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# Boundary data identification from overspecified data for elastic-plastic linear strain-hardening bodies

Thouraya Baranger, Stéphane Andrieux

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Thouraya Baranger, Stéphane Andrieux. Boundary data identification from overspecified data for elastic-plastic linear strain-hardening bodies. 12e Colloque national en calcul des structures, CSMA, May 2015, Giens, France. hal-01517305

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Submitted on 3 May 2017

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# Boundary data identification from overspecified data for elastic-plastic linear strain-hardening bodies.

T. N. Baranger<fnref n="1" />, S. Andrieux<fnref n="2" />

**Résumé** — A general solution method for boundary data identification formulated for incremental elastoplasticity is presented. The method extends previous works on Cauchy Problems for linear operators and convex hyperelasticity to the case of generalized standard materials. The issue is formulated as a minimization of an error-functional between the solutions of two well-posed elastoplastic problems. A one-parameter family of error in constitutive equation is derived based on Legendre-Fenchel residuals. An application is presented to illustrate the efficiency of the method.

**Mots clés** — Identification, Nonlinear Cauchy Problem, Legendre-Fenchel Residuals, Drucker Error, Full-field surface data.

## 1 Introduction

Exploiting overspecified boundary data on a part of a solid, for instance displacement and stress vector fields, in order to extend the mechanical fields within the solid, or to identify missing or unknown boundary conditions, remains an open issue, especially for three-dimensional applications presenting nonlinearity behaviours. Potential applications are numerous in mechanical and material sciences and in industry as well. The realized progress in the development of digital cameras and image correlation techniques (DIC) allows now to have measurement means for full field surface displacements that are cheap and easy to manage and, more important, leading to very large amounts of information that could be exploited efficiently to identify data on inaccessible boundaries.

In this communication, we propose a method that allows expanding available mechanical fields (such as stresses and displacements) available on a part of the boundary of a solid inside it and up to unreachable parts of its boundary. The concerned solid is made of an elastic, plastic linear strain hardening.

This method has been developed for other linear operators, as the Lamé operator for linear infinitesimal elasticity with various applications such as determination of contact zones, identification of inclusions or determination of the stress state on buried interfaces in [1,2,3,4]. Linear parabolic or hyperbolic operators have also been addressed with an appropriate extension for the definition of the energy error functional [5,8,9]. This promising approach dealing with this problem consist in reformulate the issue as a Cauchy or Data Completion Problem.

This paper is devoted to an extension of this method to incremental plasticity, involving dissipative and memory effects. After recalling of the framework used for the description of strain hardening plasticity and the formulation of the data completion problem, two well-posed boundary value problems are defined. Each of them has only one of the overspecified data and both having the a Neumann boundary condition on the inaccessible boundary. This last field denoted by $η$ is the unknown variable that has to be identified. Then a Drucker error-functional $J(η)$ measuring the gap between the fields solving of the above problems is build. Minimizing this functional carries out the identification of the wanted Neumann condition $η$. Illustration of the method is performed on the simple two-dimensional situation of symmetric holed plate.

<footnote n="1">¹ CNRS, LaMCoS, Université Claude Bernard Lyon1, [thouraya.baranger@univ-lyon1.fr](mailto:thouraya.baranger@univ-lyon1.fr)</footnote>

<footnote n="2">² EDF R&D, CNRS, LAMSID UMR2932, Clamart, [Stephane.Andrieux@edf.fr](mailto:Stephane.Andrieux@edf.fr)</footnote>
# 2 Formulation of the identification problem.

## 2.1 Nonlinear Cauchy problem.

Consider the framework of generalized standard materials. More specifically, the general constitutive equation studied hereafter is obtained involving the following ingredients : $\epsilon$ is the linearized strain tensor, $\epsilon^p$ the additive plastic strain, $\alpha$ a set of additional internal variables, $\sigma$ is the stress tensor, $u$ is the displacement field. We denotes by $W$ the free energy density and by $\Psi$ the dissipative pseudo potential. For the sake of simplicity we shall drop the arguments of the potentials. Given overspecified data ($U_m, F_m$) on the accessible boundary $\Gamma_m$ incremental Cauchy problem can be formulated as the following : To determine $(\Delta u, \Delta \sigma, \Delta \epsilon, \Delta \alpha)$ satisfying :

$$ egin{cases} 
\operatorname{div}[\sigma + \Delta\sigma] = 0, \quad \epsilon(u + \Delta u) = [\nabla(u + \Delta u)]^{\text{sym}} \\ 
\sigma + \Delta\sigma = \frac{\partial W}{\partial \epsilon}, \quad A + \Delta A = -\frac{\partial W}{\partial \alpha} \\ 
\sigma + \Delta\sigma \in \partial_{\epsilon^p} \Psi(\Delta\epsilon^p, \Delta\alpha), \quad A + \Delta A \in \partial_{\alpha} \Psi(\Delta\epsilon^p, \Delta\alpha) \\ 
\Delta u = \Delta U^m, \quad \Delta\sigma.n = \Delta F^m \text{ on } \Gamma_m 
\end{cases} 
\quad (1) $$

