# Mechanical Behavior of Helical Spring Made of Composite with and without Material Property Grading under Stable Load

Authors: Saeed Asiri
Journal: Journal of Nanomaterials
 (2022)

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## Abstract

Springs are one of the most popular means of mechanically storing and issuing energy, and they can be found in a wide variety of machines and products. Most springs are made of metal, and nowadays, there are many new alloys to choose from, but nonmetallic materials, such as the reinforced plastics and ceramics, have been appearing worldwide. The research problem is the performance for circular cross-sectional springs with the same geometry and manufactured with three different materials: steel, composite, and functionally graded material (FGM) under stable loading using finite element analysis. This paper intends to guide mechanical designers in considering different material options for a spring design as well as to provide a methodology through finite element analysis for selecting the most favorable material option for the application required. The findings of this research show that some feature performances of compression springs made of carbon steel are improved by using FGM and composite materials. Enhanced capabilities include higher load to failure: 1.48 times in an FGM spring and 1.1 times in a composite spring, as well as increased energy storage: 1.53 times in an FGM spring and 6.84 times in a composite spring and less weight representing only 61% in an FGM spring and 24% in a composite spring.


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## Body

## 1. Introduction

Springs are one of the most popular means of mechanically storing and issuing energy, and they can be found in a wide variety of machines and products. Most springs are made of metal, and nowadays, there are many new alloys to choose from, but nonmetallic materials, such as the reinforced plastics and the ceramics, have been appearing worldwide.The research aim focuses on contributing to the mechanical design of a spring considering the selection of its manufacturing material. In this sense, it will carry out an analysis of two springs made with two new types of materials used worldwide for manufacturing them, comparing the performance of a spring made of FGM with a spring made of composite materials, and these results will be contrasted with a traditional spring made of carbon steel. For completing the evaluation, a finite element analysis will be carried out using the commercial software ANSYS Workbench. Three computational cases will be performed with the same geometric model of a helical spring of circular cross section and subjected to compression. It will vary only on the material used in each computational case; for comparing the spring performance made of each of three materials analyzed in this study, FGM, composite material, and carbon steel were used.First, it will corroborate the computational model of carbon steel spring, comparing the finite element analysis (FEA) results with mathematical models of spring design from [1]. Then, it will validate the computational model of a carbon fiber spring (composite materials) considering the experimental results published in the international literature. These validations will allow corroborating the methodology used for the FEA of each of the cases assessed in this study and will ensure the reliability of the results issued.


## 2. Methodology

The following phases were considered for the numerical analysis of circular cross-sectional spring with steel, FGM, and composite materials.(i)
Validation analysis(ii)
Computational models


### 2.1. Validation Analysis

Some researchers [2–6] have demonstrated that the mechanical behavior of composite materials, as well as FGM, can be simulated satisfactorily. This research intends to simulate the mechanical behavior of three springs taking into account several experimental research results as well as international standards to demonstrate the reliability of the FEA results in the present investigation.According to [1, 7], the design calculation for steel spring in the case of compression load without initial tension refers to the following equations.


#### 2.1.1. Deflection of Springδ

(1)δ=8NaD3PGd4,whereNa is the number of active coils or number of active turns, D is the mean diameter of coil D=Di+Do/2 (mm), P is the load (force) acting on spring (N), G is the G modulus of rigidity (7.85×104N/mm2 for steel spring), and d is the diameter of the material (mm).


#### 2.1.2. Spring Constant (k)

(2)k=Pδ.


#### 2.1.3. Energy Stored in Spring (U)

(3)U=Pδ2.Equation (1) will be used to validate the deflection result of the numerical analysis in the case of a steel spring with a circular cross section.Equations (2) and (3) will be used to evaluate the spring performance in each of the three cases assessed in this study.


#### 2.1.4. Composite Spring

According to [8], develop an experimental investigation into the mechanical behaviors of helical composite springs made of preimpregnated fibers fabricated with fibers along with ±45° directions. Then, pack, coil, and outer braid one layer with resin-impregnated 3K carbon fiber. The present paper will focus only on the “BU” preformed composite bar structure shown by [8] (see Figure 1).Figure 1
3K carbon fiber outer braid (one layer): 37.5 cm of length.The experimental results of this research [8] considering a cylindrical helical spring with a square cross section will be used in this paper only for validation purpose to corroborate the mechanical behavior of the composite spring made of 3K carbon fiber outer braid with a “BU” preformed structure.Figure2 and Table 1 show the geometrical dimensions of the square cross section for the computational model used in the validation of composite spring.Figure 2
Geometrical dimensions of the square cross section for the validation of composite spring.Table 1
Geometrical dimensions of the square cross section for the validation of composite spring.
Lamina coiledSquare cross section (Figure2)Side length of square cross section (mm)Perimeter (mm)Carbon fiber lamina total length (cm)1a3.8415.3637.52b4.2817.123c4.7218.884d5.1620.645e5.6022.406f6.0424.167g6.4825.928h6.9227.689i7.3629.4410j7.8031.2011k8.2432.9612l8.6834.7213m9.1236.4814n9.5638.24Considering the manufacturing methodology of carbon fiber composite spring described by [8], Figure 2 shows the square cross-sectional compound for a core made of preimpregnated fibers fabricated with fibers along with ±45° directions with an outer braided one layer of 3K carbon fiber. This 3K carbon fiber lamina with a length of 37.5 cm is coiled 14 times, as shown in Table 1.The total side length of the square cross section is 10 mm, after coiling 14 times the 3K carbon fiber lamina according to the description in Table1 and considering a carbon fiber lamina thickness of 0.22 mm. Table 2 shows the properties of 3K carbon fiber that were used in the numerical simulation.Table 2
3K carbon fiber properties.
Tow sizekFilament propertiesMinimum carbon content (%)Tensile strength (GPa)Young’s modulus (GPa)Diameter (μ m)Elongation (%)Density (g cm-3)33.752317.01.41.7692Source: [9].Figure3(a) shows the experimental investigation spring model [8] in comparison with the geometrical model developed in the module ACP (pre) from ANSYS Workbench of the composite spring used for validation purposes (Figure 3(b)).Figure 3
Geometric comparison between the real coil spring and the computational model.
(a)
Experimental investigation spring model(b)
Numerical simulation spring modelConsidering the maximum compression load in the elastic range of the spring (250 kgf or 2452 N) obtained experimentally by [8]. Figure 4 shows the boundary conditions and mesh used in the numerical simulation. Spring mesh was refined until achieving the convergence. The final mesh sizing was 0.8 mm (0.031 in) using second-order solid elements.Figure 4
Boundary conditions and mesh of the computational model for validation purpose.
(a)
Boundary conditions(b)
MeshAnother aspect for validating is the failure load in the elastic range for the composite spring; hence, it will consider the results obtained experimentally by [8] related to the failure behavior of this coil spring.


