Lifetime-Oriented Structural Design Concepts- P18 docx

4.2.9.2.1.4 p-Version of the XFEM By using the XFEM with bi-linear shape functions to compute the displace-ments of the tension test shown in Figure 4.68 the resulting displacedisplace-m

Meth

Meth

Meth

Meth

Meth

Fig 4.68 Configuration of a tension test

which can be rewritten as

By integrating the function f6(x, y) analytically in the integration range x ∈

[−1.02; 1] and y ∈ [−1.01; 1] the integration value

is obtained The numerical results are shown in Figures 4.66 and 4.67

4.2.9.2.1.4 p-Version of the XFEM

By using the XFEM with bi-linear shape functions to compute the

displace-ments of the tension test shown in Figure 4.68 the resulting displacedisplace-ments u x

around the crack tip are shown in Figure 4.69 for the deformed system Theblack line indicates the asymptotic displacements of the crack surfaces in thevicinity of the crack tip It can be seen from the figure that the approximation

of the displacement field results in big differences to the analytical solution.Furthermore a constriction is noticeable which occurs at the position wherethe crack cuts the edge of the crack tip element

The results can be improved by using a refinement of the element mesh

(h-version or h-extension) as well as higher order polynomials for the shape functions (p-version or p-extension) Using a h-extension will improve

the results but the constriction is still visible but closer to the crack tip

while using a p-version for the shape functions results in the displacements

Fig 4.69 Displacementsu xfor the deformed system using bilinear shape functions

Fig 4.70 Displacementsu xfor the deformed system, left: using bi-quadratic shapefunctions, right: using quadratic hierarchical shape functions

shown in figure 4.70 In both cases the accuracy of the results is improved,but using the hierarchical shape functions the displacement field results insmaller differences and furthermore the constriction is not visible here

To find out more about this phenomenon we examine a single element andcompute the coefficients in the XFEM displacement field using the minimizingprocess of Equation 4.186

ncoll

i=1

(uanalyt(xi)− uXFEM(xi))2→ min (4.186)

Figures 4.71-4.75 show the results of equation 4.187 for different blending

el-ements Because the condition of the Partition of Unity is satisfied inside the

crack tip element the integrated error is zero and not shown here The errors

|Δu| dΩ

Fig 4.72 Differences of displacements inside the 2nd blending element

|Δu| dΩ

Fig 4.74 Differences of displacements inside the 4th blending element

for the elements that are not enriched are equal for using Lagrange mials (the standard shape functions) or Legendre polynomials (hierarchicalshape functions) Because of the fact that the shape functions defined for theblending elements span different function spaces, differences for the error ofEquation 4.187 exist

polyno-The results of the blending elements which are pictured here indicate theenormous influence of the selection of the basis to the approximation of thenear tip field The results show that the blending elements cause a big error of

the finite element analysis, because the conditions of the Partition of Unity are

not satisfied in the blending elements The fact that only some of the nodes or

Fig 4.75 Differences of displacements inside the 5th blending element

modes inside these elements are enriched by the crack tip functions, producesthis difficulty Using the hierarchical shape functions result in smaller errorsbecause the influenced region of the crack tip functions become bigger.For more details on this phenomenon we refer to [618, 334]

4.2.9.2.1.5 3D XFEM

Authored by Christian Becker and G¨ unther Meschke

Because of the variety and complexity of mechanical engineering

crack-propagation problems the powerful two-dimensional X-FEM simulation tools

have to be enhanced for three-dimensional problems In recent years theextended finite element method was succesfully applied to fully three-dimensional problems [781, 547, 780, 301] The description of the crack topol-ogy and crack propagation is more frequently described implicitly by the levelset method [606, 605]

Considering the three-dimensional extended finite element method, there

is mostly a conflict between the accuracy, applicability, realistic results andthe complexity of the numerical implementation, in particular regarding crackpropagation criteria As an example, in a two-dimensional analysis of a single

crack, the crack front is represented by a single point P Consequently, the

crack propagation is simply reduced to the determination of the crack

prop-agation angle Θ c and the crack propagation length l c Even with a crackpropagation criterion like the minimization of the total energy, this problem

is solved quite easily by introducing two additional system degrees of freedom

Θ c and l c On the contrary, in a full three-dimensional analysis of crackpropagation of a single crack, the crack front is represented by a number ofline segments bounding the crack surface It can be seen, that this problem is

of much more complexity than the two-dimensional problem

In most of the three-dimensional X-FEM implementations tetrahedral

ele-ments with linear approximations of the displacement field are used in nation with elementwise plane crack propagation (see e.g [52]) This represents

combi-the classical h-finite element approach to capture combi-the crack propagation process.

