Abstract
This work proposes a generalized Lagrangian strain function f_alpha (that depends on modified stretches) and a volumetric strain function g_alpha (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions f_alpha and g_alpha to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data.
Highlights
This work proposes a generalized Lagrangian strain function fα and a volumetric strain function gα to characterize isotropic/anisotropic strain energy functions
We show that the construction of a strain energy function that uses a full set of spectral invariants that is consistent with infinitesimal theory can be done via the use of, modified Hill’s and volumetric strain functions: A discussion on the importance of a nonlinear strain energy function that must be consistent with infinitesimal theory can be found, for example in, Rosa et al.[22] and Shariff[14]
We define a generalized strain function that is similar to the Hill’s1 strain function and a volumetric function, where they are used to characterize strain energy functions in isotropic or anisotropic elasticity. These strain functions are single variable functions that depend on an invariant with a clear physical meaning, which facilitates the construction of specific forms of the strain energy function, in the sense that a function of a single variable with a clear physical interpretation is manageable and this is indicated in “Example of specific forms of fα and gα used in experimental fitting” section; they facilitate the construction of strain energy functions that are consistent with infinitesimal elasticity as described in sections “Isotropic” to “General anisotropy”
Summary
Transversely isotropic and two-preferred direction materials, described above, the strain energy function W(e) contains, invariants of the form. In the case of the strain energy function W(G) , given in (103), for a general anisotropic material, the corresponding tangent modulus tensor (142) can be derived using the results given in Shariff[10]; due to its complex derivation, we will not derive it here
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