### 2.2. Computational Models

After validating the methodology used for simulating, three geometric models were carried out using ANSYS Workbench considering the following general features described in Table3.Table 3
General features of computational models.
FeatureComputational model123MaterialCarbon steelC≤0.3%FGM (carbon steelC≤0.3%+Al2O3)Composite material (resinepoxy+3Kcarbonfiber)Shape of spring∗Helical coil, both end groundedHelical coil, both end groundedHelical coil, both flat end ungroundCross-sectional shapeCircularCircularCircularFree length (mm)909090Cross-sectional diameter (mm)101010Mean coil diameter (mm)404040Number of active coils3.63.63.6Type of load appliedCompressionCompressionCompressionType of finite element2nd order solid element2nd order solid element2nd order solid elementSize of element (mm)0.80.80.8∗According to [10].


#### 2.2.1. Computational Model 1

Mechanical and physical properties of the material (carbon steel withC≤0.3%) used for numerical simulation are shown in Table 4.Table 4
Mechanical and physical properties of carbon steel withC≤0.3%.
PropertyValueDensity (kg/m3)7850Young’s modulus (GPa)201Poisson’s ratio0.3Tensile yield strength (MPa)250Compressive yield strength (MPa)250Source: [11].Figure5 depicts the boundary conditions and mesh used in computational model 1. It highlighted a load compression force of 981 N (220.54 lbf), which is the maximum load in the elastic range.Figure 5
Boundary conditions and mesh of the computational model 1.
(a)
Boundary conditions(b)
Mesh


#### 2.2.2. Computational Model 2

Figure6 shows the difference in compositions and properties between an ordinary composite material and FGM. There is a distinct interface between metals and ceramics in an ordinary composite material, but not in an FGM. This difference corresponds to the distribution of properties. An ordinary composite material contains a sudden change in properties at the interface, while an FGM presents a gradual change inside it.Figure 6
Material structures and properties of ordinary composite and FGM. Source: [12] Handbook of Advanced Ceramics.For developing the numerical simulation of the FGM compound by carbon steel with less than 0.5% of C in addition with aluminum oxide (Al2O3), the research developed by [13] was considered, as well as [14, 15] to determine the material properties [14]. Describe that properties (P) are dependent on the temperature and are expressed in the form
(4)P=P0P−1T+1+P1T+P2T2+P3T3,whereP0, P1, P2, and P3 are constants in the cubic fit of the material property and T is the temperature in kelvin. For this paper, T is equivalent to 298 K.Besides, reference [14] states that, to model the material properties of FGM effectively, the properties must be both temperature-dependent and position-dependent. The combination of these functions gives rise to the effective material properties of FGMs and is expressed as
(5)PeffT,ξ=PmTVmξ+PcT1−Vmξ,where Peff is the effective material property of FGM, Pm and Pc are the temperature-dependent properties of the metal and ceramic, respectively, and Vm is the volume fraction of the metal constituent of the FGM. In addition, reference [14] mentions that a simple power-law exponent of the volume fraction distribution is used to provide a measure of the amount of metal in FGMs. For an axisymmetric cylinder, the expression for the volume fraction of ceramic is
(6)Vc=r−riro−ri,where ro is the outer radius of the cylinder, ri is the inner radius, r is the radial coordinate (ri≤r≤ro), and n is the power-law index (0≤n≤∞).According to the preceding distribution, the outer surface of the cylinder is ceramic-rich and the inner surface is metal-rich. Table5 shows the modulus of elasticity of aluminum oxide (Al2O3) in Pa, considering Equation (4). Table 6 shows the Poisson’s ratio of aluminum oxide (Al2O3), considering Equation (4).Table 5
Modulus of elasticity of aluminum oxide (Al2O3).
P0P−1P1P2P3349.55×1090−3.853×10−44.027×10−7−1.673×10−10Source: [14].Table 6
Poisson’s ratio of aluminum oxide (Al2O3).
P0P−1P1P2P30.26000000Source: [14].For this paper, it was considered a power-law index (n) equivalent to 1. In this case, the mechanical and physical properties of the FGM compound by carbon steel with less than 0.5% of C in addition with aluminum oxide (Al2O3) and considering Equations (4)–(6) are shown in Table 7.Table 7
Mechanical and physical properties of FGM.
PropertyValueDensity (kg/m3)78501−r/r0+3800r/r0Young’s modulus (Pa)2.01×10111−r/r0+3.204×1011r/r0Poisson’s ratio0.31−r/r0+0.26r/r0Tensile yield strength (Pa)2.5×1081−r/r0+4×108r/r0Compressive yield strength (Pa)2.5×1081−r/r0+4×108r/r0According to [16], develop an algorithm to simulate properties of FGM for the design optimization in a dental implant for bone remodeling. In this sense and to estimate the mechanical and physical properties, which are position-dependent, an algorithm was developed to simulate this computational model in the ANSYS Workbench.Figure7 depicts the boundary conditions and mesh used in computational model 2. It highlighted a load compression force of 1450 N (325.97 lbf), which is the maximum load in the elastic range.Figure 7
Boundary conditions and mesh of computational model 2.
(a)
Boundary conditions(b)
Mesh