A more complex simulation concept applies the well-known level-set method

(see e.g [546, 324]) Within this method arbitrary crack growth is accountedfor by solving Hamilton-Jacobi-like equations A crucial point in this con-cept is that the velocity of the crack surface evolution has to be known tocapture the propagation process

Alternatively, the proposed combination of the p-FEM and the X-FEM

that was stressed in the last section (Section 4.2.9.2.1.4) and holding forhigh accuracy, is applied to the numerical simulation of three-dimensionalcrack propagation In this concept, because of the complexity of the fullythree-dimensional problem, some assumptions are introduced The underlyingfinite element formulation is a continuum brick element holding for arbitraryhigher-order shape functions (see Section 4.2.4.3)

The proposed higher order X-FEM simulation strategy holds for:

• element-wise crack propagation

• plane crack surface at the element level

With the proposed assumptions a row of questions arise concerning numericalintegration and the determination of crack propagation

Because of the jump in the enhanced strain field, standard Integration cannot be applied Hence, both parts of a cracked element have

Gauss-to be integrated separately In two dimensional analyses the well-known launey-triangulation is used A division into sub-tetrahedra in the sense ofthe Delauney-triangulation is way too complicated, therefore a fixed sub-division into six subtetrahedra is used (see Figure 4.76) Each of these sixtetrahedra may be cut by the crack plane into the sub-domains of a pentahe-dron, tetrahedron or a pyramide by either a triangular or quadrilateral crackplane (see Figure 4.77) The element quantities (•) like stiffness matrices or in-

De-ternal load vectors are generated by summing over all obtained sub-domainsthat are numerically integrated with a Gauss-integration according to the

geometrical shape of the sub-domain V i

Fig 4.76 Numerical integration in the context of theX-FEM: Subdivision of the

continuum element into six sub-tetrahedrons

Fig 4.77 Separation of a sub-tetrahedron by a plane crack segment: left:

separa-tion into two pentahedra by a quadrilateral, right: separasepara-tion into pentahedron andtetrahedron by a triangle

X-FEM In conjunction with elementwise, plane crack propagation this leads

to strong restrictions concerning the kinematics of the evolving crack surface.These restrictions are addressed in Figure 4.78

For a start, the crack front is represented by line segments at those elementfaces where the crack surface ended so far Sound elements that are neighbours

to that faces are crack candidates and are investigated in the subsequent load

steps if the crack propagates through them In Figure 4.78 the element inthe middle is neighboured by only one crack segment whereas the candidate

at the left is bounded by two segments of the crack front Because of thealready existing crack line of the middle element, the position of the newcrack segment is not fully arbitrary but can only be positioned by a rotationaround the existing crack line segment Therefore the normal vector resultingfrom the crack propagation criterion has to be modified, like it is illustrated

at the right hand side of Figure 4.78 Regarding the left element, the position

of a crack segment is already predefined by the two linear independent linesegments Therefore, it is only possible to decide when the new crack plane is

inserted The preceding thoughts occur only in the context of a C0-continuousdescription of the crack surface, using an algorithm according to e.g [52]

To gain more flexibility concerning the kinematics of the evolving crack

sur-face, the C0-continuity of the crack surface is neglected and a C0-discontinuousalgorithm according to [301] can be used In this algorithm the new crack seg-ment introduced within a crack candidate is defined by a pointPand the normalvectornobtained through the crack propagation criterion The pointPis thegeometrical mean of all existing mid-points of existing crack segments at the el-ement boundaries (see Figure 4.79) With this method the crack propagationgains a lot of flexibility by keeping the numerical implementation very simple

In cases where the gaps between neighboured crack segments are getting to wide,

a smoothing algorithm for the crack surface is provided by [302]

4.2.9.2.1.6 XFEM for Cohesive Cracks

Authored by Christian Becker and G¨ unther Meschke

First of all, the X-FEM was applicated to problems of Linear Elastic

Fracture Mechanics (LEFM) Therein, the general balance of momentum

P2 P

Fig 4.79 Definition of the crack plane by point Pand normal vectorn Pis thegeometric mean of all midpoints of the crack front lines

at the discontinuity, requiring the jump of the traction to vanish at thediscontinuity

Numerical simulations of quasi-brittle material behaviour of concrete

require the consideration of a cohesive zone by a cohesive zone modell

[240, 85, 369] Within the cohesive zone the traction vector is not vanishing,but is dependend on the energetically conjugated variable of the displacementjump [[u]]:

Because of the non-vanishing traction vector, there is an additional term ofthe internal virtual work within the weak form of balance of momentum, here,for brevity without volume or external loads:

similiar to continuum damage mechanics [422] Here, the traction vector issupposed to point only in normal direction of the crack:

The normal component of the traction vector is a function of the equivalentdisplacement jump, which in this case is simply the absolute value of thenormal component of the displacement jump:

In case of a violation of the damage criterion, the internal variable α is updated

to the current value of the equivalent displacement jump This criterion iscompleted by the well-known Kuhn-Tucker conditions

