Hencky Strain Tensor Research Articles

On a class of nonlinear isotropic thermo-elastic body that is non-Green elastic: Applications to large elastic deformations

Summary A new class of constitutive equation is proposed for isotropic thermoelastic solids, wherein the Hencky strain tensor is assumed to be a function of the Cauchy stress tensor, via a Gibbs potential. The solid is assumed to be incompressible in the referential state, but the volume can change due to differences in the temperature relative to a reference temperature. The change in volume only depends on temperature. Some restrictions are found for the Gibbs potential, resulting in a constitutive equation for isotropic solids, wherein the volume depends on temperature. Using the resulting constitutive equation, some boundary value problems are studied, considering some relatively simple distributions for the temperature, deformations and stresses.

Stress level estimates in coated or uncoated silicon nanoparticles during lithiation

This work is devoted to the modeling of the mechanical behavior of coated or uncoated silicon nanoparticles, which constitute the anode active material in some lithium ion batteries, during their first lithiation at room temperature. The lithiation process induces a large volume expansion of the particles and a high level of stresses, which can lead to the fragmentation of the particles. Several approaches are proposed in order to estimate the volume expansion and the stress levels experienced by the particles. An original semi-analytical small strain model is first presented, which adapts the solution proposed by Seck et al. (2018) of the elastic-viscoplastic composite sphere problem subjected to radial loading, to the lithiated particle problem, in particular by considering the variation of phases properties during lithiation. The lithium concentration in the silicon particle is given by a sigmoid function (called logistic function) in order to mimic the reaction front between the phases. The implementation of the approach using the Hencky strain tensor Miehe et al. (2002) is proposed to take into account the large strains experienced by the particles. A complete description of the formulation is provided and the advantages are discussed. The importance of the large strains model is established by comparing it with the small strain one concerning the predictions of the pushing-out effect and the size effect of particles on their internal stresses during lithiation. Comparisons between our simulations and experimental data from Tardif et al. (2017) measuring the operando strain experienced by the pristine silicon gives the yield stress of the lithiated silicon. In addition, carbon-coated silicon nanoparticles are finally studied. We develop original closed-form expressions to predict the maximal stresses experienced by the coating at the end of the lithiation. Those expressions are used to re-estimate the fracture stress of pyrolitic carbon, considering a critical review of both pyrolitic carbon and lithiated silicon elastic properties. Finally, the mechanical effect of the coating on silicon during lithiation is studied.

Incremental equations and propagation of small amplitude waves for isotropic incompressible elastic bodies. The case the Hencky strain tensor is a function of the Cauchy stress tensor

For a relatively new class of constitutive equation for incompressible elastic bodies, wherein the Hencky strain tensor is a function of the Cauchy stress tensor, incremental equations are obtained, where it is assumed the presence of an initial time-independent stress that causes large deformations, to which a small time-dependent stress tensor is added. That stress tensor is assumed to cause a small (incremental) time-dependent deformation. For the case of infinite media the incremental equations are solved assuming travelling waves, and the speed of such waves is obtained for different problems considering initial homogeneous distributions of stresses and strains. Some numerical calculations of the speed of such waves are presented, considering a particular constitutive equation for rubber published recently in the literature. The speed of such small amplitude waves are compared with the predictions of the classical theory of nonlinear elasticity for similar problems.

A non-linear complementary energy-based constitutive model for incompressible isotropic materials

Inverse non-linear stress–strain relations between the logarithmic Hencky strain tensor and the Cauchy stress tensor are derived for incompressible isotropic materials. The constitutive model is based on a new type of complementary energy potential that considered to be the function of the second and third invariants of the deviatoric Cauchy stress tensor in a power-law form. A proper scaling of the third deviatoric stress invariant allows separate fitting to uniaxial tension and equibiaxial tension (or uniaxial compression) data. The predictive capabilities of a four- and a six-parameter model are presented through a parameter fitting procedure using Treloar’s experimental data for rubber. The performance of the new constitutive model is also demonstrated by solving the inflation problem of a spherical shell.

