# A localized mapped damage model for orthotropic materials

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.

## 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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##### Citations

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### Cites background from "A localized mapped damage model for..."

...The most recent macro-models regard the material as a fictitious homogeneous orthotropic continuum, without making any explicit distinction between units and joints in the discrete model [2, 3, 4]....

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40 citations

### Cites background from "A localized mapped damage model for..."

...On the contrary, macro-modeling regards masonry as an equivalent homogeneous continuum, without making any distinction between units and joints in the discrete model [1,2,12]....

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34 citations

##### References

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

...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....

[...]

...[3] Hill R....

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...Figure 4b compares the proposed model with the different theories for the same data, except for n=1.97 in the Hankinson formula and f12=7.93 MPa for Norris and Tsai-Hill criteria....

[...]

...Of all the macro-mechanical failure theories for anisotropic materials, the Tsai-Hill [50] theory is the most widely used for wood....

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...Figure 4 Comparisons between numerical results and strength values obtained by Hankinson, Norris and Tsai-Hill theories (f1=78.3 MPa, f2 =2.55 MPa): a) n=1.78 and f12=6.25 MPa; b) n=1.97 and f12=7.93 MPa....

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3,097 citations

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

...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....

[...]

...[3] Hill R....

[...]

...Figure 4b compares the proposed model with the different theories for the same data, except for n=1.97 in the Hankinson formula and f12=7.93 MPa for Norris and Tsai-Hill criteria....

[...]

...Of all the macro-mechanical failure theories for anisotropic materials, the Tsai-Hill [50] theory is the most widely used for wood....

[...]

...Figure 4 Comparisons between numerical results and strength values obtained by Hankinson, Norris and Tsai-Hill theories (f1=78.3 MPa, f2 =2.55 MPa): a) n=1.78 and f12=6.25 MPa; b) n=1.97 and f12=7.93 MPa....

[...]

3,057 citations

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

...The experimental data are compared with results derived from the proposed model, in which the Drucker-Prager criterion [54] is considered in the mapped isotropic space....

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2,765 citations

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

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2,761 citations

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...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)

###### Q2. What are the future works mentioned in the paper "A localized mapped damage model for orthotropic materials" ?

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 ].