon fibers indispensable in some special industry fields: rocket and aircraft construction, shipbuilding, automotive industry.
⇑ Corresponding author. E-mail address: [email protected] (O.A. Panina).
Polymers are among the most common types of binder. They are characterized by relatively low strength, but in composites they ensure the joint work of reinforcing elements. It forms a set of composition properties, which reflect not only initial characteristics of the components, but sometimes also properties that isolated components do not possess. Since strength properties of polymers are relatively low, the resin may break during the loading of the structure, which entails a change in properties of the composite material as a whole. One of aspects of this work is to study the effect of progressive resin destruction on mechanical behavior of the composite material. Pultrusion is one of the approaches to the manufacturing of composite materials. Method of pultrusion consists in drawing impregnated with a resin fibers through a system of dies with a gradually decreasing cross section [1]. In this work a unidirectional composite, based on thermoplastic resin, reinforced with carbon fibers, and obtained by pultrusion process, was studied (Fig. 1). During the production of a composite material by means of pultrusion, various defects in microstructure arise inevitably [1]. These are defects such as cracks, delaminations, voids, fiber waviness, fiber misalignments, wrinkles, gaps and overlaps. All these defects can affect mechanical characteristics of the material [2,3]. A lot of research works have been devoted to modeling defects and considering their influence on composite behavior. Large number of existing articles is devoted to modeling of non-ideal
https://doi.org/10.1016/j.matpr.2020.01.134 2214-7853/Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the Materials Science: Composites, Alloys and Materials Chemistry.
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
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Fig. 2. (a) Periodicity cell; (b) Quarter of cell.
Fig. 1. Obtained by pultrusion unidirectional thermoplastic-based composite reinforced with carbon fibers.
adhesion between the fiber and the resin and to modeling of composite fracture taking into account the adhesion [4]. In addition to adhesion, in some studies the waviness of fibers and the destruction of the matrix in the compression experiment are taken into account [5,6]. However, the main goal of these works was to study the destruction of composite, and not the influence of these defects on the mechanical behavior of the composite, on its effective properties. The effect of adhesion on the mechanical behavior of an elastomer-based composite reinforced with particles in a uniaxial compression test was studied in [7]. Works devoted to the fiber waviness usually consider fibers waviness as the waviness of the whole layer in a certain volume of the composite [8–13]. In this paper, the defect of fiber straightness was studied at the level of an individual fiber. The objective of this work is to analyze the effect of individual fibers straightness, and not the waviness of a certain part of the composite. In addition to adhesion and fibers waviness, existing studies focus on the experimental development [14] and numerical modeling of such defects as gaps and overlaps [15]. Works devoted to the study of the influence of fibers laying direction usually compare different angles under assumption that all fibers are directed in the same way [16]. In this study, however, we intended to investigate the effect of random small deviation in the alignment of individual fibers (therefore, a model with randomly directed within certain limits of the deflection angle fibers was analyzed). Several studies are also devoted to the simulation and studying of the wrinkle defect [17,18]. These works demonstrate new modeling methods and study the effect of wrinkles on the fracture process, taking into account many other processes, such as delamination and binder cracking. In the present work a method for predicting the nonlinear behavior of a composite material and its destruction was explored, it is based on modeling the progressive destruction of the resin. Even without considering adhesion and other phenomena affecting the destruction of material, this method shows a good agreement with the experiment. To sum up, the goals of the present study are to investigate the influence of fiber straightness, laying direction, and progressive resin destruction on the mechanical behavior of the unidirectional composite. Finite element (FE) modelling is used as the main tool for numerical analyses presented in the paper.
