 Research Article
 Open Access
 Published:
Coupling local and nonlocal damage evolutions with the Thick Level Set model
Advanced Modeling and Simulation in Engineering Sciences volume 1, Article number: 16 (2014)
Abstract
Background
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 86:358380, 2011, CMAME 233:1127, 2012, IJF 174:4960, 2012). This paper concentrates on the latter advantage.
Methods
The TLS delocalization approach is formulated as a bound on the damage gradient. The nonlocal 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 nonlocal zone is the main unknown in the problem.
Results
Based on the new model, a 1D pullout test is solved both analytically and numerically. Different regimes are observed in the solution as the loading progresses: fully elastic, local damage, coupled local/nonlocal damage and, finally, purely nonlocal damage.
Conclusions
The new model introduces delocalization as an inequality allowing local damage to develop in zones whereas nonlocal damage may develop in other zones. This reduces dramatically the cost of implementation of such models compared to fully nonlocal models.
Background
Although the scope of TLS application is much wider, we consider in this paper the fracture of quasibrittle structures under quasistatic loading and under small deformation assumption. The loading is proportional to a scalar parameter. The material is modelled by a timeindependent elastodamage constitutive model with scalar damage. Due to quasistatic analysis, the loading parameter must be controlled especially when bifurcation occurs.
The TLS model was introduced in several papers [1],[2] and [3],[4]. It lies between continuum damage mechanics and fracture mechanics. Indeed, crack opening is allowed across fully damaged zones (see [2] 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 phasefield approach [5][7] or the variational approach to fracture [8],[9]. We are rather in the vein of transition from damage to fracture as in [10]. 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 main idea of the TLS for quasibrittle fracture is to bound the spatial gradient of the damage variable d, thus avoiding spurious localization. One imposes that the spatial damage distribution satisfies at all time
where Ω is the domain of interest. The choice of the function f(d) will be discussed in what follows. As damage evolves, one eventually wants to locate the crack, i.e. the zone for which d=1. However, finding the isocontour d=1 for a quantity d than cannot go beyond 1 is a tedious operation. This is where the level set ingredient comes into play. Variable d is expressed in terms of a level set ϕ as depicted in Figure 1. This relation introduces a length scale l_{c}. Finding the zone d=1, is now wellposed since the level set ϕ is not strictly limited to l_{c} but may go beyond. With the use of the surrogate variable ϕ, condition (1) may be rewritten as
where f(d) in (1) is related to d(ϕ) by f(d)=d^{′}(ϕ(d)) (the prime indicating the derivative of d with respect to ϕ). The function d(ϕ) is called the damage shape function and is the main ingredient of the TLS. Equation (2) above indicates that ϕ is a distance function in the zone where the constraint is active (we name this zone the localization zone). The evolution of a distance function has been analyzed and updating algorithm proposed in [11]. In the localization zone, the evolution of ϕ is nonlocal, indeed
The rate of change of ϕ is thus uniform on any segment aligned with ∇ϕ and the rate of d is given by $\stackrel{.}{d}={d}^{\prime}\left(\varphi \right)\stackrel{.}{\varphi}$. Such segments over which $\stackrel{.}{\varphi}$ is uniform are depicted in Figure 2. In the local zone, the evolution of ϕ stems from the evolution of d and the relation d=d(ϕ).
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 HamiltonJacobi type equation. On the contrary, damage gradient models [12][14] yield Laplacian damage type equation rising the question of proper boundary conditions.
The TLS shares some similarities with the socalled nonlocal integral approach [15],[16] 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 l_{c}. Finally, note that as l_{c} 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 shortcoming 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 nonlocal evolutions. In the literature, the possibility to combine both local and nonlocal approach is seldom discussed with the exception of the socalled morphing numerical technique [17],[18].
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 pullout is solved semianalytically 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.
Methods
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 u^{d} and the loading λ T^{d} are prescribed, respectively. The parameter λ is a loading parameter.
Small strains and displacements are assumed as well as quasistatic evolution. The current state is characterized by the displacement field u, from which the strain field $\mathit{\mathbf{\epsilon}}=\frac{1}{2}(\nabla \mathit{u}+\nabla {\mathit{u}}^{T})$ 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.
