# Stress softening and hyper-viscoelastic modeling of soft materials: insights into parameter identification

Anne-Sophie Caro-Bretelle, Sarah Iaquinta, André Chrysochoos, Grégory Chagnon

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Anne-Sophie Caro-Bretelle, Sarah Iaquinta, André Chrysochoos, Grégory Chagnon. Stress softening and hyper-viscoelastic modeling of soft materials: insights into parameter identification. Multidisciplinary Biomechanics Journal, 2025, 50th congress of the Société de Biomécanique, 10.46298/mbj.16245. hal-05191800v2

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# Stress softening and hyper-viscoelastic modeling of soft materials: insights into parameter identification

Anne-Sophie Caro<sup>a*</sup>, Sarah Iaquinta<sup>a</sup>, André Chrysochoos<sup>b</sup>, Grégory Chagnon<sup>c</sup>

a LMGC, IMT Mines Ales, Univ Montpellier, CNRS, Ales, France

b LMGC, Univ Montpellier, CNRS, Montpellier, France

c Univ. Grenoble Alpes, CNRS, UMR 5525, VetAgro Sup, Grenoble INP, TIMC, Grenoble, France

*Corresponding author: anne-sophie.caro@mines-ales.fr

Received date: 06/04/2025

Accepted date: 27/06/2025

Publication date: 27/10/2025

**Keywords:** soft materials, hyperviscoelasticity, Mullins effect, relaxation

© 2025 The Authors

Licence CC-BY 4.0

Published by Société de Biomécanique

## 1. Introduction

Soft materials, such as biological tissues, often undergo large deformations and exhibit complex mechanical behavior under cyclic loading, including stress softening (Mullins effect (Peña, Peña, and Doblaré 2009)) and time-dependent responses (viscoelasticity (Iaquinta, Chagnon, and Caro-bretelle 2024)). Accurately modeling these phenomena is essential for predictive bio-mechanics. While each mechanism has been studied individually, few models combine hyperelasticity, stress softening, and viscoelasticity within a unified framework—likely due to the increased complexity and number of parameters to identify. This study proposes a constitutive model integrating these effects (hyperviscoelasticity, stress softening for compressible materials) in a thermodynamically consistent framework based on Generalized Standard Materials and provides a detailed analysis of parameter identification and model sensitivity.

## 2. Methods

### 2.1 Constitutive model

The proposed model is developed within the Generalized Standard Materials (GSM) framework to ensure thermodynamic consistency. It combines an isochoric hyper-elastic component (Yeoh-type), a Mullins-type stress softening based on a history variable (the maximum of the first reduced invariant of the modified right Cauchy-Green tensor $\overline{C}(\overline{I}_1)$ denoted $\overline{I}_1$ and a linear viscoelastic

branch modeled by a Maxwell element (Simo 1987). Anisotropy is not considered in the present work. The Helmholtz free energy is defined in terms of the elastic and internal variables, and a Legendre transformation is used to derive the constitutive equations, leading to stress contributions from elastic, viscous, and volumetric effects. The isochoric strain energy $W_e(\overline{C})$ is modulated by a softening function $F(\overline{I}_1, \overline{I}_1)$ equal to 1 when the load is active and remains less than 1 during unload or neutral load (e.g. $\overline{I}_1 = 0$). The total second Piola-Kirchhoff stress $S$ is split into isochoric, viscous, and volumetric contributions:

$$S = 2(FP: \partial_c W_e) + P: S_v + \kappa(J - 1)C^{-1}$$

where, $P = \partial_c \overline{I}_1$ and $P = \partial_c \overline{C}$ the viscous stress that verifies this first order differential equation:

$$R: S_v + \frac{1}{\mu(\overline{C}, \overline{I}_1)} S_v = \frac{\overline{C}}{2},$$

$$R_{ijkl} = \frac{1}{2G}(\delta_{ik}\delta_{jl} + \delta_{ij}\delta_{jk}) - \frac{1}{3G}\delta_{ij}\delta_{kl}; i, j, k, l = 1..3$$

where $\mu$ is a viscosity coefficient and $G$ is the shear modulus.

### 2.2 Model parameters

The model parameters include, in addition to the hyperelastic constants, $c_1, c_2, c_3$ the shear modulus $G$, a
relaxation time $\tau$ defined as $\mu/3G$ and Mullins softening parameters $\alpha, \eta$ which controls the amplitude and rate of stress reduction under cyclic loading.