the homogenization method. In this paper the direct homogenization method was used [19,20]. In the case of a unidirectional carbon fiber tape, composite material periodicity cell is represented in the form of a regular hexagon, where fibers are located at the tops and at the center (Fig. 2(a)). Due to the symmetry, only a quarter of the cell is considered (Fig. 2(b)). Effective characteristics of the material are determined as a result of solving five boundary value problems of the elasticity theory in accordance with direct homogenization method. After that, composite material is modeled as homogeneous, with certain effective characteristics. To verify correctness of this method and selected model, a three-point bend test was simulated [21]. Behavior of the material in the three-point bend test with characteristics determined by the method described above is consistent with the experimental data (Fig. 3). The comparison in Fig. 3 illustrates that results are similar, but real material seems to be slightly softer than the simulated one. We suppose that that might be due to the presence of various structural defects in the real composite. Therefore, the influence of defects on the mechanical characteristics is investigated in the following parts of the paper. 3. The influence of fiber laying direction One of the typical defects in the structure of composite is the laying direction defect. This defect means deviation of laying direction for some of fibers from the nominal value. With a purpose to investigate the possible influence of this defect on the effective elastic moduli, a 3D model of the composite with randomly distributed fibers with randomly assigned angle deviation was developed. Each fiber is deviated from the zero direction by a random angle from a certain range (±5°, ±10°, ±15°, ±20°, ±25°, ±30°). Fiber volume concentration is assumed to be 50%. Technically, the fibers in the model are created by assigning the certain material proper-
2. An approach to simulation of composite materials: homogenization method Many composites have inclusions of micro size, and it is usually not possible to model such composite in direct accordance with its structure. In this work a unidirectional composite reinforced with carbon fibers was investigated. Diameter of fiber is 7 mm. To account for microstructure in such material effective characteristics are introduced. Effective characteristics are calculated using
Fig. 3. Comparison of simulation results in ANSYS and experimental data.
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
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ties to the elements of the mapped brick mesh – that allows easy varying of fibers distribution and alignment but leads to the not accurate shape of the fibers. However, that inaccuracy of geometrical representation of the fibers is expected to have a negligible effect on effective material characteristics (though that model should not be used for accurate calculation of stress concentrations, of course). Some of the developed models are shown in Figs. 4 and 5. Using these models, effective elastic moduli were determined. The influence of fibers laying direction deviation angle value on the value of effective elastic moduli was investigated. Results are shown in Fig. 6. As expected, due to the laying direction defect, the elastic modulus ‘‘along fibers” (Ez) decreases, and the elastic moduli in perpendicular directions increase. Results showed that deviation of fiber direction significantly affects elastic modulus ‘‘along fibers” (Ez): the Ez modulus of elasticity at a maximum fiber deflection of 30°decreased by 25% compared to a material without defect. Besides, the laying direction defect affects the elastic moduli in perpendicular directions: the Ey modulus of elasticity at a maximum fiber deflection of 30°increased by 20% compared to that for the material without this defect. However, it should be noted that such large difference in elastic moduli values is observed at fairly large angles (20–30°). Deviations of fiber laying direction at such angles have to be noticeable with an unarmed eye. The conclusion can be drawn that taking this defect into account will sig-
Fig. 4. 3D FE model of a composite material with fiber laying deviation from nominal direction ±5°, concentration 50%.
Fig. 6. Dependence of effective elastic moduli values from the maximum value of the fiber deviation angle.
nificantly improve modeling accuracy only if fiber laying direction deviation is noticeable on material samples. 4. The influence of fiber straightness defect In addition to laying direction defect, a fiber straightness defect may be present in the composite — individual fibers in the composite may be bent. To investigate the influence of this factor, a threedimensional model of a periodicity cell was developed, where one of the fibers is bent (Fig. 7). With a purpose to investigate the effect of this defect on the values of effective moduli, the curvature of bent fiber was varied and the maximum deviation of the fiber center from its nominal position was controlled. Results of effective elastic moduli calculation for models with different values of the fiber center displacement are presented in Table 1. In addition to the Table 1, Fig. 8 demonstrates the dependence of the value of ‘‘along fibers” elastic modulus on the maximum deviation of the fiber center. The results obtained indicate that with the increase in the deviation of the fiber center, that is, with an increase in its curvature, the elastic modulus ‘‘along fibers” decreases and remaining elastic
Fig. 7. Three-dimensional model of a periodicity cell with bent fiber.
