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Journal ArticleDOI

A localized mapped damage model for orthotropic materials

01 Jul 2014-Engineering Fracture Mechanics (Pergamon)-Vol. 124, pp 196-216

AbstractThis paper presents an implicit orthotropic model based on the Continuum Damage Mechanics isotropic models. A mapping relationship is established between the behaviour of the anisotropic material and that of an isotropic one. The proposed model is used to simulate the failure loci of common orthotropic materials, such as masonry, fibre-reinforced composites and wood. The damage model is combined with a crack-tracking technique to reproduce the propagation of localized cracks in the discrete FE problem. The proposed numerical model is used to simulate the mixed mode fracture in masonry members with different orientations of the brick layers.

Topics: Orthotropic material (59%), Masonry (54%), Isotropy (52%)

Summary (4 min read)

1. Introduction

  • The mechanical behaviour of anisotropic materials involves properties that vary from point to point, due to composite or heterogeneous nature, type and arrangement of constituents, presence of different phases or material defects.
  • The introduction of local or global crack-tracking techniques into the framework of standard finite elements and constitutive models [25,26] has revealed to be a satisfactory solution to some of the major drawbacks of the classical Smeared Crack Approach (SCA) [27].
  • These features of the method allow the analyst to avoid the aforementioned problems usually found in classical SCA, without increasing - 4 - excessively the implementation effort or the computational cost.
  • The model is able to predict the failure load and the cracking path in orthotropic materials subject to complex stress states.

2. Mapped Damage Model

  • The orthotropic mapping of CDM constitutive laws has been presented in References [1,24,25].
  • The basics of the method are recovered and its thermodynamic consistency is demonstrated.
  • The flexibility of the procedure for the application to generic orthotropic materials is stressed.

2.1 Definition of the Space Transformation Tensors

  • The method is based on assuming that the real anisotropic space of stresses σ and the conjugate space of strains ε have their respective image in two mapped isotropic spaces of stresses *σ and strains *ε , respectively .
  • It is important to note that the procedure may be extended to the 3-dimensional case, at the cost of providing the necessary additional strength parameters.
  • In order to circumvent this limitation, a more refined form of the stress transformation tensor was proposed by Oller et al. [23], making use of a “shape adjustment tensor”, whose purpose is to adjust correctly the isotropic criterion to the desired orthotropic one.
  • It is worth noting that the isotropic solid properties, i.e. *f and elastic constants in tensor *C , can be selected arbitrarily, since they disappear at the end of the mapping procedure to the isotropic space and back to the real one.

2.2 Underlying Damage Model

  • The isotropic CDM constitutive model considered in the mapped space considers one scalar internal variable to monitor the local damage [38,39,40,41].
  • The variable r is an internal stress-like variable representing the current damage threshold, as its value controls the size of the expanding damage surface.
  • The constitutive equation for the real orthotropic material is obtained by writing the dissipation occurring in an isothermic elasto-damageable process in the real anisotropic space.
  • The dissipation expression is obtained taking into account the first and second principles of thermodynamics.
  • All the variables in (9) are amenable to the classical thermodynamic representation [43], i.e. the free variable ε , the internal variable r and the dependent variable d(r).

2.3 Evolution of the Damage Variable and Inelastic Behaviour

  • The evolution of the damage index that has been adopted in this work is given by the exponential softening law reported in Ref. [24].
  • Two different elemental softening parameters can be specified along the material axes, to reproduce totally different fracture energies along the material directions and provide a full orthotropic softening behaviour.
  • Figure 3b shows the capability of the model to represent the softening orthotropy under uniaxial tension along x- and y-global directions.
  • The properties in the real space, referred to the material axes 1 and 2, are the same considered before.
  • In the first case, the material strength in the y-direction degrades at a faster rate than the material strength in the x-direction.

