Large deformation near a crack tip in a fiber-reinforced neo-Hookean sheet with discrete and continuous distributions of fiber orientations

https://doi.org/10.1016/j.tafmec.2021.103020Get rights and content

Highlights

  • The crack tip behavior of Neo-Hookean fiber-reinforced composite sheets is analyzed.

  • Asymptotic isotropy is discovered for specific fiber orientation distributions.

  • Asymptotic universal one-to-one anisotropic-to-isotropic mapping is found.

Abstract

We consider crack tip deformations under plane stress conditions of a Neo-Hookean sheet reinforced by Neo-Hookean fibers, whose orientation and elastic properties are described by discrete and continuous spatial distributions. The mechanical behavior of the composite is described in terms of the first and fourth invariant of the right Cauchy-Green tensor following Guo et al. [1], [2], [3]. The crack tip integrals developed in Liu and Moran [4], [5] are used to determine the coefficients of the crack tip asymptotic expansion. The von Mises distribution of orientation is analyzed. The existence of a regime of isotropic behavior, which we call asymptotic isotropy, in the region of dominance of the asymptotic fields is established for certain combinations of fiber orientations. Finally, the possibility to construct an asymptotic universal one-to-one mapping between anisotropic and isotropic Neo-Hookean plane stress response at the crack tip is discussed.

Introduction

A large class of both biological and synthetic bioinspired soft materials can be modeled as composed of a hyperelastic matrix reinforced by one or more families of hyperelastic fibers. Each family of fibers is characterized by their specific elastic properties and an orientation in space, and the overall variation in elastic properties and orientations is expressed in terms of discrete or continuous distributions. We thus refer to this class of materials as hyperelastic materials reinforced by a distribution of fibers. Examples are numerous: articular cartilage [6], human cornea [7], aneurysms [8], human brain arteries [9], muscle tissue [10], the aorta [11], aortic heart valves [12], hydrogels [13], scaffolding materials to grow load-bearing soft tissues [14], ink-printed hydrogels reinforced by cellulose fibrils [15].

Focusing on the mechanical response of arterial layers, the importance of the presence of distributions of fibers in soft materials is emphasized in [16]. In the latter, the authors include the effect of the distributed fiber orientations through a generalized structure tensor in the framework of anisotropic large-deformation elasticity. By means of numerical experiments, they show that the mechanical response of the soft-composite is strongly dependent of the fiber distribution. A highly dispersed distribution of orientations translates into a stiffer material less dependent on the main direction of alignment of the fibers, whereas a distribution concentrated around the main direction causes a highly direction-dependent response. They furthermore observe that, in the case of a single fiber orientation, fibers undergo large rotations before carrying load to align with the direction of the applied load. They thus conclude that the study of ideally aligned fibers, i.e. all parallel to one direction, is not sufficient to predict the mechanical response of fiber-reinforced soft materials. In [16], the authors adopt a continuum mechanics approach and use the strain-energy superposition assumption. Based on the volumetric-distortional decomposition of [17], [18], the strain-energy superposition assumption allows the additive split of the strain energy into a purely volumetric term and a purely distortional term [19] and it is valid for incompressible and nearly-incompressible materials [20]. In the case of fiber-reinforced soft composites, the matrix may be considered as the driver of volumetric deformations whereas fibers are responsible for the isochoric distortional part of the deformation. Thus, the contribution to the strain energy density of the two phases can be additively split by identifying the matrix contribution with the volumetric term and the fiber contribution with the distortial term of the volumetric-distortional decomposition [6], [21]. In the presence of multiple families of fibers located in a finite and discrete set of orientations and properties, the contributions of each family of fibers (meant as a set of fibers with one specific orientation and the same set of mechanical properties) are summed up to provide the distortional part of the strain-energy density [22]. In the case of a continuous distribution of orientations and/or mechanical properties, the finite sum becomes a distribution-weighted integral [6], [16], [21], [23]. Further details on the continuum mechanics approach to modeling fiber-reinforced soft materials with discrete or continuous distribution of orientations or properties can be found in the reviews of [24], [25]. Efforts at modeling the failure behavior have proposed a number of approaches, namely limiting failure criteria [26], softening models [27], micro-structural damage mechanisms [28], [29], cohesive surfaces [30], internal damage variables [31], [32], permanent deformations [33], statistical approaches [34], bond kinetics [35], phase field modeling [36] (see [37] for a thorough review on the subject).

