1. Introduction
Micro-fractures strongly influence the (seismic) wave propagation that gives rise to scattering and fracture-induced anisotropy. This phenomenon makes the derivation of accurate relationships between the micro-structure (pores, micro-cracks) and overall elastic properties of brittle materials (rocks, ceramics,…) difficult. In a fractured medium, when the dimensions of the fractures are substantially smaller than the wavelength, the wave propagation can be described by using effective-medium theories (see, for instance, [
1,
2,
3,
4] or [
5] and references therein). However, if the wavelength is of the same order as the micro-cracks’ radii, numerical simulations are the main tool of investigation.
The performances of meta-materials can be explained by their internal architectures involving stiff and strong building blocks bonded by weaker interfaces. These interfaces, which play a crucial role, enable large deformations and energy dissipation mechanisms throughout large volumes of the materials. Moreover, the mechanical behavior of these interfaces, which is typically nonlinear and governed by friction, is strongly related to their morphology (see, for instance, [
6]).
A large majority of numerical schemes treating wave propagation in materials with micro-fractures use the finite-difference (FD) method. Some of them take the cracks as secondary point sources [
7]. Others use penny-shaped weak inclusions [
8,
9] to model the micro-cracks. In order to adequately model the thickness of cracks, the finite-difference discretization has to be carried out on a small grid spacing, which generates high computational costs (both the grid spacing and the time interval have to be small to satisfy stability considerations). Additionally, when the medium contains high-contrast discontinuities (strong heterogeneities), some instability problems arise on a staggered grid [
10]. Some of them could be avoided by using the rotated staggered grid technique [
11].
In contrast, for “explicit interface” approaches the fracture is assumed to have a vanishing width across which tractions are continuous, but displacements and velocities are allowed to have jumps. One of the “explicit interface” approaches is the so-called “linear-slip displacement-discontinuity model” [
3,
12,
13,
14] offering a unified description of fractures on scales both large and small, relative to the wavelength. However, this model is linear and cannot describe the nonlinear phenomena present on the micro-cracks’ interface, such as as unilateral contact and/or friction.
To model frictional contact constraints, the classical finite element (FE) technique makes use of two (nodal values) discretization methods: the (augmented) Lagrangian method and the penalty method (see, for instance, [
15,
16,
17,
18,
19,
20,
21,
22] or [
23]). Other discretization schemes, such as mortar methods, were developed for non-matching grids [
24,
25,
26,
27]. Another class of mixed formulations encompasses the “dual mortar” methods (see, for instance, [
28,
29]). An alternative to mixed methods is using the primal formulation (in which the displacement field is the only unknown) by Nitsche’s method (see [
30,
31] and its extensions to estimate either quasi-static friction [
32,
33] or explicit dynamics [
34]).
Nitsche’s method has also been used under the guise of the “interior penalty” method within the context of discontinuous Galerkin (DG) methods (see Arnold [
35] for the earliest applications). There have been a lot of important developments in DG approaches for linear and nonlinear solid mechanics (see, for instance, [
36] for linear elasticity developments; [
37,
38,
39] for finite-strain elasticity developments; [
40] for elasto-plasticity developments; and [
41] for second-order computational homogenization). A unifying analysis of the DG method applied to elliptic problems is contained in [
17]. Recently, Truster and Masud [
42] developed a stabilized DG interface method for transient contact with Coulomb friction that extends their previous work on interphase damage modeling [
43]. To overcome the non-smoothness of the Coulomb friction model, they used an elasto-plastic regularization technique (see Simo and Laursen [
18]). This regularization needs a tangential stiffness parameter which is not always simple to capture experimentally. Truster and Masud’s approach is associated with a classical treatment (FE discretization in space and with an implicit Newmark scheme in time) of the elastodynamics equations, while a DG method is used only for the interface nonlinear conditions.
Single and multi-field versions of an h-adaptive, asynchronous space-time discontinuous Galerkin (aSDG) method for elastodynamics, proposed initially in [
44,
45], were developed by Abedi et al. [
46] to simulate dynamic crack propagation with a cohesive model. The aSDG numerical fluxes derive from Riemann solutions of the hyperbolic elastodynamic system, and are therefore more accurate than other fluxes (for instance, the centered flux [
47]), but they are restricted to isotropic elasticity. Moreover, Abedi and Haber [
48] extend the elastodynamic Riemann fluxes in order to treat interfaces subject to frictional contact constraints and use them to obtain high-resolution aSDG solutions for complex contact problems.