3. Local Crack-Tracking Technique for Damage Localization in

  • The local crack-tracking technique proposed in [26] was successfully applied to 2D three-noded standard elements with the aim of simulating the propagation of localized cracks in isotropic quasi-brittle materials.
  • The method is again based on a flag system that labels the finite elements pertaining to the crack path which may experience damage.
  • The regularization procedure according to the finite element characteristic length mentioned in Section 2.3 ensures that dissipation will be element-size independent.
  • These elements are labelled and can experience damage during the analysis.
  • The crack propagation direction is computed by considering the direction orthogonal to the corresponding first mapped stress eigenvector of each element.

4. Validation Examples

  • This section presents the validation of the proposed model by means of comparisons with experimental data of orthotropic materials.
  • Firstly, the orthotropic model is used to reproduce the directional strength of wood, the failure envelopes of composite laminates and masonry.
  • Such applications show how to set the parameters of the model and demonstrate the wide applicability of the method to different orthotropic materials.
  • - 12 - Secondly, the damage model combined with the local crack-tracking technique is used to simulate numerically the cohesive crack propagation in a benchmark uniaxial problem.
  • Finally, the FE analysis of mixed mode fracture experimental tests on brickwork masonry is presented.

4.1 Directional Strength of Wood

  • The uniaxial strength of wood elements is assessed for different orientations of the grain relative to the loading direction.
  • The results from the proposed model are compared with predictions obtained by the common strength criteria generally used for wood.
  • The walls of isotropic material between these voids form the three principal planes of the orthotropic material.
  • These results are compared with those derived by the proposed model, where the von Mises criterion is considered in the mapped isotropic space.
  • Good agreement is discovered by comparing the proposed model and the other analytical predictions.

4.2 Biaxial Failure Envelopes for Unidirectional Fibre-Reinforced Composite

  • Laminae Figure 5a shows the comparison of the failure envelope obtained using the proposed model with experimental results [52] for an unidirectional glass fibre reinforced lamina (E-Glass/LY556/HT907/DY063), with a fibre volume fraction kf =0.62, under shear stresses and normal stresses orthogonal to fibre direction.
  • The average properties of the homogenized material are obtained by the information concerning the constituents provided by Soden et al. and the basic formulae of the mixing theory [53].
  • Real shear strength has been considered equal to 61.2 MPa according to the obtained experimental value.
  • It can be observed that the model reproduces with an acceptable approximation the experimental failure envelope.
  • Drucker-Prager criterion has been considered in the mapped isotropic space, with 900 MPacf and 1500 MPatf .

4.3 Uniaxial and Biaxial Failure Envelopes for Masonry

  • The ability of the present model to reproduce the orthotropic strength of masonry is assessed through the comparison with experimental data obtained by Page [55,56].
  • Different orientations of the bed joints relative to the loading direction are considered.
  • The load is gradually increased until the ultimate conditions are reached.
  • The second strength value has been selected taking into account that, for =90°, there is a less significant experimental result with a rather pronounced deviation ( 63% ).
  • It is worth noting that for all the tests, the material properties in the 1-axis have been selected for the mapped isotropic space.

4.4 Holed strip under uniaxial traction

  • Calculations are performed with an enhanced version of the FE program COMET [62], developed at the International Center for Numerical Methods in Engineering (CIMNE, Barcelona).
  • The problem is solved incrementally in a time step-by-step manner.
  • Pre- and post-processing are done with GiD [63], also developed at CIMNE.
  • On the other hand, if the direction of cracks is evaluated by using the mapped isotropic stresses affected by orthotropy via the scaling procedure, the correct crack paths shown in Figures 8a-b-c-d are obtained.
  • Figure 9 shows the (half)-load vs. (half)-imposed vertical displacement curves obtained by the numerical analyses of strips with different angles of orthotropy.

4.5 Mixed mode fracture tests on brickwork masonry beams

  • The localized damage model is further validated by simulating numerically mixed mode fracture tests on brickwork masonry under three-point bending configuration with nonsymmetrical boundary conditions .
  • The FE simulations are compared with the experimental tests presented by Reyes et al. [64,65].
  • The stress-strain responses to uniaxial tension along different directions of the orthotropic material are shown in Figure 11.
  • Figure 13 shows the comparison between the experimental crack paths and numerical predictions for different inclinations of the bed joints.