Table 1 Research the effect of fiber straightness defect on the values of effective elastic moduli.
Fig. 5. 3D FE model of a composite material with fiber laying deviation from nominal direction ±25°, concentration 50%.
Fiber center displacement, mm
Ex , GPa
Ey , GPa
Ez , GPa
0 0.50 1.05 1.40 1.75 2.33 2.50
4.62 4.72 4.74 4.76 4.80 4.97 5.06
4.63 4.72 4.71 4.72 4.73 4.76 4.79
115.15 115.11 114.95 114.80 114.60 114.20 114.09
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
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Fig. 8. Three-dimensional model of a periodicity cell with bent fiber.
moduli increase. At maximum fiber curvature, the decrease of the modulus Ez is less than 1%. At the same time the increase of the modulus Ex is about 9.5%. The conclusion can be drawn that the fiber straightness defect does not affect the modulus ‘‘along fibers” significantly, but it increases elastic moduli in the perpendicular to fibers directions considerably. 5. Simulation of the composite behavior with progressive damage In the process of loading, the resin in the composite, due to its low strength characteristics, might be locally destroyed. Local destruction of the resin leads to the weakening of composite effective properties in this area. This local destruction affects the behavior of the whole composite structure, and particularly it leads to the gradual decrease in stiffness of the material and, therefore, to the nonlinearity of the ‘‘stress-strain” curve. Eventually, the resin destruction can lead to the destruction of the entire composite. The effect of progressive damage caused by the resin destruction was studied on the example of a three-point bending test. Two-scale simulations were performed using a FE model of the sample in the bending test and a FE model of composite periodicity cell. In the FE model of the sample subjected to the three-point bend test composite material is described by means using of effective characteristics. Effective characteristics are calculated using the model of periodicity cell that takes into account the structure of composite and incorporates the data on the characteristics of the resin and fibers. Tensile strength of the resin is assumed to be 45 MPa. In order to simulate the resin destruction, elements with a stress above 45 MPa are assigned to material with an elastic modulus of 1000 Pa (Fig. 9). To link the FE models on two scales (macro scale – the sample, micro scale – the cell of periodicity) in simulation of the three-
Fig. 9. (a) FE model of the composite cell of periodicity in the absence of the resin destruction; (b) FE model of the composite cell of periodicity with the local resin destruction. Purple color – resin; turquoise color – fiber; blue color – destroyed resin elements. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
point bending test, a submodeling procedure was used: threepoint bend test model is coarse model, model of composite periodicity cell is submodel. This approach to multi-scale simulation was previously used to simulate the mechanical behavior of threecomponent composite materials [22]. The simulation of the bending test was performed incrementally, with the load increased at each iteration. Using submodeling procedure, the load value is determined, starting from which the destruction of the resin occurs in the submodel according to the mentioned failure criteria. Starting from this load value, effective characteristics are recalculated at each load increment taking into account the destroyed resin elements in the submodel. In the specimen model elements with maximum tensile stresses are assigned to a new material with the effective characteristics calculated at each iteration. The procedure of submodeling, recalculation of effective characteristics, and changing material in the specimen model repeats at each iteration (which in the considered example corresponded to the load increment 0.5 N). Considering recalculation of effective characteristics by load increments, at the first step material with new effective characteristics №. 1 was obtained, at the second step material №. 2 was obtained, and similarly, at the n-th step, the n-th material was obtained. To simulate the progressive damage, material in the elements of specimen model was changed sequentially: on the first step material №. 1 was assigned to elements with maximum tensile stresses. On the second step material №. 2 was assigned to elements with material №. 1; and material №. 1 was assigned to elements with original material, where the maximum tensile stress was reached at this step. Similarly, on the n-th step, the n-th material was assigned to the elements with the (n-1)-th material. Subsequent replacement of material reflects different levels of resin destruction depending on the location. An example of such material replacement is shown in Fig. 10. With this procedure material in the most loaded places is sequentially updated, it becomes weaker with each new step and weakens the whole model, so affecting its mechanical behavior. A nonlinear behavior of composite is obtained by means of progressive damage simulation. In addition to obtaining a nonlinear force-deflection curve that is close the experimental one (Fig. 11), the method described allowed to obtain the strength of the composite that is very close to the experimental one (the highest point in the force-deflection curve).