#### 2.2.3. Computational Model 3

Considering the same methodology explained in Section2.1 about the validation of composite spring simulation, Figure 8 shows the geometrical dimensions of the circular cross section for computational model 3.Figure 8
Geometrical dimensions of the circular cross section for computational model 3.Table8 shows the geometrical dimensions of the circular cross section for computational model 3.Table 8
Geometrical dimensions of the circular cross section for computational model 3.
Lamina coiledCircular cross section (Figure8)Circular cross-sectional diameter (mm)Total length of carbon fiber lamina (cm)1a3.8429.52b4.283c4.724d5.165e5.606f6.047g6.488h6.929i7.3610j7.8011k8.2412l8.6813m9.1214n9.56The final diameter of the circular cross section is 10 mm, after coil 14 times the 3K carbon fiber lamina according to the description in Table8 and considering a carbon fiber lamina thickness of 0.22 mm. Table 2 shows the properties of 3K carbon fiber that will be used in the numerical simulation.Figure9 depicts the boundary conditions and mesh used in computational model 3. It highlighted a load compression force of 1070 N (240.55 lbf), which is the maximum load in the elastic range.Figure 9
Boundary conditions and mesh of computational model 3.
(a)
Boundary conditions(b)
Mesh


## 2.1. Validation Analysis

Some researchers [2–6] have demonstrated that the mechanical behavior of composite materials, as well as FGM, can be simulated satisfactorily. This research intends to simulate the mechanical behavior of three springs taking into account several experimental research results as well as international standards to demonstrate the reliability of the FEA results in the present investigation.According to [1, 7], the design calculation for steel spring in the case of compression load without initial tension refers to the following equations.


### 2.1.1. Deflection of Springδ

(1)δ=8NaD3PGd4,whereNa is the number of active coils or number of active turns, D is the mean diameter of coil D=Di+Do/2 (mm), P is the load (force) acting on spring (N), G is the G modulus of rigidity (7.85×104N/mm2 for steel spring), and d is the diameter of the material (mm).


### 2.1.2. Spring Constant (k)

(2)k=Pδ.


### 2.1.3. Energy Stored in Spring (U)

(3)U=Pδ2.Equation (1) will be used to validate the deflection result of the numerical analysis in the case of a steel spring with a circular cross section.Equations (2) and (3) will be used to evaluate the spring performance in each of the three cases assessed in this study.


### 2.1.4. Composite Spring

According to [8], develop an experimental investigation into the mechanical behaviors of helical composite springs made of preimpregnated fibers fabricated with fibers along with ±45° directions. Then, pack, coil, and outer braid one layer with resin-impregnated 3K carbon fiber. The present paper will focus only on the “BU” preformed composite bar structure shown by [8] (see Figure 1).Figure 1
3K carbon fiber outer braid (one layer): 37.5 cm of length.The experimental results of this research [8] considering a cylindrical helical spring with a square cross section will be used in this paper only for validation purpose to corroborate the mechanical behavior of the composite spring made of 3K carbon fiber outer braid with a “BU” preformed structure.Figure2 and Table 1 show the geometrical dimensions of the square cross section for the computational model used in the validation of composite spring.Figure 2
Geometrical dimensions of the square cross section for the validation of composite spring.Table 1
Geometrical dimensions of the square cross section for the validation of composite spring.
Lamina coiledSquare cross section (Figure2)Side length of square cross section (mm)Perimeter (mm)Carbon fiber lamina total length (cm)1a3.8415.3637.52b4.2817.123c4.7218.884d5.1620.645e5.6022.406f6.0424.167g6.4825.928h6.9227.689i7.3629.4410j7.8031.2011k8.2432.9612l8.6834.7213m9.1236.4814n9.5638.24Considering the manufacturing methodology of carbon fiber composite spring described by [8], Figure 2 shows the square cross-sectional compound for a core made of preimpregnated fibers fabricated with fibers along with ±45° directions with an outer braided one layer of 3K carbon fiber. This 3K carbon fiber lamina with a length of 37.5 cm is coiled 14 times, as shown in Table 1.The total side length of the square cross section is 10 mm, after coiling 14 times the 3K carbon fiber lamina according to the description in Table1 and considering a carbon fiber lamina thickness of 0.22 mm. Table 2 shows the properties of 3K carbon fiber that were used in the numerical simulation.Table 2
3K carbon fiber properties.
Tow sizekFilament propertiesMinimum carbon content (%)Tensile strength (GPa)Young’s modulus (GPa)Diameter (μ m)Elongation (%)Density (g cm-3)33.752317.01.41.7692Source: [9].Figure3(a) shows the experimental investigation spring model [8] in comparison with the geometrical model developed in the module ACP (pre) from ANSYS Workbench of the composite spring used for validation purposes (Figure 3(b)).Figure 3
Geometric comparison between the real coil spring and the computational model.
(a)
Experimental investigation spring model(b)
Numerical simulation spring modelConsidering the maximum compression load in the elastic range of the spring (250 kgf or 2452 N) obtained experimentally by [8]. Figure 4 shows the boundary conditions and mesh used in the numerical simulation. Spring mesh was refined until achieving the convergence. The final mesh sizing was 0.8 mm (0.031 in) using second-order solid elements.Figure 4
Boundary conditions and mesh of the computational model for validation purpose.
(a)
Boundary conditions(b)
MeshAnother aspect for validating is the failure load in the elastic range for the composite spring; hence, it will consider the results obtained experimentally by [8] related to the failure behavior of this coil spring.


## 2.1.1. Deflection of Springδ

(1)δ=8NaD3PGd4,whereNa is the number of active coils or number of active turns, D is the mean diameter of coil D=Di+Do/2 (mm), P is the load (force) acting on spring (N), G is the G modulus of rigidity (7.85×104N/mm2 for steel spring), and d is the diameter of the material (mm).


## 2.1.2. Spring Constant (k)

(2)k=Pδ.


## 2.1.3. Energy Stored in Spring (U)

(3)U=Pδ2.Equation (1) will be used to validate the deflection result of the numerical analysis in the case of a steel spring with a circular cross section.Equations (2) and (3) will be used to evaluate the spring performance in each of the three cases assessed in this study.


