**Principal investigators:**

Damien Vanbembroucq

Sylvain Patinet

**Members of the group:**

Tyukodi Botond (PhD 2013-16)

Our research is at the interface of physics and mechanics. Following a multi-scale approach, they aim to physically justify the mechanical constitutive relations from the modeling of the microscopic phenomena through different simulation and theoretical methods (atomistic, elastic line, continuous, depinning transition).

Our activities include the study of dissipative phenomena such as plasticity and cracking occurring in disordered materials during their mechanical loading. They can be clustered into two main themes:

1) The study of the plasticity of amorphous materials at the atomic and the mesoscopic scales.

2) The description of threshold, velocity and force fluctuations along elastic interfaces trapped in disordered media.

**From depinning transition to plastic yielding of amorphous media: A soft modes perspective**

A mesoscopic model of amorphous plasticity is discussed in the wider context of depinning models. After embedding in a d + 1 dimensional space, where the accumulated plastic strain lives along the additional dimension, the gradual plastic deformation of amorphous media can be regarded as the motion of an elastic manifold in a disordered landscape. While the associated depinning transition leads to scaling properties, the quadrupolar Eshelby interactions at play in amorphous

plasticity induce specific additional features like shear-banding and weak ergodicity break-down. The latters are shown to be controlled by the existence of soft modes of the quadrupolar interaction, the consequence of which is discussed in the context of depinning.

Botond Tyukodi, Sylvain Patinet, Stéphane Roux and Damien Vandembroucq, arXiv:1502.07694 (2016)

**Finite-size effects in a model for plasticity of amorphous composites**

We discuss the plastic behavior of an amorphous matrix reinforced by hard particles. A mesoscopic depinning-like model accounting for Eshelby elastic interactions is implemented. Only the effect of a plastic disorder is considered. Numerical results show a complex size dependence of the effective flow stress of the amorphous composite. In particular, the departure from the mixing law shows opposite trends associated to the competing effects of the matrix and the reinforcing particles, respectively. The reinforcing mechanisms and their effects on localization are discussed. Plastic strain is shown to gradually concentrate on the weakest band of the system. This correlation of the plastic behavior with the material structure is used to design a simple analytical model. The latter nicely captures reinforcement size effects in (log N/N )^1/2 , where N is the linear size of the system, observed numerically. Predictions of the effective flow stress accounting for further logarithmic corrections show a very good agreement with numerical results.

**Cracks in random brittle solids: From fiber bundles to continuum mechanics**

Statistical models are essential to get a better understanding of the role of disorder in brittle disordered solids. Fiber bundle models play a special role as a paradigm, with a very good balance of simplicity and non-trivial effects. We introduce here a variant of the fiber bundle model where the load is transferred among the fibers through a very compliant membrane. This Soft Membrane fiber bundle mode reduces to the classical Local Load Sharing fiber bundle model in 1D. Highlighting the continuum limit of the model allows to compute an equivalent toughness for the fiber bundle and hence discuss nucleation of a critical defect. The computation of the toughness allows for drawing a simple connection with crack front propagation (depinning) models.

**Soft spots and their structural signature in a metallic glass**

In a 3D model mimicking realistic Cu 64 Zr 36 metallic glass, we uncovered a direct link between the quasi-localized low-frequency vibrational modes and the local atomic packing structure. We also demonstrate that quasi-localized soft modes correlate strongly with fertile sites for shear transformations: geometrically unfavored motifs constitute the most flexible local environments that encourage soft modes and high propensity for shear transformations, whereas local configurations preferred in this alloy, i.e., the full icosahedra (around Cu) and Z16 Kasper polyhedra (around Zr), contribute the least.

Jun Ding, Sylvain Patinet, Michael L. Falk, Yongqiang Cheng and Evan Ma, PNAS 111 (2014)

**Quantitative Prediction of Effective Toughness at Random Heterogeneous Interfaces**

The propagation of an adhesive crack through an anisotropic heterogeneous interface is considered. Tuning the local toughness distribution function and spatial correlation is numerically shown to induce a transition between weak to strong pinning conditions. While the macroscopic effective toughness is given by the mean local toughness in the case of weak pinning, a systematic toughness enhancement is observed for strong pinning (the critical point of the depinning transition). A self-consistent approximation is shown to account very accurately for this evolution, without any free parameter.

Sylvain Patinet, Damien Vandembroucq and Stéphane Roux PRL 110, 165507 (2013)

**Finite size effects on crack front pinning at heterogeneous planar interfaces: Experimental, finite elements and perturbation approaches**

Understanding the role played by the microstructure of materials on their macroscopic failure properties is an important challenge in solid mechanics. Indeed, when a crack propagates at a heterogeneous brittle interface, the front is trapped by tougher regions and deforms. This pinning induces non-linearities in the crack propagation problem, even within Linear Elastic Fracture Mechanics theory, and modifies the overall failure properties of the material. For example crack front pinning by tougher places could increase the fracture resistance of multilayer structures, with interesting technological applications. Analytical perturbation approaches, based on Bueckner–Rice elastic line models, focus on the crack front perturbations, and hence allow for a description of these phenomena. Here, they are applied to experiments investigating the propagation of a purely interfacial crack in a simple toughness pattern: a single defect strip surrounded by homogeneous interface. We show that by taking into account the finite size of the body, quantitative agreement with experimental and finite elements results is achieved. In particular this method allows to predict the toughness contrast, i.e. the toughness difference between the single defect strip and its homogeneous surrounding medium. This opens the way to a more accurate use of the perturbation method to study more disordered heterogeneous materials, where the finite elements method is less adequate. From our results, we also propose a simple method to determine the adhesion energy of tough interfaces by measuring the crack front deformation induced by known interface patterns.

S. Patinet, L. Alzate, E. Barthel, D. Dalmas, D. Vandembroucq and V. Lazarus, JMPS 61 (2013)