Interdiscip Sci Comput Life Sci (2014) 6: 1–4 DOI: 10.1007/s12539-014-0206-0
Analysis of Finite Element Method on Mechanical Properties of Wheat Kernel Feng JIA, Jinshui WANG∗ , Pei FAN, Haicheng YIN,
Junjun GUAN,
Mengqing ZHOU
(College of Bioengineering, Henan University of Technology, Zhengzhou 450001, China)
Received 28 May 2014 / Revised 17 July 2014 / Accepted 11 September 2014
Abstract: The scattered cases of wheat kernels were examined for ensuring the direction of wheat kernels. Above 50% kernels of wheat were back up and about 43% were sides up. Finite element method was performed to simulate stress distributions inside a wheat kernel during storage. The distributions of fixture, force, tangential, deformation, and displacement were mapped and analyzed. The grid for wheat consisted of the model consists of 1620 nodes and 937 triangular elements. The stress of sides was about 3 times higher than the bottom. Figure 6 illustrated the displacements of wheat were distinct in different status in storage. The displacement on side status was greater than on bottom of wheat. Key words: wheat, kernel, mechanical properties, finite element method, storage.
1 Introduction The ultimate goal of the wheat industry is to produce high quality flour. To attain this goal, the mechanical properties of wheat kernel make it necessary to milling. On the other hand, wheat is the staple food in northern China, and it is also the major grain of long-term strategic reserves in China. The wheat transport many times from harvest to storage and finally to the mills in China, while the wheat kernels would be subject to different forces in a relatively long storage processes. The finite element method (FEM) is currently one of the most effective and universal methods of numerical calculation for solving partial differential equations from engineering and scientific perceptive (Fries et al., 2011; Kuna, 2013). FEM has been introduced and employed increasingly to solve heat and/or mass transfer problems (Gast´ on et al., 2002). FEM has also been used in wheat for analysis and experimental verification of wheat grain under compression loads (Zhang et al., 2010), and analysis the carbon dioxide diffusion in stored wheat (Singh et al., 1983). Moreover, the force model of wheat kernel was used on the interaction of wheat kernels under the storage conditions to build the theoretical basis of wheat stored in different locations for adjusting the storage and transportation process of wheat and, to reduce the crushed kernels of wheat during the storage and transportation. In addition, the integrity of the wheat ker∗
Corresponding author. E-mail: [email protected]; [email protected]
nels is critical for storage performance, and imperfect kernels are vulnerable to suffer microbial contamination, which change the quality of wheat. Therefore, it is necessary for the shape of wheat kernels and associated mechanical analysis. The objectives of this research were to: 1. Develop a finite-element model to predict the deformation of wheat kernels during the transport and storage processes. 2. Infer and analyze possible stress condition in wheat kernels based on the finite-element model. 3. Adjust the process during storage and transportation of wheat in order to reduce wheat grain crushing.
