Elsevier

Computers & Structures

Volume 240, November 2020, 106350
Computers & Structures

An orthotropic multi-surface damage-plasticity FE-formulation for wood: Part I – Constitutive model

https://doi.org/10.1016/j.compstruc.2020.106350Get rights and content

Highlights

  • The present study formulates a novel FE constitutive model with multi-surface plasticity and anisotropic multi-surface damage.

  • The model correctly captures the sensitivity of the material to loading-directionality, and ductile and brittle phenomena.

  • We define a set of plasticity and damage activation criteria and three damage variables corresponding to distinct stress states.

  • We formulate the algorithmic treatment of the relevant tangent constitutive matrix.

  • We show that the proposed formulation performs better than analogous existing models.

Abstract

Restoration of ancient wooden beams and design of smart wood-based structures are gaining an increasing interest in building industry. In this context, the computational challenge is to develop numerical constitutive models that account for the complex and strongly non-linear behaviour of wood. Wood is a natural composite exhibiting pronounced orthotropic behaviour, and markedly different properties along the parallel and transverse-to-the-fiber directions. It displays a strongly non-linear mechanical behavior, almost elasto-plastic at compressive loadings and elasto-damaging when subjected to tensile and shearing stresses. We devise here a novel constitutive model for wood with a multi-surface failure domain resulting from a set of plastic laws for compressive failure modes and orthotropic damage laws for tensile/shear failure modes. The advantage over existing formulations is that the coexistence of anisotropic damage and plasticity constitutive laws allows to correctly capture brittle failure induced by strain localization as well as the possible occurrence of ductile plastic behavior. Furthermore, the present contribution shows how to numerically treat the simultaneous presence of anisotropic damage and plasticity in a general algorithmic multi-surface framework. It is shown that the obtained numerical results satisfactorily fit to experimental data.

Introduction

Wood is increasingly being used in structural engineering, for instance, for the restoration of ancient deteriorated wooden beams or to realize advanced wood-based materials for the design industry, one striking example being the newborn transparent wood [1]. It displays high versatility, easy workability, and worth noting aesthetic and insulating properties. Its complex cellular-like micro-structure makes it an inhomogeneous, anisotropic and porous material with moisture-, temperature- and time-dependent behaviour [2]. Its mechanical behavior ranges from ductile to brittle regimes, depending on the loading and stress state [3], [4]. This study aims to develop a novel constitutive model that consistently captures the whole structural path of wood structures until failure.

Common features of state-of-the-art FE-models for wood structures are the assumption of material orthotropy and the recognition that both structural behavior and failure modes change depending on direction and sign of the dominant stress. In the present context, three main groups of studies can be identified: plasticity models, damage models and models combining plastic and damage constitutive laws. We briefly review the main contributions pertaining to these three classes with a special focus on wood-dedicated FE-models.

Among the contributions inherent to the first group, the earliest studies are based on orthotropic single-surface plasticity models [5], [6], [7], and combine the classical flow theory of plasticity with anisotropic failure criteria originally proposed for composites [8], [9], [10]. Single-surface plasticity models fail to capture the dependency of the structural response on load directionality. This motivated the development of multi-surface plasticity models [11], [12], [13], [14], [15], where the global yield failure surface results from multiple yield surfaces, each one associated with a specific ductile or brittle failure mode. To describe late cracking, multi-surface plasticity models are eventually coupled to cohesive-zone-models or interface elements [14], [16].

The second group of models emanate from the application of Continuum Damage Mechanics (CDM) to laminates and fiber-composites [17], [18], [19], [20], [21], [22], [23]. In this context, constitutive anisotropic damage tensors are often deduced with the aid of two tools: the concept of effective stress [24], [25] and the adoption of an equivalence principle between the strain or the strain-energy characterizing a fictitious undamaged material and the truly damaged material [18], [17]. Usually, anisotropic damage is associated with a second or fourth-order tensor [17], [26], [27], [28], [29], [19], [30]. This significantly complicates the development of the constitutive theory with respect to isotropic damage [21]. Instead of the canonical derivation of the damage constitutive tensor through a strict imposition of equivalence principles, other contributions devoted to the FE-analysis of laminated and fiber-composites [20], [22], [23] pragmatically deduce the damaged constitutive operator on a phenomenological basis. Among the studies that follow this pragmatic approach to develop FE-models specific for wood, we mention the multi-surface CDM model [31] introducing eight brittle and ductile failure modes governed by six damage variables. Though successfully applied in certain circumstances [32], [33], CDM formulations cannot consistently reproduce the plasticity-induced ductile failure such as that experimentally detectable in the presence of bolted joints [34], [35].

