率形式大变形弹塑性本构律中的超弹性条件

本文证明了若取客观应力率为 Kirchhoff 应力的 Oldroyd 导数,对于 Lame 参数λ、μ为常数的情况,率形式弹性本构律的可积条件为 λ=0。这显然表明在大变形情况下率形式弹塑性本构律与超弹性条件这两者之间在一般情况下并不协调。文中还讨论了几种弹性本构律可以近似用于大变形描述的情况。     

Finite Bending of a Rectangular Block of an Elastic Hencky Material

Hencky's elasticity model is an isotropic, finite hyperelastic equation obtained by simply replacing the Cauchy stress tensor and the infinitesimal strain tensor in the classical Hooke's law for isotropic infinitesimal elasticity with the Kirchhoff stress tensor and Hencky's logarithmic strain tensor. A study by Anand in 1979 and 1986 indicates that it is a realistic finite elasticity model that is in good accord with experimental data for a variety of engineering materials for moderate deformations. Most recently, by virtue of well-founded physical grounds and rigorous mathematical procedures it has been demonstrated by these authors that this model may be essential to achieving self-consistent Eulerian rate type theories of finite inelasticity, e.g., the J ₂-flow theory for metal plasticity, etc. Its predictions have been studied for some typical deformation modes, including extension, simple shear and torsion, etc. Here we are concerned with finite bending of a rectangular block. We show that a closed-form solution may be obtained. We present explicit expressions for the bending angle and the bending moment in terms of the maximum or minimum circumferential stretch in a general case of compressible deformations for any assigned stretch normal to the bending plane. In particular, simplified results are derived for the plane strain case and for the case of incompressibility. This revised version was published online in June 2006 with corrections to the Cover Date.     

Modeling of micropipette aspiration and optical tweezers stretching of erythrocytes with or without Malaria parasite

The erythrocytes play an important role in the human body. The healthy erythrocytes can undergo extremely large deformation while passing through small capillaries. Their infection by Malaria Plasmodium falcipurum (P.f.) will lead to capillary blockage and blood flow obstruction. Many experimental and computational methods have been applied to study the increase in stickiness and decrease in deformability of the Malaria (P.f.) infected erythrocytes. The novelty of this paper lies in the establishment of an multi-component model for investigating mechanical properties of Malaria (P.f.) infected erythrocytes, especially of their enclosed parasites. Finite element method was applied to simulate the erythrocytes’ deformation in micropipette aspiration and optical tweezers stretching using the computational software ABAQUS. The comparisons between simulations and experiments were able to quantitatively conclude the effects of stiffness and stickiness of the parasitophorous vacuole membrane on the cells’ deformation, which could not be obtained from experiments directly.     

A closed form solution for the uniaxial tension test of biological soft tissues

This paper concerns the modeling of biological soft tissues in the framework of anisotropic hyperelasticity. A closed form solution is proposed in the special case where uniaxial unconstrained tension loading is applied to a strip modeled by Holzapfel-Gasser-Ogden's (HGO) strain energy. The classical Cardano's formula is used to calculate the solution. This solution is investigated for different values of β which represents the angle between the collagen fibers and the circumferential direction. This study allows for the understanding of the reason why a one-to-one correspondence does not exist between the principal stretch and the fourth invariant of the right Cauchy-Green deformation tensor. It is demonstrated that the relationship becomes non-bijective when β is greater than a critical angle of 54.73^°. The HGO model is implemented into an in-house finite element program and numerical results are in good agreement with theoretical ones.     

Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids

Based on the Theory of Porous Media (TPM), a formulation of a fluid-saturated porous solid is presented where both constituents, the solid and the fluid, are assumed to be materially incompressible. Therefore, the so-called point of compaction exists. This deformation state is reached when all pores are closed and any further volume compression is impossible due to the incompressibility constraint of the solid skeleton material. To describe this effect, a new finite elasticity law is developed on the basis of a hyperelastic strain energy function, thus governing the constraint of material incompressibility for the solid material. Furthermore, a power function to describe deformation dependent permeability effects is introduced.After the spatial discretization of the governing field equations within the finite element method, a differential algebraic system in time arises due to the incompressibility constraint of both constituents. For the efficient numerical treatment of the strongly coupled nonlinear solid-fluid problem, a consistent linearization of the weak forms of the governing equations with respect to the unknowns must be used.     