Fig. 10. Specimen model (a quarter is analyzed due to the symmetry) in three-point bend test at the end of loading. Different colors represent different materials, which simulate different values of resin destruction.
Fig. 11. Comparison of simulation that takes into account progressive damage with experimental data.
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
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The graph in Fig. 11 shows that due to the resin destruction, the composite gradually weakens and ultimately breaks down. Obtained simulation results of composite behavior are in a good agreement with experimental data. Thus, the two-scale method described above can be used to simulate the nonlinear behavior and destruction of composite structures based on thermoplastic resins. 6. Conclusions A number of numerical simulations of the mechanical behaviour of a unidirectional fiber-reinforced composite material based on a thermoplastic resin was performed. A two-scale simulation approach based on homogenization method and submodeling technique was implemented. Investigation of possible imperfections in composite structure (fiber laying direction defect and fiber straightness defect) on the effective mechanical characteristics was conducted. A simulation of composite material behaviour taking into account progressive damage was carried out. The results of this work showed that: 1) Fiber laying direction defect leads to a substantial decrease in the ‘‘along fibers” elastic modulus and to some increase in the elastic moduli for perpendicular directions. However, the effect is significant only for large deviation angles. Based on the foregoing, it could be concluded that this defect should be considered for elastic moduli values correction if the deviation angle is noticeable in the samples of the materials. 2) Fiber straightness defect does not significantly affect the modulus ‘‘along fibers”, but it does increase the moduli in the perpendicular to fibers directions. The maximum increase in the elastic moduli values was observed for the case with the most curved fiber. The increase in elastic moduli for moderate values of curvature was found to be within 2–3%. Thus, this defect should be considered only if there are substantial prerequisites for that. 3) Simulation of the composite behaviour with progressive damage allowed to obtain the nonlinear behaviour of the material close to that obtained experimentally. It also allowed to predict the destruction of the specimen that was consistent with experimental data. That proves effectiveness of the implemented two-scale method for simulation of nonlinear behaviour of thermoplastic-based unidirectional composite structures, including prediction of their failure.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
Contents lists available at ScienceDirect
Materials Today: Proceedings journal homepage: www.elsevier.com/locate/matpr
Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing O.A. Panina ⇑, A.S. Nemov, A.Yu. Zobacheva, I.A. Kobykhno, O.V. Tolochko, V.K. Yadykin Peter the Great St. Petersburg Polytechnic University, Polytechnicheskaya 29, St. Petersburg 195251, Russia
a r t i c l e
i n f o
Article history: Received 11 December 2019 Accepted 6 January 2020 Available online xxxx Keywords: Composite Defects Mechanical characteristics Carbon fibers Thermoplastic Nonlinear simulation
a b s t r a c t The paper is dedicated to the simulation of mechanical behavior of a unidirectional thermoplastic-based composite material reinforced with carbon fibers. The influence of microstructure imperfections on mechanical characteristics of the composite material has been studied, and numerical parameters that ensure a good correlation between nonlinear two-scale simulation of progressive damage and experimental data have been identified. Results of the study revealed limited importance of accounting for such imperfections of composite microstructure as deviation in fiber laying direction and fiber curvature. The two-scale simulation approach based on submodeling technique was proved to describe the nonlinear behavior of the thermoplastic-based carbon fiber material with a reasonable accuracy. Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the Materials Science: Composites, Alloys and Materials Chemistry.