Analysis and visualization of traceless symmetric tensors. Application to the Hencky strain tensor for large strain tension–torsion

Cyclic multiaxial loadings of soft materials are usually studied throughout experiments involving machines that prescribe a combination of uniaxial tension and torsion. Due to the large strain framework, classical kinematic analyses of strain in uniaxial tension–torsion are usually very complex. Based on this observation, the present paper proposes a method to both analyze and visualize a strain measure during a duty cycle of uniaxial tension–torsion in large strain: based on the mathematical properties of the Hencky strain tensor h, the method consists in projecting h onto a well-chosen tensor basis, whose constituting elements are described in terms of physical meaning. Thanks to this decomposition, the history of h reduces to the time evolution of a 3-component vector α(t). This vector history can then be visualized as a path in 3D space, rendering very visual the complex kinematic phenomenon. As a second result, an original definition of the mean and the amplitude of a strain path, based on the theory of the Minimum Circumscribed Spheres, is proposed. This definition could be useful for fatigue studies, for instance.

A note on a new constitutive model for rubber-like solids

A new constitutive equation is proposed for rubber, based on expressing the Cauchy stress as a function of the Hencky strain tensor. The model is used for isotropic bodies considering the classical invariants of Rivlin and Spencer. The predictions for the extension and compression of a cylinder, and the inflation of a hollow sphere, are compared with the predictions of the constitutive equation for rubber by Ogden.

Multiaxial fatigue experiments for elastomers based on true strain invariants

For decades, multiaxial fatigue tests have been proposed in order to derive models for fatigue life prediction of elastomers. Nevertheless, the importance of quantifying the multiaxiality has not sufficiently been discussed. In this paper, a new method is proposed to handle the multiaxiality in elastomers fatigue testing. It is based on the pair $$(K_2,K_3)$$ , two invariants of the Hencky (true) strain tensor. Thanks to the example of the simultaneous uniaxial tension-torsion fatigue of a Styrene Butadiene Rubber, we demonstrate that it is possible to prescribe a constant multiaxiality level throughout different tests. The method consists in fixing the multiaxiality indicator $$K_3$$ , and then varying the strain level with the intensity of distorsion indicator $$K_2$$ . Results show that the fatigue lives seem to unify with $$K_2$$ regardless of the value of $$K_3$$ . They also demonstrate that the initiation crack angle depends on the multiaxiality indicator $$K_3$$ but not on $$K_2$$ . Finally, the relevance of the approach is acknowledged by comparing the pair $$(K_2,K_3)$$ to the classical largest principal stretch ratio and biaxiality factor.

Some Universal Solutions for a Class of Incompressible Elastic Body that is Not Green Elastic: The Case of Large Elastic Deformations

Summary Some universal solutions are studied for a new class of elastic bodies, wherein the Hencky strain tensor is assumed to be a function of the Kirchhoff stress tensor, considering in particular the case of assuming the bodies to be isotropic and incompressible. It is shown that the families of universal solutions found in the classical theory of nonlinear elasticity, are also universal solutions for this new type of constitutive equation.

Gibbs free energy based representation formula within the context of implicit constitutive relations for elastic solids

We derive a representation formula for a class of solids described by implicit constitutive relations between the Cauchy stress tensor and the Hencky strain tensor. Using a thermodynamic framework, we show that the Hencky strain tensor can be obtained as the derivative of the specific Gibbs free energy with respect to a stress tensor related to the Cauchy stress tensor. Unlike previous studies that have considered implicit relations between the Cauchy stress tensor and the Hencky strain we work with quantities that allow us to split the deformation into two parts. One part is connected to deformations that change the volume and the other to deformations where volume is preserved. Such a decomposition allows us to clearly characterise the interplay between the corresponding parts of the stress tensor, and to identify additional restrictions regarding the admissible formulae for the Gibbs free energy. We also show that if the constitutive relations of this type are linearised under the small strain assumption, then one can transparently obtain linearised models with density/pressure/stress dependent elastic moduli in a natural manner.

Fusing the Seth–Hill strain tensors to fit compressible elastic material responses in the nonlinear regime

Strain energy densities based on the Seth–Hill strain tensors are often used to describe the hyperelastic mechanical behaviours of isotropic, transversely isotropic and orthotropic materials for relatively large deformations. Since one parameter distinguishes which strain tensor of the Seth–Hill family is used, one has in theory the possibility to fit the material response in the nonlinear regime. Most often for compressible deformations however, this parameter is selected such that the Hencky strain tensor is recovered, because it yields rather physical stress-strain responses. Hence, the response in the nonlinear regime is in practise not often tailored to match experimental data. To ensure that elastic responses in the nonlinear regime can more accurately be controlled, this contribution proposes three generalisations that combine several Seth–Hill strain tensors. The generalisations are formulated such that the stress-strain responses for infinitesimal deformations remain unchanged. Consequently, the identification of the Young’s moduli, Poisson’s ratios and shear moduli is not affected. 3D finite element simulations are performed for isotropy and orthotropy, with an emphasis on the identification of the new material parameters.