4.2.9.2.2 Strong Discontinuity Approach and Enhanced Assumed Strain

Authored by J¨ orn Mosler

An overview of the Strong Discontinuity Approach (SDA) is given in thissection The SDA is characterized by the incorporation of strong discontinu-ities, i.e discontinuous displacement fields, into standard displacement-basedfinite elements by means of the Enhanced Assumed Strain (EAS) concept.The fundamentals of the SDA are illustrated and compared to those of othermodels based on discontinuous deformation mappings The main part of thiscontribution deals with the numerical implementation of the SDA Besides theoriginal finite element formulation of the SDA, a more recently proposed algo-rithmic framework which avoids the use of the static condensation technique

is presented as well This section follows to a large extent [554]

4.2.9.2.2.1 Kinematics: Modeling Embedded Strong Discontinuities

In this section, a review of the kinematics associated with the Strong continuity Approach (SDA) is given For the sake of simplicity, attention isrestricted to a geometrically linearized framework Further details on the ge-ometrically exact SDA can be found in [555]

Dis-Fundamentals

According to Simo & Oliver [745, 598], the SDA is characterized by adisplacement field of the type

u = ¯ u + ru (Hs− ϕ), with ¯u ∈ C ∞ (Ω,R3), ϕ ∈ C ∞ (Ω, R). (4.203)

Here, Hs is the Heaviside function, ru the displacement discontinuity

and ϕ is a smooth ramp function which allows to prescribe the Dirichlet

boundary conditions in terms of ¯u (see [745, 598]) This will be described later.

Applying the generalized derivative D to the Heaviside function (see [766, 767]) which results in the identity DHs = N δs, the linearized strains arecomputed from Equation (4.203) as

X0

(4.207)

holds Hence, in contrast to the displacement field associated with the X-FEM,

the kinematics corresponding to Ω − and Ω+ are not completely independent

of one another For further details, refer to [412, 556] It is straightforward toshow that identity (4.207) is also fulfilled in the three-dimensional case

Numerical Implementation

Referring to the finite element method and focusing on constant strain

tri-angular elements which are cut by means of a planar surface ∂sΩ for now, Simo

& Oliver[745] proposed an approximation of the displacement field (4.203)

i the nodal displacements at node i, nnodethe number of nodes of

the respective finite element, Xe∗the node that connects the two sides of the

element which are cut by ∂sΩ and he represents the distance from Xe∗ to the

opposite side, with unit vector m i(see Figure 4.80) According to Figure 4.80,

with the nodal coordinates Xei Hence, by combining Eqs (4.208)3and

Equa-tion (4.203), the displacements at node i with coordinates Xei are computedas

u(Xei) = ¯u(Xei) ∀Xe

Consequently, as mentioned before, the function ϕ allows to prescribe the

Dirichletboundary conditions in terms of ¯u Clearly, N i and ϕ are linear and

continuous functions and Conditions (4.209) are fulfilled for the interpolation

function N ∗ associated with node Xe∗ Thus, the identity

holds (see [598]) Originally, the ramp function (4.211) was proposed for thedesign of a numerical length scale, cf [596] The extension to higher orderelements is straightforward Applying the interpolation conditions and con-

sidering the most general case, ϕ is designed according to

denotes the summation over all nodes

of the respective finite element belonging to ¯Ω+ For bi-linear and bi-quadraticshape functions see [559] (Figure 4.81)

According to [746, 745], the incorporation of the discontinuous displacementfield into the finite element formulation is achieved by adopting the EASconcept, cf [747, 742] Hence, the enriched displacement field is modeled in anincompatible fashion Consequently, it is admissible to neglect the gradient of

Interestingly, the local decomposition (4.213) is similar to the additive split

ε = εe+ εpused in standard plasticity models, cf [556] It it noteworthy thatalthough most SDA models in the literature are based on the assumption

∂ru/∂X =0, the more general case can be derived in a relatively forward manner, cf [499]

straight-4.2.9.2.2.2 Numerical Implementation

This section contains different algorithmic formulations of the strong continuity approach In Subsection 4.2.9.2.2.2, the original finite elementmodel proposed by Simo et al [746, 745] is summarized first (see also [598]).This implementation is based on the nowadays classical EAS concept andthe static condensation technique Starting from this SDA implementation,

dis-a recently suggested dis-algorithmic formuldis-ations is discussed next This methodavoids the static condensation technique and results in linearized constitutiveequations formally identical to those of classical continuum models such asstandard plasticity theory

Numerical Implementation Based on Static Condensation Technique

The additive decomposition of the strain tensor (4.213) is formally identical

to the corresponding additive split of the by now standard EAS concept (see[747, 742]) Hence, the original implementation of the strong discontinuityapproach as proposed by Simo and co-workers was based on the algorithmicformulation of the EAS concept, cf [746, 745, 600]

According to Simo & Rifai [747], the starting point of the EAS method

is represented by the two-field functional

holds (see [598]) Originally, the ramp function (4.211) was proposed for thedesign of a numerical length scale, cf [596] The extension to higher orderelements is straightforward... straightforward Applying the interpolation conditions and con-

sidering the most general case, ϕ is designed according to

denotes the summation over all nodes

of the respective finite