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Finite Elements for the Treatment of the Inextensibility Constraint

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Finite Elements for the Treatment of the Inextensibility Constraint Comput Mech DOI 10.1007/s00466-017-1437-9 ORIGINAL PAPER Fiber-reinforced materials: finite elements for the treatment of the inextensibility constraint Ferdinando Auricchio1 · Giulia Scalet1 · Peter Wriggers2 Received: 5 April 2017 / Accepted: 29 May 2017 © Springer-Verlag GmbH Germany 2017 Abstract The present paper proposes a numerical frame- 1 Introduction work for the analysis of problems involving fiber-reinforced anisotropic materials. Specifically, isotropic linear elastic The use of fiber-reinforcements for the development of novel solids, reinforced by a single family of inextensible fibers, and efficient materials can be traced back to the 1960s are considered. The kinematic constraint equation of inex- and, to date, it is an active research topic. Such materi- tensibility in the fiber direction leads to the presence of als are composed of a matrix, reinforced by one or more an undetermined fiber stress in the constitutive equations. families of fibers which are systematically arranged in the To avoid locking-phenomena in the numerical solution due matrix itself. The interest towards fiber-reinforced mate- to the presence of the constraint, mixed finite elements rials stems from the mechanical properties of the fibers, based on the Lagrange multiplier, perturbed Lagrangian, and which offer a significant increase in structural efficiency. penalty method are proposed. Several boundary-value prob- Particularly, these materials possess high resistance and lems under plane strain conditions are solved and numerical stiffness, despite the low specific weight, together with an results are compared to analytical solutions, whenever the anisotropic mechanical behavior determined by the fiber derivation is possible. The performed simulations allow to distribution. Typical examples are carbon fiber-reinforced assess the performance of the proposed finite elements and epoxy resins, boron fiber-reinforced aluminium, and nylon- to discuss several features of the developed formulations or steel-reinforced rubbers [1]. Thanks to such important concerning the effective approximation for the displacement features, fiber-reinforced materials are largely employed in and fiber stress fields, mesh convergence, and sensitivity to several fields as mechanical components (e.g., soft actuators), penalty parameters. aeronautical and aerospace parts (e.g., fuselage, radomes), civil engineering structures (e.g., pile foundations massif), Keywords Fiber-reinforced material · Inextensible fibers · or home products (e.g., hoses), and in the domain of biolog- Anisotropic material · Finite element analysis · Mixed finite ical materials studied in biomechanics (e.g., bones, vegetal element method tissues) [2]. Given the widespread use of fiber-reinforced materials, a B Giulia Scalet comprehensive understanding of their behavior is fundamen- [email protected] tal for design and production purposes. Extensive theoretical Ferdinando Auricchio surveys focusing on the mechanics of fiber-reinforced solids [email protected] are available from the literature, e.g., [1,3–5], in addition to Peter Wriggers works investigating the wave propagation problem, e.g., [6]. [email protected] A common approach is to consider the fiber-reinforced material as an elastic continuum which is internally con- 1 Department of Civil Engineering and Architecture, University of Pavia, via Ferrata 3, 27100 Pavia, Italy strained. In particular, constraints of incompressibility and/or inextensibility have been treated theoretically in both small 2 Institut für Kontinuumsmechanik, Gottfried Wilhelm Leibniz Universität Hannover, Appelstraße 11, 30167 Hannover, and large strains [1,7–17]. The presence of these constraints Germany in the formulation determines new features which are not 123 Comput Mech encountered in classical elasticity and which require special the penalty parameter is addressed. The simulations pro- attention in the theoretical and computational formulation. vide interesting results to understand the consequences of From the numerical point of view, the presence of con- constraining the extensional deformation in the fiber direc- straints leads to locking-phenomena in the numerical solution tion. and appropriate numerical methods have to be adopted. The paper is organized as follows. Section 2 presents an Specifically, in the finite element framework, standard dis- overview of the continuum problem. Then, Sect.3 presents placement interpolations cannot be used and ad-hoc inter- the proposed constraint formulations, while Sect. 4 derives polations need to be adopted [18]. This has been largely the related finite element formulations. Section 5 presents discussed in solid and fluid mechanics in case of incom- the numerical results. Finally, summary and conclusions are pressibility or contact at both small and finite strains [19]. given in Sect. 6. For incompressible and inextensible fiber-reinforced mate- rials mixed finite elements based on a penalty formulation have been proposed for small deformations and tested on sim- 2 Continuum formulation ple beam problems under plane stress in [13]. Simacek and Kaliakin [20] developed the governing equations for incom- This section is dedicated to a brief overview of the continuum pressible plates exhibiting a direction of inextensibility. The equations for a three-dimensional body made of isotropic paper by Hayes and Horgan [9] presents a formulation of a linear elastic material reinforced by a single family of parallel class of mixed boundary-value problems for isotropic and inextensible fibers. The reader is referred to extensive surveys transversely isotropic materials and establishes a uniqueness as [1,3–5] for further details. theorem which provides necessary and sufficient conditions for uniqueness of solution to the mixed problems for homo- geneous materials. 