5. Conclusions

  • A novel methodology has been presented to simulate numerically the tensile crack propagation in orthotropic materials.
  • The different behaviours along the material axes can be reproduced by means of a very simple formulation, taking advantage of the well-known isotropic damage models.
  • The model can be used for the analysis of different orthotropic materials, such as wood, fibre reinforced composites and masonry.
  • The numerical results are in a very good agreement with the experimental ones.
  • Since the computational costs is limited, it can be used in large scale computations [47,68,69].

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- 1 -
A LOCALIZED MAPPED DAMAGE MODEL FOR
ORTHOTROPIC MATERIALS
Luca Pelà
a *
, Miguel Cervera
a
, Sergio Oller
a
, Michele Chiumenti
a
a
CIMNE, Internacional Center for Numerical Method in Engineering, Technical
University of Catalonia (UPC), Campus Norte, Jordi Girona 1-3, 08034 Barcelona,
Spain.
Abstract - This paper presents an implicit orthotropic model based on the Continuum
Damage Mechanics isotropic models. A mapping relationship is established between the
behaviour of the anisotropic material and that of an isotropic one. The proposed model
is used to simulate the failure loci of common orthotropic materials, such as masonry,
fibre-reinforced composites and wood. The damage model is combined with a crack-
tracking technique to reproduce the propagation of localized cracks in the discrete FE
problem. The proposed numerical model is used to simulate the mixed mode fracture in
masonry members with different orientations of the brick layers.
Keywords: Continuum Damage Mechanics; Orthotropy; Transformation Tensor;
Fracture; Crack-tracking; Masonry.
1. Introduction
The mechanical behaviour of anisotropic materials involves properties that vary from
point to point, due to composite or heterogeneous nature, type and arrangement of
constituents, presence of different phases or material defects. A macroscopic continuum
model aimed at the phenomenological description of anisotropic materials should
account for i) the elastic anisotropy, ii) the strength anisotropy (or yield anisotropy, in
case of ductile materials) and iii) the brittleness (or softening) anisotropy [
1].
Several materials can be considered, with an acceptable degree of approximation, to be
orthotropic, even though some of them are not so in the whole range of behaviour.
*
Corresponding author: Luca Pelà, Universitat Politècnica de Catalunya BarcelonaTech, Department of
Construction Engineering, C/ Jordi Girona, 1-3 (Module C1 - Office 206-B), 08034 Barcelona Spain.
Phone: +34 93 401 10 36, Fax: +34 93 405 41 35.
E-mail addresses:
luca.pela@upc.edu (Luca Pelà), miguel.cervera@upc.edu (Miguel Cervera),
sergio.oller@upc.edu (Sergio Oller), michele@cimne.upc.edu (Michele Chiumenti).
*Manuscript
Click here to view linked References