## 2.1.4. Composite Spring

According to [8], develop an experimental investigation into the mechanical behaviors of helical composite springs made of preimpregnated fibers fabricated with fibers along with ±45° directions. Then, pack, coil, and outer braid one layer with resin-impregnated 3K carbon fiber. The present paper will focus only on the “BU” preformed composite bar structure shown by [8] (see Figure 1).Figure 1
3K carbon fiber outer braid (one layer): 37.5 cm of length.The experimental results of this research [8] considering a cylindrical helical spring with a square cross section will be used in this paper only for validation purpose to corroborate the mechanical behavior of the composite spring made of 3K carbon fiber outer braid with a “BU” preformed structure.Figure2 and Table 1 show the geometrical dimensions of the square cross section for the computational model used in the validation of composite spring.Figure 2
Geometrical dimensions of the square cross section for the validation of composite spring.Table 1
Geometrical dimensions of the square cross section for the validation of composite spring.
Lamina coiledSquare cross section (Figure2)Side length of square cross section (mm)Perimeter (mm)Carbon fiber lamina total length (cm)1a3.8415.3637.52b4.2817.123c4.7218.884d5.1620.645e5.6022.406f6.0424.167g6.4825.928h6.9227.689i7.3629.4410j7.8031.2011k8.2432.9612l8.6834.7213m9.1236.4814n9.5638.24Considering the manufacturing methodology of carbon fiber composite spring described by [8], Figure 2 shows the square cross-sectional compound for a core made of preimpregnated fibers fabricated with fibers along with ±45° directions with an outer braided one layer of 3K carbon fiber. This 3K carbon fiber lamina with a length of 37.5 cm is coiled 14 times, as shown in Table 1.The total side length of the square cross section is 10 mm, after coiling 14 times the 3K carbon fiber lamina according to the description in Table1 and considering a carbon fiber lamina thickness of 0.22 mm. Table 2 shows the properties of 3K carbon fiber that were used in the numerical simulation.Table 2
3K carbon fiber properties.
Tow sizekFilament propertiesMinimum carbon content (%)Tensile strength (GPa)Young’s modulus (GPa)Diameter (μ m)Elongation (%)Density (g cm-3)33.752317.01.41.7692Source: [9].Figure3(a) shows the experimental investigation spring model [8] in comparison with the geometrical model developed in the module ACP (pre) from ANSYS Workbench of the composite spring used for validation purposes (Figure 3(b)).Figure 3
Geometric comparison between the real coil spring and the computational model.
(a)
Experimental investigation spring model(b)
Numerical simulation spring modelConsidering the maximum compression load in the elastic range of the spring (250 kgf or 2452 N) obtained experimentally by [8]. Figure 4 shows the boundary conditions and mesh used in the numerical simulation. Spring mesh was refined until achieving the convergence. The final mesh sizing was 0.8 mm (0.031 in) using second-order solid elements.Figure 4
Boundary conditions and mesh of the computational model for validation purpose.
(a)
Boundary conditions(b)
MeshAnother aspect for validating is the failure load in the elastic range for the composite spring; hence, it will consider the results obtained experimentally by [8] related to the failure behavior of this coil spring.


## 2.2. Computational Models

After validating the methodology used for simulating, three geometric models were carried out using ANSYS Workbench considering the following general features described in Table3.Table 3
General features of computational models.
FeatureComputational model123MaterialCarbon steelC≤0.3%FGM (carbon steelC≤0.3%+Al2O3)Composite material (resinepoxy+3Kcarbonfiber)Shape of spring∗Helical coil, both end groundedHelical coil, both end groundedHelical coil, both flat end ungroundCross-sectional shapeCircularCircularCircularFree length (mm)909090Cross-sectional diameter (mm)101010Mean coil diameter (mm)404040Number of active coils3.63.63.6Type of load appliedCompressionCompressionCompressionType of finite element2nd order solid element2nd order solid element2nd order solid elementSize of element (mm)0.80.80.8∗According to [10].


### 2.2.1. Computational Model 1

Mechanical and physical properties of the material (carbon steel withC≤0.3%) used for numerical simulation are shown in Table 4.Table 4
Mechanical and physical properties of carbon steel withC≤0.3%.
PropertyValueDensity (kg/m3)7850Young’s modulus (GPa)201Poisson’s ratio0.3Tensile yield strength (MPa)250Compressive yield strength (MPa)250Source: [11].Figure5 depicts the boundary conditions and mesh used in computational model 1. It highlighted a load compression force of 981 N (220.54 lbf), which is the maximum load in the elastic range.Figure 5
Boundary conditions and mesh of the computational model 1.
(a)
Boundary conditions(b)
Mesh