1. Introduction Development of new reliable and cost-effective structures in mechanical engineering requires the use of materials with high physical-mechanical, technological and operational properties: high strength, heat resistance, corrosion resistance, resistance to crack propagation, low density. Today, the majority of natural and artificial materials do not meet the requirements of industry. Therefore, there is a need in use of composite materials which are characterized by a combination of unique properties: lightness, strength, sufficiently high rigidity, non-magnetic, radio transparency, etc. For a specific product it is often necessary to create a unique composite material that meets certain requirements by selection of appropriate components, choice of reinforcing pattern and of volumetric ratio of the reinforcing component, and by implementing an appropriate manufacturing process. Composite materials usually consist of reinforcing elements and a binder. One of the most promising reinforcing fillers is carbon fibers. Carbon fibers are characterized by high tensile strength, low weight, and low coefficient of thermal expansion, which makes carbon fibers indispensable in some special industry fields: rocket and aircraft construction, shipbuilding, automotive industry.
⇑ Corresponding author. E-mail address: [email protected] (O.A. Panina).
Polymers are among the most common types of binder. They are characterized by relatively low strength, but in composites they ensure the joint work of reinforcing elements. It forms a set of composition properties, which reflect not only initial characteristics of the components, but sometimes also properties that isolated components do not possess. Since strength properties of polymers are relatively low, the resin may break during the loading of the structure, which entails a change in properties of the composite material as a whole. One of aspects of this work is to study the effect of progressive resin destruction on mechanical behavior of the composite material. Pultrusion is one of the approaches to the manufacturing of composite materials. Method of pultrusion consists in drawing impregnated with a resin fibers through a system of dies with a gradually decreasing cross section [1]. In this work a unidirectional composite, based on thermoplastic resin, reinforced with carbon fibers, and obtained by pultrusion process, was studied (Fig. 1). During the production of a composite material by means of pultrusion, various defects in microstructure arise inevitably [1]. These are defects such as cracks, delaminations, voids, fiber waviness, fiber misalignments, wrinkles, gaps and overlaps. All these defects can affect mechanical characteristics of the material [2,3]. A lot of research works have been devoted to modeling defects and considering their influence on composite behavior. Large number of existing articles is devoted to modeling of non-ideal
https://doi.org/10.1016/j.matpr.2020.01.134 2214-7853/Ó 2020 Elsevier Ltd. All rights reserved. Selection and Peer-review under responsibility of the scientific committee of the Materials Science: Composites, Alloys and Materials Chemistry.
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
2
O.A. Panina et al. / Materials Today: Proceedings xxx (xxxx) xxx
Fig. 2. (a) Periodicity cell; (b) Quarter of cell.
Fig. 1. Obtained by pultrusion unidirectional thermoplastic-based composite reinforced with carbon fibers.
adhesion between the fiber and the resin and to modeling of composite fracture taking into account the adhesion [4]. In addition to adhesion, in some studies the waviness of fibers and the destruction of the matrix in the compression experiment are taken into account [5,6]. However, the main goal of these works was to study the destruction of composite, and not the influence of these defects on the mechanical behavior of the composite, on its effective properties. The effect of adhesion on the mechanical behavior of an elastomer-based composite reinforced with particles in a uniaxial compression test was studied in [7]. Works devoted to the fiber waviness usually consider fibers waviness as the waviness of the whole layer in a certain volume of the composite [8–13]. In this paper, the defect of fiber straightness was studied at the level of an individual fiber. The objective of this work is to analyze the effect of individual fibers straightness, and not the waviness of a certain part of the composite. In addition to adhesion and fibers waviness, existing studies focus on the experimental development [14] and numerical modeling of such defects as gaps and overlaps [15]. Works devoted to the study of the influence of fibers laying direction usually compare different angles under assumption that all fibers are directed in the same way [16]. In this study, however, we intended to investigate the effect of random small deviation in the alignment of individual fibers (therefore, a model with randomly directed within certain limits of the deflection angle fibers was analyzed). Several studies are also devoted to the simulation and studying of the wrinkle defect [17,18]. These works demonstrate new modeling methods and study the effect of wrinkles on the fracture process, taking into account many other processes, such as delamination and binder cracking. In the present work a method for predicting the nonlinear behavior of a composite material and its destruction was explored, it is based on modeling the progressive destruction of the resin. Even without considering adhesion and other phenomena affecting the destruction of material, this method shows a good agreement with the experiment. To sum up, the goals of the present study are to investigate the influence of fiber straightness, laying direction, and progressive resin destruction on the mechanical behavior of the unidirectional composite. Finite element (FE) modelling is used as the main tool for numerical analyses presented in the paper.