- 2 -
Modelling the elastic orthotropy does not present big difficulties, since it is possible to
use the general elasticity theory [
2]. On the other hand, the need to model the strength
and nonlinear orthotropic behaviour requires the formulation of adequate constitutive
laws, which can be based on such theories as plasticity or damage. In particular,
although several failure functions have been proposed, the choice of a suitable
orthotropic criterion still remains a complex task.
One of the more popular attempts to formulate orthotropic yield functions for metals in
the field of plasticity theory is due to Hill [3,4], who succeeded in extending the von
Mises [
5] isotropic model to the orthotropic case. The main limitation of this theory is
the impossibility of modelling materials that present a behaviour which not only
depends on the second invariant of the stress tensor, i.e. the case of geomaterials or
composite materials. On the other hand, Hoffman [
6] and Tsai-Wu [7] orthotropic yield
criteria are useful tools for the failure prediction of composite materials.
For the description of incompressible plastic anisotropy, not only yield functions [
8] and
phenomenological plastic potentials [
9] have been proposed over the years. Other
formulation strategies have been developed, related to general transformations based on
theory of tensor representation [10,11]. A particular case of this general theory, which is
based on linearly transformed stress components, has received more attention. This
special case is of practical importance because convex formulations can be easily
developed and, thus, stability in numerical simulations is ensured. Linear
transformations on the stress tensor were first introduced by Sobotka [12] and Boehler
and Sawczuck [
13]. For plane stress and orthotropic material symmetry, Barlat and Lian
[
14] combined the principal values of these transformed stress tensors with an isotropic
yield function. Barlat et al. [
15] applied this method to a full stress state and Karafillis
and Boyce [
16] generalized it as the so-called isotropic plasticity equivalent theory with
a more general yield function and a linear transformation that can accommodate other
material symmetries. Betten [17,18] introduced the concept of mapped stress tensor to
express the behaviour of an anisotropic material by means of an equivalent isotropic
solid (mapped isotropic problem). The same approach was later refined by Oller et al.
[
19,20,21,22,23] with the definition of transformation tensors to relate the stress and
strain tensors of the orthotropic space to those of a mapped space, in which the isotropic
criterion is defined. The stress and strain transformation tensors are symmetric and
rank-four and establish a one-to-one mapping of the stress/strain components defined in
one space into the other and vice versa (
Figure 1). The constitutive law and the damage

- 3 -
criterion are explicitly expressed only in the isotropic mapped space. In this way, it is
possible to use standard isotropic models in calculations, with all the related
computational benefits, while the information concerning the real orthotropic properties
of the material is included in the transformation tensor. The parameters that define the
transformation tensor can be calibrated from adequate experimental tests. The
implementation of this theory into the framework of the standard FE codes is
straightforward.
The aforementioned approach based on mapped tensors was principally addressed to
Plasticity problems. Recently it has been extended to Continuum Damage Mechanics
(CDM) constitutive laws by Pelà et al. [
24,25] and applied to the study of masonry
structures. This paper explores the application of the model also to generic orthotropic
materials. The underlying theory applied to CDM is recovered and its theoretical
consistency and flexibility to different applications are stressed. The proposed mapped
damage model is then used to simulate the failure loci of masonry, fibre-reinforced
composites and wood.
The main novelty of this research is the combination of the mapped damage model with
the local crack-tracking technique proposed by Cervera et al. [
26]. The purpose of this
improvement of the original approach is the FE analysis of tensile cracking phenomena
in orthotropic materials. The combination of the mapped tensor theory with a crack-
tracking algorithm poses some issues that are addressed in this paper.
The introduction of local or global crack-tracking techniques into the framework of
standard finite elements and constitutive models [
25,26] has revealed to be a
satisfactory solution to some of the major drawbacks of the classical Smeared Crack
Approach (SCA) [
27]. In addition to modelling the tensile damage as a smeared
quantity spreading over large regions of the FE mesh, the SCA presents other well-
known disadvantages. Firstly, the smeared damage propagation depends on mesh-size
and mesh-bias, with a consequent lack of objectivity in the numerical results when
different spatial discretizations are considerd. Secondly, crack locking can be observed
especially in bending problems, when the advancing flexural crack experiences a
sudden “about-turn” [26].
The crack-tracking procedure labels the finite elements which can damage and prevents
the others from failing. A correction of spurious changes of crack propagation direction
is carried out. These features of the method allow the analyst to avoid the
aforementioned problems usually found in classical SCA, without increasing

