A spectral approach for nonlinear transversely isotropic elastic bodies, for a new class of constitutive equation: Applications to rock mechanics

Shariff, M. H. B. M.; Bustamante, R.

doi:10.1007/s00707-020-02797-2

Cited by 9 publications

(10 citation statements)

References 59 publications

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“…We compare our theory with the axial compression experiment of Jin et al [15] on compressible rectangular slabs of transversely isotropic Marcellus shale. Although the measured experimental strains are infinitesimal, the stress-strain behaviour is very mildly nonlinear [45]. Since, the strains are infinitesimal, the Cauchy and nominal stress are indistinguishable.…”

Section: Isotropicmentioning

confidence: 96%

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A Generalized Strain Approach to Anisotropic Elasticity

Shariff

2021

Preprint

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This work proposes a generalized Lagrangian strain function fα (that depends on modified stretches) and a volumetric strain function gα (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 fα and gα 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.

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Section: Isotropicmentioning

confidence: 96%

“…It can be easily shown that, in the incompressible limit, as ν zp → 0.5 and 1 − ν a − ν p → 0 [45] the ground-state bulk modulus χ → ∞. It is clear from (65) that, since,…”

Section: Infinitesimal Strainmentioning

confidence: 98%

A Generalized Strain Approach to Anisotropic Elasticity

Shariff

2021

Preprint

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“…We compare our theory with the axial compression experiment of Jin et al 51 on compressible rectangular slabs of transversely isotropic Marcellus shale. Although the measured experimental strains are infinitesimal, the stress-strain behaviour is very mildly nonlinear 42 . Since, the strains are infinitesimal, the Cauchy and nominal stress are indistinguishable.…”

Section: Example Of Specific Forms Of and Used In Experimental Fittingmentioning

confidence: 97%

“…We can express ( 58 ) in the form where The infinitesimal hydrostatic stress The ground-state bulk modulus then takes the form It can be easily shown that, in the incompressible limit, as and 42 the ground-state bulk modulus . It is clear from ( 65 ) that, since, exists, we have In spectral form, …”

Section: Transversely Isotropic With a Unit Preferred Directionmentioning

confidence: 99%

A generalized strain approach to anisotropic elasticity

Shariff

2022

Sci Rep

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This work proposes a generalized Lagrangian strain function $$f_\alpha$$ f α (that depends on modified stretches) and a volumetric strain function $$g_\alpha$$ g α (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 $$f_\alpha$$ f α and $$g_\alpha$$ g α 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.

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“…The above subclass of constitutive equation has many potential applications in the modeling of the mechanical behavior of concrete [15], rock [13,14,[16][17][18], some metal alloys [31], and in stress concentration [12].…”

Section: Constitutive Relations and Equationsmentioning

confidence: 99%

A beam theory for a new class of constitutive relation for elastic bodies

Bustamante

2022

Mathematics and Mechanics of Solids

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A theory for two-dimensional (2D) beams is presented for the case of considering new constitutive equations, wherein the linearized (infinitesimal) strain tensor is assumed to be a nonlinear function of the Cauchy stress tensor. An incremental formulation is developed to solve numerically the resultant equations. Three constitutive equations for elastic bodies are considered: a model wherein we have strain-limiting behavior, a nonlinear model for rock, and a bimodular constitutive equation for rock (which can also be considered as a nonlinear model for rock). Two boundary value problems are studied: the deformation of a cantilever beam with a point load, and a three-point beam with a point load applied on the middle. The numerical results are compared with the predictions of the classical theory for beams for the case of linearized elastic solids. The results for the stresses can be notoriously different, when comparing the predictions of the nonlinear constitutive equations with the results obtained using the linearized theory of elasticity.

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A spectral approach for nonlinear transversely isotropic elastic bodies, for a new class of constitutive equation: Applications to rock mechanics

Cited by 9 publications

References 59 publications

A Generalized Strain Approach to Anisotropic Elasticity

A Generalized Strain Approach to Anisotropic Elasticity

A generalized strain approach to anisotropic elasticity

A beam theory for a new class of constitutive relation for elastic bodies

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