Coupling local and non-local damage evolutions with the Thick Level Set model
© Moës et al.; licensee Springer. 2014
Received: 25 April 2014
Accepted: 9 September 2014
Published: 15 October 2014
The Thick Level Set model (TLS) is a recent method to delocalize local constitutive models suffering spurious localization. It has two major advantages compared to other delocalization methods. The first one is that the transition from localization to fracture is taken into account in the model. The second one is that the delocalization only acts when and where needed. In other words, the TLS has no effect when the local model is stable. The former advantage was already detailed in several papers (IJNME 86:358-380, 2011, CMAME 233:11-27, 2012, IJF 174:49-60, 2012). This paper concentrates on the latter advantage.
The TLS delocalization approach is formulated as a bound on the damage gradient. The non-local zone is defined as the zone where the bound is met whereas the local zone is defined as the zone where it is not met. The boundary (localization front) between the local and non-local zone is the main unknown in the problem.
Based on the new model, a 1D pull-out test is solved both analytically and numerically. Different regimes are observed in the solution as the loading progresses: fully elastic, local damage, coupled local/non-local damage and, finally, purely non-local damage.
The new model introduces delocalization as an inequality allowing local damage to develop in zones whereas non-local damage may develop in other zones. This reduces dramatically the cost of implementation of such models compared to fully non-local models.
KeywordsDamage Delocalization Non-local damage models Level set TLS
Although the scope of TLS application is much wider, we consider in this paper the fracture of quasi-brittle structures under quasi-static loading and under small deformation assumption. The loading is proportional to a scalar parameter. The material is modelled by a time-independent elasto-damage constitutive model with scalar damage. Due to quasi-static analysis, the loading parameter must be controlled especially when bifurcation occurs.
The TLS model was introduced in several papers , and ,. It lies between continuum damage mechanics and fracture mechanics. Indeed, crack opening is allowed across fully damaged zones (see  for instance). The fully damaged zone is located by a level set. Let us note that the description above is different from a diffuse vision of the crack in which crack opening is not explicitly modeled as in the phase-field approach - or the variational approach to fracture ,. We are rather in the vein of transition from damage to fracture as in . However, the TLS will not be in need of a cohesive zone to perform the transition. The model can be considered as a continuous transition from damage to fracture.
The delocalization (1) used in the TLS is different from existing delocalization techniques. Indeed, it directly uses the norm of the damage gradient. It is thus a Hamilton-Jacobi type equation. On the contrary, damage gradient models - yield Laplacian damage type equation rising the question of proper boundary conditions.
The TLS shares some similarities with the so-called non-local integral approach , in which weighted averages are performed over segments (1D), disks (2D) and spheres (3D) of fixed size. In the TLS approach, however, weighted averages are always performed on segments (Figure 2) whatever the dimension of the body and over a length which is not fixed in time but evolves from zero to a maximum length lc. Finally, note that as lc is the minimal distance between a point where d=0 and a fully damaged point, d=1, it plays the role of the fracture process zone size.
After this quick introduction of the TLS, we get to the objective of the paper. In previous TLS paper, the delocalization condition (2) was considered as an equality on the whole domain. It meant that d was zero on the domain except in zone where the gradient norm was fixed. The short-coming of this view was that uniform or smooth damage field (because of damage hardening for instance) could not be modeled prior to localization. The inequality analyzed in this paper allows a combination of local and non-local evolutions. In the literature, the possibility to combine both local and non-local approach is seldom discussed with the exception of the so-called morphing numerical technique ,.
The paper is organized as follows. The TLS concept with the inequality constraint discussed above are detailed in the first section. Next, the TLS boundary value problem is set up and a dissipation analysis is carried out. A 1D pull-out is solved semi-analytically to show the main feature of the TLS solution. This 1D test is then solved numerically with the TLS to observe the influence of the parameters choice in the model. A conclusion and perspectives end the paper.
We consider a solid body occupying a domain Ω. The external surface ∂ Ω is composed of two parts ∂Ω u u and ∂Ω u T on which the displacements λ ud and the loading λ Td are prescribed, respectively. The parameter λ is a loading parameter.
Small strains and displacements are assumed as well as quasi-static evolution. The current state is characterized by the displacement field u, from which the strain field is derived. The current state is also characterized by an internal scalar variable, the damage denoted d. In this paper, we will not consider other internal variables.
The potential ψ is assumed for now at least convex with respect to 𝛜. The need for other properties will be discussed later.
Such model was already considered in  for dissymmetric tension-compression evolution. We emphasize the fact that the TLS description is not restricted to associated damage models.
Finding the subdomain where d=1 is equivalent to find the subdomain whose boundary is the iso-contour ϕ=lc.
The boundary Γc defines the crack faces. Figure 2 shows a typical scenario of a crack appearing inside the localization zone.
Note that the volume measure of Ωc may be zero. This information is part of the solution process. We expect different shapes of Ωc in comminution and brittle crack propagation.
The fact that damage is related to a variable satisfying the eikonal equation, the cornerstone of the level set technology , explains why the damage model is coined Thick Level Set. In the non-local zone, damage is modeled over a thick layer in terms of level sets.
Non-local damage evolution boils down to decomposing Ω+ into a set of independent segments and finding a value over each of them.
The above indicates that duality is preserved through the averaging technique. This is not often the case in delocalization techniques as discussed in .
