Graduate Category: Engineering and Technology Degree Level: MS ...
Graduate Category: Engineering and Technology Degree Level: MS. Abstract ID# 998
Mechanical Properties Approach Composite Material and Simulation of Leaves Microstructure Gongdai Liu, Hamid Nayeb-Hashemi, Ranajay Ghosh
Abstract • In this research, we mimic the venous morphology of a typical plant leaf into a fiber composite structure where the veins are replaced by stiff fibers and the rest of the leaf is idealized as an elastic perfectly plastic polymeric matrix. The variegated venations found in nature are idealized into three principal fibers – the central mid-fiber corresponding to the mid-rib, straight parallel secondary fibers attached to the mid-fiber representing the secondary veins and then another set of parallel fibers emanating from the secondary fibers mimicking the tertiary veins of a typical leaf. We carry out finite element (FE) based computational investigation of the mechanical properties such as Young’s moduli, Poisson’s ratio and yield stress under uniaxial loading of the resultant composite structures and study the effect of different fiber architectures. We find significant effect of fiber inclination on the overall mechanical properties of the composites with higher fiber angles transitioning the composite increasingly into a matrixdominated response. •
Northeastern University, College of Engineering, Mechanicas and Design Computational Models and Result 1. Close Cell Composite material
2. Close Cell Composite material d
Fig. 2. Position of θ.
Conclusion
a
b
c
d
Fig. 3 Computational model of leaf composite at 𝜽 = 𝟓𝟎𝟎 (a) with only secondary fibers and no closed cells (b) with one closed cell (c) two closed cells and (d) three closed cells.
Fig. 5, Model with tertiary fibers. In this section we considered the effect of tertiary fibers on the structural response of the leaf composite. The diameter of fibers has been kept constant at 1.4 cm and the overall dimensions of the composite remains the same as the previous simulation. The fiber angle varies from 0 to 40 degrees in order to prevent interactions between tertiary fibers as the angle of the secondary fibers are changed. Overall for this set of simulations the model with open cells discussed in previous section (Fig. 5(a)) was used as the basic structure, but tertiary fibers were added to the model, as shown in Fig.5
In this case, Using our computational model it is found that as the fiber angle changes from 00 to 500 , the effective Young’s modulus in the longitudinal direction steadily increases for all fiber architectures. However, fiber architectures apparently has little effect on its transverse elastic modulus. a
b
c
d
e
f
Methods
• Computational analysis were carried out using a commercially available FE software ABAQUS® (Dassault Systemes). The leaf veins were modeled as stiff fibers embedded in a much softer surrounding matrix. For all simulations the Young’s modulus for the fiber was set to 𝐸𝑓 =65Gpa and its Poisson’s ratio was taken as 𝜗𝑓 =0.23. The matrix was modeled as an elastic perfectly plastic material with 𝐸𝑚 =3.5Gpa, 𝜗𝑚 =0.25, 𝜎𝑚 =45MPa. Where 𝐸𝑚 is the Young’s modulus of the matrix, 𝜗𝑚 is the matrix Poisson’s ratio and 𝜎𝑚 is the yield stress of the matrix.
f
Fig. 7, Parametric comparison of (a) longitudinal Young’s modulus normalized by matrix Young’s modulus (b) transverse Young’s modulus normalized by matrix Young’s modulus (c) longitudinal yield stress nrmalized by matrix yield stress(b) transverse yield stress normalized by matrix yield stress (e) Poisson’s Ratio for whole structure in elastic regime (f) Poisson’s Ratio for whole structure in plastic regime.
