# Manual Functional Differential Geometry

Especially, it is assumed to be positively homogeneous. The importance of a generalized metric has been emphasized by many authors [2], [5], [7]. The purpose of the present paper is to investigate the function without the assumption of homogeneity from another point of view.

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Source J. Kyoto Univ. Zentralblatt MATH identifier Subjects Primary: 53B Finsler spaces and generalizations areal metrics Secondary: 53C Finsler spaces and generalizations areal metrics [See also 58B20].

## Faculty of Mechanics and Mathematics

Okubo, Katsumi. Differential geometry of generalized lagrangian functions. Abstract Article info and citation First page Abstract There are many generalizations of Finsler geometry. Article information Source J. The comparison between the CSE model and the uniaxial tension test data of rabbit abdominal skins is shown in Figure 2. Figure 2. Comparison between CSE Models and uniaxial tension tests for rabbit abdominal skins. Porcine liver tissues, composed of liver lobules and connective tissues, are transversely isotropic with the principal axis along the direction of the lobule.

Uniaxial tension and compression tests of porcine liver tissues, using cylindrical specimens, have been conducted by Chui et al. Uniaxial tension experimental data averaging over five porcine liver tissue specimens in both longitudinal and transverse directions have been used to fit the uniaxial tension models 53 and Two sets of constitutive constants have been solved by an iterative least square method.

The comparison between the anisotropic CSE model and the uniaxial tension test data of porcine liver tissues is shown in Figure 3. Heart walls consist of three distinct layers: the endocardium, the myocardium, and the epicardium. The ventricular myocardium is the main functional tissue of the heart wall. Biaxial loading is closer to the actual loading condition of passive ventricular myocardial tissues than uniaxial loading. Biaxial tension tests of human passive ventricular myocardial tissues have been conducted by Sommer et al.

The Cauchy stress-stretch equibiaxial tension experimental data in the mean-fiber direction and the cross-fiber direction has been averaged among 26 specimens, respectively. The averaged equibiaxial tension experimental data have been used to fit the biaxial tension models 65 and 66 and the two. Figure 3. Comparison between CSE models and uniaxial tension tests for porcine liver tissues.

## Functional Differential Geometry

The comparison between the anisotropic CSE models and the averaged equibiaxial tension test data of human passive ventricular myocardial tissues is shown in Figure 4. Triaxial shear tests for porcine myocardial tissues demonstrate the orthotropic behavior of myocardial tissues by Dokos et al. The passive myocardial tissue can therefore be modeled as an orthotropic material with three orthogonal directions: fiber, sheet, and normal.

The fiber refers to muscle fiber, the sheet covers a network of collagen fibers, and the normal direction is perpendicular to both muscular and collagen fiber directions [24]. The triaxial shear tests for human myocardial tissues have also been conducted by Sommer et al. Human myocardial tissues have also been modeled as orthotropic materials with three orthogonal directions: fiber, sheet, and normal. The data for triaxial perpendicular shear TP-Shear to fiber direction are averaged between those of fiber-sheet mode and fiber-normal mode, the data for TP-Shear to sheet direction are averaged between those of sheet-fiber mode and sheet-normal mode, and the data for TP-Shear to normal direction are averaged between those of normal-fiber mode and normal-sheet mode.

The triaxial shear experimental data of ventricular myocardial tissues in fiber, sheet, and normal directions have been used to fit the CSE model for triaxial shear 73 and three sets of constitutive constants have been resolved by an iterative least square. Figure 4. Comparison between CSE models and biaxial tension tests for human myocardium. The comparison between the CSE model and the triaxial shear test data of human passive myocardial tissues is shown in Figure 5. In anisotropic CSE constitutive models, four constitutive constants are needed to describe anisotropic finite deformations in a preferred direction.

Based on the ISF condition 49 , one of the four constitutive constants can be treated as a dependent constant. The first constant, c 1, i , is treated as a dependent constant since the last three constants, c 2, i , c 3, i , and c 4, i , are contained in the anisotropic elasticity tensors 39 and The ISF condition may be applied during extraction of constitutive constants for curve fittings of uniaxial tension and biaxial tension tests. There are two methods for curve fitting experimental data with the ISF condition. In the first method, three unknowns, c 2, i , c 3, i , and c 4, i , are to be solved by the iterative least square method and c 1, i is eliminated from a model by the ISF condition In the second method, four unknowns, c 1, i , c 2, i , c 3, i , and c 4, i , are to be solved by the iterative least square method and the ISF condition 49 is simply subtracted from a model.

The constitutive constants for modeling rabbit abdominal skins RAS and porcine liver tissues PLT in uniaxial tension, human myocardial tissues HMT in biaxial tension and triaxial shear are collected in Table 1. Extraction of constitutive constants for uniaxial tensions in longitudinal L and transverse T directions, biaxial tensions in the mean-fiber MF and the cross-fiber CF.

Figure 5. Comparison between CSE model and triaxial shear tests for human myocardium. Table 1. Constitutive constants for different SBTs.

Extraction of constitutive constants for triaxial shear tests in fiber F , sheet S , and normal N directions was conducted directly from the model since the ISF condition is automatically satisfied. In the triaxial shear model 73 , the extracted second constitutive constants, however, are the sums of and. For the three orthogonal directions of fiber, sheet, and normal, they are , , and , respectively. The data for triaxial perpendicular shear of fiber are averaged between those of fiber-sheet mode and fiber-normal mode.