### 2.2.2. Computational Model 2

Figure6 shows the difference in compositions and properties between an ordinary composite material and FGM. There is a distinct interface between metals and ceramics in an ordinary composite material, but not in an FGM. This difference corresponds to the distribution of properties. An ordinary composite material contains a sudden change in properties at the interface, while an FGM presents a gradual change inside it.Figure 6
Material structures and properties of ordinary composite and FGM. Source: [12] Handbook of Advanced Ceramics.For developing the numerical simulation of the FGM compound by carbon steel with less than 0.5% of C in addition with aluminum oxide (Al2O3), the research developed by [13] was considered, as well as [14, 15] to determine the material properties [14]. Describe that properties (P) are dependent on the temperature and are expressed in the form
(4)P=P0P−1T+1+P1T+P2T2+P3T3,whereP0, P1, P2, and P3 are constants in the cubic fit of the material property and T is the temperature in kelvin. For this paper, T is equivalent to 298 K.Besides, reference [14] states that, to model the material properties of FGM effectively, the properties must be both temperature-dependent and position-dependent. The combination of these functions gives rise to the effective material properties of FGMs and is expressed as
(5)PeffT,ξ=PmTVmξ+PcT1−Vmξ,where Peff is the effective material property of FGM, Pm and Pc are the temperature-dependent properties of the metal and ceramic, respectively, and Vm is the volume fraction of the metal constituent of the FGM. In addition, reference [14] mentions that a simple power-law exponent of the volume fraction distribution is used to provide a measure of the amount of metal in FGMs. For an axisymmetric cylinder, the expression for the volume fraction of ceramic is
(6)Vc=r−riro−ri,where ro is the outer radius of the cylinder, ri is the inner radius, r is the radial coordinate (ri≤r≤ro), and n is the power-law index (0≤n≤∞).According to the preceding distribution, the outer surface of the cylinder is ceramic-rich and the inner surface is metal-rich. Table5 shows the modulus of elasticity of aluminum oxide (Al2O3) in Pa, considering Equation (4). Table 6 shows the Poisson’s ratio of aluminum oxide (Al2O3), considering Equation (4).Table 5
Modulus of elasticity of aluminum oxide (Al2O3).
P0P−1P1P2P3349.55×1090−3.853×10−44.027×10−7−1.673×10−10Source: [14].Table 6
Poisson’s ratio of aluminum oxide (Al2O3).
P0P−1P1P2P30.26000000Source: [14].For this paper, it was considered a power-law index (n) equivalent to 1. In this case, the mechanical and physical properties of the FGM compound by carbon steel with less than 0.5% of C in addition with aluminum oxide (Al2O3) and considering Equations (4)–(6) are shown in Table 7.Table 7
Mechanical and physical properties of FGM.
PropertyValueDensity (kg/m3)78501−r/r0+3800r/r0Young’s modulus (Pa)2.01×10111−r/r0+3.204×1011r/r0Poisson’s ratio0.31−r/r0+0.26r/r0Tensile yield strength (Pa)2.5×1081−r/r0+4×108r/r0Compressive yield strength (Pa)2.5×1081−r/r0+4×108r/r0According to [16], develop an algorithm to simulate properties of FGM for the design optimization in a dental implant for bone remodeling. In this sense and to estimate the mechanical and physical properties, which are position-dependent, an algorithm was developed to simulate this computational model in the ANSYS Workbench.Figure7 depicts the boundary conditions and mesh used in computational model 2. It highlighted a load compression force of 1450 N (325.97 lbf), which is the maximum load in the elastic range.Figure 7
Boundary conditions and mesh of computational model 2.
(a)
Boundary conditions(b)
Mesh


### 2.2.3. Computational Model 3

Considering the same methodology explained in Section2.1 about the validation of composite spring simulation, Figure 8 shows the geometrical dimensions of the circular cross section for computational model 3.Figure 8
Geometrical dimensions of the circular cross section for computational model 3.Table8 shows the geometrical dimensions of the circular cross section for computational model 3.Table 8
Geometrical dimensions of the circular cross section for computational model 3.
Lamina coiledCircular cross section (Figure8)Circular cross-sectional diameter (mm)Total length of carbon fiber lamina (cm)1a3.8429.52b4.283c4.724d5.165e5.606f6.047g6.488h6.929i7.3610j7.8011k8.2412l8.6813m9.1214n9.56The final diameter of the circular cross section is 10 mm, after coil 14 times the 3K carbon fiber lamina according to the description in Table8 and considering a carbon fiber lamina thickness of 0.22 mm. Table 2 shows the properties of 3K carbon fiber that will be used in the numerical simulation.Figure9 depicts the boundary conditions and mesh used in computational model 3. It highlighted a load compression force of 1070 N (240.55 lbf), which is the maximum load in the elastic range.Figure 9
Boundary conditions and mesh of computational model 3.
(a)
Boundary conditions(b)
Mesh


## 2.2.1. Computational Model 1

Mechanical and physical properties of the material (carbon steel withC≤0.3%) used for numerical simulation are shown in Table 4.Table 4
Mechanical and physical properties of carbon steel withC≤0.3%.
PropertyValueDensity (kg/m3)7850Young’s modulus (GPa)201Poisson’s ratio0.3Tensile yield strength (MPa)250Compressive yield strength (MPa)250Source: [11].Figure5 depicts the boundary conditions and mesh used in computational model 1. It highlighted a load compression force of 981 N (220.54 lbf), which is the maximum load in the elastic range.Figure 5
Boundary conditions and mesh of the computational model 1.
(a)
Boundary conditions(b)
Mesh


## 2.2.2. Computational Model 2

Figure6 shows the difference in compositions and properties between an ordinary composite material and FGM. There is a distinct interface between metals and ceramics in an ordinary composite material, but not in an FGM. This difference corresponds to the distribution of properties. An ordinary composite material contains a sudden change in properties at the interface, while an FGM presents a gradual change inside it.Figure 6
Material structures and properties of ordinary composite and FGM. Source: [12] Handbook of Advanced Ceramics.For developing the numerical simulation of the FGM compound by carbon steel with less than 0.5% of C in addition with aluminum oxide (Al2O3), the research developed by [13] was considered, as well as [14, 15] to determine the material properties [14]. Describe that properties (P) are dependent on the temperature and are expressed in the form
(4)P=P0P−1T+1+P1T+P2T2+P3T3,whereP0, P1, P2, and P3 are constants in the cubic fit of the material property and T is the temperature in kelvin. For this paper, T is equivalent to 298 K.Besides, reference [14] states that, to model the material properties of FGM effectively, the properties must be both temperature-dependent and position-dependent. The combination of these functions gives rise to the effective material properties of FGMs and is expressed as
(5)PeffT,ξ=PmTVmξ+PcT1−Vmξ,where Peff is the effective material property of FGM, Pm and Pc are the temperature-dependent properties of the metal and ceramic, respectively, and Vm is the volume fraction of the metal constituent of the FGM. In addition, reference [14] mentions that a simple power-law exponent of the volume fraction distribution is used to provide a measure of the amount of metal in FGMs. For an axisymmetric cylinder, the expression for the volume fraction of ceramic is
(6)Vc=r−riro−ri,where ro is the outer radius of the cylinder, ri is the inner radius, r is the radial coordinate (ri≤r≤ro), and n is the power-law index (0≤n≤∞).According to the preceding distribution, the outer surface of the cylinder is ceramic-rich and the inner surface is metal-rich. Table5 shows the modulus of elasticity of aluminum oxide (Al2O3) in Pa, considering Equation (4). Table 6 shows the Poisson’s ratio of aluminum oxide (Al2O3), considering Equation (4).Table 5
Modulus of elasticity of aluminum oxide (Al2O3).
P0P−1P1P2P3349.55×1090−3.853×10−44.027×10−7−1.673×10−10Source: [14].Table 6
Poisson’s ratio of aluminum oxide (Al2O3).
P0P−1P1P2P30.26000000Source: [14].For this paper, it was considered a power-law index (n) equivalent to 1. In this case, the mechanical and physical properties of the FGM compound by carbon steel with less than 0.5% of C in addition with aluminum oxide (Al2O3) and considering Equations (4)–(6) are shown in Table 7.Table 7
Mechanical and physical properties of FGM.
PropertyValueDensity (kg/m3)78501−r/r0+3800r/r0Young’s modulus (Pa)2.01×10111−r/r0+3.204×1011r/r0Poisson’s ratio0.31−r/r0+0.26r/r0Tensile yield strength (Pa)2.5×1081−r/r0+4×108r/r0Compressive yield strength (Pa)2.5×1081−r/r0+4×108r/r0According to [16], develop an algorithm to simulate properties of FGM for the design optimization in a dental implant for bone remodeling. In this sense and to estimate the mechanical and physical properties, which are position-dependent, an algorithm was developed to simulate this computational model in the ANSYS Workbench.Figure7 depicts the boundary conditions and mesh used in computational model 2. It highlighted a load compression force of 1450 N (325.97 lbf), which is the maximum load in the elastic range.Figure 7
Boundary conditions and mesh of computational model 2.
(a)
Boundary conditions(b)
Mesh