- 4 -
excessively the implementation effort or the computational cost. Crack-tracking
algorithms are also employed in E-FEM and X-FEM to establish which elements lie in
the discontinuity path and need to be enriched [30]. Despite the wide diffusion of the
aforementioned procedures, it is worth noting that the introduction of mixed approaches
in the field of Computational Failure Mechanics does not require any crack-tracking
method [
31,32,33].
Benchmark numerical examples are presented to check the capability of the numerical
model to reproduce the correct crack paths in a material with different inclinations of
the axes of orthotropy. The FE simulation of mixed mode fracture experimental tests on
brickwork masonry members is discussed. The model is able to predict the failure load
and the cracking path in orthotropic materials subject to complex stress states.
The material is modelled by considering a macro-scale approach and it is represented as
a homogeneous continuum. No distinction is made among components if a composite
material, e.g. FRP or masonry, is analysed. An alternative treatment is the use of any
theory of homogenization [
34,35].
Notation. Tensor notation is used in this paper. The material coordinate system, which
coincides with the principal axes of orthotropy of the solid, is denoted by axes 1 and 2
in the two-dimensional case, see
Figure 2. Tensors and vectors referred to that local
coordinate system are marked by apex
. The angle
indicates the inclination
between the material and the global coordinate systems (xy) and it is measured counter
clockwise from the x-axis to the 1-axis. Finally, apex (
) is assigned to variables related
to the mapped isotropic space.
2. Mapped Damage Model
The orthotropic mapping of CDM constitutive laws has been presented in References
[
1,24,25]. In this section, the basics of the method are recovered and its thermodynamic
consistency is demonstrated. The flexibility of the procedure for the application to
generic orthotropic materials is stressed.
2.1 Definition of the Space Transformation Tensors
The method is based on assuming that the real anisotropic space of stresses
σ
and the
conjugate space of strains
ε
have their respective image in two mapped isotropic spaces
of stresses
*
σ
and strains
*
ε
, respectively (Figure 1). The relationship between these
spaces is defined by

- 5 -
*
:
σ
σ A σ
or
(1)
*
:
ε
ε A ε
or
ij ijkl kl
A

(2)
where
ijkl
A
σ
A
and
ijkl
A
ε
A
are the transformation tensors, for stresses and strains,
respectively, relating the mapped and real spaces. These rank four-tensors embody
directly the elastic and strength anisotropy of the material. Since the symmetry of the
Cauchy stress tensor both in the anisotropic and isotropic spaces is required, it follows
that
ijkl jikl jilk
A A A

. The symmetry of the four-rank transformation tensor is also
necessary, hence
ijkl klij
AA

.
The assumption of a strain space transformation tensor [
21,22,23], in addition to the
definition of the stress space transformation tensor, allows for no-proportionality
between the strength and the elastic modulus for each material direction. For this
reason, the adopted methodology has been also termed isotropic mapped model for
non-proportional materials” [21]. This feature of the method avoids the basic
assumption of elastic strains uniqueness for both the real and mapped spaces made in
previous works [
19,20]. In fact, that situation would introduce a limitation in the
anisotropic mapped theory, because it would result that
11 1 22 2 12 12
f E f E f G
(
ii
f
and
i
E
are the uniaxial strengths and the Young’s moduli referred to i-axes, whereas
ij
f
and
ij
G
are the pure shear strength and the shear modulus).
In this work, the material is assumed to be initially orthotropic and under in-plane stress
conditions. There are different alternatives to define the tensor
σ
A
for this case, see for
instance Betten [
17], Oller et al. [21,22] and Car et al. [36,37]. In this context, the stress
space transformation tensor in the material coordinate system (axes 1 and 2, see
Figure
2
) is:
1111 11 11
2222 22 22
1212 1221 12 12
2112 2121 12 12
1122 1112 1121
2211 2212 2221
1211 1222 2111 2122
2
2
0
0
0
A f f
A f f
A A f f
A A f f
AAA
A A A
A A A A







(3)

