WYPiWYG hyperelasticity without inversion formula: Application to passive ventricular myocardium

[EN] WYPiWYG hyperelasticity is a family of computational procedures for determining the stored energy density of soft materials. Instead of assuming the global analytical shape of these functions (the model), they are computed solving numerically the differential equations of a complete set of experimental tests that uniquely define the material behavior. WYPiWYG hyperelasticity traditionally uses an inversion formula to solve the differential equations, which limits the possible types of tests employed in the procedure. In this work we introduce a new method that does not need an inversion formula and that can be used with any type of tests. We apply the new procedure to determine the stored energy function of passive ventricular myocardium from five experimental simple shear tests.Partial financial support for this work has been given by grant DPI2015-69801-R from the Direccion General de Proyectos de Investigation of the Ministerio de Economia y Competitividad of Spain. FJM also acknowledges the support of the Department of Mechanical and Aerospace Engineering of University of Florida during the sabbatical period in which this paper was finished and that of Ministerio de Educacion, Cultura y Deporte of Spain for the financial support for that stay under grant PRX15/00065Latorre, M.; Montáns, FJ. (2017). WYPiWYG hyperelasticity without inversion formula: Application to passive ventricular myocardium. Computers & Structures. 185:47-58. https://doi.org/10.1016/j.compstruc.2017.03.001475818

Hyperelasticity of Soft Tissues and Related Inverse Problems

International audienceIn this chapter, we are interested in the constitutive equations used to model macroscopically the mechanical function of soft tissues. After reviewing some basics about nonlinear finite–strain constitutive relations, we present recent developments of experimental biomechanics and inverse methods aimed at quantifying consti-tutive parameters of soft tissues. A focus is given to in vitro characterization of hyperelastic parameters based on full-field data that can be collected with digital image correlation systems during the experimental tests. The specific use of these data for membrane-like tissues is first illustrated through the example of bulge inflation tests carried out onto pieces of aortic aneurysms. Then an inverse method, based on the principle of virtual power, is introduced to estimate regional variations of material parameters for more general applications

The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issues and rank-one convexity

We investigate a family of isotropic volumetric-isochoric decoupled strain energies $$F\mapsto W_{_{\rm eH}}(F):=\widehat{W}_{_{\rm eH}}(U):=\left\{\begin{array}{lll} \frac{\mu}{k}\,e^{k\,\|{\rm dev}_n\log {U}\|^2}+\frac{\kappa}{{2\, {\widehat{k}}}}\,e^{\widehat{k}\,[{ \rm tr}(\log U)]^2}&\text{if}& { \rm det} F>0,\\ +\infty &\text{if} &{ \rm det} F\leq 0, \end{array}\right.\quad$$ based on the Hencky-logarithmic (true, natural) strain tensor $\log U$, where $\mu>0$ is the infinitesimal shear modulus, $\kappa=\frac{2\mu+3\lambda}{3}>0$ is the infinitesimal bulk modulus with $\lambda$ the first Lam\'{e} constant, $k,\widehat{k}$ are dimensionless parameters, $F=\nabla \varphi$ is the gradient of deformation, $U=\sqrt{F^T F}$ is the right stretch tensor and ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n} {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the strain tensor $\log U$. For small elastic strains, $W_{_{\rm eH}}$ approximates the classical quadratic Hencky strain energy $$F\mapsto W_{_{\rm H}}(F):=\widehat{W}_{_{\rm H}}(U):={\mu}\,\|{\rm dev}_n\log U\|^2+\frac{\kappa}{2}\,[{\rm tr}(\log U)]^2,$$ which is not everywhere rank-one convex. In plane elastostatics, i.e. $n=2$, we prove the everywhere rank-one convexity of the proposed family $W_{_{\rm eH}}$, for $k\geq \frac{1}{4}$ and $\widehat{k}\geq \frac{1}{8}$. Moreover, we show that the corresponding Cauchy (true)-stress-true-strain relation is invertible for $n=2,3$ and we show the monotonicity of the Cauchy (true) stress tensor as a function of the true strain tensor in a domain of bounded distortions. We also prove that the rank-one convexity of the energies belonging to the family $W_{_{\rm eH}}$ is not preserved in dimension $n=3$

Computation of Cauchy heterogeneous stress field in a cruciform specimen subjected to equibiaxial tensile within parameter identification of isotropic hyperelastic materials

Heterogeneous stress and strain fields have been investigated by Finite Element Method (FEM) in a cruciform specimen holed at the center and subjected to equibiaxial tensile. The stress field is zero at the boundary of the hole; it is a useful boundary condition to compute local stress field. Also, the heterogeneity proves out to be an advantage in order to increase the variety of deformation states. So, a digital image correlation (DIC) system could provide the local deformations, and the corresponding stress field was optimized and adapted to the specimen geometry. Indeed, on the basis of FE results, the heterogeneous Cauchy stress field has been computed analytically in a sub-core region of the s ample. As a result, the local strain and stress fields may be related; so that, the material parameters of isotropic and incompressible rubber-like materials could be identified from experimental data arising from a single heterogeneous test. Besides, the key ideas have been highlighted in order to solve the inverse problem related to the identification procedure