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.

A comparative study of Maxwell viscoelasticity at large strains and rotations

We present a new Eulerian large-strain model for Maxwell viscoelasticity using a logarithmic co-rotational stress rate and the Hencky strain tensor. This model is compared to the small-strain model without co-rotational terms and a formulation using the Jaumann stress rate. Homogeneous isothermal simple shear is examined for Weissenberg numbers in the interval [0.1; 10]. Significant differences in shear stress and energy evolution occur at Weissenberg numbers >0.1 and shear strains >0.5. In this parameter range, the Maxwell–Jaumann model dissipates elastic energy erroneously and thus should not be used. The small-strain model ignores finite transformations, frame indifference and self-consistency. As a result, it overestimates shear stresses compared to the new model and entails significant errors in the energy budget. Our large-strain model provides an energetically consistent approach to simulating non-coaxial viscoelastic deformation at large strains and rotations.

Geometry of Logarithmic Strain Measures in Solid Mechanics

AbstractWe consider the two logarithmic strain measures$$\begin{array}{ll} {\omega_{\mathrm{iso}}} = ||{{\mathrm dev}_n {\mathrm log} U} || = ||{{\mathrm dev}_n {\mathrm log} \sqrt{F^TF}}|| \quad \text{ and } \quad \\ {\omega_{\mathrm{vol}}} = |{{\mathrm tr}({\mathrm log} U)} = |{{\mathrm tr}({\mathrm log}\sqrt{F^TF})}| = |{\mathrm log}({\mathrm det} U)|\,,\end{array}$$ωiso=||devnlogU||=||devnlogFTF||andωvol=|tr(logU)=|tr(logFTF)|=|log(detU)|,which are isotropic invariants of the Hencky strain tensor$${\log U}$$logU, and show that they can be uniquely characterized by purely geometric methods based on the geodesic distance on the general linear group$${{\rm GL}(n)}$$GL(n). Here,$${F}$$Fis the deformation gradient,$${U=\sqrt{F^TF}}$$U=FTFis the right Biot-stretch tensor,$${\log}$$logdenotes the principal matrix logarithm,$${\| \cdot \|}$$‖·‖is the Frobenius matrix norm,$${\rm tr}$$tris the trace operator and$${{\text dev}_n X = X- \frac{1}{n} \,{\text tr}(X)\cdot {\mathbb{1}}}$$devnX=X-1ntr(X)·1is the$${n}$$n-dimensional deviator of$${X\in{\mathbb {R}}^{n \times n}}$$X∈Rn×n. This characterization identifies the Hencky (or true) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor$${\varepsilon={\text sym}\nabla u}$$ε=sym∇u, which is the symmetric part of the displacement gradient$${\nabla u}$$∇u, and reveals a close geometric relation between the classical quadratic isotropic energy potential$$\mu {\| {\text dev}_n {\text sym} \nabla u \|}^2 + \frac{\kappa}{2}{[{\text tr}({\text sym} \nabla u)]}^2 = \mu {\| {\text dev}_n \varepsilon \|}^2 + \frac{\kappa}{2} {[{\text tr} (\varepsilon)]}^2$$μ‖devnsym∇u‖2+κ2[tr(sym∇u)]2=μ‖devnε‖2+κ2[tr(ε)]2in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy$$\mu {\| {\text dev}_n log U \|}^2 + \frac{\kappa}{2}{[{\text tr}(log U)]}^2 = \mu {\omega_{{\text iso}}^2} + \frac{\kappa}{2}{\omega_{{\text vol}}^2},$$μ‖devnlogU‖2+κ2[tr(logU)]2=μωiso2+κ2ωvol2,where$${\mu}$$μis the shear modulus and$${\kappa}$$κdenotes the bulk modulus. Our deduction involves a new fundamental logarithmic minimization property of the orthogonal polar factor$${R}$$R, where$${F=RU}$$F=RUis the polar decomposition of$${F}$$F. We also contrast our approach with prior attempts to establish the logarithmic Hencky strain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity.