the homogenization method. In this paper the direct homogenization method was used [19,20]. In the case of a unidirectional carbon fiber tape, composite material periodicity cell is represented in the form of a regular hexagon, where fibers are located at the tops and at the center (Fig. 2(a)). Due to the symmetry, only a quarter of the cell is considered (Fig. 2(b)). Effective characteristics of the material are determined as a result of solving five boundary value problems of the elasticity theory in accordance with direct homogenization method. After that, composite material is modeled as homogeneous, with certain effective characteristics. To verify correctness of this method and selected model, a three-point bend test was simulated [21]. Behavior of the material in the three-point bend test with characteristics determined by the method described above is consistent with the experimental data (Fig. 3). The comparison in Fig. 3 illustrates that results are similar, but real material seems to be slightly softer than the simulated one. We suppose that that might be due to the presence of various structural defects in the real composite. Therefore, the influence of defects on the mechanical characteristics is investigated in the following parts of the paper. 3. The influence of fiber laying direction One of the typical defects in the structure of composite is the laying direction defect. This defect means deviation of laying direction for some of fibers from the nominal value. With a purpose to investigate the possible influence of this defect on the effective elastic moduli, a 3D model of the composite with randomly distributed fibers with randomly assigned angle deviation was developed. Each fiber is deviated from the zero direction by a random angle from a certain range (±5°, ±10°, ±15°, ±20°, ±25°, ±30°). Fiber volume concentration is assumed to be 50%. Technically, the fibers in the model are created by assigning the certain material proper-
2. An approach to simulation of composite materials: homogenization method Many composites have inclusions of micro size, and it is usually not possible to model such composite in direct accordance with its structure. In this work a unidirectional composite reinforced with carbon fibers was investigated. Diameter of fiber is 7 mm. To account for microstructure in such material effective characteristics are introduced. Effective characteristics are calculated using
Fig. 3. Comparison of simulation results in ANSYS and experimental data.
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
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ties to the elements of the mapped brick mesh – that allows easy varying of fibers distribution and alignment but leads to the not accurate shape of the fibers. However, that inaccuracy of geometrical representation of the fibers is expected to have a negligible effect on effective material characteristics (though that model should not be used for accurate calculation of stress concentrations, of course). Some of the developed models are shown in Figs. 4 and 5. Using these models, effective elastic moduli were determined. The influence of fibers laying direction deviation angle value on the value of effective elastic moduli was investigated. Results are shown in Fig. 6. As expected, due to the laying direction defect, the elastic modulus ‘‘along fibers” (Ez) decreases, and the elastic moduli in perpendicular directions increase. Results showed that deviation of fiber direction significantly affects elastic modulus ‘‘along fibers” (Ez): the Ez modulus of elasticity at a maximum fiber deflection of 30°decreased by 25% compared to a material without defect. Besides, the laying direction defect affects the elastic moduli in perpendicular directions: the Ey modulus of elasticity at a maximum fiber deflection of 30°increased by 20% compared to that for the material without this defect. However, it should be noted that such large difference in elastic moduli values is observed at fairly large angles (20–30°). Deviations of fiber laying direction at such angles have to be noticeable with an unarmed eye. The conclusion can be drawn that taking this defect into account will sig-
Fig. 4. 3D FE model of a composite material with fiber laying deviation from nominal direction ±5°, concentration 50%.