Citations
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Journal ArticleDOI
TL;DR: A comprehensive review of the existing modeling strategies for masonry structures, as well as a novel classification of these strategies are presented, which attempts to make some order on the wide scientific production on this field.
Abstract: Masonry structures, although classically suitable to withstand gravitational loads, are sensibly vulnerable if subjected to extraordinary actions such as earthquakes, exhibiting cracks even for events of moderate intensity compared to other structural typologies like as reinforced concrete or steel buildings. In the last half-century, the scientific community devoted a consistent effort to the computational analysis of masonry structures in order to develop tools for the prediction (and the assessment) of their structural behavior. Given the complexity of the mechanics of masonry, different approaches and scales of representation of the mechanical behavior of masonry, as well as different strategies of analysis, have been proposed. In this paper, a comprehensive review of the existing modeling strategies for masonry structures, as well as a novel classification of these strategies are presented. Although a fully coherent collocation of all the modeling approaches is substantially impossible due to the peculiar features of each solution proposed, this classification attempts to make some order on the wide scientific production on this field. The modeling strategies are herein classified into four main categories: block-based models, continuum models, geometry-based models, and macroelement models. Each category is comprehensively reviewed. The future challenges of computational analysis of masonry structures are also discussed.

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Journal ArticleDOI
TL;DR: A novel tracking algorithm that can simulate cracking starting at any point of the mesh and propagating along one or two orientations is proposed, which allows the simulation of structural case-studies experiencing multiple cracking.
Abstract: Tracking algorithms constitute an efficient numerical technique for modelling fracture in quasi-brittle materials. They succeed in representing localized cracks in the numerical model without mesh-induced directional bias. Currently available tracking algorithms have an important limitation: cracking originates either from the boundary of the discretized domain or from predefined "crack-root" elements and then propagates along one orientation. This paper aims to circumvent this drawback by proposing a novel tracking algorithm that can simulate cracking starting at any point of the mesh and propagating along one or two orientations. This enhancement allows the simulation of structural case-studies experiencing multiple cracking. The proposed approach is validated through the simulation of a benchmark example and an experimentally tested structural frame under in-plane loading. Mesh-bias independency of the numerical solution, computational cost and predicted collapse mechanisms with and without the tracking algorithm are discussed.

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Journal ArticleDOI
Abstract: A novel damage mechanics-based continuous micro-model for the analysis of masonry-walls is presented and compared with other two well-known discrete micro-models The discrete micro-models discretize masonry micro-structure with nonlinear interfaces for mortar-joints, and continuum elements for units The proposed continuous micro-model discretizes both units and mortar-joints with continuum elements, making use of a tension/compression damage model, here refined to properly reproduce the nonlinear response under shear and to control the dilatancy The three investigated models are validated against experimental results They all prove to be similarly effective, with the proposed model being less time-consuming, due to the efficient format of the damage model Critical issues for these types of micro-models are analysed carefully, such as the accuracy in predicting the failure load and collapse mechanism, the computational efficiency and the level of approximation given by a 2D plane-stress assumption

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Abstract: This work presents a multiscale method based on computational homogenization for the analysis of general heterogeneous thick shell structures, with special focus on periodic brick-masonry walls. The proposed method is designed for the analysis of shells whose micro-structure is heterogeneous in the in-plane directions, but initially homogeneous in the shell-thickness direction, a structural topology that can be found in single-leaf brick masonry walls. Under this assumption, this work proposes an efficient homogenization scheme where both the macro-scale and the micro-scale are described by the same shell theory. The proposed method is then applied to the analysis of out-of-plane loaded brick-masonry walls, and compared to experimental and micro-modeling results.

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Abstract: This paper extends the use of crack-tracking techniques within the smeared crack approach for the numerical simulation of cohesive–frictional damage on quasi-brittle materials. The mechanical behaviour is described by an isotropic damage model with a Mohr–Coulomb failure surface. The correct crack propagation among the two alternative fracture planes proposed by the Mohr–Coulomb theory is selected with the use of an energy criterion based on the total elastic strain energy. The simulation of three benchmark problems of mixed-mode fracture in concrete demonstrates that the proposed methodology can reproduce the material’s frictional characteristics, showing robustness, as well as mesh-size and mesh-bias independence.

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    [...]

