A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy

Benítez, José María; Montáns, Francisco J.

doi:10.1016/j.jnnfm.2018.08.011

Cited by 16 publications

(6 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since there is no known closed form, and accuracy in the evaluation is extremely important [51], several approximants have been proposed in the last years [52,53,54,55,56,57]. Comparisons are given in [55] and [44]. The most accurate approximant is given in [44], using a spline-based approach.…”

Section: Discussionmentioning

confidence: 99%

“…or a description employing the Inverse Langevin function are better for a global representation of the chain behavior, then, these functions may be best-fitted to the obtained P ch (λ ch ) behavior. Moreover, for the extrapolation domain when using B-splines, any of these functions may be appended at the end of the interpolation domain following a similar approach as that employed in [44] for the Inverse Langevin function.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Data-driven, structure-based hyperelastic manifolds: A macro-micro-macro approach to reverse-engineer the chain behavior and perform efficient simulations of polymers

Amores

Benítez

Montáns

2020

Computers & Structures

Self Cite

View full text Add to dashboard Cite

Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

Data-driven, structure-based hyperelastic manifolds: A macro-micro-macro approach to reverse-engineer the chain behavior and perform efficient simulations of polymers

Amores

Benítez

Montáns

2020

Computers & Structures

Self Cite

View full text Add to dashboard Cite

“…The Cohen approximation (Cohen 1991) has a maximum relative error of 0.0494 and improved approximations with maximum relative errors approaching 10 -4 Arruda 2015, Jedynak 2018) have been reported with the recent work of Marchi and Arruda (2019) detailing approximations with maximum relative errors less than 10 -6 . Benitez and Montáns (2018) utilized discretization and interpolation via cubic splines to obtain highly accurate numerical results; a maximum relative error of the order of 10 -11 can be achieved with the use of 10,000 points.…”

Section: Introductionmentioning

confidence: 99%

Analytical approximations for the inverse Langevin function via linearization, error approximation, and iteration

Howard

2020

Rheol Acta

View full text Add to dashboard Cite

This paper details an analytical framework, based on an intermediate function, which facilitates analytical approximations for the inverse Langevin function—a function without an explicit analytical form. The approximations have relative error bounds that are typically much lower than those reported in the literature and which can be made arbitrarily small. Results include convergent series expansions in terms of polynomials and sinusoids which have modest relative error bounds and convergence properties but are convergent over the domain of the inverse Langevin function. An important advance is to use error approximations, and then iterative relationships, which allow simple initial approximations for the inverse Langevin function, with modest relative errors, to generate approximations with arbitrarily low relative errors. One example is that of an initial approximating function, with a relative error bound of 0.00969, which yields relative error bounds of 2.77 × 10−6 and 2.66 × 10−16 after the use of first-order error approximation and then first-order iteration. Functions with much lower error bounds are possible and are detailed. First- and second-order Taylor series can be used to simplify the error- and iteration-based approximations.

show abstract

“…Furthermore, it is difficult to accurately evaluate the inverse Langevin function because of the asymptotic behavior near locking. Thus, some studies are dedicated to this issue [ 38 , 39 , 40 , 41 , 42 , 43 , 44 ]. However, in the present case, it is relevant to separate the Gaussian linear zone from the nonlinear locking one.…”

Section: Non-affine Model With Three Parametersmentioning

confidence: 99%

Using the Mooney Space to Characterize the Non-Affine Behavior of Elastomers

Moreno-Corrales,

Sanz-Gómez,

Benítez

et al. 2024

Materials

Self Cite

View full text Add to dashboard Cite

The formulation of the entropic statistical theory and the related neo-Hookean model has been a major advance in the modeling of rubber-like materials, but the failure to explain some experimental observations such as the slope in Mooney plots resulted in hundreds of micromechanical and phenomenological models. The origin of the difficulties, the reason for the apparent need for the second invariant, and the reason for the relative success of models based on the Valanis–Landel decomposition have been recently explained. From that insight, a new micro–macro chain stretch connection using the stretch tensor (instead of the right Cauchy–Green deformation tensor) has been proposed and supported both theoretically and from experimental data. A simple three-parameter model using this connection has been suggested. The purpose of this work is to provide further insight into the model, to provide an analytical expression for the Gaussian contribution, and to provide a simple procedure to obtain the parameters from a tensile test using the Mooney space or the Mooney–Rivlin constants. From different papers, a wide variety of experimental tests on different materials and loading conditions have been selected to demonstrate that the simple model calibrated only from a tensile test provides accurate predictions for a wide variety of elastomers under different deformation levels and multiaxial patterns.

show abstract

A simple and efficient numerical procedure to compute the inverse Langevin function with high accuracy

Cited by 16 publications

References 28 publications

Data-driven, structure-based hyperelastic manifolds: A macro-micro-macro approach to reverse-engineer the chain behavior and perform efficient simulations of polymers

Data-driven, structure-based hyperelastic manifolds: A macro-micro-macro approach to reverse-engineer the chain behavior and perform efficient simulations of polymers

Analytical approximations for the inverse Langevin function via linearization, error approximation, and iteration

Using the Mooney Space to Characterize the Non-Affine Behavior of Elastomers

Contact Info

Product

Resources

About