Anisotropic stress softening of residually stressed solids

Shariff, M. H. B. M.

doi:10.1098/rspa.2021.0289

Cited by 6 publications

(17 citation statements)

References 40 publications

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“…where the α i are scalar functions of the invariants of τ and G = M ⊗ M , and the brackets notation denotes the symmetric part of a tensor: D s = (D + D T )/2 for any tensor D. Note that terms such as G τ s , and other higher-order terms in τ , do not appear as they can be expressed using the terms already present in (46). Some of the coefficients α i depend on each other due to the symmetry of the elasticity moduli.…”

Section: Linearisation For Small Strain and Small Rotationmentioning

confidence: 99%

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Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy

Mukherjee¹,

Destrade²,

Gower³

2022

Preprint

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Real-world solids, such as rocks, soft tissues, and engineering materials, are often under some form of stress. Most real materials are also, to some degree, anisotropic due to their microstructure, a characteristic often called the 'texture anisotropy'. This anisotropy can stem from preferential grain alignment in polycrystalline materials, aligned micro-cracks, or structural reinforcement, such as collagen bundles in biological tissues, steel rods in prestressed concrete and reinforcing fibres in composites. Here we establish a framework for initially stressed solids with transverse texture anisotropy. We consider that the strain energy per unit mass of the reference is an explicit function of the elastic deformation gradient, the initial stress tensor, and the texture anisotropy. We determine the corresponding constitutive relations and develop examples of nonlinear strain energies which depend explicitly on the initial stress and direction of texture anisotropy. As an application, we then employ these models to analyse the stress distribution of an inflated initially stressed cylinder with texture anisotropy, and the tension of a welded metal plate. We also deduce the elastic moduli needed to describe linear elasticity from stress reference with transverse texture anisotropy. As an example we show how to measure the stress with small-amplitude shear waves.

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Section: Linearisation For Small Strain and Small Rotationmentioning

confidence: 99%

“…Among other relevant works, we mention those by Merodio et al [35] and Shariff et al [47] for initially stressed isotropic materials, and by Ogden and Singh [42] and Shariff [46] for initially stressed structurally anisotropic materials, some of which do not satisfy ISRI [14,15].…”

Section: Introductionmentioning

confidence: 99%

Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy

Mukherjee¹,

Destrade²,

Gower³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In general, Hill’s strain invariants do not depend explicitly on right Cauchy-Green tensor and their 1st and 2nd order derivatives with respect to can only be obtained via spectral derivative formulae that are recently developed (see, for example references 8 – 11 ) and, in view of this, the author believes that anisotropic/isotropic strain energy functions that are characterized by Hill’s generalized strain functions (to the best of the author’s knowledge) do not exist in the literature. This motivates the author to develop infinitesimal-consistent anisotropic/isotropic finite strain energy functions that are based on the generalized Hill’s strain function and the development requires proposing modified Hill’s and volumetric strain functions; it also requires a spectral approach based on the author’s previous work on anisotropic spectral models (see, for example references 12 – 15 ) that used spectral invariants with a clear physical meaning. The advantages of spectral invariants over classical invariants 16 in constitutive modelling are described, for example, in Shariff and Merodio 17 , hence we will not elaborate them here.…”

Section: Introductionmentioning

confidence: 99%

“…In future, there might be a possibility to connect our approach to the WYPiWYG approach. Using an approach similar to that given in references 8 , 9 , 15 , 23 , our proposed model may be extended to model dissipative materials such as those discussed in 24 – 26 . Our proposed model may also be possibly extended to model strain gradient materials (see, for example references 26 , 27 ) via a similar approach to that of Soltadtos et al 28 .…”

Section: Introductionmentioning

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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“…In general, Hill's strain invariants do not depend explicitly on C and their 1st and 2nd order derivatives with respect to C can only be obtained via spectral derivative formulae that are recently developed (see, for example references [34,35,41,43]) and, in view of this, the author believes that anisotropic/isotropic strain energy functions that are characterized by Hill's generalized strain functions (to the best of the author's knowledge) do not exist in the literature. This motivates the author to develop infinitesimal-consistent anisotropic/isotropic finite strain energy functions that are based on the generalized Hill's strain function and the development requires proposing modified Hill's and volumetric strain functions; it also requires a spectral approach based on the author's previous work on anisotropic spectral models (see, for example references [39,40,42,47]) that used spectral invariants with a clear physical meaning. The advantages of spectral invariants over classical invariants [48] in constitutive modelling are described, for example, in Shariff and Merodio [44], hence we will not elaborate them here.…”

Section: Introductionmentioning

confidence: 99%

A Generalized Strain Approach to Anisotropic Elasticity

Shariff

2021

Preprint

View full text Add to dashboard Cite

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.

show abstract

Anisotropic stress softening of residually stressed solids

Cited by 6 publications

References 40 publications

Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy

Representing the stress and strain energy of elastic solids with initial stress and transverse texture anisotropy

A generalized strain approach to anisotropic elasticity

A Generalized Strain Approach to Anisotropic Elasticity

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