## 2.2 Identification method.

The first step of the solution method is to split this problem into two well-posed incremental problems $\Delta P_1$ and $\Delta P_2$ :

$$ egin{cases} 
\operatorname{div}[\sigma + \Delta\sigma_i] = 0, \quad \epsilon(u + \Delta u_i) = [\nabla(u + \Delta u_i)]^{\text{sym}} \\ 
\sigma + \Delta\sigma_i = \frac{\partial W}{\partial \epsilon}, \quad A + \Delta A_i = -\frac{\partial W}{\partial \alpha} \\ 
\sigma + \Delta\sigma_i \in \partial_{\epsilon^p} \Psi(\Delta\epsilon_i^p, \Delta\alpha_i) \\ 
\text{for } i = 1, 2 
\end{cases} 
\quad \text{with} 
\quad 
\begin{cases} 
(\Delta P_1) \quad \begin{cases} 
\Delta u_1 = \Delta U^m \text{ on } \Gamma_m \\ 
\Delta\sigma_1.n = \Delta\eta \text{ on } \Gamma_u 
\end{cases} \\ 
(\Delta P_2) \quad \begin{cases} 
\Delta\sigma_2.n = \Delta F^m \text{ on } \Gamma_m \\ 
\Delta\sigma_2.n = \Delta\eta \text{ on } \Gamma_u 
\end{cases} 
\end{cases} 
\quad (2) $$

The building of errors between the two states $\Delta e_1(\Delta u_1, \Delta \sigma_1, \Delta \epsilon_1^p, \Delta \alpha_1)$ and $\Delta e_2(\Delta u_2, \Delta \sigma_2, \Delta \epsilon_2^p, \Delta \alpha_2)$ relies now on the convexity property of the free energy and dissipation potential, by using the Legendre-Fenchel inequalities related to each of them, we can then derive several errors with suitable properties. They are positive quantities and whenever they vanish then the distance between the two state variable increments vanish together with the distance of their dual counterparts. The linear strain-hardening function is defined by $R(\gamma) = -H\gamma + \sigma_0$, where $\gamma$ is the accumulated plastic strain, so $\alpha = \gamma$ and $A = -H\gamma$, the free energy is defined by :

$$ W(\epsilon, \epsilon^p, \alpha) = \frac{1}{2}(\epsilon - \epsilon^p) : \mathbb{C} : (\epsilon - \epsilon^p) + \int_0^\alpha (R(\beta) - \sigma_0)d\beta \quad (3) $$

In this case, the error can be build by combining the error in free energy and the error in dissipation via a parameter $0 \le \chi \le 1$.

$$ 
\begin{aligned} 
\mathcal{E}_{\chi} = (1 - \chi) & [(\Delta\sigma_1 - \Delta\sigma_2) : (\Delta\epsilon_1^e - \Delta\epsilon_2^e) - (R(\alpha + \Delta\alpha_1) - R(\alpha + \Delta\alpha_2)).(\Delta\alpha_1 - \Delta\alpha_2)] \\ 
& + \chi [(\Delta\sigma_1 - \Delta\sigma_2) : (\Delta\epsilon_1^p - \Delta\epsilon_2^p) + (R(\alpha + \Delta\alpha_1) - R(\alpha + \Delta\alpha_2)).(\Delta\alpha_1 - \Delta\alpha_2)] 
\end{aligned} 
\quad (4) $$

$$ J_{\chi}(\Delta\eta) = \int_{\Omega} \mathcal{E}_{\chi}(\Delta e_{1}(\Delta\eta)), \Delta e_{2}(\Delta\eta)) d\Omega \quad (5) $$

Equipped with this error in energy that can be also called error in constitutive equations, we can then define the general error as a function of $\Delta\eta$ to be minimized in order to get the solution of the data completion problem. The Drücker error $J_{1/2}(\Delta\eta)$ is the only one that can be computed by boundary integration on the whole external surface of the body reduced to $\Gamma_m$ only (as it is the case in linear elasticity and hyperelasticity), thanks to the virtual power principle. This feature has been largely exploited previously to improve the global performance of the solution algorithm for linear Cauchy Problem.