It appears however that the analysis of the behavior of the elastic fields at the crack tip remains unaddressed. In [4], a set of crack tip contour integrals is presented to evaluate the parameters determining the stress and deformation fields at the crack tip in an isotropic hyperelastic material under conditions of plane stress or plane strain. The integrals are then extended in [5] to anisotropic fiber-reinforced soft materials with fibers oriented in one or two directions. Building upon this previous work, the crack tip contour integrals approach is applied in this paper to the analysis of the crack tip large deformation fields in Neo-Hookean sheets reinforced by discrete and continuous distributions of fibers under conditions of plane stress.

This paper is organized as follows. In Section 2, the governing equations of a Neo-Hookean matrix reinforced by discrete and continuous distributions of fibers are introduced for general 3D deformation fields. The model is then restricted to plane stress conditions and its main equations are provided in Section 3. The asymptotic boundary value problem and its solution are introduced in Section 4. Section 5 briefly introduces the crack tip integrals used for the evaluation of the coefficients of the asymptotic expansion derived in Section 4. Numerical results for discrete and continuous distributions of fibers are presented and analyzed in Section 6. Conclusions are finally discussed in Section 7.

Section snippets

Governing equations

In the presence of multiple families of fibers, their properties and orientation are described by a distribution function f(a0,κ). In general, for a given family of fibers the vectora0=sinϕcosψe1+sinϕsinψe2+cosϕe3represents the fiber orientation in the material reference (undeformed) configuration in a three-dimensional space spanned by the unit vectors e1,e2,e3 using the spherical coordinates ϕ and θ (see Fig. 1). On the other hand, the parameter κ in the distribution function f(a0,κ)

Plane stress

Given that many applications of highly stretchable fiber-reinforced polymers are in the form of thin sheets [39], [40], [41] and that tissues as well can be modeled as such [16], [23], [25], we restrict our analysis to this structural configuration and thus, following [42], to plane stress conditions. It is worth to remark here that the assumption of plane stress is key for all the derivations and results presented in this work. Extension to axisymmetric and/or generic 3D configurations is thus

Asymptotic boundary value problem

We analyze the case of an edge crack lying on the x1 axis, as depicted in Fig. 1b1

Asymptotically path-independent interaction integral

Following [5], it is possible to evaluate all the crack tip parameters by introducing asymptotically path-independent interaction integrals of the typeIyαxi=limΓ0ΓPαjy,jauxn1-Pαjy,1auxnj-Pjauxyα,1njdΓ,α=1,2.

In the previous equation, yaux is an auxiliary field satisfying the asymptotic boundary value equations and Pjaux is the conjugated auxiliary stress evaluated as Pjaux=y,kauxDkj, where Dkj are the components of the matrixD=c55c45c45c44.

It has been shown [5] that the latter is related to

FEM model

The crack tip fields are evaluated for an edge crack in an initially rectangular strip. The strip has dimensions L0×H0 with L0=4H0 and the initial length of the crack is 0.25L0. The strip is subject to far-field Mode I boundary conditions of the type y1=x1|x2=±H02,y2=λSx2|x2=±H02, where λS is the far-field stretch in the x2 direction. The stiffness of the matrix μ is equal to 1 MPa while the fiber-to-matrix stiffness ratio is a parameter of the analysis. The governing Eq. (18) are solved using

Conclusions

In this paper, we consider the large deformations at the crack tip of Neo-Hookean fiber reinforced composite sheet, where the orientation and elastic properties of the reinforcement are described by discrete and continuous spatial distributions. We adopt the material behavior introduced in [1], [2], [3] and use the crack tip integrals developed in [4], [5] to evaluate the coefficients of the asymptotic expansion of the elastic fields at the crack tip. It is observed that the canonical

CRediT authorship contribution statement

Luca Di Stasioa: Conceptualization, Data curation, Formal analysis, Methodology, Visualization, Writing - original draft, Writing - review & editing. Yin Liua: Conceptualization, Data curation, Formal analysis, Methodology, Software, Validation, Visualization, Writing - review & editing. Brian Morana: Conceptualization, Data curation, Formal analysis, Methodology, Funding acquisition, Project administration, Resources, Supervision, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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  • Cited by (3)

    • Crack tip fields in a fiber-reinforced hyperelastic sheet: Competing roles of fiber and matrix stiffening

      2022, Mechanics Research Communications
      Citation Excerpt :

      Widely-used models for the fibers include a composite based model (neo-Hookean fibers) [4], a standard reinforcing model [5–7] and the Holzapfel-Gasser-Ogden (HGO) model [8]. For the crack tip fields in fiber-reinforced hyperelastic materials, the neo-Hookean sheets reinforced by the fibers characterized by the composite-based model and standard reinforced model have been studied [9–11]. The effect of multiple families of nonlinear fibers on the crack tip fields was also investigated [12].

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