2 Material and methods 2.1
Material
Wheat kernels of Zhengmai 9023 variety were used in this study. The lengths of wheat (a, b, c=6.36, 3.23, 2.86) along three principle axes were referenced by Kang and Delwiche (Kang and Delwiche, 1999). Random sample of 1000 wheat kernels were detected based on their shapes, which were divided in three classes: the intact, deformed, and broken. The 500 kernels of wheat were scattered on the hard/soft surface from a height of 1 meter. And wheat kernels were tested the face up of wheat on the surface on the base of stress analysis. When the moisture content of the wheat was 9.6%, the elastic modulus of the wheat was 5.5 × 109 Pa, Poisson’s ratio was 0.4 (Li et al., 2007). Wheat grain moisture was at about 10%, endosperm showed
2
Interdiscip Sci Comput Life Sci (2014) 6: 1–4
brittle materials and failure mode was brittle fracture, so the maximum tensile stress theory was used in this work. 2.2
Finite element model and analysis
Commercial finite element analysis software Solidworks 2012 was used to determine the Finite Element Model of wheat kernel. The shape of the wheat kernel was assumed to be spherical while that of the intact kernel was considered to be prolate spheroidal (Kang and Delwiche, 1999). Nevertheless, in any given plane out of a three-node triangular elements, respectively, threenode i, j and m, triangular unit displacement variables x and y directions were u and v. The shape function of three nodes were Si , Sj , Sm , and the node displacement components of three-node were Uix , Uiy , Ujx , Ujy , Umx , Umy . So the displacement of triangular unit variable could be expressed as follows, ⎡ u v
=
S0
0
Sj
0
Sm
0
Si
0
Sj
0
Uix
⎥ ⎢ ⎢U ⎥ ⎢ iy ⎥ ⎥ 0 ⎢ ⎢ Ujx ⎥ ⎥ ⎢ ⎥ Sm ⎢ ⎢ Ujy ⎥ ⎥ ⎢ ⎣Umx ⎦ Umy
(1)
εyy
⎡
1 ⎢ ⎢v ⎣ 0
v 1 0
0
1 [Uix (Uiy − Umy ) + Ujx (Umy − Uiy ) + 2 Umx (Uiy − Ujy )] Then the matrix form of (4) was
where A =
⎡ ⎡
⎤
⎡ εxx βi ⎢ ⎥ ⎢ ⎢ εyy ⎥ = 1 ⎢ 0 ⎣ ⎦ 2A ⎣ γxy δi
0U βj
0
βm
0
δi
0
δj
0
βi
δj
βj
δm
Uix
⎤
⎥ ⎤⎢ ⎢ Uiy ⎥ ⎢ ⎥ ⎥ ⎥⎢ U jx ⎢ ⎥ ⎢ δm ⎥ ⎦⎢U ⎥ jy ⎥ ⎥ βm ⎢ ⎢ ⎥ ⎣Umx ⎦ Umy (5)
The strain energy was 1 1 Λ(e) = εT vεdV = U T B T vBU dV 2 V 2 V
(6)
where V was the volume of element. The differential of displacement nodes Uk ,
∂Λ(e) 1 ∂ T T = U B vBU dV ∂Uk ∂Uk 2 V (k = 1, 2, 3, 4, 5, 6)
⎤⎡
⎤
εxx ⎥⎢ ⎥ 0 ⎥ ⎢ εyy ⎥ ⎦⎣ ⎦ 1−v γxy 2
(3)
where σxx , σyy , γxy showed the strains and shear stress of correspondence of εxx , εyy , γxy , E was Modulus of elasticity, and v was Poisson’s ratio. Then ∂ ∂u = (Si Uix + Sj Ujx + Sm Umx ) ∂x ∂x 1 (βi Uix + βj Ujx + βm Umx ) = 2A
εxx =
(4)
(7)
(2)
γxy
where εT showed the transpose matrix of strain at any point within the object, and εxx , εyy , γxy showed strain components of in x or y direction of ε and shear stress. The general form of Hooke’s law was ⎡ ⎤ σxx ⎢ ⎥ ⎢σyy ⎥ = E ⎣ ⎦ 1 − v2 τxy
γxy
⎤
For plane stress, the strain state of the object can be used at any point in three independent variables, that was
εT = εxx
∂ ∂v = (Si Uiy + Sj Ujy + Sm Umy ) ∂y ∂y 1 (δi Uiy + δj Ujy + δm Umy ) = 2A ∂v 1 ∂u = = (δi Uix + δj Ujx + = ∂y ∂x 2A δm Umx + βi Uiy + βj Ujy + βm Umy )
εyy =
The expression of stiffness matrix was B T vBdV = V B T vB K (e) =
(8)
V
2.3
Boundary condition and load
The boundary condition was that the kernel surface maintains an equilibrium stress. The height of wheat stack was calculated in accordance with 5 m, Range top wheat force was about 0.8-1.0 N.