## 2.2.3. Computational Model 3

Considering the same methodology explained in Section2.1 about the validation of composite spring simulation, Figure 8 shows the geometrical dimensions of the circular cross section for computational model 3.Figure 8
Geometrical dimensions of the circular cross section for computational model 3.Table8 shows the geometrical dimensions of the circular cross section for computational model 3.Table 8
Geometrical dimensions of the circular cross section for computational model 3.
Lamina coiledCircular cross section (Figure8)Circular cross-sectional diameter (mm)Total length of carbon fiber lamina (cm)1a3.8429.52b4.283c4.724d5.165e5.606f6.047g6.488h6.929i7.3610j7.8011k8.2412l8.6813m9.1214n9.56The final diameter of the circular cross section is 10 mm, after coil 14 times the 3K carbon fiber lamina according to the description in Table8 and considering a carbon fiber lamina thickness of 0.22 mm. Table 2 shows the properties of 3K carbon fiber that will be used in the numerical simulation.Figure9 depicts the boundary conditions and mesh used in computational model 3. It highlighted a load compression force of 1070 N (240.55 lbf), which is the maximum load in the elastic range.Figure 9
Boundary conditions and mesh of computational model 3.
(a)
Boundary conditions(b)
Mesh


## 3. Results and Discussion

### 3.1. Validation Analysis

Figure10 shows the maximum displacement in the elastic range for the carbon steel spring considering a compression load of 981 N (220.54 lbf).Figure 10
Steel spring maximum displacement in the elastic range.
(b)
Final condition(a)
Initial conditionsEquation (1) represents the deflection of the spring according to international standard [1]. Considering Equation (1), the theoretical maximum deflection in the elastic range for the carbon steel spring is 2.3 mm, and the FEA result in Figure 10 shows a maximum displacement equivalent to 2.4 mm; in consequence, the relative error is 4.19%. Hence, results about displacement demonstrate that numerical simulation of the steel spring can represent very well the spring theoretical behavior. Figure 11 shows the maximum displacement in the elastic range for the composite spring with a square cross section.Figure 11
Composite spring maximum displacement in the elastic range.
(a)
Initial conditions(b)
Final conditionConsidering the data issued by [8], the maximum deflection for the composite spring during the experiment in the elastic range is 16.23 mm, and the FEA result in Figure 11 shows a maximum displacement equivalent to 17.12 mm; hence, the relative error is 5.48%.When the experimental behavior of failure (Figure12(b)) is compared with results obtained by FEA (Figure 12(a)), it can be noticed that there is a correspondence in the failure location.Figure 12
Comparison of failure between FEA simulation and experimental behavior of composite spring.
(a)
Numerical simulation of failure(b)
Experimental behavior of failureThe maximum stress ratio (1.2) in FEA results matches the experimental location of a crack at the opening end of the coil spring. With this compressive load (1452 N), the spring in the numerical simulation exhibits an orange color in most coils of spring that represent a stress ratio near to one, being the value of 1 the limit of the stress ratio for the failure in the elastic range. Stress ratio values equal to or more than one represent a failure by yield strength.In consequence, results about not only displacement but also failure behavior demonstrate that numerical simulation of the composite spring can represent very well the composite spring behavior during experimental investigation.


### 3.2. Computational Models

#### 3.2.1. Computational Model 1

Figure10 shows the steel spring maximum displacement in the elastic range, and this displacement was validated with Equations (1) and (2) that represents the theoretical behavior.Figure13 exhibits the stress ratio obtained in the numerical simulation, considering that stress ratio equivalent or more than 1 represents the failure in the elastic range of the material.Figure 13
Failure at maximum displacement in the elastic range for computational model 1.In this case, Figure13 shows a maximum stress ratio equivalent to 1.025, which means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (981 N) is considered as the maximum load in the elastic range.


#### 3.2.2. Computational Model 2

Figure14 shows the FGM spring maximum displacement in the elastic range. It can be noticed that a maximum displacement is equivalent to 2.48 mm considering a compressive load of 1450 N (325.97 lbf).Figure 14
Maximum displacement in the elastic range for computational model 2.
(a)
Initial conditions(b)
Final conditionFigure15 shows the stress ratio obtained in the numerical simulation, with a maximum stress ratio equivalent to 1.03; it means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (1450 N) is considered as the maximum load in the elastic range.Figure 15
Failure at maximum displacement in the elastic range for computational model 2.