Regarding the material model, we consider a free energy ψ(𝛜,d) from which the stress tensor σ and local energy release rate Y may be derived
The potential ψ is assumed for now at least convex with respect to 𝛜. The need for other properties will be discussed later.
Regarding the timeindependent damage evolution, we consider a function y depending on damage and strain history (through e) such that
where
and Y_{c} is some threshold. We believe the above formalism encompasses most of the damage models in the literature. To be even more general, one may consider two relations of the kind (5): one for damage in tension and a second one for damage in compression and then combine these damages into d. One has a socalled associated damage model when the y variable is Y. In this case, damage evolution is expressed in terms of the dissipation potential φ^{*}(Y) which is the indicator function of Y−Y_{c}≤0:
Such model was already considered in [1] for dissymmetric tensioncompression evolution. We emphasize the fact that the TLS description is not restricted to associated damage models.
What we have described so far is a purely local damage model. This type of model is known to suffer spurious localizations meaning that the damage gradient may become infinite. The main idea of the TLS approach is to bound damage gradient as expressed in (1). In the TLS model, damage is allowed to go to 1 (but not beyond of course). The location of a crack (or fully degraded zones like in comminution problems) is defined by the set of points for which d=1. Numerically speaking, finding the set of points for which d=1 knowing that d may not go beyond 1 is not very practical. This is why the TLS expresses damage in terms of a surrogate variable ϕ whose values are not limited as depicted in Figure 1. We assume the following regularity on d(ϕ)
Finding the subdomain where d=1 is equivalent to find the subdomain whose boundary is the isocontour ϕ=l_{c}.
In terms of the surrogate variable, ϕ, condition (1) reads
provided f(d) is given by
For instance, if d is linear with respect to ϕ, the gradient of damage will be bounded by a constant
whereas for more complex function d(ϕ), the bound depends on the level of damage. For instance, for the profile shown in Figure 1, we have
For a general power law with n≥1, we obtain
Whether local or nonlocal constitutive model should be used at a point x is based on condition (9).
The first condition is the major novelty of this paper, compared to previous paper on the TLS. At any time t, the domain may thus be decomposed into three nonoverlapping zones : a local zone Ω^{−}, a nonlocal zone Ω^{+} and a fully damaged zone Ω_{c}
We define also the boundary Γ_{c} of the fully damaged zone and the interface Γ between the local and nonlocal zones.
The boundary Γ_{c} defines the crack faces. Figure 2 shows a typical scenario of a crack appearing inside the localization zone.
Note that the volume measure of Ω_{c} may be zero. This information is part of the solution process. We expect different shapes of Ω_{c} in comminution and brittle crack propagation.
Eikonal equation
Condition, ║∇ϕ(x)║=1 is a nonlinear firstorder partial differential equation. It is called an eikonal equation and belongs to the HamiltonJacobi equation family. Among the possible solution satisfying ║∇ϕ(x)║=1, we will pick the one corresponding to the vanishing viscosity solution [19]. It is characterized by
where d(x,y) is the length of the shortest path connecting x and y inside Ω^{+}. The value of ϕ at x∈Ω^{+} can be thought as the minimal fare to go from Γ to x. The fare being the sum of the initial fare ϕ(y) plus the mileage from y to x. Damage on Ω^{+} is thus fully determined from values on Γ. A 1D example of ϕ satisfying the eikonal on a segment [ b,d] is given in Figure 3.
The fact that damage is related to a variable satisfying the eikonal equation, the cornerstone of the level set technology [11], explains why the damage model is coined Thick Level Set. In the nonlocal zone, damage is modeled over a thick layer in terms of level sets.
Damage evolution
In the local zone, Ω^{−}, damage evolution is local and given by (5). In the nonlocal zone, Ω^{+}, damage rate is related to $\stackrel{.}{\varphi}$ by
where $\stackrel{.}{\varphi}$ is uniform on segments aligned with ∇ ϕ, see Equation (3). We denote this space as $\mathcal{A}$:
Nonlocal damage evolution boils down to decomposing Ω^{+} into a set of independent segments and finding a value $\stackrel{.}{\varphi}$ over each of them.
As in [2], we suggest to introduce averaged quantities, $\stackrel{\u2013}{y},\stackrel{\u2013}{\stackrel{.}{d}}$ over each segments. This may be expressed by a projection operation.
We note that the averages satisfy the following property
The above indicates that duality is preserved through the averaging technique. This is not often the case in delocalization techniques as discussed in [20].