The aim of this study was to develop a robust and accurate (fully) DG method for solving the elasto-dynamics’ equations with nonlinear boundary conditions (as friction and/or contact) on a set of interfaces (as internal micro-cracks for instance). This numerical method can be used to determine the effective properties of the damaged materials via a numerical up-scaling homogenization technique by analyzing the wave propagation (speed, amplitude, wavelength) in a cracked material, as in [
49], or to study the dissipation properties of meta-materials that exhibit many frictional interfaces. The applications we have in mind concern brittle materials (as ceramics and rocks), but other elastic materials, such as metals, could also be considered.
The principal original aspects of the proposed numerical scheme lay in the interplay between the leapfrog scheme for the time discretization and the augmented Lagrangian algorithm for solving the associated non-linear problems. Concerning the space discretization, a Galerkin discontinuous method was chosen for its accuracy and efficient parallelization of the computations. Even if for the bulk elements several choices of the numerical flux could be made, the centered flux was preferred for the numerical implementation. The flux in micro-cracks boundaries is computed by solving two non-linear and non-smooth variational problem without any regularization technique. In this way the interfaces’ conditions are modeled simply by different flux choices for the adjacent elements without any specific geometric treatment of the micro-cracks. Since the proposed (augmented) Lagrangian algorithm is related only to the interfaces’ degrees of freedom, the additional computational effort in modeling the nonlinear interfaces is not important.
Let us outline the content of the paper. The elastodynamics problem in a domain with interfaces (cracks) in frictional unilateral contact is stated in
Section 2, whereas in
Section 3 the proposed numerical method is introduced. The leapfrog time discretization splits the elastodynamics problem into two problems: velocity and stress problems. After that, the nonlinear boundary conditions are written as two variational inequalities involving fluxes at the interfaces (micro-cracks). The unilateral condition is associated with the velocity problem, and the friction law relates to the stress problem. By using the DG method for space discretization, the bulk elements are independent of the contact interfaces, which means that classical choices of the flux can be made. However, the fluxes on the interfaces, related to the two nonlinear variational inequalities, have to be found through a numerical iterative algorithm, such as the (augmented) Lagrangian approach. In
Section 4, the numerical method was tested (stability, mesh and time step sensitivity) through two model problems for which analytical solutions exist. Finally, we illustrate how our DG method may be used to investigate more complex wave propagation phenomena, such as blast-wave propagation in a ceramic block with an anisotropic crack distribution.
2. Problem Statement
Let
be an elastic domain which contains a set of interfaces on its boundary. To model a cracked solid, these interfaces (a set of micro-cracks) are located on the internal boundary
(see
Figure 1 for a schematic representation), but other configurations could also be considered. We are looking for the displacement
and the stress tensor
(here
is the space of symmetric
matrices) solution of the following equations:
where
is the small strain tensor,
are the volume forces and
is the fourth order tensor of elastic coefficients. If we denote by
the velocity field and by
the compliance tensor, then (
2) reads
Let
n be the normal
-oriented from − to + sides as defined on
Figure 1. We define the jump
of
by the difference
. The boundary
of
will be divided into two parts: the internal boundary
, and the “loading” boundary, which is the union of two disjoints parts
and
, i.e.,
. On the external boundary
we impose the displacement and the stress vector:
while on the internal boundary
we consider unilateral contact conditions with Coulomb friction.
The non-penetration Signorini conditions read
while
represent the (isotropic) Coulomb friction conditions, with
being the Coulomb coefficient. We have used here the normal and tangential decomposition
,
.
Let us write the Coulomb friction law (
7) as a variational inequality. This will be useful in developing the numerical approach. For that, let us consider the set of admissible stresses
Then (
7) is equivalent with
We complete the field equations and the boundary conditions with the initial conditions
The initial and boundary problem can be formulated now as follows: Find the displacement
(or equivalently the velocity
), the stress
, the solution of (
1)–(
3) with the external boundary conditions (
5), the nonlinear internal boundary conditions (
6) and (
7) and the initial conditions (
10).
4. Testing the Numerical Scheme
In order to test the above mentioned algorithms, we considered two examples for which we could constructed an exact solution and compute the absolute error of our numerical schemes. The first one concerns the unilateral contact and is related principally to the velocity problem. The second one focuses on the frictional contact and concerns mainly the stress problem.