#### 3.2.3. Computational Model 3

Figure16 exhibits the composite spring maximum displacement in the elastic range. It can be noticed that a maximum displacement is equivalent to 15.07 mm (0.59 in) considering a compressive load of 1070 N (240.55 lbf).Figure 16
Maximum displacement in the elastic range for computational model 3.
(a)
Initial conditions(b)
Final conditionFigure17 shows the stress ratio obtained in the numerical simulation, obtaining a maximum stress ratio equivalent to 1.05; it means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (1070 N) is considered as the maximum load in the elastic range.Figure 17
Failure at maximum displacement in the elastic range for computational model 3.Table9 shows the performance comparison between these three computational models, which will help to better understand how new development in materials such as FGM and composite can enhance the capability and potential use of springs.Table 9
Comparison of spring performance.
Computational model1 (plain steel)2 (FGM)3 (composite)Failure load (N)98114501070Deflectionδ (mm)2.402.4815.07Spring constantk (N/mm)408.75584.6871.00Weight of spring (g)307.0187.774.6Energy stored in springU (J)1177.201798.008052.45Specific energy stored in spring (J/g)3.839.58108.07As shown in Table9, springs made of FGM and composite outperform springs made of carbon steel in some characteristics. Regarding load-to-failure performance, an FGM spring outperforms a carbon steel spring 1.48 times, while a carbon fiber spring outperforms its carbon steel counterpart by 1.1 times.Regarding weight and for industrial applications that require it, such as in the manufacturing of vehicles and automobiles [17, 18], this performance characteristic also shows its advantages for FGM and composite springs. As shown in Table 9, the weight of an FGM spring represents only 61% of the mass of a carbon steel spring, while its similar one made of composite has a weight equivalent to 24% of the weight carbon steel spring [19–21].Another favorable feature is about the energy storage in the coil spring, where again both springs, FGM and composite, outperform their carbon steel counterpart by 1.53 times and 6.84 times, respectively. In the same sense, the characteristic about specific energy stored in the coil spring that relates the energy stored in the spring with the weight of the spring demonstrates that a composite spring can store more energy per mass unity, 11.3 times more than an FGM spring and 28.2 times more than a steel spring.On the other hand, for applications that require high stiffness or higher spring constant, the best option is an FGM spring, followed by a carbon steel spring, and at last a composite material spring. This characteristic intrinsically linked to the deflection of the spring is modifiable in a composite material spring by selecting laminates of carbon fiber with mechanical properties superior to those used in the present work [22, 23].


## 3.1. Validation Analysis

Figure10 shows the maximum displacement in the elastic range for the carbon steel spring considering a compression load of 981 N (220.54 lbf).Figure 10
Steel spring maximum displacement in the elastic range.
(b)
Final condition(a)
Initial conditionsEquation (1) represents the deflection of the spring according to international standard [1]. Considering Equation (1), the theoretical maximum deflection in the elastic range for the carbon steel spring is 2.3 mm, and the FEA result in Figure 10 shows a maximum displacement equivalent to 2.4 mm; in consequence, the relative error is 4.19%. Hence, results about displacement demonstrate that numerical simulation of the steel spring can represent very well the spring theoretical behavior. Figure 11 shows the maximum displacement in the elastic range for the composite spring with a square cross section.Figure 11
Composite spring maximum displacement in the elastic range.
(a)
Initial conditions(b)
Final conditionConsidering the data issued by [8], the maximum deflection for the composite spring during the experiment in the elastic range is 16.23 mm, and the FEA result in Figure 11 shows a maximum displacement equivalent to 17.12 mm; hence, the relative error is 5.48%.When the experimental behavior of failure (Figure12(b)) is compared with results obtained by FEA (Figure 12(a)), it can be noticed that there is a correspondence in the failure location.Figure 12
Comparison of failure between FEA simulation and experimental behavior of composite spring.
(a)
Numerical simulation of failure(b)
Experimental behavior of failureThe maximum stress ratio (1.2) in FEA results matches the experimental location of a crack at the opening end of the coil spring. With this compressive load (1452 N), the spring in the numerical simulation exhibits an orange color in most coils of spring that represent a stress ratio near to one, being the value of 1 the limit of the stress ratio for the failure in the elastic range. Stress ratio values equal to or more than one represent a failure by yield strength.In consequence, results about not only displacement but also failure behavior demonstrate that numerical simulation of the composite spring can represent very well the composite spring behavior during experimental investigation.


## 3.2. Computational Models

### 3.2.1. Computational Model 1

Figure10 shows the steel spring maximum displacement in the elastic range, and this displacement was validated with Equations (1) and (2) that represents the theoretical behavior.Figure13 exhibits the stress ratio obtained in the numerical simulation, considering that stress ratio equivalent or more than 1 represents the failure in the elastic range of the material.Figure 13
Failure at maximum displacement in the elastic range for computational model 1.In this case, Figure13 shows a maximum stress ratio equivalent to 1.025, which means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (981 N) is considered as the maximum load in the elastic range.


### 3.2.2. Computational Model 2

Figure14 shows the FGM spring maximum displacement in the elastic range. It can be noticed that a maximum displacement is equivalent to 2.48 mm considering a compressive load of 1450 N (325.97 lbf).Figure 14
Maximum displacement in the elastic range for computational model 2.
(a)
Initial conditions(b)
Final conditionFigure15 shows the stress ratio obtained in the numerical simulation, with a maximum stress ratio equivalent to 1.03; it means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (1450 N) is considered as the maximum load in the elastic range.Figure 15
Failure at maximum displacement in the elastic range for computational model 2.


