A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity

Linder, Christian; Tkachuk, Mykola; Miehé, Christian

doi:10.1016/j.jmps.2011.05.005

Cited by 119 publications

(53 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…However, this approach also has a very high computational cost and has not been extended to anisotropic materials. Finally, affine continuum models have been modified to include a non-affinity parameter with respect to the affine stretch ( Itskov et al, 2010;Linder et al, 2011;Unterberger et al, 2013 ).…”

Section: Fiber Network Modelmentioning

confidence: 99%

A multiscale micromechanical model of needlepunched nonwoven fabrics

Martínez-Hergueta

Ridruejo

González

et al. 2016

International Journal of Solids and Structures

View full text Add to dashboard Cite

Section: Fiber Network Modelmentioning

confidence: 99%

A multiscale micromechanical model of needlepunched nonwoven fabrics

Martínez-Hergueta

Ridruejo

González

et al. 2016

International Journal of Solids and Structures

View full text Add to dashboard Cite

“…Elastomers, hydrogels and soft biological tissues, non-woven fabrics are all on the microscopic level composed of elongated one-dimensional elements, the fibers. When these soft materials are subjected to a macroscopic strain, the underlying microstructure undergoes a peculiar deformation [1]. One can expect that in the case when the material is stretched in a certain direction the fibers initially oriented closely to it will accordingly elongate.…”

Section: The Maximal Advance Path Constraintmentioning

confidence: 99%

Homogenization of random elastic networks with non‐affine kinematics

Tkachuk

Linder

2012

Proc Appl Math and Mech

Self Cite

View full text Add to dashboard Cite

This contribution concerns the mechanics of materials with random network microstructures. It develops a general homogenization approach that allows to link the microscopic deformation of fibers in the disordered network with the macroscopic response of the continuous solid. This link is established by a novel micro-macro relation based on the kinematics of maximal advance paths that constrains the unknown microscopic stretch of fibers with respect to the macroscopic strain. This relation accounts for the topology of the network, in particular, its connectivity and takes for the tetrafunctional networks a clearly interpretable tensorial form. In line with the principle of the minimum averaged free energy the elastic response of the network is obtained by the relaxation of the variable fiber stretch subjected to the kinematic constraint. Network microstructures are commonly encountered in materials of artificial as well as natural origin. Elastomers, hydrogels and soft biological tissues, non-woven fabrics are all on the microscopic level composed of elongated one-dimensional elements, the fibers. When these soft materials are subjected to a macroscopic strain, the underlying microstructure undergoes a peculiar deformation [1]. One can expect that in the case when the material is stretched in a certain direction the fibers initially oriented closely to it will accordingly elongate. A straightforward assumption of affine kinematics is taken by the classical models of rubber elasticity [2,3] and has again been recently adopted for the modeling of biological gels [4]. Nevertheless, as pointed out in [5], the affine stretch assumption is not generally valid and does not explain the experimentally observable relation of the elastic stress upon the strain like the difference in stiffening of rubber-like materials in uniaxial and equibiaxial tension. The the non-affine microsphere model proposed in [5] suggests certain variations of stretch constrained by a specific relation to the macroscopic deformation. The present work is inspired by the approach to the homogenization of elastic networks developed in [5]. The main contribution is the novel construction of the kinematic constraints [6,7] and an averaged description of the network. The fibers are identified by their initial orientation λ 0 uniformly distributed over the unit sphere S 0 . The deformation of the network is described by a vector-valued stretch function λ(λ 0 ). The variation of this unknown microstretch λ is constrained with respect to the macroscopic strain by a kinematic relation established by considering the maximal advance paths. At each junction on the path attached to f fibers with the orientations λ i 0 as shown in Fig. 1 it propagates by the one of the remaining f − 1 fibers with the maximal advance in a certain direction l 0 . Such fibers in the path have orientation vectors λ m 0 with the distributionThe higher is the functionality of the network the more aligned is the path so that the average advance in the path ξ m = (f − 2)/f with ξ m = λ ...

show abstract

“…In the paper by Hossain and Steinmann (2012), the authors summarized various micromechanical and phenomenological models. Other interesting review articles are written by Boyce and Arruda (2000), Elias-Zuniga and Beatty (2002), Böl and Reese (2006), Gillibert et al (2010), Kroon (2011), Linder et al (2011), Gloria et al (2012 and Jedynak (2013, 2014). These papers present also a significant contribution to the modeling of rubber-like materials.…”

Section: Introductionmentioning

confidence: 99%

Approximation of the inverse Langevin function revisited

Jedynak

2014

Rheol Acta

View full text Add to dashboard Cite

The main purpose of this paper is to provide an easy-to-use approximation formula for the inverse Langevin function. The mathematical complexity of this function makes it unfeasible for an analytical manipulation and inconvenient for computer simulation. This situation has motivated a series of papers directed on its approximation. The best known solution is given by Cohen. It is used in a lot of statistically based models of rubber-like materials. The formula is derived from rounded Padé approximation [3/2]. The main idea of the presented approach in this paper relies on improvement of the precision of approximation formula for the inverse Langevin function by using multipoint Padé approximation method. We focused our study strongly on obtaining a simple and accurate approximation. It is assumed that the proposed approximation formula may be considered a useful tool for verification of the results obtained in other ways. Our results are supported by investigating several numerical examples. The paper also presents a few applications of computer software named Mathematica which can be used to calculate symbolically one point Padé approximants and numerically multipoint Padé approximants. Using this software, we showed also how to compute higher order derivatives of the inverse function in a simple and elegant way. This issue was discussed by Itskov et al.

show abstract

A micromechanically motivated diffusion-based transient network model and its incorporation into finite rubber viscoelasticity

Cited by 119 publications

References 61 publications

A multiscale micromechanical model of needlepunched nonwoven fabrics

A multiscale micromechanical model of needlepunched nonwoven fabrics

Homogenization of random elastic networks with non‐affine kinematics

Approximation of the inverse Langevin function revisited

Contact Info

Product

Resources

About