3 Results and discussion 3.1
Shape statistics of wheat seeds and the anisotropic of after wheat scattered
Statistical results wheat seed shape showed that the around 20% wheat kernels were deformed, while near 75% wheat kernels were intact and about 5% were broken. As shown from Fig. 1, the results of the scattered cases of wheat kernels were estimated the rates of the scattered on hard/soft surface. The back up of wheats were 56.67% on hard surface and 49.33% on soft surface.
Interdiscip Sci Comput Life Sci (2014) 6: 1–4
3
5.67 37.67 7.67
56.67 56.
(a)
(b)
(c)
(d)
(a) 7.33 49.3 49.33
00 43.00
Fig. 3
(b) The back up of wheat
The side up of wheat The ventralcanal up of wheat
Fig. 1
The image of fixture and force of wheat kernel were shown using the model of finite element. (a) Fixed ventralcanal of wheat, forced from the back of wheat; (b) Fixed side of wheat, forced from the other side of wheat; (c) The image of fixture of (a); (d) The image of fixture of (c).
The scattered cases of wheat on hard-surface (a) and soft-surface (b).
The side up of wheats were 37.67% on hard surface and 43.00% on soft surface. The only about 6-7% of wheat kernels were ventralcanal side up on the hard/soft surface. 3.2
The mechanical analysis of wheat
The mesh was automatically optimized upon created by the Solidworks 2012. A finite-element discretization of the kernel was shown in Fig. 2. The grid for wheat in this model consisted of 1620 nodes and 937 triangular elements (Fig. 2). The force of model was 0.8 N, and two kinds of fixture and force of wheat kernel using this model of finite element were shown in Fig. 3.
(a)
(b)
Fig. 4
Fig. 2
3.3
The finite element meshing of wheat kernel.
The deformation and stress analysis of model of wheat kernel
Figure 4 showed the deformation of wheat using the model-SimulationXpress at pressure in two different directions. It also showed the different deformation of
The deformation of wheat using the modelSimulationXpress. The notes of (a) and (b) were same as in Fig. 3.
wheat in different directions. When the ventralcanal was located in the bottom, the larger deformation of wheat kernel compared with the wheat side located in the bottom. The stress and displacement of wheat were displayed in Fig. 5 and Fig. 6. It could be seen that the stress of bottom was higher than others in Fig. 5(a), and the stress of side was higher than others in Fig. 5 (b). Also
4
Interdiscip Sci Comput Life Sci (2014) 6: 1–4 von mises (M/m2)
von mises (M/m2)
269, 738.8 247, 431.7 225, 124.7 202, 817.7 180, 510.7 158, 203.7 135, 896.6 113, 589.6 91, 282.6 68, 975.6 46, 668.6 24, 361.5 2, 054.5
509, 551.0 467, 647.4 425, 783.7 383, 920.1 342, 056.4 300, 192.8 258, 329.1 216, 465.4 174, 601.8 132, 738.1 90, 874.5 49, 010.8 147.2 77, 147 2
(a)
Fig. 5
(b)
The stress of wheat using the model-SimulationXpress. The notes of (a) and (b) were same as in Fig. 3. URES (mm)
URES (mm) UR 7.899e−005 7.241e−005 6.582e−005 5.924e−005 5.266e−005 4.608e−005 3.949e−005 3.291e−005 2.633e−005 1.975e−005 1.316e−005 6.582e−006 1.000e−030
1.104e−004 1.012e−004 9.197e−005 8.277e−005 7.358e−005 6.438e−005 5.518e−005 4.599e−005 3.679e−005 2.759e−005 1.839e−005 9.197e−006 11.000e 000e−030
(a)
Fig. 6
(b)
The displacement of wheat using the model-SimulationXpress. The notes of (a) and (b) were same as in Fig. 3.
the stress of sides was about 3 times higher than bottom. This indicated that the form of damage to wheat was the same, and the stress concentration was located in wheat ventralcanal. Figure 6 illustrates the displacements of wheat were different in different status in storage. Mathematical models showed that the displacement of wheat on side status was greater than ventralcanal. Ultimately, a better understanding of the mechanical properties of wheat kernel through the current model would lead to a clearer understanding of the damage forms of wheat kernel.