Introduction The structure of the leaf can be tremendously variegated both in function and morphology, making it an exciting area of recent research [1-8]. Among a variety of morphological traits that distinguish leaves, the venation patterns can be both strikingly different while simultaneously preserving broad universality in their organization making them an important variable in explaining certain aspects of leaf variations [9-13]. At the broadest structural level, the venation typically consist of a main fiber (or a midrib) which holds the leaf in contact with the rest of the tree structure and the many secondary veins emanating from it which can further support tertiary veins attached to them. Interestingly, from a mechanical point of view, these veins can be visualized as stiff fibers reinforcing a softer surrounding matter constituting the rest of the leaves resulting in a composite structure. This makes them an important template for materials design since composite structures have several wellknown advantages, which include high strength, low weight, and good fatigue and corrosion resistance
e
a
d
b
e
c
f
Fig. 4. Parametric comparison of (a) normalized longitudinal Young’s modulus and (b) transverse Young’s modulus respect to the matrix (c) normalized longitudinal yield stress and transverse yield stress respect to the matrix yield stress (e) Poisson’s Ratio 𝝑𝒙𝒚 in the elastic regime and (f) in the plastic regime.
Fig. 6, a: first kind of tertiary fiber model. b: second kind of tertiary fiber model. c: third kind of tertiary fiber model. d: fourth kind of tertiary fiber model. e: fiveth kind of tertiary fiber model . As in the previous case, we find that with an increase in angular position of the secondary fibers, the longitudinal Young’s modulus increases sharply whereas transverse counterpart shows an opposite trend. Furthermore, the gains in longitudinal Young’s modulus in composites with tertiary fibers seem to be insignificant compared to composites with just only secondary fibers. A similar conclusions could be drawn for Poisson’s ratio in composites with just secondary fibers and composites having tertiary fibers as well.
In this paper, FE based simulations were used to obtain Young’s modulus, Poisson’s ratio (elastic and plastic regime) and yield strength of biomimetic leaf like composite structures under uniaxial loading in two directions. Several types of simplified fiber architectures were considered mimicking the variety of venation patterns in leafs. The overall volume fraction of fibers and matrix was kept fixed for comparison purposes. From our extensive parametric studies we found that angle of the secondary fibers with the central main fiber plays a crucial role in determining the properties of the structure. The elastic modulus of the composite in longitudinal direction increases as the angle of the secondary fiber respect to the main fiber decreases. Opposite behavior is observed in the transvers direction. We also found that within these broad trends, closed cell fiber architecture significantly enhance the elastic modulus in the longitudinal direction but has little effect on the transverse direction. For a composite loaded in the longitudinal direction, Poisson’s ratio also strongly depends on the secondary fiber angle. Poisson’s ratio is significantly lower for the closed cell composites. The yield stress of the structure also showed significant variation with fiber architecture and angle. In this case as well, the changing fiber angle plays an important role in decreasing yield stress in both directions as matrix domination becomes significant at higher fiber angles. Here Again the yield stress in both directions is higher for the closed cell composites. The effects of various tertiary fiber morphologies on the overall mechanical properties of leaf like composite are also investigated. The results showed that for a constant fiber volume fraction, the elastic properties of such composites may not be significantly affected by the presence of tertiary fibers. Indicating that desired elastic properties could be achieved just be having secondary fibers. However, the yield stress of the composites is significantly enhanced by having tertiary fibers in the composites as it produces more obstacle to dislocation motion and thus strengthening composites in both directions.
References • • • • • • •
• •
• Fig. 1. Strain-stress plot for whole structure a
b
c
• •
[1] Osnas, J. L., Lichstein, J. W., Reich, P. B., & Pacala, S. W. (2013). Global leaf trait relationships: mass, area, and the leaf economics spectrum.Science, 340(6133), 741-744. [2] Westoby, M., Reich, P. B., & Wright, I. J. (2013). Understanding ecological variation across species: area‐based vs mass‐based expression of leaf traits. New Phytologist, 199(2), 322-323. [3] Bensmihen, S., Hanna, A.I., Langlade, N.B., Micol, J.L., Bangham, A. & Coen, E. 2008. Mutational spaces for leaf shape and size.HFSP J. 2: 110–120. [4] Royer, D.L., Meyerson, L.A., Robertson, K.M. & Adams, J.M. 2009. Phenotypic plasticity of leaf shape along a temperature gradient inAcer rubrum. PLoS ONE 4: e7653. [5] Koenig, D. & Sinha, N. 2010. Evolution of leaf shape: a pattern emerges. Curr. Top. Dev. Biol. 91: 169– 183. [6] Tsukaya, Hirokazu. "Leaf shape: genetic controls and environmental factors." International Journal of Developmental Biology 49, no. 5/6 (2005): 547.