Plane stress conditions (i.e., ) in an isotropic homogeneous elastic body are assumed in all cases. The two-dimensional rectangular domain has an internal interface ( ). In what follows we have chosen the material data, associated with ceramics, to be GPa, and . On the geometric domain (chosen to have m, ) we have considered three meshes: a coarse one (300 triangles, 185 vertexes and 4 segments on ), a medium one (1252 triangles, 695 vertexes and 8 segments on ) and a fine one (4914 triangles 2594, vertexes and 16 segments on ). In all the computations we have chosen the degree of polynomials to be in the construction of the discontinuous Galerkin space .
4.1. Unilateral Contact
The first problem was designed to analyze the unilateral contact and to test the numerical approach of the nonlinear velocity problem. For that we have chosen the following boundary conditions: at and , vanishing shear stress and normal displacement were imposed, and a stress-free condition was considered at . For , we imposed a smooth pulse of time length and amplitude on the velocity field with where if and otherwise. At the internal boundary , we imposed a frictionless non-penetration Signorini condition (), and at the initial state , we supposed that the elastic body was at rest () and stress-free (). The compressive wave, generated by the loading boundary condition at , was propagating through and after the reflection at became a traction wave that generated the separation (jump in normal displacement) in of the two elastic domains.
The initial and boundary conditions were chosen such that we dealt with unidimensional behavior (i.e., ), and one can easily deduce a first-order hyperbolic system for and with an associated speed wave . The time interval of interest will be , with , such that the wave which starts at has the time to reflect at and then to reflect again at .
We can compute the exact solution (
) on the fault
using the method of characteristics. Until the wave reflected at
arrives on the fault, there is no jump in the velocity field and we get:
After that, we deal with a tractional wave, the velocity is positive on the right side of
and the stress is vanishing:
Choice of parameters. The iterative algorithm for the velocity problem stops when the relative error is less than the tolerance . The choices of and of the Uzawa parameter are related. The best choice we found for the numerical parameter in the Lagrangian approach was . For this choice of the convergence is rapid at a tolerance around . If the tolerance is larger than , spurious oscillations could appear and the algorithm is no longer stable. For the algorithm is stable but the error is more important than for . For a tolerance smaller than , the computational time increases without any significant decrease in the error. In the following computations we have chosen .
Stability. There is no theoretical stability condition (CFL condition) for the leapfrog DG scheme. Only a heuristic one (
18) is given in [
52] for the 3D computations. That is why we have numerically checked the stability of the proposed numerical scheme to see how the leapfrog DG scheme is affected by our Lagrangian approach of the unilateral contact conditions. After several tests, we found that the the numerical scheme is stable for a CFL less than 0.108172 for the coarse mesh, less than 0.101463 for the medium mesh and less than 0.109394 for the fine mesh. These values have to be compared with the maximum CFL coefficient founded for the DG leap-frog scheme without any unilateral conditions (0.274515 for the coarse mesh, 0.282538 for the medium mesh and 0.289262 for the fine mesh). We can conclude that for a good stability the CFL coefficient has to be less than 0.1.
To analyze the proposed numerical scheme, we have focused on the error of the unilateral contact condition. For that we have computed the
absolute error of the displacement jump
where
is the displacement amplitude, and the
absolute error of the stress
on the crack
, given by
where
is the stress amplitude.
Mesh sensitivity. In
Figure 2, we have plotted the time evolution of the normalized averaged stress
(left) and of the normalized averaged displacement jump
(right) in
for three different meshes versus the exact solution. All the computations have been done with a fixed time step
(corresponding to a CFL of 0.0178 for the coarse mesh, 0.0379 for the medium mesh and 0.0752 for the finer mesh). We see that the compression pulse travels through the interface
without any perturbation (no displacement jump), but after the reflection at
the traction wave will generate a separation (displacement jump). We remark that the exact solution and the computed one are very close. To see the difference we have zoomed in on the end of the stress pulse during the compression phase of the interface
and the beginning of the displacement jump generated by the reflected wave. We see in both zooms that the numerical solutions corresponding to the medium and fine mesh are very close to the exact one. To see that more precisely, we have plotted in
Figure 3 the stress absolute error
(left) and the displacement jump absolute error
(right) for the three meshes. As before, we remark that the errors of medium and fine meshes are very small. Moreover the error of the fine mesh is of the same order as the error associated with the wave propagator leapfrog-DG method (see “Fine mesh no crack” in
Figure 3) without any unilateral conditions.
4.2. Frictional Slip
In the second test we wanted to see how the numerical scheme works for the frictional slip and to test the numerical approach of the stress problem. For that we have considered an elastic body under a spherical compressive stress acting on all the boundaries ( on ); at and the tangential velocity was vanishing (i.e., ). We also imposed a vanishing tangential stress ( ) at , and for we considered a loading tangential pulse (i.e., ), with ). Here is the stress amplitude and was defined in the previous subsection. At the interface we considered a Coulomb friction law (). The frictional coefficient was chosen to be . At , the elastic body was at rest () and under a spherical pressure .