The local constitutive model, (5), is then expressed in terms of the nonlocal quantities
where we have assumed Y_{c} uniform (if not it needs to be averaged by formula (25)). Finally, we write the relation giving $\stackrel{.}{\varphi}$ in terms of $\stackrel{\u2013}{\stackrel{.}{d}}$:
To end this section we illustrate the average formula on the 1D example depicted in Figure 3. Averages are given by
TLS boundary value problem
We are now able to define the boundary value problem. The set of admissible displacements is given by
The fact that the fully damage zones are removed from the domain is important. It allows the displacement to be discontinuous across Γ_{c}. Regarding the regularity of the displacement, we request that the energy, i.e. integral of ψ over Ω\Ω_{c}, is finite. This space is not simply H^{1} as in elasticity since the stiffness is possibly vanishing on Γ_{c} boundary [21].
Regarding the ϕ variable, it is required to be continuous over Ω and belong to the set . The admissible set for ϕ is denoted K.
The continuity requirement on ϕ leads to a Hadamard compatibility condition on the moving boundary Γ. Let us define the jump of a quantity f across Γ by
The exponent −/+ placed on some quantities f defined at x on Γ has the following meaning
where n is the outward normal to Ω^{+}. With these notations we have
where v_{n} is the normal velocity of Γ counted positively along n. This gives the respective evolution of domains Ω^{+} and Ω^{−}.
Potential energy of the domain is given by :
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 ${\mathcal{U}}_{0}$. It has the same definition as except that u is set to zero on ∂ Ω_{ u }.
Assuming, that at time t, the spatial distribution of ϕ of the two volumes Ω^{+} and Ω^{−} is known, the displacement field u is the field that solves the stationarity of the potential energy:
This means;
For simplicity, we assume that the boundary Γ_{c} is traction free (no contact on crack faces). The equilibrium (39) yields the following local equations
We stress the fact that Γ_{c} denotes the boundary of the fully damage zone and thus in case of a crack Γ_{c} indicates both crack lips. To complete the set of equations to be solved for a known damage distribution, we add the stress definition and kinematic relations
Finally, we need to add damage evolution equations in the local zone (5) and nonlocal zone (28).
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.
Taking into account the conservation law for the total energy during the evolution of the system, the total dissipation associated with the loading rate $\stackrel{.}{\lambda}$ is:
In the above, we did not consider the energy inside Ω_{c} because it is assumed to be zero. Indeed, no compression is considered in this zone (see (43)).
Using Leibniz formula for the time derivative of moving domains as well as the relation:
we obtain
Integrating the domain integral by parts in the second line above and using the equations characterizing the equilibrium state, we get
During the propagation of the interface, perfect contact is assumed on Γ, that is the displacement jump across Γ must be zero at all time. As a consequence, the derivative along the moving interface of the displacement jump must be zero, [22], yielding the so called first Hadamard compatibility condition between the front velocity v_{n} and the jump in material velocities ${\left[\stackrel{.}{\mathit{u}}\right]}_{\Gamma}$:
Equation (51), now becomes
where P is the Eshelby tensor
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.
Since normal stress and displacement are continuous across Γ, the product of the jump in stress and strain across Γ is zero, [23],[24]:
Let ψ_{d}(ϵ) be the density of free energy for a given value of damage and let ${\psi}_{\mathrm{d}}^{*}\left(\mathit{\sigma}\right)$ be its dual by the LegendreFenchel transform. Since the couples (ϵ^{+},σ^{+}) and (ϵ^{−},ϵ^{−}) do satisfy the constitutive model (4), we have
Summing the two relations above and using (55), we have
Since both terms above are greater or equal to zero (classical property of convex analysis, see [25]), we have
This implies that the couples (ϵ^{+},σ^{−}) and (ϵ^{−},σ^{+}) do satisfy the constitutive model. Assuming that the convex potential ψ_{d}(ϵ) is such that the stress associated to any strain is unique, we have
The continuity of the strain and displacement across Γ leads to the continuity of the displacement gradient
leading finally to the continuity of the Eshelby tensor.
Dissipation is thus reduced to
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 Ω_{c} (and thus a negative velocity v_{n}). 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 Γ_{c}. This regularity must be assessed from the nonlocal constitutive model condition: $\stackrel{}{y}{Y}_{\mathrm{c}}\le 0$.
Note that dissipation may also be written
where $\stackrel{}{Y}$ is defined by (25) (y replaced by Y). The above expression exhibits the duality between $\stackrel{}{Y}$ and $\stackrel{}{\stackrel{.}{d}}$ in the localization zone.
Results