Acknowledgements This work was supported by the general science and technology research projects of Zhengzhou (N2013G0077), the science and technology research key project in Henan province department of education (No. 13A550166), the Natural Science Foundation of China (No. 31371850) and doctor fund of Henan University of Technology (No. 2012BS013).
References [1] Fries T.P., Zilian A., Mo¨es N. 2011. Extended Finite Element Method. International Journal for Numerical
Methods in Engineering 86, 403-403. [2] Gast´ on A.a.L., Abalone R.M., Giner S.A. 2002. Wheat drying kinetics. Diffusivities for sphere and ellipsoid by finite elements. Journal of Food Engineering 52, 313322. [3] Kang S., Delwiche S. 1999. Moisture diffusion modeling of wheat kernels during soaking. Transactions of the ASAE-American Society of Agricultural Engineers 42, 1359-1366. [4] Kuna M. 2013. Finite Element Method, Finite Elements in Fracture Mechanics: Theory - Numerics Applications, Springer, Heidelberg. pp. 153-192. [5] Li X., Gao L., Ma F. 2007. analysis of fintie element method on mechanical properties of corn seed. Transactions of the Chinese Society for Agricultural Machinery 38, 64-67, 72. [6] Singh D., Muir W., Sinha R. 1983. Finite element modelling of carbon dioxide diffusion in stored wheat. Canadian Agricultural Engineering 25, 149-152. [7] Zhang K., Huang J., Yang M., Zhang F., Huang X., Zhao C. 2010. Finite element analysis and experimental verification of wheat grain under compression loads. Transactions of the Chinese Society of Agricultural Engineering 26, 352-356.
Analysis of Finite Element Method on Mechanical Properties of Wheat Kernel Feng JIA, Jinshui WANG∗ , Pei FAN, Haicheng YIN,
Junjun GUAN,
Mengqing ZHOU
(College of Bioengineering, Henan University of Technology, Zhengzhou 450001, China)
Received 28 May 2014 / Revised 17 July 2014 / Accepted 11 September 2014
Abstract: The scattered cases of wheat kernels were examined for ensuring the direction of wheat kernels. Above 50% kernels of wheat were back up and about 43% were sides up. Finite element method was performed to simulate stress distributions inside a wheat kernel during storage. The distributions of fixture, force, tangential, deformation, and displacement were mapped and analyzed. The grid for wheat consisted of the model consists of 1620 nodes and 937 triangular elements. The stress of sides was about 3 times higher than the bottom. Figure 6 illustrated the displacements of wheat were distinct in different status in storage. The displacement on side status was greater than on bottom of wheat. Key words: wheat, kernel, mechanical properties, finite element method, storage.