О ГЕОМЕТРИЧЕСКИ НЕЛИНЕЙНЫХ ОПРЕДЕЛЯЮЩИХ СООТНОШЕНИЯХ УПРУГОГО МАТЕРИАЛА

In the field of solid mechanics it is often necessary to use constitutive (physical) relations in the rate form, for example, for formulation of the boundary value problem in the rate form with contact conditions when a contact area is a priori unknown and changes in a deformation process. The paper considers some questions of formulating geometrically nonlinear constitutive relations of elastic material in the rate form and the relationship of these equations and the constitutive relations in the finite form. In many existing elastic and elasto-plastic models Hooke's law (written in terms of the actual configuration) is used as the constitutive relation. As a rule, the rate measure of the strain is deformation rate tensor (the symmetrical part of velocity gradient) and the stress rate measure is some objective derivative (convective or corotational) of weighted Kirchhoff stress tensor. The use of the convective derivative leads to certain difficulties, for example, analysis of evolution stress state in a deformable basis is complicated; consequently, convective derivatives are excluded from consideration. Using a stress tensor corotational derivative instead of the material derivative (derivative by time) allows satisfying the principle of material indifference (the principle of independence of constitutive relation from the choice of reference frame), however a choice of the derivative type can be implemented by many ways. An arbitrary selection of stress corotational derivative leads to undesirable effects: stress oscillations for the simple shear monotonic loading (for example, for Jaumann derivative), “not closed” stress trajectories and nonzero stress work for a closed deformation trajectory. In papers of A. Meyers, H. Xiao, O. Bruhns corotational derivative (with logarithmic spin or logspin) was proposed.The derivative of the right Hencky's strain tensor is exactly equal to deformation rate tensor. When using this derivative in the constitutive relation the described effects are missing, on the basis of this remarked logspin authors state the logspin exclusivity and recommend it only for using in rate form constitutive relations. The article shows full compliance between Hooke's law in the rate and finite form under conditions: existence of the material basis in which properties of the body (in this case - elastic) remain unchanged and the use of the same type corotational derivative for stress and strain measures. Computational experiments for illustration of the proved assertion about the equivalence of different Hooke's law forms were conducted: various measures of strain state and their corotational derivatives are considered; a kinematic loading in a closed cycle (in the space of deformations) is applied. It is shown for isotropic elastic materials that stress trajectories are closed. When using work-conjugacy stress and strain measures the dissipation energy is absent. Thus it is possible to question the exclusive choice of logarithmic spin and the above-mentioned commonly used measures of stress-strain state. In formulating and solving the problems of solid mechanics there should be an opportunity to use different stress-strain measures and their objective derivatives, the choice of measures and constitutive relations should be justified from the point of physical analysis of the process.

How to identify a hyperelastic constitutive equation for rubber-like materials with multiaxial tension–torsion experiments

This paper discusses the different approaches that can be used to determine the strain energy density of a given rubber-like material based on tension–torsion experimental results. More precisely, the aim is to answer the question: how to handle the measured macroscopic quantities, i.e. load and torque, to determine the constitutive equation with the less possible assumptions? The method initially proposed by Penn and Kearsley [Trans. Soc. Rheol. 20 (1976) 227–238] is adopted: the strain energy derivatives with respect to kinematical quantities have to be calculated in terms of the measured load and torque. Here, we propose to consider different sets of kinematical quantities to overcome the incoherence encountered with the classical Cauchy–Green strain invariants I1 and I2. Two new sets are considered: the principal stretch ratios and two specific invariants of the logarithmic (true) Hencky strain tensor. The corresponding derivations coupled with new experimental results permit (i) to calculate the Cauchy stress tensor on the outer surface of the cylindrical samples, and (ii) to demonstrate that a well-conditioned set of kinematical quantities must be adopted to determine the strain energy density. It is proved here that the principal stretch ratios are good candidates to express and determine the strain energy density with tension–torsion experiments.