We consider a 1D axisymmetric fiber pullout depicted in Figure 4. The fiber of radius r_{ i } is considered rigid and infinitely long. It is pulled out of a clamped circular domain of radius r_{ e }=r_{ i }+L. The only nonzero stress component is the shear stress τ satisfying the following equilibrium conditions
The only nonzero strain is the shear strain, derivative of the displacement along the fiber direction
We consider the following free energy density involving some hardening function h(d), satisfying h(1)=0. The shear stiffness is denoted μ and Y_{c} is also a material parameter.
So, state laws read
The local evolution model is given by
The condition Y=Y_{c} reduces to
Let us now be more precise on the type of function g(d) we will be considering. Basically, we are interested by C^{1} positive concave functions with a maximum value at some damage d_{c}<1 :
We shall use in what follows
The corresponding stress strain curve is given in Figure 5. We will now search for the complete solution linking the (nondimensional) shear stress T needed to move by a (nondimensional) displacement U the fiber:
Four regimes will be observed. They are depicted in Figure 6: elastic, local damage, local and nonlocal damage and finally purely nonlocal damage. The first two regimes may be solved analytically whereas the two last one may not. We however pursue as much as possible the analytical path. Next section is devoted to a numerical solver.
Pure elastic regime: T∈ [ 0,TA=1]
The displacement solution is given by
We thus have a linear relationship between the stress and displacement
Local damage regime: T∈ [ 1,TB]
When T reaches 1 local damage starts around the fiber. Its distribution is obtained by combining (64) and (69)
This distribution of local damage is acceptable provided the condition below holds true
The norm of the damage gradient is maximum at r=r_{ i } and of value
where d_{ i }=d(r_{ i }). Considering a general power law damage profile (13), the condition (76) is
Let us denote by ${d}_{i}^{B}$ the smallest value of damage for which the condition above is violated and T_{ B } the corresponding loading. Due to the fact that g^{′}(d_{c})=0, it is clear that ${d}_{i}^{B}$ will be slightly lower than d_{c}. We note that as the material length gets bigger with respect to r_{ i }, nonlocality (violation of (78)) will step in for smaller and smaller damage d_{ i }. Considering the choice (71), we get the condition
For instance if n=1, we get
As a numerical application, with d_{c}=0.5 and l_{c}/r_{ i }=0.1, we get ${d}_{i}^{B}=0.47$.
Combined local and nonlocal damage regime: T∈ [ TB,TC]
For a loading higher than T_{ B }, nonlocal damage will develop close to the fiber. Let [ r_{ i },r_{ l }] be the current extension of the nonlocal damage zone in which damage ranges from d_{ i } to d_{ l } following:
The condition for the nonlocal zone to grow is $\stackrel{}{Y}={Y}_{\mathrm{c}}$, i.e.
Using, (81), we may rewrite it as
Now, using (69), we get
Since loading is rising, so does local d_{ l } damage at r=r_{ l }, following
Given T, system (84)(85) returns unknowns d_{ i } and d_{ l } as well as the extent of the nonlocal 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.
Nonlocal damage regime: T decreases from TCto 0
There is no solution of the problem for a loading higher than T_{ C }. When the loading decreases below T_{ C }, there is of course a possible elastic solution. Another possible solution is the further development of the nonlocal damage zone (while damage in the local zone no longer evolves since loading is decreasing). The system of equations to solve still involves (84)
Equation (85) is now different and reads
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
The displacement of the fiber is given by
As the damage around the fiber goes to 1, the integrand goes to infinity. But, at the same time the loading goes to zero. Let us study the limit of the fiber displacement for the loading going to zero. The loading is given by (84) recalled below
Due to the property of g(d), (70), we have
So
Finally, we have
Note that this property does not depend on the choice of d(ϕ). Going back to the displacement expression, (88), we have
where C is a finite constant and
assuming a power law asymptotic behavior of ϕ(d) as d goes to 1:
Finally, we get
We conclude that there exists three regimes of delocalization.
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 [20].
Numerical solve
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 t_{n+1} must satisfy the following equations.
Kinematics and equilibrium on ] r _{ i } , r _{ e } [.
State laws and d ( ϕ ) relation on ] r _{ i } , r _{ e } [.
Nonlocal evolution law on $]{r}_{i},{r}_{l}^{n+1}[$ .
Local evolution law on $]{r}_{l}^{n+1},{r}_{e}[$ .
Regarding space discretization, the segment ]r_{ i },r_{ e }[ is discretized with a set of finite elements. Initially, the nonlocal zone is empty and we proceed with a classical NewtonRaphson scheme depicted in the solver flowchart without nonlocal zone.
Solver flowchart without nonlocal zone