Within the context of elasto-damaging-plastic FE-formulations for wood, recent contributions are the model [36] for cyclic loads and [37] for large plastic deformations, which make use of elasto-damaging-plastic orthotropic constitutive laws. However, they are restricted to isotropic damage, thus failing to capture the orthotropy-dependent structure of the damaging process in wood.

The present study can be ascribed to the category of FE-models for wood combining plasticity and damage. It proposes a novel elasto-damaging plastic constitutive model that combines all the relevant advantages of the aforementioned models, such as the multi-surface feature that allows to take into account loading-directionality, the possibility to consistently capture ductile and brittle failure, while retaining the orthotropy of both the elastic constitutive tensor and the damaging process. This results in a set of plastic activation criteria and specific damage variables for distinct stress states. The present plasticity constitutive law draws inspiration from the multi-surface plasticity model [11], [12], [13], while the orthotropic Continuum Damage Model we have developed is an extension of the formulations [20], [22], [23], [38]. We emphasize that the algorithmic treatment of the proposed constitutive model is expectedly more complex than classic FE-oriented anisotropic damage-plasticity models [27], [28], [29] since, here, the tangent constitutive matrix has to comply with multi-surface criteria.

The remainder of this paper is structured as follows. Section 2 presents the basic statements of the proposed elastic–plastic multi-surface model with anisotropic damage, and the continuum damage mechanics settings, with special focus on plane stress for orthotropic materials. To simplify the development and the derivation of the constitutive law, the present contribution is restricted to the plane stress case. In this circumstance, we adopt two plastic activation functions and three damage variables each one evolving according to a non-associative exponential law drawn from previous authors’papers on elastodamaging FE-models for concrete [39]. The multi-surface plasticity model [13], we drew inspiration from, is three-dimensional. However, for the sake of simplicity, the present constitutive model is restricted to stress states pertaining to the radial and the longitudinal directions. In order to extend the proposed constitutive model to three-dimensional stress states, new multi-dimensional damage evolution laws should be devised consistently with the experimentally observed behavior. For instance, it has been observed [40] that the behavior in the radial-tangential plane is polar orthotropic and exhibits a marked shear coupling effect.

Section 3 illustrates the main features of the proposed multi-surface failure criterion. We purposely distinguish between ductile and brittle modes, the former ones being governed by a plastic activation criteria, and the latter modes by damage laws. Moreover, each failure mode is governed by consistency laws that change depending on the stress state, namely several activation functions are available that differentiate depending on values taken from the longitudinal, radial and shear stress components. In Section 4, we formulate the constitutive laws and the consistency loading/unloading conditions in the rate form. This allows us to introduce the elasto-damage and elasto-damage plastic tangent tensors. The algorithmic solution of the problem of the integration of the rate constitutive laws is investigated in Section 5. The salient operational steps are displayed in suitable boxes where the purely computational aspects are shown. The procedure extends to the current case the classic closest point projection algorithm for multi-surface plasticity [41]. Section 6 is devoted to applications of the proposed numerical framework to a bolted joint test for which the simultaneous availability of ductile and brittle modes in the constitutive FE-model reveals particularly crucial. Results have been obtained with a parallelized Fortran code originally developed by the Authors.

Notation. Table 1 collects the most relevant symbols used throughout the paper. The main theoretical laws are formulated adopting tensor notation. However, Voigt notation will be purposely used to simplify the mathematical expression of the salient entities and exemplify certain operational steps.

Section snippets

Orthotropic elasto-damaging plastic constitutive law

The aim of this section is to formulate an elasto-plastic multi-surface model with anisotropic damage for wood FE-modeling. We aim to retaining the highest simplicity of the constitutive law to be derived, especially in view of its FE-implementation.

