El papel del maestro en la transmisión de valores dentro del aula

stress locking as a function of the regularisation parameter can be observed (Wells and Sluys, 2000). Stress locking in that case becomes more severe askapproaches zero, closer to a true discontinuity. In other cases, askapproaches zero, the robust-ness of the procedure is reduced (Wells and Sluys, 1999). For practical application, it is not possible to use very small values ofkrelative to the element size.

4.4 Comparison of the EAS-based model and smeared crack formulations

COMPARISON OF THE EAS-BASED MODEL AND SMEARED CRACK

FORMULATIONS 45

The only difference to the smeared crack formulation for a single crack is the matrix G, which is a measure of the element size and reflects the element geometry. For the symmetric approach (G = G^∗, leading to a symmetric D˜ matrix), the only dif-ference with smeared crack formulations is the inclusion of the scalar Ae/Ve in the tangent, which is a measure of element size. This avoids the need to adjust the hard-ening modulus element-wise, since the element length scale is already included in the formulation. In light of this equivalence of the embedded discontinuity formu-lation and classical smeared crack models, it must be concluded that many of the difficulties which dog classical smeared crack models will persist. The embedded discontinuity model has not been implemented in a continuum format since this in-volves the direct introduction of element length scales into the constitutive model, which is in conflict with the requirements set out in chapter 1.

The material tangent can be further refined for the Von Mises model developed in chapter 3, for which the direction of the displacement jump is fixed and can be described by a scalar in two-dimensions. In two dimensions, the displacement jump vectorαis described by:

α=ζm (4.40)

wheremis the direction of the displacement jump (orthogonal to the discontinuity normal) andζis the magnitude of the displacement jump. This allows the displace-ment jump at a discontinuity to be described by a scalar quantity. The enhanced degrees of freedomαcan be reduced to a scalar by substitutingGwithGmandG^∗ withG^∗min the weak governing equations and the linearised equations. Following the same procedures as for equations (4.35) to (4.38), equation (4.38) reduces to:

D˜ = ^D

e(−^Gm) (nem)^TD^e (nem)^TD^e(−^Gm) +Te

. (4.41)

Note the similarity of the expression in equation (4.41) to the plastic tangent for non-associative plasticity. The termnemthat contains the normal components to the dis-continuity and the jump direction is analogous to the gradient of the yield surface, the term−^Gmis analogous to the gradient of the plastic potential and the scalarT_ecan be considered as the hardening modulus (it is a scalar since it is assumed that the di-rection of the displacement jumpmis fixed, reducing the discrete constitutive model at a discontinuity to one-dimensional relationship). Defining the ‘elasto-plastic’ tan-gent as:

D^ep=D^e−^D˜ =D^e− ^D

e(−^Gm) (nem)^TD^e (nem)^TD^e(−^Gm) +Te

(4.42) shows that the embedded discontinuity model based on incompatible strain modes for Von Mises plasticity could be implemented in an equivalent form as a continuum

plasticity model with a standard Galerkin finite element procedure. The tangent ma-trix shown in equation (4.42) was derived by Borja (2000) without following the EAS formulation. This can be done by considering the gradient of the yield surface f as being equal to:

∂f

∂σ =nem (4.43)

and the gradient of the plastic potentialgas:

∂g

∂σ =−^{Gm. (4.44)}

Equation (4.44) implies that the plastic strain field is equal to:

ε^p=−ζ(Gm) =−^Gα (4.45)

and therefore the elastic strain field is equal to:

ε^e=Ba+ζ(Gm) =Ba+Gα. (4.46)

It is clear then that the plastic strain field is made a function of the element size and position of the discontinuity through the formation of the matrixG. Therefore, rather than adjusting the hardening modulus per element (as is done for smeared crack models), the plastic strain field is dependent on the element size and discontinuity position.

4.5 Finite element implementation

The use of incompatible strain modes leads to a formulation which can be easily implemented in a conventional finite element framework. The static condensation procedure means that the structure of the global system of equations is identical to that for standardC⁰compatible finite elements. There are however some important implementation aspects not yet addressed. These include whether or not disconti-nuity paths should be continuous, integration schemes, how the normal vector is determined and when exactly in the solution procedure discontinuities should be in-troduced. These issues are addressed in this section. Following from assumptions in the variational formulation, only constant strain elements are considered.