1.
initialization: u ^{0}=d ^{0}=T ^{0}=0

2.
elastic step: find the load step for which damage starts

3.
load step T ^{n+1}=T ^{n}+Δ T

4.
iterations initialization k=0

5.
solve linear system (110) to find Δ u

6.
after the first iteration adapt the load step so that the maximum damage increment is d _{inc}
 7.

8.
if residual ≤ tol, go to 9 else go to 5

9.
if ║∇ϕ ^{n+1}║≤1, go to 3, else go to solver flowchart with nonlocal zone
The linear problem at each iteration reads: find $\mathrm{\Delta u}\in \mathcal{U}$ such that:
where the right hand side is the residual at iteration k. Once the displacement correction is obtained, the local update of the fields is computed from
whereas tangent operators are obtained by
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 nonlocal zone is placed and the solver flowchart with nonlocal zone is used.
Solver flowchart with nonlocal zone

1.
initialization: ${r}_{l}^{0}={r}_{i}$

2.
increase nonlocal zone: ${r}_{l}^{n+1}={r}_{l}^{n}+\Delta {r}_{l}$

3.
iterations initialization: k=0

4.
linear solve: solve (119) to find Δ u,Δ T,Δ ϕ

5.
load update: T ^{k+1}=T ^{k}+Δ T

6.
update in local zone (111)(118), and nonlocal zone (120)(124)

7.
if residual ≤ tol, go to 8, else go to 4

8.
if domain not fully broken (d(r _{ i })<1), go to 2, else go to 9

9.
end
The extent of the nonlocal zone is imposed and one tries to find a continuous displacement and damage field satisfying the problem.
The linear symmetric problem to be solved at each iteration when the nonlocal zone is not empty is to find $\mathrm{\in u}\Delta \mathcal{U},\phantom{\rule{2.77626pt}{0ex}}\mathrm{\Delta \varphi}\in \mathcal{A},\mathrm{\Delta T}\in R$ such that
where η^{k} is evaluated following (116). The update in the local zone follows (111)(118) whereas in the nonlocal zone we have
Is is interesting to note the difference between the two solver flowcharts. When the nonlocal 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 nonlocal zone is not empty, the linear solve involves both displacement and damage (or more precisely the surrogate ϕ variable) increment in the nonlocal zone (local zone being treated as before).
The mesh is built so that it is much finer in the localization zone. Node j is located at a position x(j) given by
where N is the number of elements considered. Results will be shown for the following mechanical parameters:
and numerical parameters
Regarding parameter Δ r_{ l }, the nonlocal zone is advanced by one element at a time or smaller when damage gets close to 1 at r_{ i }. This is done in order to capture the full loaddisplacement curve. The formula used in the simulation is
where h^{*} is the size of the element adjacent to the nonlocal zone at time step n. The initial (n=0) nonlocal zone needs to be more than oneelement for convergence. Between 5 and 10 elements are used.
As a final remark on the solver flowchart with the nonlocal zone, we noticed that in nonlocal 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.
Discussion
In Figure 6, the forcedisplacement curve in the case n = 2 is shown. The figure indicates the different regime of the solution (pure elastic, local damage, coupled and pure nonlocal damage). Note that snapback is taken into account automatically since the loading is not imposed but an unknown in the numerical scheme. Profiles of ϕ and damage along the radius at different loads are depicted in Figure 7.
Figure 8 shows the influence of the delocalization parameter n. Plots confirm the analytical limit results (97). As long as damage is purely local, all curves are superposed. As nonlocality steps in, the delocalization parameters n plays a role.
Finally, in order to show the insensitivity of the model with respect to the discretization parameters N, d_{inc}, we show Figure 9 the influence of the choice of the N parameter (for the case n=2 and d_{inc}=0.02). In Figure 10, we show the influence of d_{inc} (for the case n=2 and N=50). Note that as expected, parameter d_{inc} has only an influence when damage is purely local (rising part of the curve). For both figures, a zoom was used. Otherwise, curves cannot be distinguished.
Conclusions
The Thick Level Set damage model allows coupling local damage evolution in some part of the domain to a nonlocal damage evolution in the localization zone. Damage gradient is bounded. The bound is reached in the nonlocal 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 semianalytical 1D solution has been developed showing different regimes in the solution (elastic, local damage, coupled local and nonlocal damage and finally pure nonlocal 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.
Authors’ contributions
All authors contributed to the main ideas in the paper: the way to couple local and nonlocal 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.
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Acknowledgements
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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Moës, N., Stolz, C. & Chevaugeon, N. Coupling local and nonlocal damage evolutions with the Thick Level Set model. Adv. Model. and Simul. in Eng. Sci. 1, 16 (2014). https://doi.org/10.1186/s4032301400162
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Keywords
 Damage
 Delocalization
 Nonlocal damage models
 Level set
 TLS