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A gradient-enhanced cohesive model

Nunziante Valoroso

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Nunziante Valoroso. A gradient-enhanced cohesive model. 12e Colloque national en calcul des structures, CSMA, May 2015, Giens, France. hal-01706202

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Submitted on 10 Feb 2018

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# A gradient-enhanced cohesive model

Nunziante Valoroso

Università di Napoli Parthenope, nunziantekar@uniparthenope.it

**Abstract** — A novel implementation of the cohesive zone concept that makes use of the gradient enhancement is discussed. The model can be shown to fit in the non-local generalized standard approach, whereby the framework of standard materials is extended to include the gradients of internal variables. A representative numerical example is shown that demonstrates the capabilities of the proposed approach.

**Keywords** — Cohesive zone model, gradient enhancement.

## 1 Introduction

Gradient enhancements have become increasingly popular in the last decades for dealing with problems in mechanics suffering from spurious mesh sensitivity induced by strain softening. Many proposals exist in this sense and various regularization techniques have been presented and successfully applied to study localization and fracture. In short, the idea underlying almost all such techniques is that of using some extended constitutive equations in which information about the material microstructure is introduced via a length scale-related parameter. From a computational standpoint, once spatial gradients and/or length scales are included in the constitutive equations the latter are no longer defined at the quadrature point level but they are established at a larger scale, i.e. the scale of the structural model. The presence of gradient or averaging operators in the constitutive equations enforces a greater regularity of strains, stresses and internal variables and the resulting solution will be globally smoothed through elements.

## 2 Cohesive damage model

In this work reference is made to an interface cohesive model originally proposed in [1]. When considering the two-parameter version for mode-I case, the model is governed by the following relationships :

$$ 
\begin{align} 
t &= (1 - D)h^{+} \langle [[u]] \rangle_{+} + h^{-} \langle [[u]] \rangle_{-} \nonumber \\ 
Y &= \frac{1}{2} h^{+} \langle [[u]] \rangle_{+}^{2} \tag{1} \\ 
f(D) \le 0; \quad \dot{D} \ge 0; \quad f(D) \dot{D} = 0 \nonumber 
\end{align} 
$$

where $[[u]]$ is the displacement jump in the direction normal to the interface, $h^+$ is the undamaged interface stiffness in tension, $t$ and $Y$ denote the interface traction and the (local) damage-driving force, respectively, $D \in [0, 1]$ is the scalar damage variable, and $f$ is the (local) damage function. The impenetrability constraint is introduced in penalty form via the stiffness coefficient $h^-$ and by distinguishing between the positive $\langle \cdot \rangle_+$ and negative part $\langle \cdot \rangle_-$ of the displacement jump.

The gradient enhancement can be introduced using different arguments, see e.g. [2] among others. However, it can be shown that for the case at hand one ends up with a complementarity problem which has the same formal structure as $(1)_3$, the only modification concerning the damage function that now has to include the laplacian of damage, e.g. by appending the laplacian to the local damage function $f$:

$$ g(D) = f(D) + k \nabla^2 D \tag{2} $$

In the above relationship $k$ is a model parameter with physical dimensions of a force that can be related to the characteristic length parameter. Owing to the presence of spatial derivatives in the damage function (2), the computation of the damage state requires the solution of a boundary value problem.
# 3 The DCB problem

To demonstrate the current model capabilities we consider the DCB problem sketched in Figure 1. The right-end of the structure is free, whilst on the left side two supports are prescribed and an increasing vertical displacement is imposed via arc-length control. The gradient-enhanced cohesive model outlined in the previous section is taken as the constitutive law for dedicated zero-thickness interface elements that have been implemented as a part of a customized version of the FE code FEAP [3].

<figure><img src="image_2.png" /><figcaption>Figure 1 – Schematic of the DCB problem</figcaption></figure>

The main results of computations are summarized in Figure 2, where is depicted the load-deflection curve obtained at varying *k* whilst keeping constant the local model parameters, i.e. the fracture energy and the cohesive stress $t_c$.

<figure><img src="image_3.png" /><figcaption>Figure 2 – DCB problem. Load-deflection curves</figcaption></figure>

Figure 2 clearly shows that in this case the macroscopic effect of the diffusion-like parameter *k* that comes into the picture with the gradient enhancement is very localized on the global response curve and amounts to a progressive modification of the global peak force.

The observation of this effect prompts for two considerations. First, the changes introduced in the global response by the presence of the gradient enhancement cannot be obtained by changing only the local cohesive parameters. Second, the inverse identification of the gradient-related parameter does not strictly require unconventional measurements such as full-field displacements as data set since the effect of the gradient enhancement is quite apparent even on the global load-displacement curve.

# References

[1] N. Valoroso and L. Champaney. A damage-mechanics-based approach for modelling decohesion in adhesively bonded assemblies. *Engineering Fracture Mechanics*, 73(18):2774–2801, 2006.

[2] Quoc-Son Nguyen and S. Andrieux. The non-local generalized standard approach : a consistent gradient theory. *Comptes Rendus Mécanique*, 333(2):139 – 145, 2005.

[3] R. L. Taylor. FEAP - A Finite Element Analysis Program, Programmer manual. University of California at Berkeley, 2011. Available from: http://www.ce.berkeley.edu/feap/.