# 3 Application.

The proposed solution method has been implemented on the following example in two dimensions. A perforated plate with unknown pressure applied on the hole boundary is loaded in its plane with a known
force going from zero to a maximum value $F_{max}$ and unloaded back to zero. The displacements on the top and bottom parts of the boundary (free of any loading) are available at various steps of the loading and unloading history. The loading part and the unloading part are divided into 5 increments. The constitutive material of the plate is an isotropic elastoplastic material with linear hardening. It is characterized by the Young modulus $E = 69000 \text{ MPa}$, the Poisson ratio $\nu = 0.33$, the yield stress $\sigma_0 = 500 \text{ MPa}$ and a strain-hardening modulus $H = 0.2 \text{ MPa}$. For each step, the incremental Cauchy Problem is solved by minimizing the Drücker error $J_{1/2}$, starting from the mechanical state obtained at the end of the preceding step. The two problems $\Delta P_1$ and $\Delta P_2$ are solved by the finite element method and the gradient of the error function is computed by finite differences (because the adjoint method cannot be directly applied here as the dissipation pseudo-potential is not twice differentiable). The average number of iterations for the minimization process varies from 1000 to 2000 iterations.

Results are illustrated on the following figures where various identified mechanical fields within the solid are displayed. Figures 1a, 1c, 1b and 1d show the exact and identified Von Mises stresses and figures 1e, 1f, 1g and 1h show the exact and identified plastic strain tensor components at the end of the loading step and then the unloading one. Comparing the identified data with the exact ones, we observe good agreement for the reconstructed displacements, stresses and plastic strains, even "far away" from the boundary where the data are available.

# 4 Conclusion.

We presented a general method for the solution of the Cauchy problem for incremental elastic linear strain-hardening plasticity. Based on the splitting of the problem into two well-posed evolution problems, the derivation of a one-parameter family of energy errors, and the minimization of it, the method has shown that the precision for the identification of unknown boundary data and of the plastic strain field is very good. Work under progress is devoted to numerical three-dimensional applications and evaluation of the effects of noise in the Cauchy data on the identifications' previsions.

# Références

[1] S. Andrieux, T. Baranger, *An energy error-based method for the resolution of the Cauchy problem in 3D linear elasticity*, CMAME 197 (2008) 902-920.

[2] S. Andrieux, T. N. Baranger, *Emerging crack front identification from tangential surface displacements*, Comptes Rendus Mécanique 340 (8) (2012) 565-574.

[3] S. Andrieux, T. N. Baranger, *Three-dimensional recovery of stress intensity factors and energy release rates from surface full-field displacements*, Int. Journal of Solids and Structures 50 (10) (2013) 1523-1537.

[4] T. N. Baranger S. Andrieux, *Constitutive law gap functionals for solving the Cauchy problem for linear elliptic PDE*, Applied Mathematics and Computation 218 (2011) 1970-1989.

[5] T. N. Baranger, S. Andrieux, R. Rischette, *Combined energy method and regularization to solve the Cauchy problem for the heat equation*, Inverse Problems in Science and Engineering 22 (1) (2014) 199-212.

[6] J. Hadamard, *Lectures on Cauchy's problem in linear partial differential equations*, Dover, New York, 1923.

[7] P. Ladeveze, A. Chouaki, *Application of a posteriori error estimation for structural model updating*, Inverse Problems 15 (1) (1999) 49.

[8] R. Rischette, T. N. Baranger, N. Debit, *Numerical analysis of an energy-like minimization method to solve Cauchy problem with noisy data*, J. of Computational and Applied Mathematics 235 (2011) 3257-3269.

[9] R. Rischette, T. N. Baranger, S. Andrieux, *Regularization of the noisy Cauchy problem solution approximated by an energy-like method*, Int. J. for Num. Methods in Engineering 95 (4) (2013) 271-287.
<figure><img src="image_2.png" /><figcaption>(a) Identified $\sigma_{eq}$ at the end of loading.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>(b) Reference $\sigma_{eq}$ at the end of loading.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(c) Identified $\sigma_{eq}$ at the end of unloading.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(d) Reference $\sigma_{eq}$ at the end of unloading.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(e) Identified $\varepsilon_{xx}^p$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(f) Reference $\varepsilon_{xx}^p$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(g) Identified $\varepsilon_{yy}^p$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(h) Reference $\varepsilon_{yy}^p$.</figcaption></figure>

<caption>FIGURE 1 – Von Mises stress $\sigma_{eq}$ and plastic strain components ($\varepsilon_{xx}^p, \varepsilon_{yy}^p$).</caption>