**7** Publications

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Proc Math Phys Eng Sci 2021 Feb 10;477(2246):20200617. Epub 2021 Feb 10.

Mathematics, Mechanics, and Materials Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan.

There are two familiar constructions of a developable surface from a space curve. The tangent developable is a ruled surface for which the rulings are tangent to the curve at each point and relative to this surface the absolute value of the geodesic curvature of the curve equals the curvature . The alternative construction is the rectifying developable. The geodesic curvature of the curve relative to any such surface vanishes. We show that there is a family of developable surfaces that can be generated from a curve, one surface for each function that is defined on the curve and satisfies || ≤ , and that the geodesic curvature of the curve relative to each such constructed surface satisfies = .

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http://dx.doi.org/10.1098/rspa.2020.0617 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC8317982 | PMC |

February 2021

Bull Math Biol 2016 11 19;78(11):2135-2164. Epub 2016 Oct 19.

Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, 60660, USA.

The microscopic structure and anisotropy of plant cell walls greatly influence the mechanical properties, morphogenesis, and growth of plant cells and tissues. The microscopic structure and properties of cell walls are determined by the orientation and mechanical properties of the cellulose microfibrils and the mechanical properties of the cell wall matrix. Viewing the shape of a plant cell as a square prism with the axis aligning with the primary direction of expansion and growth, the orientation of the microfibrils within the side walls, i.e. the parts of the cell walls on the sides of the cells, is known. However, not much is known about their orientation at the upper and lower ends of the cell. Here we investigate the impact of the orientation of cellulose microfibrils within the upper and lower parts of the plant cell walls by solving the equations of linear elasticity numerically. Three different scenarios for the orientation of the microfibrils are considered. We also distinguish between the microstructure in the side walls given by microfibrils perpendicular to the main direction of the expansion and the situation where the microfibrils are rotated through the wall thickness. The macroscopic elastic properties of the cell wall are obtained using homogenization theory from the microscopic description of the elastic properties of the cell wall microfibrils and wall matrix. It is found that the orientation of the microfibrils in the upper and lower parts of the cell walls affects the expansion of the cell in the lateral directions and is particularly important in the case of forces acting on plant cell walls and tissues.

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http://dx.doi.org/10.1007/s11538-016-0207-8 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC5090020 | PMC |

November 2016

Appl Mech Rev 2014 Sep 29;66(5):0508021-5080216. Epub 2014 May 29.

Professor Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology , 1919-1 Tancha, Onna-son, Kunigami-gun , Okinawa, Japan 904-0495 e-mail:

Transport theorems, such as that named after Reynolds, are an important tool in the field of continuum physics. Recently, Seguin and Fried used Harrison's theory of differential chains to establish a transport theorem valid for evolving domains that may become irregular. Evolving irregular domains occur in many different physical settings, such as phase transitions or fracture. Here, emphasizing concepts over technicalities, we present Harrison's theory of differential chains and the results of Seguin and Fried in a way meant to be accessible to researchers in continuum physics. We also show how the transport theorem applies to three concrete examples and approximate the resulting terms numerically. Furthermore, we discuss how the transport theorem might be used to weaken certain basic assumptions underlying the description of continua and the challenges associated with doing so.

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http://dx.doi.org/10.1115/1.4026910 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4240036 | PMC |

September 2014

Arch Ration Mech Anal 2012 Dec;206(3):1039-1072

Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Tel.: 514-398-2998, ,

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. By considering inertial forces on par with other forces, these laws hold relative to any frame of reference, inertial or noninertial. Finally, we introduce a simple set of constitutive relations to govern the interactions between rods and find restrictions necessary and sufficient for these laws to be consistent with thermodynamics. Our framework provides a foundation for a statistical mechanical derivation of the macroscopic balance laws governing liquid crystals.

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http://dx.doi.org/10.1007/s00205-012-0550-3 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3679949 | PMC |

December 2012

Arch Ration Mech Anal 2013 Jan;207(1):1-37

Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC H3A 2K6, Tel.: 514-398-2998, ,

Working on a state space determined by considering a discrete system of rigid rods, we use nonequilibrium statistical mechanics to derive macroscopic balance laws for liquid crystals. A probability function that satisfies the Liouville equation serves as the starting point for deriving each macroscopic balance. The terms appearing in the derived balances are interpreted as expected values and explicit formulas for these terms are obtained. Among the list of derived balances appear two, the tensor moment of inertia balance and the mesofluctuation balance, that are not standard in previously proposed macroscopic theories for liquid crystals but which have precedents in other theories for structured media.

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http://dx.doi.org/10.1007/s00205-012-0551-2 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3611664 | PMC |

January 2013

J Math Biol 2014 Feb 7;68(3):647-65. Epub 2013 Feb 7.

Department of Mathematics and Statistics, 805 Sherbrooke St. W., Montreal, QC, H3A 2K6, Canada,

The Canham-Helfrich free-energy density for a lipid bilayer has drawn considerable attention. Aside from the mean and Gaussian curvatures, this free-energy density involves a spontaneous mean-curvature that encompasses information regarding the preferred, natural shape of the lipid bilayer. We use a straightforward microphysical argument to derive the Canham-Helfrich free-energy density. Our derivation (1) provides a justification for the common assertion that spontaneous curvature originates primarily from asymmetry between the leaflets comprising a bilayer and (2) furnishes expressions for the splay and saddle-splay moduli in terms of derivatives of the underlying potential.

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http://dx.doi.org/10.1007/s00285-013-0647-9 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3694997 | PMC |

February 2014

Biomech Model Mechanobiol 2013 Oct 6;12(5):997-1017. Epub 2012 Dec 6.

Department of Mechanical Engineering, McGill University, Montréal, QC, H3A 0C3, Canada.

Continuum mechanical tools are used to describe the deformation, energy density, and material symmetry of a lipid bilayer with spontaneous curvature. In contrast to conventional approaches in which lipid bilayers are modeled by material surfaces, here we rely on a three-dimensional approach in which a lipid bilayer is modeling by a shell-like body with finite thickness. In this setting, the interface between the leaflets of a lipid bilayer is assumed to coincide with the mid-surface of the corresponding shell-like body. The three-dimensional deformation gradient is found to involve the curvature tensors of the mid-surface in the spontaneous and the deformed states, the deformation gradient of the mid-surface, and the transverse deformation. Attention is also given to the coherency of the leaflets and to the area compatibility of the closed lipid bilayers (i.e., vesicles). A hyperelastic constitutive theory for lipid bilayers in the liquid phase is developed. In combination, the requirements of frame indifference and material symmetry yield a representation for the energy density of a lipid bilayer. This representation shows that three scalar invariants suffice to describe the constitutive response of a lipid bilayer exhibiting in-plane fluidity and transverse isotropy. In addition to exploring the geometrical and physical properties of these invariants, fundamental constitutively associated kinematical quantities are emphasized. On this basis, the effect on the energy density of assuming that the lipid bilayer is incompressible is considered. Lastly, a dimension reduction argument is used to extract an areal energy density per unit area from the three-dimensional energy density. This step explains the origin of spontaneous curvature in the areal energy density. Importantly, along with a standard contribution associated with the natural curvature of the lipid bilayer, our analysis indicates that constitutive asymmetry between the leaflets of the lipid bilayer gives rise to a secondary contribution to the spontaneous curvature.

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http://dx.doi.org/10.1007/s10237-012-0459-7 | DOI Listing |

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3695068 | PMC |

October 2013

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