2.1 Kinematics and equilibrium equations The present work is motivated by the current interest in the formulation of novel modeling and numerical approaches We consider a three-dimensional deformable continuous ⊂ R3 for the analysis of constrained fiber-reinforced materials. The body, consisting of a domain and a boundary ∂ ∂ ∂ aim of this work is to develop a computational framework , subdivided in turn into u and t , representing for the phenomenological theory of isotropic linear elastic respectively the boundary where displacements and surface ∂ ∪ ∂ = ∂ materials reinforced by a single family of parallel inexten- tractions are imposed, such that u t . sible fibers. These materials offer the framework to easily Assuming small strains, the problem is expressed by the understand the physical problem and to analyze it from the following equilibrium equation: theoretical and numerical point of view. In this way we are able to propose a complete approach to the analysis of divσ + b = 0 in (1) these materials, that can be extended to more complex situa- tions. We recall, in fact, that finite strain approaches, finding where σ is the symmetric Cauchy stress tensor and b is the applications in soft tissue biomechanics and in the mechan- body force per unit volume vector. The symmetric infinites- ics of fiber-reinforced rubber-like materials, have been also imal strain tensor ε is defined as follows: largely studied, e.g., [21–25], but are out of the present scope. 1 ε =∇su = [∇u + (∇u)T ] (2) According to our purpose, we carefully investigate differ- 2 ent novel approaches to fulfill the inextensibility constraint, particularly, the method of Lagrange multipliers and a u being the displacement vector and ∇s the symmetric relaxed version, i.e., the perturbed Lagrangian method. To gradient operator. We introduce the following boundary con- develop the mixed finite elements, different ansatz func- ditions: tions for both the displacement and the Lagrange multi- plier fields are considered. As a further comparison, the u = u on ∂u penalty method is adopted to fulfill approximately the (3) σ = ∂ constraint. Several boundary-value problems under plane n t on t strain conditions are considered to analyze the perfor- mance of the proposed formulations. For each test, dif- where u and t are the assigned displacement and surface force ferent fiber directions and mesh refinements are consid- field, respectively, and n is the outward normal vector. ered. The proposed finite element formulations are dis- The formulation is completed by the constitutive and fiber cussed and compared to analytical solutions, computed constraint equations, as described in the following subsec- whenever possible. Moreover, the sensitivity analysis to tions. 123 Comput Mech 2.2 Constitutive equation Y We focus on isotropic linear elastic matrix materials for which the strain energy W iso is defined as: iso W (ε) = W(I1(ε), I2(ε)) (4) where I1 and I2 are the first and second invariant of the strain tensor, defined together with their derivatives as: I = tr(ε), ∂ I /∂ε = I 1 1 1 2 2 I2 = tr(ε) − tr(ε ) ,∂I2/∂ε = I1I − ε 2 Fig. 1 Schematic representation of a two-dimensional continuum body made of isotropic linear elastic material reinforced by a single I being the second order identity tensor. The energy can be family of parallel fibers whose direction is described by the unit vec- defined as follows: tor a 1 W iso(ε) = λ(tr(ε))2 + μtr(ε2) (5) 2 We assume that the material does not extend in the fiber direction a, i.e., the fibers are inextensible. This means that where λ and μ are the Lamé constants. the kinematic constraint is the following: We obtain the following expression for the stress tensor σ : a · εa = M : ε = tr(ε · M) = 0(10) ∂W iso σ = = λI I + 2με (6) ∂ε 1 3 Constraint formulation and for the fourth-order elasticity tensor D: To treat the inextensible constraint (10), we propose to ∂σ adopt the following approaches: (1) the method of Lagrange D = = λ(I ⊗ I) + μII ∂ε 2 (7) multiplier, (2) a relaxed version based on the perturbed Lagrangian, and (3) a penalty method. This implies to define where ⊗ represents the tensor outer product and the total strain energy W tot in the following general form: tot iso f I I + I ˆ I W = W + W (11) II = (8) 2 where W iso is the isotropic strain energy defined in Eq. (5), f with and ˆ being defined as (A B)C = ACBT and while W is the strain energy related to the inextensibil- (A ˆ B)C = ACT BT .
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