Torsion in nonlinearly elastic incompressible circular cylinders

We consider a variable torsion deformation for incompressible right-circular cylinders and investigate the possibility of equilibrium states other than simple torsion in isotropic hyperelasticity. Our problem formulation also provides generalized versions of Rivlin's universal relations in torsion. In this more general setting, it is shown that simple torsion persists as the only solution for large classes of strain energies. When one allows for more general boundary conditions, variable torsion solutions are possible for special forms of the stored energy function. We derive such a form and develop general results.     

Influence of random uncertainties of anisotropic fibrous model parameters on arterial pressure estimation

This paper deals with a stochastic approach based on the principle of the maximum entropy to investigate the effect of the parameter random uncertainties on the arterial pressure. Motivated by a hyperelastic, anisotropic, and incompressible constitutive law with fiber families, the uncertain parameters describing the mechanical behavior are considered. Based on the available information, the probability density functions are attributed to every random variable to describe the dispersion of the model parameters. Numerous realizations are carried out, and the corresponding arterial pressure results are compared with the human non-invasive clinical data recorded over a mean cardiac cycle. Furthermore, the Monte Carlo simulations are performed, the convergence of the probabilistic model is proven. The different realizations are useful to define a reliable confidence region, in which the probability to have a realization is equal to 95%. It is shown through the obtained results that the error in the estimation of the arterial pressure can reach 35% when the estimation of the model parameters is subjected to an uncertainty ratio of 5%. Finally, a sensitivity analysis is performed to identify the constitutive law relevant parameters for better understanding and characterization of the arterial wall mechanical behaviors.     

Mechanical Characterization of Biological Membranes with Moiré Techniques and Multi-Point Simulated Annealing

This paper describes an hybrid procedure for mechanical characterization of biological membranes. The in-plane displacement field of a glutaraldehyde treated bovine pericardium patch obtained with an equi-biaxial tension test is measured with intrinsic moiré and then compared with finite element predictions. Preliminary analysis of moiré patterns observed in the experiments justifies the assumption of the constitutive model based on transversely isotropic hyperelasticity. In order to determine the 16 hyperelastic constants included in the constitutive model and the fiber orientation, the difference Ω between displacement values measured with moiré and their counterpart determined numerically is minimized by means of multi-level and multi-point simulated annealing. Results clearly demonstrate the efficiency of the identification procedure presented in this research: in fact, residual difference between experimental data and numerical values of in-plane displacements is less than 2%. In order to validate the entire identification process, another experimental test is conducted by inflating the same specimen. Out-of-plane displacements, now measured with projection moiré, are compared with predictions of a new finite element model reproducing the experimental test. The 16 hyper-elastic constants previously determined are given in input to the inflation test FE model. Remarkably, experimental and numerical results are again in excellent agreement: maximum percent error on w-displacement is less than 3%.     

Fracture toughness from the standpoint of softening hyperelasticity

Fracture toughness of brittle materials is calibrated in experiments where a sample with a preexisting crack/notch is loaded up to a critical point of the onset of static instability. Experiments with ceramics, for example, exhibit a pronounced dependence of the toughness on the sharpness of the crack/notch: the sharper is the crack the lower is the toughness. These experimental results are not entirely compatible with the original Griffith theory of brittle fracture where the crack sharpness is of minor importance.¹To explain the experimental observations qualitatively we simulate tension of a thin plate with a small crack of a finite and varying sharpness. In simulations, we introduce the average bond energy as a limiter for the stored energy of the Hookean solid. The energy limiter induces softening, indicating material failure. Thus, elasticity with softening allows capturing the critical point of the onset of static instability of the cracked plate, which corresponds to the onset of the failure propagation at the tip of the crack. In numerical simulations we find, in agreement with experiments, that the magnitude of the fracture toughness cannot be determined uniquely because it depends on the sharpness of the crack: the sharper is the crack, the lower is the toughness.Based on the obtained results, we argue that a stable magnitude of the toughness of brittle materials can only be reached when a characteristic size of the crack tip is comparable with a characteristic length of the material microstructure, e.g. grain size, atomic distance, etc. In other words, the toughness can be calibrated only under conditions where the hypothesis of continuum fails.