Mechanical Properties Approach Composite Material and Simulation of Leaves Microstructure Gongdai Liu, Hamid Nayeb-Hashemi, Ranajay Ghosh
Abstract • In this research, we mimic the venous morphology of a typical plant leaf into a fiber composite structure where the veins are replaced by stiff fibers and the rest of the leaf is idealized as an elastic perfectly plastic polymeric matrix. The variegated venations found in nature are idealized into three principal fibers – the central mid-fiber corresponding to the mid-rib, straight parallel secondary fibers attached to the mid-fiber representing the secondary veins and then another set of parallel fibers emanating from the secondary fibers mimicking the tertiary veins of a typical leaf. We carry out finite element (FE) based computational investigation of the mechanical properties such as Young’s moduli, Poisson’s ratio and yield stress under uniaxial loading of the resultant composite structures and study the effect of different fiber architectures. We find significant effect of fiber inclination on the overall mechanical properties of the composites with higher fiber angles transitioning the composite increasingly into a matrixdominated response. •
Northeastern University, College of Engineering, Mechanicas and Design Computational Models and Result 1. Close Cell Composite material
2. Close Cell Composite material d
Fig. 2. Position of θ.
Conclusion
a
b
c
d
Fig. 3 Computational model of leaf composite at 𝜽 = 𝟓𝟎𝟎 (a) with only secondary fibers and no closed cells (b) with one closed cell (c) two closed cells and (d) three closed cells.
Fig. 5, Model with tertiary fibers. In this section we considered the effect of tertiary fibers on the structural response of the leaf composite. The diameter of fibers has been kept constant at 1.4 cm and the overall dimensions of the composite remains the same as the previous simulation. The fiber angle varies from 0 to 40 degrees in order to prevent interactions between tertiary fibers as the angle of the secondary fibers are changed. Overall for this set of simulations the model with open cells discussed in previous section (Fig. 5(a)) was used as the basic structure, but tertiary fibers were added to the model, as shown in Fig.5
In this case, Using our computational model it is found that as the fiber angle changes from 00 to 500 , the effective Young’s modulus in the longitudinal direction steadily increases for all fiber architectures. However, fiber architectures apparently has little effect on its transverse elastic modulus. a
b
c
d
e
f
Methods
• Computational analysis were carried out using a commercially available FE software ABAQUS® (Dassault Systemes). The leaf veins were modeled as stiff fibers embedded in a much softer surrounding matrix. For all simulations the Young’s modulus for the fiber was set to 𝐸𝑓 =65Gpa and its Poisson’s ratio was taken as 𝜗𝑓 =0.23. The matrix was modeled as an elastic perfectly plastic material with 𝐸𝑚 =3.5Gpa, 𝜗𝑚 =0.25, 𝜎𝑚 =45MPa. Where 𝐸𝑚 is the Young’s modulus of the matrix, 𝜗𝑚 is the matrix Poisson’s ratio and 𝜎𝑚 is the yield stress of the matrix.
f
Fig. 7, Parametric comparison of (a) longitudinal Young’s modulus normalized by matrix Young’s modulus (b) transverse Young’s modulus normalized by matrix Young’s modulus (c) longitudinal yield stress nrmalized by matrix yield stress(b) transverse yield stress normalized by matrix yield stress (e) Poisson’s Ratio for whole structure in elastic regime (f) Poisson’s Ratio for whole structure in plastic regime.
Introduction The structure of the leaf can be tremendously variegated both in function and morphology, making it an exciting area of recent research [1-8]. Among a variety of morphological traits that distinguish leaves, the venation patterns can be both strikingly different while simultaneously preserving broad universality in their organization making them an important variable in explaining certain aspects of leaf variations [9-13]. At the broadest structural level, the venation typically consist of a main fiber (or a midrib) which holds the leaf in contact with the rest of the tree structure and the many secondary veins emanating from it which can further support tertiary veins attached to them. Interestingly, from a mechanical point of view, these veins can be visualized as stiff fibers reinforcing a softer surrounding matter constituting the rest of the leaves resulting in a composite structure. This makes them an important template for materials design since composite structures have several wellknown advantages, which include high strength, low weight, and good fatigue and corrosion resistance
e
a
d
b
e
c
f
Fig. 4. Parametric comparison of (a) normalized longitudinal Young’s modulus and (b) transverse Young’s modulus respect to the matrix (c) normalized longitudinal yield stress and transverse yield stress respect to the matrix yield stress (e) Poisson’s Ratio 𝝑𝒙𝒚 in the elastic regime and (f) in the plastic regime.