1 Introduction The ultimate goal of the wheat industry is to produce high quality flour. To attain this goal, the mechanical properties of wheat kernel make it necessary to milling. On the other hand, wheat is the staple food in northern China, and it is also the major grain of long-term strategic reserves in China. The wheat transport many times from harvest to storage and finally to the mills in China, while the wheat kernels would be subject to different forces in a relatively long storage processes. The finite element method (FEM) is currently one of the most effective and universal methods of numerical calculation for solving partial differential equations from engineering and scientific perceptive (Fries et al., 2011; Kuna, 2013). FEM has been introduced and employed increasingly to solve heat and/or mass transfer problems (Gast´ on et al., 2002). FEM has also been used in wheat for analysis and experimental verification of wheat grain under compression loads (Zhang et al., 2010), and analysis the carbon dioxide diffusion in stored wheat (Singh et al., 1983). Moreover, the force model of wheat kernel was used on the interaction of wheat kernels under the storage conditions to build the theoretical basis of wheat stored in different locations for adjusting the storage and transportation process of wheat and, to reduce the crushed kernels of wheat during the storage and transportation. In addition, the integrity of the wheat ker∗
Corresponding author. E-mail: [email protected]; [email protected]
nels is critical for storage performance, and imperfect kernels are vulnerable to suffer microbial contamination, which change the quality of wheat. Therefore, it is necessary for the shape of wheat kernels and associated mechanical analysis. The objectives of this research were to: 1. Develop a finite-element model to predict the deformation of wheat kernels during the transport and storage processes. 2. Infer and analyze possible stress condition in wheat kernels based on the finite-element model. 3. Adjust the process during storage and transportation of wheat in order to reduce wheat grain crushing.
2 Material and methods 2.1
Material
Wheat kernels of Zhengmai 9023 variety were used in this study. The lengths of wheat (a, b, c=6.36, 3.23, 2.86) along three principle axes were referenced by Kang and Delwiche (Kang and Delwiche, 1999). Random sample of 1000 wheat kernels were detected based on their shapes, which were divided in three classes: the intact, deformed, and broken. The 500 kernels of wheat were scattered on the hard/soft surface from a height of 1 meter. And wheat kernels were tested the face up of wheat on the surface on the base of stress analysis. When the moisture content of the wheat was 9.6%, the elastic modulus of the wheat was 5.5 × 109 Pa, Poisson’s ratio was 0.4 (Li et al., 2007). Wheat grain moisture was at about 10%, endosperm showed
2
Interdiscip Sci Comput Life Sci (2014) 6: 1–4
brittle materials and failure mode was brittle fracture, so the maximum tensile stress theory was used in this work. 2.2
Finite element model and analysis
Commercial finite element analysis software Solidworks 2012 was used to determine the Finite Element Model of wheat kernel. The shape of the wheat kernel was assumed to be spherical while that of the intact kernel was considered to be prolate spheroidal (Kang and Delwiche, 1999). Nevertheless, in any given plane out of a three-node triangular elements, respectively, threenode i, j and m, triangular unit displacement variables x and y directions were u and v. The shape function of three nodes were Si , Sj , Sm , and the node displacement components of three-node were Uix , Uiy , Ujx , Ujy , Umx , Umy . So the displacement of triangular unit variable could be expressed as follows, ⎡ u v
