Hencky Model Research Articles

Fourth‐order phase field model with spectral decomposition for simulating fracture in hyperelastic material

AbstractWe present a fourth‐order phase field model for fracture behavior simulations of hyperelastic material undergoing finite deformation. Governing equations of the fourth‐order phase field model consist of the biharmonic operator of the phase field, which requires the second‐order derivatives of shape function. Therefore, a 5 × 5 Jacobian matrix of isoparametric transformation is constructed. Neo‐Hooken model and Hencky model are adopted as the material constitutive models. The spectral decomposition of stored strain energy is used to distinguish the contributions of tension and compression, and the corresponding stress tensor and constitutive tensors are derived, and subsequently, the numerical framework of modeling fracture with the fourth‐order phase field model is implemented in details. Several typical numerical examples are conducted to demonstrate the robustness and effectiveness of the fourth‐order phase field model in simulating the fracture phenomenon of rubber‐like materials.

Concretization of nonlinear constitutive relations by results of uniaxial compression and indentation experiments

A method for the experimental specification of the variant of the Hencky–Murnaghan constitutive relations is proposed. A series of compression and indentation experiments of a number of specimens made of highly elastic materials is carried out on the experimental stand. An iterative procedure for determining constants from experiments on inhomogeneous deformation is developed. The convergence of the iterative process is investigated. The values of the constants of the Hencky–Murnaghan model are found for materials of two types. It is found that the Hencky–Murnaghan model describes the results of indentation experiments on a wider range of deformations than the Hencky model.

A two-parameter strain energy function for brain matter: An extension of the Hencky model to incorporate locking

By just replacing the infinitesimal strains by logarithmic strains, the Hencky strain energy has proven to extend successfully the infinitesimal framework to moderately large strains, as those found in brain. However, as polymers and soft tissues, brain presents an important strain-stiffening towards locking. Based on both observations, in this paper we propose a simple two-parameter isotropic strain energy function for representing the inviscid (conservative) behavior of brain matter. The two parameters of the model are the Young modulus (or alternatively the shear modulus) and the locking stretch during a test. Through a comparison with experimental data, we show that with this simple model, employing the two material parameters directly measured from a tensile test, we capture the qualitative aspects and quantitative behavior of brain mater in tension, compression and simple shear tests with good accuracy. Statement of SignificanceThis paper shows a simple mathematical model capable of reproducing qualitative aspects and quantitative behavior of brain matter in tension, compression and simple shear tests with good accuracy. The model is governed by only two parameters, namely Young's modulus (or alternatively the shear modulus) and the locking stretch.

The Effect of the Size of the Sample on Results of Indentation Tests

The results of the numerical solution of the problem about interaction between spherical stamp and weakly compressible elastic specimen are investigated. The nonlinear generalization of linear elastic Hencky model is used as a constitutive relation. The results of the indentation problem solution are in good agreement with experimental data. The tests were performed on the kinematical loading fixture. The influence of geometrical parameters of specimen during indentation test on stress strain state and macro response are investigated.

The Physically Nonlinear Model of an Elastic Material and Its Identification

This work is devoted to the new variant of relations between the energetically conjugated Hencky strain tensor and corotational Kirchhoff stress tensor. The elastic energy is represented as a third-order polynomial of the Hencky tensor containing five material constants. Unlike the Almansi tensor in the Murnaghan model, the Hencky tensor allows assigning a clear physical meaning to material constants. Linear part of the constitutive relation represents the Hencky model and contains the bulk modulus and the shear modulus. The two extra constants express nonlinear effects at a purely volumetric strain and a purely isochoric strain, whereas the third constant takes into account the possible deviation from the similarity of the deviators of the Hencky stress and strain tensors. The resulting relations are naturally generalized for incompressible materials. In this case, the overall number of constants decreases from five to two. The designed test unit was used for a compression test of prismatic specimens made of incompressible material. The proposed version of the relations is in good agreement with the experimental data on the compression of rubber samples.