Fig. 6, a: first kind of tertiary fiber model. b: second kind of tertiary fiber model. c: third kind of tertiary fiber model. d: fourth kind of tertiary fiber model. e: fiveth kind of tertiary fiber model . As in the previous case, we find that with an increase in angular position of the secondary fibers, the longitudinal Young’s modulus increases sharply whereas transverse counterpart shows an opposite trend. Furthermore, the gains in longitudinal Young’s modulus in composites with tertiary fibers seem to be insignificant compared to composites with just only secondary fibers. A similar conclusions could be drawn for Poisson’s ratio in composites with just secondary fibers and composites having tertiary fibers as well.
In this paper, FE based simulations were used to obtain Young’s modulus, Poisson’s ratio (elastic and plastic regime) and yield strength of biomimetic leaf like composite structures under uniaxial loading in two directions. Several types of simplified fiber architectures were considered mimicking the variety of venation patterns in leafs. The overall volume fraction of fibers and matrix was kept fixed for comparison purposes. From our extensive parametric studies we found that angle of the secondary fibers with the central main fiber plays a crucial role in determining the properties of the structure. The elastic modulus of the composite in longitudinal direction increases as the angle of the secondary fiber respect to the main fiber decreases. Opposite behavior is observed in the transvers direction. We also found that within these broad trends, closed cell fiber architecture significantly enhance the elastic modulus in the longitudinal direction but has little effect on the transverse direction. For a composite loaded in the longitudinal direction, Poisson’s ratio also strongly depends on the secondary fiber angle. Poisson’s ratio is significantly lower for the closed cell composites. The yield stress of the structure also showed significant variation with fiber architecture and angle. In this case as well, the changing fiber angle plays an important role in decreasing yield stress in both directions as matrix domination becomes significant at higher fiber angles. Here Again the yield stress in both directions is higher for the closed cell composites. The effects of various tertiary fiber morphologies on the overall mechanical properties of leaf like composite are also investigated. The results showed that for a constant fiber volume fraction, the elastic properties of such composites may not be significantly affected by the presence of tertiary fibers. Indicating that desired elastic properties could be achieved just be having secondary fibers. However, the yield stress of the composites is significantly enhanced by having tertiary fibers in the composites as it produces more obstacle to dislocation motion and thus strengthening composites in both directions.
References • • • • • • •
• •
• Fig. 1. Strain-stress plot for whole structure a
b
c
• •
[1] Osnas, J. L., Lichstein, J. W., Reich, P. B., & Pacala, S. W. (2013). Global leaf trait relationships: mass, area, and the leaf economics spectrum.Science, 340(6133), 741-744. [2] Westoby, M., Reich, P. B., & Wright, I. J. (2013). Understanding ecological variation across species: area‐based vs mass‐based expression of leaf traits. New Phytologist, 199(2), 322-323. [3] Bensmihen, S., Hanna, A.I., Langlade, N.B., Micol, J.L., Bangham, A. & Coen, E. 2008. Mutational spaces for leaf shape and size.HFSP J. 2: 110–120. [4] Royer, D.L., Meyerson, L.A., Robertson, K.M. & Adams, J.M. 2009. Phenotypic plasticity of leaf shape along a temperature gradient inAcer rubrum. PLoS ONE 4: e7653. [5] Koenig, D. & Sinha, N. 2010. Evolution of leaf shape: a pattern emerges. Curr. Top. Dev. Biol. 91: 169– 183. [6] Tsukaya, Hirokazu. "Leaf shape: genetic controls and environmental factors." International Journal of Developmental Biology 49, no. 5/6 (2005): 547.