=
S0
0
Sj
0
Sm
0
Si
0
Sj
0
Uix
⎥ ⎢ ⎢U ⎥ ⎢ iy ⎥ ⎥ 0 ⎢ ⎢ Ujx ⎥ ⎥ ⎢ ⎥ Sm ⎢ ⎢ Ujy ⎥ ⎥ ⎢ ⎣Umx ⎦ Umy
(1)
εyy
⎡
1 ⎢ ⎢v ⎣ 0
v 1 0
0
1 [Uix (Uiy − Umy ) + Ujx (Umy − Uiy ) + 2 Umx (Uiy − Ujy )] Then the matrix form of (4) was
where A =
⎡ ⎡
⎤
⎡ εxx βi ⎢ ⎥ ⎢ ⎢ εyy ⎥ = 1 ⎢ 0 ⎣ ⎦ 2A ⎣ γxy δi
0U βj
0
βm
0
δi
0
δj
0
βi
δj
βj
δm
Uix
⎤
⎥ ⎤⎢ ⎢ Uiy ⎥ ⎢ ⎥ ⎥ ⎥⎢ U jx ⎢ ⎥ ⎢ δm ⎥ ⎦⎢U ⎥ jy ⎥ ⎥ βm ⎢ ⎢ ⎥ ⎣Umx ⎦ Umy (5)
The strain energy was 1 1 Λ(e) = εT vεdV = U T B T vBU dV 2 V 2 V
(6)
where V was the volume of element. The differential of displacement nodes Uk ,
∂Λ(e) 1 ∂ T T = U B vBU dV ∂Uk ∂Uk 2 V (k = 1, 2, 3, 4, 5, 6)
⎤⎡
⎤
εxx ⎥⎢ ⎥ 0 ⎥ ⎢ εyy ⎥ ⎦⎣ ⎦ 1−v γxy 2
(3)
where σxx , σyy , γxy showed the strains and shear stress of correspondence of εxx , εyy , γxy , E was Modulus of elasticity, and v was Poisson’s ratio. Then ∂ ∂u = (Si Uix + Sj Ujx + Sm Umx ) ∂x ∂x 1 (βi Uix + βj Ujx + βm Umx ) = 2A
εxx =
(4)
(7)
(2)
γxy
where εT showed the transpose matrix of strain at any point within the object, and εxx , εyy , γxy showed strain components of in x or y direction of ε and shear stress. The general form of Hooke’s law was ⎡ ⎤ σxx ⎢ ⎥ ⎢σyy ⎥ = E ⎣ ⎦ 1 − v2 τxy
γxy
⎤
For plane stress, the strain state of the object can be used at any point in three independent variables, that was
εT = εxx
∂ ∂v = (Si Uiy + Sj Ujy + Sm Umy ) ∂y ∂y 1 (δi Uiy + δj Ujy + δm Umy ) = 2A ∂v 1 ∂u = = (δi Uix + δj Ujx + = ∂y ∂x 2A δm Umx + βi Uiy + βj Ujy + βm Umy )
εyy =
The expression of stiffness matrix was B T vBdV = V B T vB K (e) =
(8)
V
2.3
Boundary condition and load
The boundary condition was that the kernel surface maintains an equilibrium stress. The height of wheat stack was calculated in accordance with 5 m, Range top wheat force was about 0.8-1.0 N.
3 Results and discussion 3.1
Shape statistics of wheat seeds and the anisotropic of after wheat scattered
Statistical results wheat seed shape showed that the around 20% wheat kernels were deformed, while near 75% wheat kernels were intact and about 5% were broken. As shown from Fig. 1, the results of the scattered cases of wheat kernels were estimated the rates of the scattered on hard/soft surface. The back up of wheats were 56.67% on hard surface and 49.33% on soft surface.
Interdiscip Sci Comput Life Sci (2014) 6: 1–4
3
5.67 37.67 7.67
56.67 56.
(a)
(b)
(c)
(d)
(a) 7.33 49.3 49.33
00 43.00
Fig. 3
(b) The back up of wheat
The side up of wheat The ventralcanal up of wheat
Fig. 1
The image of fixture and force of wheat kernel were shown using the model of finite element. (a) Fixed ventralcanal of wheat, forced from the back of wheat; (b) Fixed side of wheat, forced from the other side of wheat; (c) The image of fixture of (a); (d) The image of fixture of (c).
The scattered cases of wheat on hard-surface (a) and soft-surface (b).
The side up of wheats were 37.67% on hard surface and 43.00% on soft surface. The only about 6-7% of wheat kernels were ventralcanal side up on the hard/soft surface. 3.2
The mechanical analysis of wheat
The mesh was automatically optimized upon created by the Solidworks 2012. A finite-element discretization of the kernel was shown in Fig. 2. The grid for wheat in this model consisted of 1620 nodes and 937 triangular elements (Fig. 2). The force of model was 0.8 N, and two kinds of fixture and force of wheat kernel using this model of finite element were shown in Fig. 3.
(a)
(b)
Fig. 4
Fig. 2
3.3
The finite element meshing of wheat kernel.