## 2.3 Sensitivity analysis

To identify the most influential parameters, we performed a Sobol sensitivity analysis on $c_{i, i=1,2,3}$, $\tau$, $G$, $\alpha$ and $\eta$ in the case of cyclic uniaxial tensile and relaxation test. After a convergence study on the Sobol indices with respect to the size of the dataset (Yang 2011), a quasi Monte Carlo Sobol sequence with $2^{12}$ samples was used. Four mechanical indicators were extracted from the stress/elongation response: stress slopes ($S_\theta, S_u$), hysteresis area ($HA$), and from the stress/time response: initial relaxation slope ($S_r$) and stress relaxation ratio ($RR$) (Figure 1). To further investigate parameter identifiability, we also plan to compute local identifiability metrics based on the Hessian of the cost function. This includes the identifiability index and the normalized parameter interaction matrix, as introduced by (Barick et al. 2020). These quantities provide insight into the local curvature of the cost function and quantify both the sensitivity and the potential correlations between parameters, offering a complementary perspective to the global Sobol analysis.

# 3. Results and discussion

## 3.1 Model behavior

Simulations reproduce characteristic features of soft material response under cyclic uniaxial loading and relaxation: hysteresis, stress softening, and relaxation (Figure 1). The model captures the evolution of mechanical response across cycles, validating the role of $F(I_1, I_1^*)$ and $\mu(I_1^*)$.

## 3.2 Sensitivity results

The Sobol indices reveal that hyperelastic parameters strongly influences the initial slope ($S_\theta$) and the hysteresis area ($HA$), while $\tau$, $G$ dominate the stress slope ($S_r$) and also contributes to the hysteresis area ($HA$). The Mullins effect parameters govern the stress softening via the unloading slope $S_u$. Parameter interactions are minimal, supporting the possibility of individual identifiability. These results provide valuable guidance for designing targeted experimental protocols for robust parameter identification.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Model response to a loading-relaxation-unloading cycle (top) and corresponding first and total Sobol sensitivity indices (bottom).</figcaption></figure>

This conclusion is further supported by the local identifiability analysis based on the Hessian of the cost function. The computed identifiability indices for each phase (loading, relaxation, unloading) remain below commonly accepted thresholds (<3), suggesting that the corresponding parameter combinations are correctly identifiable.

# 4. Conclusions

The proposed model combines hyperelasticity, viscoelasticity, and the Mullins effect in a thermodynamically consistent way, using 7 parameters. It offers a balance between realism and simplicity, relying on physical principles. A Yeoh-type hyperelastic formulation captures large strains, a scalar variable models stress softening, and a simple viscoelastic branch accounts for relaxation. Each parameter shows clear and separate influence, as confirmed by sensitivity analyses. Future work will address anisotropy.

## Conflict of Interest Statement

None.

## Contributor Roles

ASC: Conceptualization, Methodology, Writing original draft; SA: Conceptualization, Methodology, Writing
original draft, AC: Conceptualization, Methodology, Writing-review, GC: Conceptualization, Methodology.

## Fundings

This project is supported by Horizon Europe EIC. Pathfinder project PELVITRACK (PELVIC Floor Evaluation live TRACKing), 2025-2029, Grant ID 101186212.

## References

Barick, M. C., Y. Gaillard, A., Lejeune, F., Amiot, & Richard, F. (2020). On the uniqueness of intrinsic viscoelastic properties of materials extracted from nanoindentation using FEMU. *International Journal of Solids and Structures*, 202, 929–946. https://doi.org/10.1016/j.ijsolstr.2020.03.015

Iaquinta, S., Chagnon, G., & Caro-bretelle, A.-S. (2024). Hyper-viscoelastic stress-softening modeling of pig perineal skin. *Multidisciplinary Biomechanics Journal*, 1, 199–201. https://doi.org/10.46298/mbj.14506

Peña, E., Peña, J. A. & Doblaré, M. (2009). On the Mullins effect and hysteresis of fibered biological materials: A Comparison between continuous and discontinuous damage models. *International Journal of Solids and Structures*, 46(7–8), 1727–1735. https://doi.org/10.1016/j.ijsolstr.2008.12.015

Simo, J. C. (1987). On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. *Compt. Meth. Appl. Mech. Engng*, 60, 153–173.

Yang, J. (2011). Convergence and uncertainty analyses in Monte-Carlo based sensitivity analysis. *Environmental Modelling & Software*, 26(4), 444–457. https://doi.org/10.1016/j.envsoft.2010.10.007