Methodology for Inclusion of Cracks in Concrete Material Model
The major methodologies used in finite element modelling of fracture in material are namely: a) the discrete and b) the smeared approaches (Gerstle W, et al.). In a smeared model, cracks are represented by changing the constitutive properties of the finite elements rather than changing the topography of the finite element meshing (Rashid) whereas the discrete model treats a crack as a geometric entity (Ngo et al.). The material model proposed in this paper is based on the smeared methodology. To achieve an accurate representation of cracked concrete material, it is imperative that the cracked plane is established. The methodology for the proposed material model is employed to find the crack plane in an element based on the principal stresses at the instant that cracking in the element occurs. In dynamic problems, where structures experience large displacements, the relativity of the crack plane to the element axis must be preserved as the element is displaced in three-dimensional space. The steps necessary to account for cracking of an element and its displacement in three-dimensional space are briefly described in this section followed by a brief discussion on how the compression failure of concrete is considered.
In Figure 2, the sequences of steps taken to establish the fracture of each element is shown. The first step is to verify if the element of concern has been cracked in previous time steps. Depending on the crack status of the element the following steps are taken,
1. If the element has never cracked before:
a. The maximum principal stress is found to establish if element is under tensile normal stress,
b. If the maximum principal stress is greater than tensile strength of material then the Jacobi iterative method is used (Golub G. et al.) to find the rotation tensor that relates the global to principal axes. This tensor transforms the element stress tensor to the principal stress tensor (cracked plane stress tensor); the same tensor can be used to relate the cracked plane to the element axis,
c. The tensile principal stress that exceeds the tensile strength of material is set to zero (cracking has occurred); the stress perpendicular to crack plane is released and reapplied to the structure as residual loads,
d. The tensile strength of the material in the element is set to zero,
e. The Cracked and Ever-Cracked flags are activated,
f. The principal stress (tensile crack component removed) tensor is rotated back to the element axis using the rotation tensor from step “b”,
2. If the element has cracked at any point during static or dynamic loading stages:
a. The rotation tensor from step “b” is updated by using an average element spin (rotational velocity of the element) tensor in each time step,
b. The updated rotation tensor is then used to rotate the stresses in the element axis to the cracked plane,
c. If stress perpendicular to the crack plane is positive (tensile) then the value of the stress in that direction is set to zero,
d. The crack plane stress (tensile removed) tensor is rotated back to the element axis using the rotation tensor from step “a”,
e. The Cracked flag is activated.
When the status of the element regarding the crack is established, a compressive failure using a failure model as shown in Figure 3 is checked. Failure of the element is based on the element hydrostatic pressure and deviatoric stresses. For concrete, it is proposed to use a hydrodynamic pressure – dependent material model in conjunction with a failure/damage model. Since the reinforcing steel is explicitly modelled, the need for assumptions regarding the use of a mixture rule is avoided. In this proposed material model, a pressure-dependent flow rule is defined with the attendant parabolic form of the yield function for compression as:
<P = J2 -[«0 + a1 f] (1)
where p is the hydrostatic stress (pressure), J2 is the second invariant of the deviatoric stress tensor, a0 and a2 are the constants that are defined based on the concrete material. At yield <p=0 and J2=^y/3 where ay is the yield point corresponding to hydrostatic stress p in the concrete. The basic specification data for the concrete is obtained in references (Bangash, M.) and (Winter, G. et al.).
If J2 is less than [а0+а1р], then no compression failure in the element has occurred. In cases, when J2 is slightly larger than [а0+а1р], to account for a relatively small ductility in concrete, some ductility might be defined as a percentage of the yield (shaded area in Figure 3). For these cases the deviatoric stresses are scaled down using equation (2).
If J2 is significantly larger than [ао+а;р], the element is considered failed in compression. Theoretically, as hydrostatic pressure increases, yield stress of the material increase. However, two cap models are considered to limit the extent at which the yield strength can be increased (Figure 3).
These cap models ensure that the material properties will degrade after it has experienced large hydrostatic pressures. This is necessary due to the fact that voids in concrete material collapse and micro cracks are formed in the material.
For reversible loadings such as blast or impact, it is important to limit the strength of the element after it has passed the softening hydrostatic pressure (location I on Figure 3) to the minimum value it experienced during the previous loading phases. As an example if an element experiences hydrostatic pressures up to the value of location II on Figure 3 and the elliptical cap model is chosen, then the yield strength of that element is limited to ay. This mechanism will ensure that elements that have previously experienced softening in their material strength will not carry loads beyond their set limits.
Figure 2: Methodology Flowchart
Yield C urve
Elliptical and linear cap models
Start of material softening
An equation of state is employed for the proposed concrete material that relates the volumetric strain to the hydrostatic pressure in each element. This equation of state is used as part of the proposed material model in a form of a table lookup. The equation of state for the proposed concrete material is schematically shown in Figure 4.