The deformation and stress analysis of model of wheat kernel
Figure 4 showed the deformation of wheat using the model-SimulationXpress at pressure in two different directions. It also showed the different deformation of
The deformation of wheat using the modelSimulationXpress. The notes of (a) and (b) were same as in Fig. 3.
wheat in different directions. When the ventralcanal was located in the bottom, the larger deformation of wheat kernel compared with the wheat side located in the bottom. The stress and displacement of wheat were displayed in Fig. 5 and Fig. 6. It could be seen that the stress of bottom was higher than others in Fig. 5(a), and the stress of side was higher than others in Fig. 5 (b). Also
4
Interdiscip Sci Comput Life Sci (2014) 6: 1–4 von mises (M/m2)
von mises (M/m2)
269, 738.8 247, 431.7 225, 124.7 202, 817.7 180, 510.7 158, 203.7 135, 896.6 113, 589.6 91, 282.6 68, 975.6 46, 668.6 24, 361.5 2, 054.5
509, 551.0 467, 647.4 425, 783.7 383, 920.1 342, 056.4 300, 192.8 258, 329.1 216, 465.4 174, 601.8 132, 738.1 90, 874.5 49, 010.8 147.2 77, 147 2
(a)
Fig. 5
(b)
The stress of wheat using the model-SimulationXpress. The notes of (a) and (b) were same as in Fig. 3. URES (mm)
URES (mm) UR 7.899e−005 7.241e−005 6.582e−005 5.924e−005 5.266e−005 4.608e−005 3.949e−005 3.291e−005 2.633e−005 1.975e−005 1.316e−005 6.582e−006 1.000e−030
1.104e−004 1.012e−004 9.197e−005 8.277e−005 7.358e−005 6.438e−005 5.518e−005 4.599e−005 3.679e−005 2.759e−005 1.839e−005 9.197e−006 11.000e 000e−030
(a)
Fig. 6
(b)
The displacement of wheat using the model-SimulationXpress. The notes of (a) and (b) were same as in Fig. 3.
the stress of sides was about 3 times higher than bottom. This indicated that the form of damage to wheat was the same, and the stress concentration was located in wheat ventralcanal. Figure 6 illustrates the displacements of wheat were different in different status in storage. Mathematical models showed that the displacement of wheat on side status was greater than ventralcanal. Ultimately, a better understanding of the mechanical properties of wheat kernel through the current model would lead to a clearer understanding of the damage forms of wheat kernel.
Acknowledgements This work was supported by the general science and technology research projects of Zhengzhou (N2013G0077), the science and technology research key project in Henan province department of education (No. 13A550166), the Natural Science Foundation of China (No. 31371850) and doctor fund of Henan University of Technology (No. 2012BS013).
References [1] Fries T.P., Zilian A., Mo¨es N. 2011. Extended Finite Element Method. International Journal for Numerical
Methods in Engineering 86, 403-403. [2] Gast´ on A.a.L., Abalone R.M., Giner S.A. 2002. Wheat drying kinetics. Diffusivities for sphere and ellipsoid by finite elements. Journal of Food Engineering 52, 313322. [3] Kang S., Delwiche S. 1999. Moisture diffusion modeling of wheat kernels during soaking. Transactions of the ASAE-American Society of Agricultural Engineers 42, 1359-1366. [4] Kuna M. 2013. Finite Element Method, Finite Elements in Fracture Mechanics: Theory - Numerics Applications, Springer, Heidelberg. pp. 153-192. [5] Li X., Gao L., Ma F. 2007. analysis of fintie element method on mechanical properties of corn seed. Transactions of the Chinese Society for Agricultural Machinery 38, 64-67, 72. [6] Singh D., Muir W., Sinha R. 1983. Finite element modelling of carbon dioxide diffusion in stored wheat. Canadian Agricultural Engineering 25, 149-152. [7] Zhang K., Huang J., Yang M., Zhang F., Huang X., Zhao C. 2010. Finite element analysis and experimental verification of wheat grain under compression loads. Transactions of the Chinese Society of Agricultural Engineering 26, 352-356.