Experimental determination of the parameters of the nonlinearly elastic Hencky model

This work presents a variant of physically nonlinear constitutive elasticity relations derived using the Hencky strain tensor. A test plant for determining the model’s constants is designed. These constants for weakly compressible materials are derived by compression tests of prismatic elastomeric specimens. The results of solving the problem of indentation of a rigid spherical stamp into a cylindrical specimen are provided. The suggested constitutive relations were used with the constants found by the compression tests. In addition, calculations were performed according to the relations known for the Hencky material and to the hypoelasticity relations with the neutral derivative of the stress tensor. To stabilize the calculation procedures, the Hencky tensor and its derivative were expanded by the monotonous parameter in a series by the Cauchy strain tensor degrees without any restrictions for allowable strain values. A numerical algorithm was built for tracking the variable area of contact. The calculation procedures of discretizing the stated problem by the finite elements method and step-by-step loading were implemented as an applied software suite. One of the integral characteristics of the process considered was the indentation curve defined as the dependence of the stamp immersion effort on the stamp immersion depth. The curves derived by computation are compared with the data of the indentation test conducted on the designed plant. It has been shown that the calculations according to the Hencky model and hypoelasticity relations provide a fair description of the curve to strains of around 30%. The suggested physical nonlinear model allows describing the continued strain growth to a sufficient degree of accuracy.

Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling

In order to get detailed information about the mechanical behavior of pantographic elementary substructure and elements, small-scale specimens were sintered using polyamide powder, constituted by three orthogonal pairs of beams interconnected through pivots forming pantographic cells. The mechanical properties of interconnecting pivots and constituting beams are investigated by comparing experimental evidence with an enhanced Piola–Hencky model. The careful agreement between experimental and predicted results allows us to estimate: (i) the macro-shear stiffness of interconnecting pivots (corresponding to micro-torsional stiffness), (ii) extensional stiffness and (iii) bending stiffness of constituting beams.

A revision of results for standard models in elasto-perfect-plasticity theory

We consider two most studied standard models in the theory of elasto-plasticity in arbitrary dimension $$d\ge 2$$ , namely, the Hencky model and the Prandtl–Reuss model subjected to the von Mises condition. There are many available results for these models—from the existence and the regularity theory up to the relatively sharp identification of the plastic strain in the natural function/measure space setting. In this paper we shall proceed further and improve some of known estimates in order to identify sharply the plastic strain. More specifically, we rigorously improve the integrability of the displacement and the velocity (which was known only under a nonnatural assumption that the Cauchy stress is bounded), show the BMO estimates for the stress and finally also the Morrey-like estimates for the plastic strain. In addition, we shall provide the whole theory up to the boundary. As an immediate consequence of such improved estimates, we provide a sharper identification of the plastic strain than that known up to date. In particular, in two dimensional setting, we show that the plastic strain can be point-wisely characterized in terms of the stresses everywhere although the stress is possibly discontinuous and thus the natural duality pairing in the space of measures could be violated.

Hencky's model for elastomer forming process

In the numerical simulation of elastomer forming process, Henckys isotropic hyperelastic material model can guarantee relatively accurate prediction of strain range in terms of large deformations. It is shown, that this material model prolongate Hooke's law from the area of infinitesimal strains to the area of moderate ones. New representation of the fourth-order elasticity tensor for Hencky's hyperelastic isotropic material is obtained, it possesses both minor symmetries, and the major symmetry. Constitutive relations of considered model is implemented into MSC.Marc code. By calculating and fitting curves, the polyurethane elastomer material constants are selected. Simulation of equipment for elastomer sheet forming are considered.

The Exponentiated Hencky Strain Energy in Modeling Tire Derived Material for Moderately Large Deformations