### 3.2.3. Computational Model 3

Figure16 exhibits the composite spring maximum displacement in the elastic range. It can be noticed that a maximum displacement is equivalent to 15.07 mm (0.59 in) considering a compressive load of 1070 N (240.55 lbf).Figure 16
Maximum displacement in the elastic range for computational model 3.
(a)
Initial conditions(b)
Final conditionFigure17 shows the stress ratio obtained in the numerical simulation, obtaining a maximum stress ratio equivalent to 1.05; it means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (1070 N) is considered as the maximum load in the elastic range.Figure 17
Failure at maximum displacement in the elastic range for computational model 3.Table9 shows the performance comparison between these three computational models, which will help to better understand how new development in materials such as FGM and composite can enhance the capability and potential use of springs.Table 9
Comparison of spring performance.
Computational model1 (plain steel)2 (FGM)3 (composite)Failure load (N)98114501070Deflectionδ (mm)2.402.4815.07Spring constantk (N/mm)408.75584.6871.00Weight of spring (g)307.0187.774.6Energy stored in springU (J)1177.201798.008052.45Specific energy stored in spring (J/g)3.839.58108.07As shown in Table9, springs made of FGM and composite outperform springs made of carbon steel in some characteristics. Regarding load-to-failure performance, an FGM spring outperforms a carbon steel spring 1.48 times, while a carbon fiber spring outperforms its carbon steel counterpart by 1.1 times.Regarding weight and for industrial applications that require it, such as in the manufacturing of vehicles and automobiles [17, 18], this performance characteristic also shows its advantages for FGM and composite springs. As shown in Table 9, the weight of an FGM spring represents only 61% of the mass of a carbon steel spring, while its similar one made of composite has a weight equivalent to 24% of the weight carbon steel spring [19–21].Another favorable feature is about the energy storage in the coil spring, where again both springs, FGM and composite, outperform their carbon steel counterpart by 1.53 times and 6.84 times, respectively. In the same sense, the characteristic about specific energy stored in the coil spring that relates the energy stored in the spring with the weight of the spring demonstrates that a composite spring can store more energy per mass unity, 11.3 times more than an FGM spring and 28.2 times more than a steel spring.On the other hand, for applications that require high stiffness or higher spring constant, the best option is an FGM spring, followed by a carbon steel spring, and at last a composite material spring. This characteristic intrinsically linked to the deflection of the spring is modifiable in a composite material spring by selecting laminates of carbon fiber with mechanical properties superior to those used in the present work [22, 23].


## 3.2.1. Computational Model 1

Figure10 shows the steel spring maximum displacement in the elastic range, and this displacement was validated with Equations (1) and (2) that represents the theoretical behavior.Figure13 exhibits the stress ratio obtained in the numerical simulation, considering that stress ratio equivalent or more than 1 represents the failure in the elastic range of the material.Figure 13
Failure at maximum displacement in the elastic range for computational model 1.In this case, Figure13 shows a maximum stress ratio equivalent to 1.025, which means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (981 N) is considered as the maximum load in the elastic range.


## 3.2.2. Computational Model 2

Figure14 shows the FGM spring maximum displacement in the elastic range. It can be noticed that a maximum displacement is equivalent to 2.48 mm considering a compressive load of 1450 N (325.97 lbf).Figure 14
Maximum displacement in the elastic range for computational model 2.
(a)
Initial conditions(b)
Final conditionFigure15 shows the stress ratio obtained in the numerical simulation, with a maximum stress ratio equivalent to 1.03; it means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (1450 N) is considered as the maximum load in the elastic range.Figure 15
Failure at maximum displacement in the elastic range for computational model 2.


## 3.2.3. Computational Model 3

Figure16 exhibits the composite spring maximum displacement in the elastic range. It can be noticed that a maximum displacement is equivalent to 15.07 mm (0.59 in) considering a compressive load of 1070 N (240.55 lbf).Figure 16
Maximum displacement in the elastic range for computational model 3.
(a)
Initial conditions(b)
Final conditionFigure17 shows the stress ratio obtained in the numerical simulation, obtaining a maximum stress ratio equivalent to 1.05; it means that the material reaches the yield strength; hence, the compression load used in this numerical simulation (1070 N) is considered as the maximum load in the elastic range.Figure 17
Failure at maximum displacement in the elastic range for computational model 3.Table9 shows the performance comparison between these three computational models, which will help to better understand how new development in materials such as FGM and composite can enhance the capability and potential use of springs.Table 9
Comparison of spring performance.
Computational model1 (plain steel)2 (FGM)3 (composite)Failure load (N)98114501070Deflectionδ (mm)2.402.4815.07Spring constantk (N/mm)408.75584.6871.00Weight of spring (g)307.0187.774.6Energy stored in springU (J)1177.201798.008052.45Specific energy stored in spring (J/g)3.839.58108.07As shown in Table9, springs made of FGM and composite outperform springs made of carbon steel in some characteristics. Regarding load-to-failure performance, an FGM spring outperforms a carbon steel spring 1.48 times, while a carbon fiber spring outperforms its carbon steel counterpart by 1.1 times.Regarding weight and for industrial applications that require it, such as in the manufacturing of vehicles and automobiles [17, 18], this performance characteristic also shows its advantages for FGM and composite springs. As shown in Table 9, the weight of an FGM spring represents only 61% of the mass of a carbon steel spring, while its similar one made of composite has a weight equivalent to 24% of the weight carbon steel spring [19–21].Another favorable feature is about the energy storage in the coil spring, where again both springs, FGM and composite, outperform their carbon steel counterpart by 1.53 times and 6.84 times, respectively. In the same sense, the characteristic about specific energy stored in the coil spring that relates the energy stored in the spring with the weight of the spring demonstrates that a composite spring can store more energy per mass unity, 11.3 times more than an FGM spring and 28.2 times more than a steel spring.On the other hand, for applications that require high stiffness or higher spring constant, the best option is an FGM spring, followed by a carbon steel spring, and at last a composite material spring. This characteristic intrinsically linked to the deflection of the spring is modifiable in a composite material spring by selecting laminates of carbon fiber with mechanical properties superior to those used in the present work [22, 23].


## 4. Conclusions

Three computational models were carried out to compare the spring performance considering three different kinds of materials: carbon steel, FGM compound by carbon steel+aluminum oxide, and composite material. Validation of these computational models was confirmed considering international standards and international publications.The results of this work show that some feature performances of compression springs made of carbon steel are improved when using FGM and composite materials. Enhanced capabilities include higher load to failure (1.48 times in an FGM spring and 1.1 times in a composite spring) as well as increased energy storage (1.53 times in an FGM spring and 6.84 times in a composite spring) and less weight (representing 61% in weight in an FGM spring and 24% in a composite spring).



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*Source: 9652966-2022-03-21.xml*