Fig. 6. Dependence of effective elastic moduli values from the maximum value of the fiber deviation angle.
nificantly improve modeling accuracy only if fiber laying direction deviation is noticeable on material samples. 4. The influence of fiber straightness defect In addition to laying direction defect, a fiber straightness defect may be present in the composite — individual fibers in the composite may be bent. To investigate the influence of this factor, a threedimensional model of a periodicity cell was developed, where one of the fibers is bent (Fig. 7). With a purpose to investigate the effect of this defect on the values of effective moduli, the curvature of bent fiber was varied and the maximum deviation of the fiber center from its nominal position was controlled. Results of effective elastic moduli calculation for models with different values of the fiber center displacement are presented in Table 1. In addition to the Table 1, Fig. 8 demonstrates the dependence of the value of ‘‘along fibers” elastic modulus on the maximum deviation of the fiber center. The results obtained indicate that with the increase in the deviation of the fiber center, that is, with an increase in its curvature, the elastic modulus ‘‘along fibers” decreases and remaining elastic
Fig. 7. Three-dimensional model of a periodicity cell with bent fiber.
Table 1 Research the effect of fiber straightness defect on the values of effective elastic moduli.
Fig. 5. 3D FE model of a composite material with fiber laying deviation from nominal direction ±25°, concentration 50%.
Fiber center displacement, mm
Ex , GPa
Ey , GPa
Ez , GPa
0 0.50 1.05 1.40 1.75 2.33 2.50
4.62 4.72 4.74 4.76 4.80 4.97 5.06
4.63 4.72 4.71 4.72 4.73 4.76 4.79
115.15 115.11 114.95 114.80 114.60 114.20 114.09
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
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Fig. 8. Three-dimensional model of a periodicity cell with bent fiber.
moduli increase. At maximum fiber curvature, the decrease of the modulus Ez is less than 1%. At the same time the increase of the modulus Ex is about 9.5%. The conclusion can be drawn that the fiber straightness defect does not affect the modulus ‘‘along fibers” significantly, but it increases elastic moduli in the perpendicular to fibers directions considerably. 5. Simulation of the composite behavior with progressive damage In the process of loading, the resin in the composite, due to its low strength characteristics, might be locally destroyed. Local destruction of the resin leads to the weakening of composite effective properties in this area. This local destruction affects the behavior of the whole composite structure, and particularly it leads to the gradual decrease in stiffness of the material and, therefore, to the nonlinearity of the ‘‘stress-strain” curve. Eventually, the resin destruction can lead to the destruction of the entire composite. The effect of progressive damage caused by the resin destruction was studied on the example of a three-point bending test. Two-scale simulations were performed using a FE model of the sample in the bending test and a FE model of composite periodicity cell. In the FE model of the sample subjected to the three-point bend test composite material is described by means using of effective characteristics. Effective characteristics are calculated using the model of periodicity cell that takes into account the structure of composite and incorporates the data on the characteristics of the resin and fibers. Tensile strength of the resin is assumed to be 45 MPa. In order to simulate the resin destruction, elements with a stress above 45 MPa are assigned to material with an elastic modulus of 1000 Pa (Fig. 9). To link the FE models on two scales (macro scale – the sample, micro scale – the cell of periodicity) in simulation of the three-
Fig. 9. (a) FE model of the composite cell of periodicity in the absence of the resin destruction; (b) FE model of the composite cell of periodicity with the local resin destruction. Purple color – resin; turquoise color – fiber; blue color – destroyed resin elements. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
point bending test, a submodeling procedure was used: threepoint bend test model is coarse model, model of composite periodicity cell is submodel. This approach to multi-scale simulation was previously used to simulate the mechanical behavior of threecomponent composite materials [22]. The simulation of the bending test was performed incrementally, with the load increased at each iteration. Using submodeling procedure, the load value is determined, starting from which the destruction of the resin occurs in the submodel according to the mentioned failure criteria. Starting from this load value, effective characteristics are recalculated at each load increment taking into account the destroyed resin elements in the submodel. In the specimen model elements with maximum tensile stresses are assigned to a new material with the effective characteristics calculated at each iteration. The procedure of submodeling, recalculation of effective characteristics, and changing material in the specimen model repeats at each iteration (which in the considered example corresponded to the load increment 0.5 N). Considering recalculation of effective characteristics by load increments, at the first step material with new effective characteristics №. 