A Common Multiscale Feature of the Deformation Mechanisms of a Semicrystalline Polymer

The deformation mechanisms of a semicrystalline polymer (SCP) subjected to tensile loading were experimentally studied from nano to micro and macro length scales. The paper’s focus is on the development of anisotropic features at all scales during deformation. At the nanoscale, original results are presented based on in-situ SAXS (small-angle X-ray scattering) measurements. At the microscale, previously obtained results using in-situ ISLT (incoherent steady light transport) and postmortem SRXTM (synchrotron radiation X-ray tomographic microscopy) are briefly summarized. For all these techniques, original data treatments were carried out for quantifying the evolution of anisotropy in the material’s microstructure. At the macroscopic scale, a 3D digital image correlation technique was used for determining all the components of the Hencky strain tensor in the center of the necking region. New results are presented in this field for the studied SCP in terms of the quantity D = eL/|eT|: ratio of the longitudinal strain (along the drawing axis) to the transverse strain (perpendicular to the drawing axis). The main result of this study lies in a plot of variable D as a function of eL. This plot evidences the same three clearly distinct regimes as those obtained using SAXS, ISLT, and SRXTM experiments to measure anisotropy quantitatively at the microstructural level. This result proves that everything occurring at the microscale gives a signature at the macroscale and hence opens up new routes for the modeling of the mechanical behavior of such materials.

Simulation of strain-induced anisotropy for polymers with weighting functions

The alignment of polymer chains is a well-known microstructural evolution effect due to straining of polymers. This has a drastic influence on the macroscopic properties of the initially isotropic material, such as a pronounced strength in the loading direction of stretched films. For modeling the effect of strain-induced anisotropy, a macroscopic constitutive model is developed in this paper. Within a thermodynamic framework, an additive decomposition of the logarithmic Hencky strain tensor into elastic and inelastic parts is used to formulate the constitutive equations. As a key idea, weighting functions are introduced to represent a strain-softening/hardening effect to account for induced anisotropy. These functions represent the ratio between the total strain rate (representing the actual loading direction) and a structural tensor (representing the stretched polymer chains). In this way, we introduce material parameters as a sum of weighted direction-related quantities. The numerical implementation of the resulting set of constitutive is used to identify material parameters based on experimental data, exhibiting strain-induced anisotropy. In the finite-element examples, we simulate the cold-forming of amorphous thermoplastic films below the glass transition temperature subjected to different re-loading directions.

A fully coupled elastoviscoplastic damage model at finite strains for mineral filled semi-crystalline polymer

A phenomenological finite strain non-associated elastoviscoplastic model coupled with damage is proposed in order to simulate the behaviour of a 20% mineral filled semi-crystalline polymer for a large strain rate range under several loading conditions. In the proposed model, the direct relation between the logarithmic rate of the Eulerian Hencky strain tensor (work-conjugate of the Cauchy stress) with the additively decomposition of the stretching tensor in an elastic and a viscoplastic part is used. The viscoplastic formulation is developed with the association of a pressure dependent yield surface coupled with damage to take the rate and the pressure dependency into account. In order to capture the non-isochoric deformation involving expansion under tensile loading and compaction under compression loading, a viscoplastic potential is developed. This model is implemented in a user-material subroutine in an implicit finite element code with a fully implicit viscoplastic return scheme for solid and shell elements. The particular case of shell elements is dealt with by using a second iteration loop to ensure the zero-normal-stress condition. All the material parameters are identified from experimental tests carried out at several kinds and speed loadings. Numerical responses of the proposed model are close to the experiments for several kinds and speed loadings.

Thermodynamic consistent modeling of polymer curing coupled to visco–elasticity at large strains

We develop a macroscopic constitutive model for temperature-dependent visco–elastic effects accompanied by curing, which are important phenomena in production processes. Within a thermodynamic framework we use an additive ternary decomposition of the logarithmic Hencky strain tensor into mechanical, thermal and chemical parts. Based on the concept of stoichiometric mass fractions for resin, curing agent and solidified material the bulk compression modulus as well as the bulk heat- and shrinking dilatation coefficients are derived and compared with ad hoc assumptions from the literature. Moreover, we use the amount of heat generated during differential scanning calorimetry until completion of the chemical reactions, to define the chemical energy. As a major result, the resulting latent heat of curing occurring in the heat-conduction equation derived in our approach reveals an ad hoc approach from the literature as a special case. In addition, thermodynamic consistency of the model will be proved, and the numerical implementation of the constitutive equations into a finite-element program is described. In the examples we illustrate the characteristic behaviour of the model, such as shrinking due to curing and temperature dependence and simulate the deep drawing of a spherical part with the finite-element-method.