This work presents a hyperviscoelastic model, based on the Hencky-logarithmic strain tensor, to model the response of a tire derived material (TDM) undergoing moderately large deformations. The TDM is a composite made by cold forging a mix of rubber fibers and grains, obtained by grinding scrap tires, and polyurethane binder. The mechanical properties are highly influenced by the presence of voids associated with the granular composition and low tensile strength due to the weak connection at the grain–matrix interface. For these reasons, TDM use is restricted to applications involving a limited range of deformations. Experimental tests show that a central feature of the response is connected to highly nonlinear behavior of the material under volumetric deformation which conventional hyperelastic models fail in predicting. The strain energy function presented here is a variant of the exponentiated Hencky strain energy, which for moderate strains is as good as the quadratic Hencky model and in the large strain region improves several important features from a mathematical point of view. The proposed form of the exponentiated Hencky energy possesses a set of parameters uniquely determined in the infinitesimal strain regime and an orthogonal set of parameters to determine the nonlinear response. The hyperelastic model is additionally incorporated in a finite deformation viscoelasticity framework that accounts for the two main dissipation mechanisms in TDMs, one at the microscale level and one at the macroscale level. The new model is capable of predicting different deformation modes in a certain range of frequency and amplitude with a unique set of parameters with most of them having a clear physical meaning. This translates into an important advantage with respect to overcoming the difficulties related to finding a unique set of optimal material parameters as are usually encountered fitting the polynomial forms of strain energies. Moreover, by comparing the predictions from the proposed constitutive model with experimental data we conclude that the new constitutive model gives accurate prediction.

Relaxation of the Hencky model in perfect plasticity

In this paper we give a full proof of the relaxation of the Hencky model in perfect plasticity, under suitable assumptions for the domain and the Dirichlet boundary.

Об устойчивости процесса полярно-симметричного деформирования тел из разупрочняющихся материалов

Special case of continuum mechanical systems is considered. It is believed that deforming is carried out under conditions of polar symmetry of stresses and strains. Also it is assumed that material properties are described by Hencky model with softening under nonpositivity of volume deformation. Then union curve has region decreasing to zero. Aforementioned conditions are realized in such problems as expansion of spherical cavity in softening space and deforming of thick-walled spherical vessel by equable external pressure (it maybe bathyscaphe which is gradually submerged to the deep). Based on the Lagrange formalism integral quadratic functional is investigated. This functional is increment of total potential energy in the form of Lagrangian for mentioned problems. This study allows to formulate conditions of buckling for active loading which changes quasistatically. For considered problems sets of possible deformations are obtained. These possible deformations perturb the equilibrium position and do not break kinematic constraints. Obtained sets of possible deformations allow to write criterion of buckling of deformation process in explicit form for mentioned problems. It is established that only with sufficiently developed softening zone buckling of deformation process is possible.

A variational constitutive update algorithm for a set of isotropic hyperelastic–viscoplastic material models

In three-dimensional formulations of plasticity, the rate of plastic deformation is usually decomposed in direction and amplitude. In the most simple cases of von Mises type flows, this allows to assume a known radial flow direction and computations are reduced to the determination of the amplitude of the plastification. In the context of a variational formulation of finite plasticity, it is verified that this decomposition between constitutive and kinematic aspects is accomplished by the choice of a quadratic elastic potential (Hencky model). The aim of this paper is to show that a wide set of simple hyperelastic–plastic isotropic models not restricted to quadratic elastic behavior can be constructed by relaxing the classical decomposition amplitude/direction by the sum of spectral quantities. In addition, this approach allows for a unified and efficient numerical implementation in which a particular hyperelastic–viscoplastic behavior is obtained by a specific choice of the elastic, plastic and dissipative (pseudo)potentials.

Rheological behaviour of chestnuts under compression tests

SummaryModuli of deformability, maximum stress, maximum strain and fracture work have been determined for chestnuts as functions of moisture content. Fresh and dehydrated chestnuts obtained by convective drying, as well as frozen chestnuts, have been tested. The samples were cut in prisms. Uniaxial compression tests in a Universal Testing Machine were employed. Hencky's model was selected as the most appropriate one. Dried chestnuts at 65 °C show a non‐linear behaviour of the modulus of deformability and maximum stress with the moisture content under 0.25 kg kg−1 [dry basis (d.b.)], while both parameters are practically constant for upper values. The modulus of deformability and the maximum stress have been correlated with the aw, obtaining a decreasing linear behaviour below aw values around 0.82. The higher maximum strain values were found at moisture contents around 0.41 kg kg−1 (d.b.), and it stays practically constant below 0.25 kg kg−1 (d.b.). Defrosted chestnuts show lower modulus of deformability and maximum stress values, but the maximum strain values are higher. This fact reveals a lesser firmness in their structures compared with fresh fruits.