TLS boundary value problem
The fact that the fully damage zones are removed from the domain is important. It allows the displacement to be discontinuous across Γc. Regarding the regularity of the displacement, we request that the energy, i.e. integral of ψ over Ω\Ωc, is finite. This space is not simply H1 as in elasticity since the stiffness is possibly vanishing on Γc boundary .
where vn is the normal velocity of Γ counted positively along n. This gives the respective evolution of domains Ω+ and Ω−.
Note that the same free energy expression, ψ, is used over Ω+ and Ω−. In what follows, n is the outward normal vector to Ω on ∂ Ω and to Ω+ on ∂ Ω+. The set of admissible displacements variations is denoted as . It has the same definition as except that u is set to zero on ∂ Ω u .
Dissipation analysis and fields regularity
The goal of this section is to analyze the expression of the dissipation as well as looking at the fields regularity across the boundary Γ.
In the above, we did not consider the energy inside Ωc because it is assumed to be zero. Indeed, no compression is considered in this zone (see (43)).
The first term is the dissipation created by the interface propagation. We show now that due to damage continuity on Γ this term is zero.
The dissipation must be positive. For classical models in which Y is positive, this implies that damage may only grow. Damage growth will create a growth of the fully damaged zone Ωc (and thus a negative velocity vn). The last term in (62) is thus automatically positive. Whether this term is zero or not depends on the regularity of ψ on the boundary Γc. This regularity must be assessed from the non-local constitutive model condition: .
where is defined by (25) (y replaced by Y). The above expression exhibits the duality between and in the localization zone.
Pure elastic regime: T∈ [ 0,TA=1]
Local damage regime: T∈ [ 1,TB]
As a numerical application, with dc=0.5 and lc/r i =0.1, we get .
Combined local and non-local damage regime: T∈ [ TB,TC]
Given T, system (84)-(85) returns unknowns d i and d l as well as the extent of the non-local zone r l =r(d l ,d i ). We note that for T=T B , we have d l =d i and r l =r i . Let T C be the loading above which the system has no solution.
Non-local damage regime: T decreases from TCto 0
Indeed the damage at r l did not change from its value at load T C because the load has been decreasing afterwards.
Analysis of the displacement of the fiber
When n<2, the fiber displacement must be zero for total failure. When n=2, there exists a limit value of fiber displacement before total failure and when n>2, it takes an infinite displacement before total failure. It is interesting to note that these three regimes also exist in gradient damage models .
Last section gave some insight on the different regimes in the solution. In order to plot the solution, we detail here a 1D numerical solver. This code is rather ad hoc for 1D problem, since we force the advance of the Γ boundary and find the corresponding loading and fields. General 2D and 3D solvers will be detailed in a forthcoming paper. We search for the solution at a set of discrete times. Consider the solution known at time t n , the solution at time tn+1 must satisfy the following equations.
Regarding space discretization, the segment ]r i ,r e [ is discretized with a set of finite elements. Initially, the non-local zone is empty and we proceed with a classical Newton-Raphson scheme depicted in the solver flowchart without non-local zone.
initialization: u 0=d 0=T 0=0
elastic step: find the load step for which damage starts
load step T n+1=T n +Δ T
iterations initialization k=0
solve linear system (110) to find Δ u
after the first iteration adapt the load step so that the maximum damage increment is d inc
if residual ≤ tol, go to 9 else go to 5
if ║∇ϕ n+1║≤1, go to 3, else go to solver flowchart with non-local zone
At the end of each load step, the gradient of the level set is computed. If it is below 1 everywhere the next load step is applied. If not, a non-local zone is placed and the solver flowchart with non-local zone is used.
increase non-local zone:
iterations initialization: k=0
linear solve: solve (119) to find Δ u,Δ T,Δ ϕ
load update: T k+1=T k +Δ T
if residual ≤ tol, go to 8, else go to 4
if domain not fully broken (d(r i )<1), go to 2, else go to 9
The extent of the non-local zone is imposed and one tries to find a continuous displacement and damage field satisfying the problem.
Is is interesting to note the difference between the two solver flowcharts. When the non-local zone is empty, the linear solve deals only with displacement increments and the local update deals with the damage variable. On the contrary, when the non-local zone is not empty, the linear solve involves both displacement and damage (or more precisely the surrogate ϕ variable) increment in the non-local zone (local zone being treated as before).
where h* is the size of the element adjacent to the non-local zone at time step n. The initial (n=0) non-local zone needs to be more than one-element for convergence. Between 5 and 10 elements are used.
As a final remark on the solver flowchart with the non-local zone, we noticed that in non-local zone update, it was more efficient (reduced number of iterations) to take Δ ϕ as the one ensuring damage continuity rather that picking the one coming from step 3.
The Thick Level Set damage model allows coupling local damage evolution in some part of the domain to a non-local damage evolution in the localization zone. Damage gradient is bounded. The bound is reached in the non-local zone (localization zone) and not reached in the local one. The localization zone boundary is the main unknown in the model. It evolves ensuring damage continuity. A semi-analytical 1D solution has been developed showing different regimes in the solution (elastic, local damage, coupled local and non-local damage and finally pure non-local damage). The solution was plotted using a numerical scheme. This numerical scheme is ad hoc for the 1D problem considered. The corresponding numerical implementation for 2D and 3D cases will be the subject of a forthcoming publication.
All authors contributed to the main ideas in the paper: the way to couple local and non-local evolutions of damage. NM came up with the analytical solution. NM and NC designed the 1D code to plot the results. All authors read and approved the final manuscript.
The support of the ERC Advanced Grant XLS no 291102 is greatfully acknowledged. Professor Antonio Huerta is also acknowledged for his advice.
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