Wood is commonly regarded as an orthotropic fiber-composite with three directions of symmetry. Moreover, it is possible to assume that the microcracks inducing damage are almost parallel or perpendicular to the fibres and that the damage process

Multi-surface failure criterion

We formulate here a unified general expression for the generic failure surface. To simulate the relevant failure modes, we adopt the multi-surface plasticity model [11] for ductile failure. On the other hand, quasi-brittle failure modes, such as failure modes in tension and shear, are modelled by extending the multi-surface orthotropic damage laws [22] for composites, and further developed [23]. In order to model the different behaviour under compressive loading along the perpendicular-to-fiber

Rate form of the multi-surface anisotropic elasto-plastic-damage constitutive model

The present section addresses the problem of the numerical integration of the proposed elasto-plastic-damage stress strain law in the presence of orthotropic elasticity and damage where the failure surface results from the envelop of the five failure surfaces Fi. The numerical procedure extends Simo’s elasto-plastic-damage operator-split framework [27], [28] as described in Box 1.

Box 1: Elasto-plastic damage scheme
Elastic partPlastic partDamage part
ε̇=su̇ε̇=0ε̇=0
ḋ=0ḋ=0ḋθ=λ̇iθdGiθτ¯iθτ¯̇iθ
r

Algorithmic multi-surface elasto-damaging plastic anisotropic law

The loading history is associated with the fictitious time 0,T and further subdivided into N non overlapping intervals [tn,tn+1]. Within each time increment, we iteratively integrate the equilibrium equations along with the constitutive equations. The process is in fact strain-driven as the history of strains tε=q(t) is assumed to be known. The algorithmic treatment of the elasto-plastic part of the constitutive model is based on the classic two-steps scheme which consists of the

Applications

In the present section, first, we show how the proposed constitutive model performs in the presence of simple loading cases (Section 6.1). Then, in Section 6.2, we simulate an engineering test, named “embedment test”, especially conceived to detect the strength of bolted joints.

Conclusions

We have devised a novel anisotropic constitutive model whose peculiarity is that it is based on anisotropic damage and that the failure domain is built from multiple damaging and plastic surfaces. This constitutive model has been shown to satisfactorily capture the complex non-linear behavior and its strong dependence on loading-directionality. We have also illustrated the salient aspects of the algorithmic formulation inherent to the proposed multi-surface criterium.

With respect to

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.

Acknowledgement

The first and second authors gratefully acknowledge the support of the research fund PRIN: Progetti di Ricerca di Rilevante Interesse Nazionale (Bando 2015) Prot.2015LYYXA8.

References (53)

  • O. Allix et al.

    A delay damage mesomodel of laminates under dynamic loading: basic aspects and identification issues

    Comput Struct

    (2003)
  • P. Maimí et al.

    A continuum damage model for composite laminates: Part I - Constitutive model

    Mech Mater

    (2007)
  • J. Simo et al.

    Strain-and stress-based continuum damage models – I. Formulation

    Int J Solids Struct

    (1987)
  • J. Simo et al.

    Strain-and stress-based continuum damage models – II

    Comput Aspects, Math Comput Model

    (1987)
  • J. Ju

    On energy-based coupled elastoplastic damage theories: Constitutive modeling and computational aspects

    Int J Solids Struct

    (1989)
  • M. Gharib et al.

    Three-dimensional constitutive modelling of arbitrarily orientated timber based on continuum damage mechanics

    Finite Elem Anal Des

    (2017)
  • N. Orlando et al.

    End-repair of timber beams with laterally-loaded glued-in rods: Experimental trials and failure prediction through modelling

    Constr Build Mater

    (2019)
  • R. Rowlands et al.

    Single-and multiple-bolted joints in orthotropic materials

    Composites

    (1982)
  • L.F. Sirumbal-Zapata et al.

    A three-dimensional plasticity-damage constitutive model for timber under cyclic loads

    Comput Struct

    (2018)
  • A. Khennane et al.

    Numerical modelling of ductile damage evolution in tensile and bending tests of timber structures

    Mech Mater

    (2014)
  • P. Maimí et al.

    A continuum damage model for composite laminates: Part II - Computational implementation and validation

    Mech Mater

    (2007)
  • E. Benvenuti

    Mesh-size-objective XFEM for regularized continuous–discontinuous transition

    Finite Elem Anal Des

    (2011)
  • A. Shipsha et al.

    Shear coupling effects on stress and strain distributions in wood subjected to transverse compression

    Compos Sci Technol

    (2007)
  • P. Ladevèze et al.

    A mesomodel for localisation and damage computation in laminates

    Comput Methods Appl Mech Eng

    (2000)
  • U. Cicekli et al.

    A plasticity and anisotropic damage model for plain concrete

    Int J Plast

    (2007)
  • K. Persson

    Micromechanical modelling of wood and fibre properties

    (2000)
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