As an example for the fiber direction, we have and. Similarly, the two other equations can be established for the sheet and normal directions. Combining equations together for three orthogonal directions yields. Note that the values of , , and are used to plot Figure 5 while the values of for triaxial shear tests listed in Table 1 are the final solution of equation In the anisotropic CSE functional, and instead of and are named and used albeit since the summation of invariant components or results in the corresponding invariant or.

Moreover, the use of defined in 19 rather than in 13 captures anisotropic transverse deformations and avoids the unnecessary calculation of inverse right Cauchy-Green tensor. In the anisotropic CSE functional 36 , the first term, , represents the work done of normal stress and translational deformation. The second term, , describes the work done of shear stress and rotational deformation. The third term, , captures the work done of stress for finite deformations with different anisotropic degrees of ellipsoidal deformations. With the fourth-order anisotropic elasticity tensor 38 , the most general fourth-order elasticity tensor for isotropic hyperelastic materials can be recovered by the mappings of , , , …, and.

Substituting the first and second order derivatives of invariants 6 , 7 , 8 , and 9 into 77 , simplifying, and rearranging produces. The three constitutive constants, , and are generally used for modeling finite deformations of isotropic hyperelastic materials. The two constitutive constants, and , determine the isotropic elasticity tensor 78 with the eight parameters 79 through In uniaxial tension tests for SBTs, not only does the stress-stretch data in the tensile direction need to be captured but also at least one other orthogonal stretch must be measured for incompressible SBTs since needs to be evaluated for 19 in the CSE model, and transverse deformation effects on longitudinal deformations are indispensable in uniaxial tension tests.

With uniaxial tension measurements in the direction, for example, we also need to capture the corresponding transverse stretch or and use the incompressible condition to determine the other stretch or , respectively since and are generally not equivalent due to orthotropy. In constitutive modeling of the transverse uniaxial tension test of porcine liver tissues, the tensile stretch is and the shortenings of fibers and cross-fibers are and , respectively. Considering the anisotropic compression test results, the anisotropic contraction is simulated with and the constitutive constants have been obtained and listed in the PLT-T row in Table 1.

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If the isotropic contraction of were assumed, the constitutive constants would be and in which they are quite different from the PLT-T row of results in Table 1. Thus, it is essential to measure both the longitudinal and transverse stretches in uniaxial tension tests for incompressible SBTs. In biaxial tension tests, it is very difficult to maintain both uniform force distribution and uniform normal deformations.

In general, the stress-stretch curves in biaxial tension tests from cruciform specimens to square specimens are not accurate. For biaxial tension tests of square specimens, the stress as a function of stretch is generally overestimated. The overestimation, the correction factor, and the inverse finite element method regarding biaxial tension tests have been studied by Nolan and McGarry [26].

In the CSE models for biaxial tension tests, shear coupling effects do exist, making curve fittings harder.

Without simplifications, at least four arguments, , , , and , have to be measured, causing material characterizations to be more complex. In triaxial shear tests, according to different orthogonal fiber reinforcement orientations, shear deformations have been classified as longitudinal shear, perpendicular shear, and transverse shear by Destrade, Horgan, and Murphy [27].

The three orthogonal directions, fiber, sheet, and normal, for myocardial tissues generate six deformation modes: fiber-sheet mode, fibernormal mode, sheet-normal mode, sheet-fiber mode, normal-fiber mode, and normal-sheet mode. In the CSE model for triaxial shear tests, the longitudinal shear does not contribute any anisotropy, only the perpendicular shear elongates fiber reinforcements, and the perpendicular shear and transverse shear are coupled together.

Without shear coupling effects, there would be three different experimental curves in the six deformation modes. The triaxial shear tests in six deformation modes for both porcine myocardial tissues by Dokos et al. An anisotropic CSE functional, for isothermal processes, is postulated to be balanced with its stress work done, constructing a PDE. A three-term particular solution, which is essentially composed of ICGs, is particularly grouped by differential geometry to capture the three fundamental deformations.

In a preferred direction i, the. The anisotropic CSE constitutive models for uniaxial tension, biaxial tension, and triaxial shear tests have been derived. The constitutive constants have been solved by an iterative least square method for the uniaxial tension tests of rabbit abdominal skins and porcine liver tissues, the biaxial tension, and the triaxial shear tests of human ventricular myocardial tissues.

With the newly defined second invariant component, the anisotropic CSE models capture the transverse effects in uniaxial tension deformations and the coupling effects between the perpendicular shear and transverse shear in triaxial shear deformations. For anisotropic CSE models, the first constitutive constant, , can be treated as a dependent constant due to the ISF condition if necessary and the last three constitutive constants, , , and , are independent constants, defining the anisotropic elasticity tensor.

For isotropic CSE models, the three constitutive constants, , , and , are needed for modeling finite deformations of isotropic hyperelastic materials. The two constitutive constants, and , are required for the isotropic elasticity tensor.

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Chen and Kenneth B. Margulies for their collaboration in earlier studies of rat hearts through ex-vivo tests of diastolic pressure as a function of balloon volume. He would also like to express his gratitude to Jianming and Jiesi Zhao for their discussion and assistance. Journals by Subject. Journals by Title. Author s Fuzhang Zhao. The anisotropic CSE PDE is generally solved by the Lie group and the ICGs through curvatures of elasticity tensor are particularly grouped by differential geometry, representing three general deformations: preferred translational deformations, preferred rotational deformations, and preferred powers of ellipsoidal deformations.

The anisotropic CSE constitutive models have been curve-fitted for uniaxial tension tests of rabbit abdominal skins and porcine liver tissues, and biaxial tension and triaxial shear tests of human ventricular myocardial tissues.