The tangential stress condition at
generates a shear wave which will propagate into the body, arriving at the frictional interface
. Since the amplitude of the shear wave
is larger than the frictional threshold
, the slab will start to slip and part of the pulse is transmitted on the right side, while the other part will be reflected on left side of the interface. Let us notice that the problem has an one-dimensional solution with
and the problem can be reduced to a first order hyperbolic system for
and
with the associated speed velocity
. The time interval of interest will be
, with
, such that the wave which starts at
has the time to arrive at
and capture the switches no-slip/slip and slip/no-slip. As before, we computed the exact solution on the fault
using the method of characteristics. If we denote by
and
the instances when
, with
, then the two slabs will slip (
) during the time interval
, while in the rest of the time no slip occurs (
). The analytical solution can be computed for each time interval:
Choice of parameters. The iterative algorithm for the stress problem stops when the relative error is less than the tolerance (related to the Uzawa parameter ). The best choice we found for the numerical parameter in the Lagrangian approach was . For this choice of , the convergence was rapid at a tolerance around . If the tolerance is larger than , spurious oscillations could appear, but the algorithm is still stable. For tolerance smaller that , the computational time increases without any significant decrease in the error. In the following computations we have chosen .
Stability. As before, we numerically checked the stability of the proposed numerical scheme to see how the leapfrog DG scheme is affected by our Lagrangian approach of the frictional condition. After several tests, we found that the numerical scheme is stable for a CFL coefficient less than 0.128649 for the corse mesh, less than 0.136483 for the medium mesh and less than 0.13556 for the fine mesh. These values have to be compared with the maximum CFL coefficient found for the DG leap-frog scheme without any frictional conditions (around 0.28; see the previous subsection). We can conclude that we have to choose a time step for the frictional stability such that the CFL coefficient is less than 0.12.
To see how the proposed numerical scheme approaches the frictional boundary condition, we have computed the absolute error of the tangential velocity jump (or slip rate)
:
where
is the velocity amplitude, and the absolute error of the tangential stress
on the crack
, is given by
Mesh sensitivity. In
Figure 4, we have plotted the time evolution of the normalized averaged stress
(left) and of the normalized averaged velocity jump
(right) in
for three different meshes against the exact solution. All the computations have been done with a fixed time step
(corresponding to a CFL of 0.0289 for the coarse mesh, 0.0614 for the medium mesh and 0.122 for the finer mesh). We can observe a plateau in the loading pulse due to the activation of the friction conditions, generating a frictional slip and a wave reflection. We remark on the very good accuracy of the proposed numerical scheme (numerical and analytical solutions are superposed). To see the difference, we have zoomed in on the end of the stress pulse. As before, we can observe an improvement in the numerical solution with regard to the mesh refinement: the numerical solutions corresponding to the medium and fine meshes are very close to the exact one. Only small spurious oscillations are present at the front of the wave. To see that more precisely, we have plotted in
Figure 5 the stress absolute error
(left) and the slip rate absolute error
(right) for the three meshes. As before, we remark that the errors of medium and fine meshes are very small. In contrast, with the unilateral condition, discussed earlier, the error of the fine mesh is larger than the error associated with the wave propagator leapfrog-DG method (see “Fine mesh no crack” in
Figure 5) without any frictional conditions.
6. Conclusions
The qualitative and quantitative investigation of the wave propagation in (damaged) materials with a nonlinear micro-structure (micro-cracks in frictional contact) needs robust, efficient and accurate numerical schemes. This paper proposes a new method based on the interplay between the leapfrog scheme (for the time discretization) and the augmented Lagrangian algorithm (to solve the associated non-linear problems). For an efficient parallelization of the computations, a DG method was used for the space discretization. Since the Lagrangian algorithm concerns only the degrees of freedom associated with the interfaces, the additional computational effort is small with respect to that needed for the wave propagation in the same domain without any interfaces.
This numerical method was tested through two model problems for which other analytical solutions exist. In both cases we analyzed the stability and the mesh sensitivity. To illustrate the numerical scheme, the wave generated by a blast in a cracked material with multiple interfaces has been analyzed.
Future developments of this work could include 3D computations for engineering applications, involving intensive parallel computing techniques, to investigate the wave propagation in elastic solids with micro-cracks (cracked solid) for a better understanding of dynamic damage in brittle materials or in architected meta-materials.