4.5.1 Introduction of a discontinuity and numerical integration

Incompatible modes are added to an element at the end of a loading step if the cri-terion for the development of a discontinuity is met. The criteria for different mod-els were presented in chapter 3. Discontinuities are introduced only at the end of a loading step for two reasons. Firstly, it is undesirable to introduce a discontinuity

FINITE ELEMENT IMPLEMENTATION 47

into a non-equilibrium state (as would be the case if discontinuities were allowed to develop during the iterative procedure). Secondly, introducing discontinuities at the end of a loading step preserves the quadratic convergence behaviour of the full Newton-Raphson solution scheme.

Three-noded triangular and four-noded tetrahedral elements which are crossed by a discontinuity are numerically integrated with a one-point integration scheme.

Since the strain field is constant, the location of the integration point is of no con-sequence. At the integration point, the stress in the continuum is calculated from the continuum strain and the tractions at a discontinuity are calculated from the dis-placement jump. History variables need to be stored for both the continuum (if an inelastic continuum model is used) and discontinuity interface parts.

4.5.2 Path continuity

The question of whether or not continuity of discontinuities in a geometric sense should be enforced is not trivial when the discontinuities are included as incompat-ible modes. Since the displacement jump is not continuous across element bound-aries, there is no theoretical reason why path continuity must be enforced. There are however practical considerations which influence whether or not path continuity should be enforced. The undesirable aspect of path continuity is that the element-local nature of the calculation is lost, since the placement of a discontinuity within an element is affected by the position of discontinuities in neighbouring elements.

For the symmetric formulation, enforcement of discontinuity-path continuity is necessary to reasonably calculate the energy dissipated in failure since the lation is dependent on the area of a discontinuity. For the non-symmetric formu-lation however, in many cases enforcement of path continuity is not essential since the formulation is independent of the area of a discontinuity. This is an important advantage for three-dimensional calculations where it is not possible to enforce ge-ometric continuity of flat planes in the three-dimensional space. In implementation, complex kinematic interactions between elements in three dimensions often mean that more than one discontinuity must develop at one time, making enforcement of continuity impossible. For two dimensions, a lack of geometric continuity leads to a significantly simpler algorithm since it is not necessary to ‘trace’ discontinuity tips during a calculation. In cases where path continuity is not enforced, discontinuities pass through the centroid of an element. It will be shown through numerical ex-amples that enforcement of path continuity can lead to improved performance with respect to mesh objectivity, although it is not always robust. Enforcement of path continuity with the non-symmetric approach has an effect on the functionϕ_ewhich attempts to ensure that the appropriate nodes of an element separate, as discussed in section 4.3.1.

4.5.3 Choosing the appropriate solution of the normal vector

The choice of the normal vector for the mode-I constitutive model is straightforward since it comes from the major tensile principal stress direction. The only difficulty arises in the case of a hydrostatic stress state. However, this is rare in practical calcu-lations and was never encountered. For plasticity-based models, the normal vector to a discontinuity comes from an analysis of the acoustic tensor. It was mentioned in section 3.2.5 that analysis of the acoustic tensor leads to multiple solutions for the normal vector to a discontinuity. The difficulty in implementation is to choose the ap-propriate normal. One option is to restrict the range of possible orientations a priori by visual observation of a problem. When path continuity is enforced, the appro-priate normal comes from the continuity requirement, although an initial direction must be specified for the first discontinuity. The situation is more difficult in three dimensions when path continuity cannot be enforced and elements tend to localise in

‘blocks’ due to the kinematic interaction. The numerical examples in this chapter use a simplified version of the method used by Wells and Sluys (2001a) for choosing the appropriate normal direction. The method fits well within finite element procedures since it is carried out entirely at element level.

The derivatives of the yield function with respect to stresses prescribe the direc-tion of the displacement jump relative to a discontinuity plane for the plasticity-based constitutive model in section 3.2. This information is used in choosing the appropri-ate normal vector. If, at the end of load step, the criterion for the introduction of a discontinuity is met, each possible solution for the normal vectorn is calculated.

For each possible normaln, a discontinuity plane that passes through the centroid of the element is constructed and the incremental nodal displacements for the next load step (or from the previous step) are averaged for the nodes on each side of the discontinuity. The incremental displacements provide an indication of the deforma-tion mode of the element. The normal direcdeforma-tion which results in the greatest relative difference in incremental displacements on each side of the discontinuity in the direc-tion of the displacement jump (specified by the constitutive model) is chosen as the appropriate normal vector. This method provides an estimate of which normal direc-tion will result in the largest displacement jump in the following step. The numerical example this chapter using the Von Mises model use this procedure for determining the normal direction. For the Von Mises model, the displacement jump is tangential to the discontinuity plane, so the relative difference in incremental displacements, parallel to the discontinuity, on each side of the discontinuity are compared.