1 was obtained, at the second step material №. 2 was obtained, and similarly, at the n-th step, the n-th material was obtained. To simulate the progressive damage, material in the elements of specimen model was changed sequentially: on the first step material №. 1 was assigned to elements with maximum tensile stresses. On the second step material №. 2 was assigned to elements with material №. 1; and material №. 1 was assigned to elements with original material, where the maximum tensile stress was reached at this step. Similarly, on the n-th step, the n-th material was assigned to the elements with the (n-1)-th material. Subsequent replacement of material reflects different levels of resin destruction depending on the location. An example of such material replacement is shown in Fig. 10. With this procedure material in the most loaded places is sequentially updated, it becomes weaker with each new step and weakens the whole model, so affecting its mechanical behavior. A nonlinear behavior of composite is obtained by means of progressive damage simulation. In addition to obtaining a nonlinear force-deflection curve that is close the experimental one (Fig. 11), the method described allowed to obtain the strength of the composite that is very close to the experimental one (the highest point in the force-deflection curve).
Fig. 10. Specimen model (a quarter is analyzed due to the symmetry) in three-point bend test at the end of loading. Different colors represent different materials, which simulate different values of resin destruction.
Fig. 11. Comparison of simulation that takes into account progressive damage with experimental data.
Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134
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The graph in Fig. 11 shows that due to the resin destruction, the composite gradually weakens and ultimately breaks down. Obtained simulation results of composite behavior are in a good agreement with experimental data. Thus, the two-scale method described above can be used to simulate the nonlinear behavior and destruction of composite structures based on thermoplastic resins. 6. Conclusions A number of numerical simulations of the mechanical behaviour of a unidirectional fiber-reinforced composite material based on a thermoplastic resin was performed. A two-scale simulation approach based on homogenization method and submodeling technique was implemented. Investigation of possible imperfections in composite structure (fiber laying direction defect and fiber straightness defect) on the effective mechanical characteristics was conducted. A simulation of composite material behaviour taking into account progressive damage was carried out. The results of this work showed that: 1) Fiber laying direction defect leads to a substantial decrease in the ‘‘along fibers” elastic modulus and to some increase in the elastic moduli for perpendicular directions. However, the effect is significant only for large deviation angles. Based on the foregoing, it could be concluded that this defect should be considered for elastic moduli values correction if the deviation angle is noticeable in the samples of the materials. 2) Fiber straightness defect does not significantly affect the modulus ‘‘along fibers”, but it does increase the moduli in the perpendicular to fibers directions. The maximum increase in the elastic moduli values was observed for the case with the most curved fiber. The increase in elastic moduli for moderate values of curvature was found to be within 2–3%. Thus, this defect should be considered only if there are substantial prerequisites for that. 3) Simulation of the composite behaviour with progressive damage allowed to obtain the nonlinear behaviour of the material close to that obtained experimentally. It also allowed to predict the destruction of the specimen that was consistent with experimental data. That proves effectiveness of the implemented two-scale method for simulation of nonlinear behaviour of thermoplastic-based unidirectional composite structures, including prediction of their failure.
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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Please cite this article as: O. A. Panina, A. S. Nemov, A. Y. Zobacheva et al., Numerical analysis of mechanical behavior of unidirectional thermoplastic-based carbon fiber composite for 3D-printing, Materials Today: Proceedings, https://doi.org/10.1016/j.matpr.2020.01.134