SECCIÓN PRIMERA DEL RECURSO DE REVOCACIÓN

The implementation of Hashin’s criteria for damage initiation in Abaqus uses a combination of both the 1973 and 1980 formulations, suggested by Matzenmiller et al [117,118], presented below. The work of Matzenmiller et al also provides the basis of the post-damage evolution material response.

ST = transverse shear strength SL = longitudinal shear strength.

39 α = formulation selection: this parameter takes a value between 1 or 0. If given a non-zero value, the software includes the shear term in fibre tension from the 1980 criterion, with the contribution of this term to the material behaviour determined by the value of this coefficient.

F_jⁱ = failure criteria status, with the superscript i and subscript j referring to loading direction (tensile or compressive) and material component (matrix or fibre) respectively.

Damage is initiated when the value this parameter reaches unity.

It must be noted that the stresses in the damage initiation criteria, as it is referred to in the Abaqus documentation [118], are components of the effective stress tensor produced by the relationship: σ = Mσt, where σt is the true stress and M is the damage operator:

Where d_i are internal damage variables that reflect the current state of fibre (f), matrix (m) and shear (s) damage. These are derived from the damage variables corresponding to each of the four failure modes, thus:

Prior to the initiation of damage, M is equal to the identity matrix, so that the effective stress equals the true stress. The damage variables equal zero at the onset of damage, and increase up to unity as the material degrades, with a value of 1 indicating that a component has fully degraded. During the damage evolution phase of the material response, the damage operator becomes significant in the initiation of damage in the other material modes, as the effective stress is increased.

40 The internal damage variables are also used to modify the post-damage initiation material response. The material response is given by σ = Cdε, where ε is the strain and Cd is the stiffness matrix for the damaged material:

𝑪_𝒅 = 1 𝐷 �

𝐸₁�1 − 𝑑_𝑓� 𝐸₁𝜈₂₁�1 − 𝑑_𝑓�(1 − 𝑑_𝑚) 0 𝐸₂𝜈₁₂�1 − 𝑑_𝑓�(1 − 𝑑_𝑚) 𝐸₂(1 − 𝑑𝑚) 0

0 0 𝐺𝐷(1 − 𝑑𝑠)

�

(6)

Where:

𝐷 = 1 − 𝜈12𝜈21�1 − 𝑑𝑓�(1 − 𝑑𝑚) (7)

The value of the damage variables is calculated via a bilinear stress-displacement relationship using a characteristic length to eliminate mesh dependency during material softening. This principle is not dissimilar to the bilinear response of a typical cohesive zone model, such as that proposed by Camanho & Davila [75]. The use of a characteristic length results in an equivalent stress-displacement response, as shown in Figure 7. This length, L^c, is based on the element geometry and formulation, and is the length of a line across a typical first-order element, and half this length for a typical second-order element in a mesh.

The positive slope of this curve corresponds to the linear elastic material behaviour prior to damage initiation. The negative gradient region is produced by evolution of the respective damage variables for each mode, in accordance to the following equations. The equivalent

Figure 7: Equivalent stress-displacement response for damage evolution in a fibre-reinforced material [118]

Image hidden due to copyright restrictions.

Please refer to [118], figure 24.3.3-1

41 displacement (which is not a crack, but actually analogous to strain) and stress is calculated separately for each damage mode as follows:

Fibre failure:

The coefficient α in the terms for fibre tension is the same as defined previously for adjusting the contribution of shear to this failure mode. < > is the Macaulay bracket operator, defined as 〈𝑥〉 = (𝑥 + |𝑥|)/2.

The evolution of the various damage parameters in the post-damage initiation regime (δeq≥ δeq0

) is given by:

𝑑 = 𝛿_𝑒𝑒^𝑓 �𝛿_𝑒𝑒− 𝛿_𝑒𝑒⁰ � 𝛿𝑒𝑒�𝛿_𝑒𝑒^𝑓 − 𝛿_𝑒𝑒⁰ �

(10)

42 Where δeq0

is the initial equivalent displacement at which the initiation criterion for a given failure mode is met, and δeqf

is the displacement at which the material is completely degraded for this failure mode. This relationship is presented in Figure 8. The value of the initial equivalent displacement is governed by the elastic properties and strength parameters for the various modes. The equivalent displacement at failure may be entered directly, but generally, this value is determined from the energy dissipated at failure for each mode, G^c, which corresponds the area beneath the complete stress-displacement curve in Figure 7. Each energy value represents the fracture toughness for each material component under transverse and longitudinal loading. The data from equation (10) for each damage mechanism links back into (4) and (6) as the load increases and damage propagates.

Maimi et al [64] discuss a number of methods of determining these energies, including the use of a standard double-cantilever beam test [119] to find the tranverse fracture toughness under tension G₂₊, and the novel compact tension and compression techniques of Pinho et al [120] to find the fracture toughness of the fibres (longitudinal direction) under tension G1+

and compression G1-. Maimi et al also propose an approximation for the transverse compressive fracture energy G2-, based on the mode-2 transverse fracture energy G6 (which itself may be found using an end-notched flexure test [121]). The strength and energy data is entered into Abaqus as required, and the above procedures are handled automatically as the solution progresses.

Abaqus also allows for damage stabilisation to overcome convergence difficulties in the implicit solver when using material models with softening and stiffness degradation. This technique applies a small, artificial viscosity to the damage variable, causing the stiffness matrix for the softening material to be positive. Provided this viscosity is small compared to

Figure 8: Relationship between damage variable and equivalent displacement [118]

Image hidden due to copyright restrictions.

Please refer to [118], figure 24.3.3-2

43 the time increment, convergence is usually improved without compromising the accuracy of the results [118].

It is essential to note that damage for fibre-reinforced materials in Abaqus is only compatible with plane stress elements [118]. As a result, solid elements, which employ a 3-dimensional state of stress cannot be used with this option, and researchers wishing to simulate progressive damage and failure with solid elements are required to produce their own sub-routines in order to do so [42,52]. Use of a sub-routine is also necessary if the user wishes to use a different or more refined failure theory.