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Abstract: A theory is suggested which describes, on a macroscopic scale, the yielding and plastic flow of an anisotropic metal. The type of anisotropy considered is that resulting from preferred orientation. A yield criterion is postulated on general grounds which is similar in form to the Huber-Mises criterion for isotropic metals, but which contains six parameters specifying the state of anisotropy. By using von Mises' concept (1928) of a plastic potential, associated relations are then found between the stress and strain-increment tensors. The theory is applied to experiments of Korber & Hoff (1928) on the necking under uniaxial tension of thin strips cut from rolled sheet. It is shown, in full agreement with experimental data, that there are generally two, equally possible, necking directions whose orientation depends on the angle between the strip axis and the rolling direction. As a second example, pure torsion of a thin-walled cylinder is analyzed. With increasing twist anisotropy is developed. In accordance with recent observations by Swift (1947), the theory predicts changes in length of the cylinder. The theory is also applied to determine the earing positions in cups deep-drawn from rolled sheet.

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    [...]

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01 May 1983
Abstract: A fracture theory for a heterogenous aggregate material which exhibits a gradual strain-softening due to microcracking and contains aggregate pieces that are not necessarily small compared to structural dimensions is developed. Only Mode I is considered. The fracture is modeled as a blunt smeard crack band, which is justified by the random nature of the microstructure. Simple triaxial stress-strain relations which model the strain-softening and describe the effect of gradual microcracking in the crack band are derived. It is shown that it is easier to use compliance rather than stiffness matrices and that it suffices to adjust a single diagonal term of the complicance matrix. The limiting case of this matrix for complete (continuous) cracking is shown to be identical to the inverse of the well-known stiffness matrix for a perfectly cracked material. The material fracture properties are characterized by only three parameters—fracture energy, uniaxial strength limit and width of the crack band (fracture process zone), while the strain-softening modulus is a function of these parameters. A method of determining the fracture energy from measured complete stres-strain relations is also given. Triaxial stress effects on fracture can be taken into account. The theory is verified by comparisons with numerous experimental data from the literature. Satisfactory fits of maximum load data as well as resistance curves are achieved and values of the three material parameters involved, namely the fracture energy, the strength, and the width of crack band front, are determined from test data. The optimum value of the latter width is found to be about 3 aggregate sizes, which is also justified as the minimum acceptable for a homogeneous continuum modeling. The method of implementing the theory in a finite element code is also indicated, and rules for achieving objectivity of results with regard to the analyst's choice of element size are given. Finally, a simple formula is derived to predict from the tensile strength and aggregate size the fracture energy, as well as the strain-softening modulus. A statistical analysis of the errors reveals a drastic improvement compared to the linear fracture theory as well as the strength theory. The applicability of fracture mechanics to concrete is thus solidly established.

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  • ...normalized with respect to the finite element characteristic length in order to ensure the FEM solution mesh-independency [45,46]....

    [...]


Journal ArticleDOI
Abstract: An operationally simple strength criterion for anisotropic materials is developed from a scalar function of two strength tensors. Differing from existing quadratic approximations of failure surfaces, the present theory satisfies the invariant requirements of coordinate transforma tion, treats interaction terms as independent components, takes into account the difference in strengths due to positive and negative stresses, and can be specialized to account for different material symmetries, multi-dimensional space, and multi-axial stresses. The measured off-axis uniaxial and pure shear data are shown to be in good agreement with the predicted values based on the present theory.

2,761 citations


"A localized mapped damage model for..." refers background in this paper

  • ...On the other hand, Hoffman [6] and Tsai-Wu [7] orthotropic yield criteria are useful tools for the failure prediction of composite materials....

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Frequently Asked Questions (2)
Q1. What have the authors contributed in "A localized mapped damage model for orthotropic materials" ?

This paper presents an implicit orthotropic model based on the Continuum Damage Mechanics isotropic models. 

A major advantage lies in the possibility of adjusting an isotropic criterion to the particular behaviour of the orthotropic material. Complex orthotropic damage threshold surfaces can be built by using simpler and well-known isotropic ones, hence avoiding the complex anisotropic yield functions normally adopted in Plasticity. The model can be used for the analysis of different orthotropic materials, such as wood, fibre reinforced composites and masonry. Since the computational costs is limited, it can be used in large scale computations [ 47,68,69 ].