Global regularity of the elastic fields of a power-law model on Lipschitz domains

In this paper, we study the global regularity of the displacement and stress fields of a nonlinear elastic model of power-law type. It is assumed that the underlying domains are Lipschitz domains which satisfy an additional geometric condition near those points, where the type of the boundary conditions changes. The proof of the global regularity result relies on a difference quotient technique. Finally, a global regularity result for the stress fields of the elastic, perfect plastic Hencky model is derived. This model appears as a limit model of the power-law model. Copyright © 2006 John Wiley & Sons, Ltd.

Hencky Strain and Hencky Model: Extending History and Ongoing Tradition

A finite strain measure and a Hookean type finite hyperelastic model based on it were introduced by H. Hencky (1928, 1929) about seventy‐five years ago. About tweenty‐five years later, an objective Eulerian equation of rate form was established by C. Truesdell (1952, 1955) for a rate type extension of Hooke’s law to finite deformations. Originating from these, a history is constantly extending and a tradition is continuing to develop in constitutive modeling of elastic and inelastic behaviour at finite deformations. In a short story consisting of five parts, we would like to relate selected episodes and events in some relevant aspects. Our main objective is to disclose unexpected, perhaps interesting, intrinsic relationships between these independent creations by two great masters and to expound how and why these relationships play essential roles in understanding and clarifying fundamental issues in Eulerian rate theories of finite elastoplasticity

Hencky's elasticity model and linear stress-strain relations in isotropic finite hyperelasticity

Hencky's elasticity model is an isotropic finite elasticity model assuming a linear relation between the Kirchhoff stress tensor and the Hencky or logarithmic strain tensor. It is a direct generalization of the classical Hooke's law for isotropic infinitesimal elasticity by replacing the Cauchy stress tensor and the infinitesmal strain tensor with the foregoing stress and strain tensors. A simple, straightforward proof is presented to show that Hencky's elasticity model is exactly a hyperelasticity model, derivable from a quadratic potential function of the Hencky strain tensor. Generally, Hill's isotropic linear hyperelastic relation between any given Doyle-Ericksen or Seth-Hill strain tensor and its work-conjugate stress tensor is studied. A straightforward, explicit expression of this general relation is derived in terms of the Kirchhoff stress and left Cauchy-Green strain tensors. Certain remarkable properties of Hencky's model are indicated from both theorectical and experimental points of view.

An Adaptive Finite Element Method for Problems in Perfect Plasticity

AbstractWe present recent progress in the development of an self‐adaptive finite element solver for problems in elastoplasticity. The adaptive mesh generation is based on a posteriori error estimates which are derived via duality arguments yielding bounds for functionals of the error. The application of this approach for the stationary Hencky model is well understood. Its formal extension to the non‐stationary Prandtl‐Reuss model within the context of an implicit load‐stepping process works surprisingly well. This effect is theoretically explained.

Reliable solution of an elasto-plastic Reissner-Mindlin beam for Hencky's model with uncertain yield function

We apply the method of reliable solutions to the bending problem for an elasto-plastic beam, considering the yield function of the von Mises type with uncertain coefficients. The compatibility method is used to find the moments and shear forces. Then we solve a maximization problem for these quantities with respect to the uncertain input data.

The obstacle problem for an elastoplastic body

The obstacle problem for elastoplastic bodies is considered within the framework of general existence results for unilateral problems recently presented by Baiocchiet al. Two models of plasticity are considered: one is based on a displacement-plastic strain formulation and the second, a specialization of the first, is the standard Hencky model. Existence theorems are given for the Neumann problem for a body constrained to lie on or above the half-space {x∈ℝ3:x3≤0}. For hardening materials the displacements are sought in the Sobolev spaceH1(Ω, ℝ3) while for perfectly plastic materials they are sought in BD(Ω), the space of functions of bounded deformation. Conditions for the existence of solutions are given in terms of compatibility and safe load conditions on applied loads.