Drucker Stability Research Articles

A hyper-viscoelastic constitutive model for elastomers: A case study of hydrogenated nitrile butadiene rubber and polychloroprene rubber

Elastomer materials are widely used as shock absorbers in a variety of practical applications including the automotive, aerospace, marine industries and structures. The straightforward Hooke’s law is insufficient to adequately describe the behavior of elastomeric materials due to their nonlinear elastic and viscous properties. This study proposes a hyper-viscoelastic model to describe the mechanical behavior of Hydrogenated Nitrile Butadiene Rubber (HNBR) 62 Duro shore A and Polychloroprene Rubber (PCR) 55 Duro shore A. Six distinct hyperelastic models—Yeoh, Rivlin, Arruda-Boyce, Mooney-Rivlin, Neo-Hookean, and Ogden— are compared herein to explain the nonlinear elastic behavior of elastomers. While the time-dependent characteristics of the considered materials have been described using the four parameters Generalised Maxwell (GM) model. The material parameters of models are determined using the least square fit (LSF) optimization algorithm from the uniaxial, planar, and stress relaxation test data. The stability and suitability of each hyperelastic model is assessed using the Drucker stability. The accuracy of each model is represented by Root Mean Square Error (RMSE) of curve fitting between theoretical and experimental data. On this basis, the Ogden order three ( N = 3) model and the four-parameter GM models are selected which well agreed with the experimental test data and they are integrated to generate the hyper-viscoelastic constitutive model. In addition, the hyper-viscoelastic model is used to simulate the HNBR and PCR dampers under shock load. The numerically generated response is finally compared with the experimental test result of natural rubber (NR) 60 IRH from the existing literature to confirm accuracy of the proposed model.

A Novel Approach for Optimization of Soft Material Constitutive Model Parameters Based on a Genetic Algorithm and Drucker's Stability Criterion.

The growing interest in soft materials to develop flexible devices involves the need to create accurate methodologies to determine parameter values of constitutive models to improve their modeling. In this work, a novel approach for the optimization of constitutive model parameters is presented, which consists of using a genetic algorithm (GA) to obtain a set of solutions from data of uniaxial tensile tests, which are later used to simulate the mechanical test using finite element analysis (FEA) software to find an optimal solution considering Drucker's stability criterion. This approach was applied to the elastomer Ecoflex 00-30 considering the Warner and Yeoh models and Rivlin's phenomenological theory. The correlation between the experimental and the predicted data by the models was determined using the root mean squared error (RMSE), where the found parameter sets provided a close fit to the experimental data with RMSE values of 0.022 (ANSYS) and 0.024 (ABAQUS) for Warner's model, while for Yeoh's model were 0.014 (ANSYS) and 0.012 (ABAQUS). It was found that the best parameter values accurately follow the experimental material behavior using FEA. The proposed GA not only optimizes the material parameters but also has a high reproducibility level with average RMSE values of 0.024 for Warner's model and 0.009 for Yeoh's model, fulfilling Drucker's stability criterion.

Measurement of the hyperelastic properties of 72 normal homogeneous and heterogeneous ex vivo breast tissue samples

The mechanical properties of normal soft tissues, including breast tissue, have been of interest to the biomedical research community as there are many clinical and industrial applications that can benefit from quantitative information characterizing such properties. For instance, computer assisted surgery planning, elastography for breast cancer diagnosis, and bra design can all involve biomechanical modeling of the breast to predict its deformation or stress distribution. It is known that most biological soft tissues, including breast tissue, exhibit nonlinear mechanical response over large strains. As such, it is necessary to model such tissues as hyperelastic. In this work, we used indentation testing to estimate the hyperelastic parameters of 4 models (3rd order Ogden, 5-term polynomial, Veronda-Westman and Yeoh) estimated from 72 healthy ex vivo breast tissue samples covering adipose, fibroglandular, and mixed tissue. All estimated parameter sets were confirmed to represent stable material using Drucker's stability criterion. We observed that all three tissue types were statistically similar solidifying the use of homogenous breast modelling over large strain simulation.

Investigation of Loss of Consciousness Induced by Gravity Acceleration on the Human Brain

Traumatic Brain Injuries (TBI’s) are any disorder in a brain’s functionality that can be caused by numerous reasons, including motor-vehicle crashes, falls, and assaults. Impractical in-vivo head injury experiments compel bio- engineers to develop a robust, accurate, and efficient computer model. In this study, bovine brain samples were tested under a confined compression testing machine. Consequently, the result from unconfined compression tests, at quasi-static strain rates of e=0.0004 s-1, e=0.008 s-1, and e=0.4 s-1, and a stress relaxation test under unconfined uniaxial compression with a e=0.67 s-1 ramp rate were utilized for fitting brain tissue model. The tissue model employs Drucker stability criteria and conventional hyperelastic models. A finite element model was also developed and validated by experimental data to examine the experiments’ friction effect. Furthermore, the extracted brain tissue model was employed in a 3D head injury model. The 3D model was employed to examine the effect of +Gz acceleration on the human brain and present damage threshold based on loss of consciousness in HIC and Maximum Brain Pressure criteria. It is shown that the relative difference between simulation results at friction coefficient of μ=0.5 and μ=0.0 are less than 20%, and the ramp rate variation has a slight effect on normalized shear modulus. Moreover, Head modeling results revealed that the Maximum Brain Pressure ≥ 3.1 KPa and HIC ≥ 30 are a representation of loss of consciousness.

Hyperelastic Material Characterization: How the Change in Mooney-Rivlin Parameter Values Effect the Model Curve

Mooney-Rivlin is the most frequently used model from all models used for mechanical characterization of the hyperelestic materials. Simplicity, applicability in a large rage of strains are the key reasons for regular use of this model. However, depending on the number of parameters, the Mooney model can take several forms. While, nine parameter being the highest order noticed, two parameter model is the most commonly found form in the current research domain. Since two parameter model used repetitively, we investigated the effect of incremental change in two material constant values one at a time, on model curve. As Drucker Stability Criterion is governing the extreme values of material parameters, changes in the model curves are discussed related to it. Resultant effects on stress-strain curves due to change in parameter values were examined and physical effect on the characterization is interpreted accordingly.

Failure Criteria for Isotropic Rocks Using a Smooth Approximation of Modified Mohr–Coulomb Failure Function

It is reported that strength of rocks is a function of three-dimensional (3D) stress state as observed from the large sets of experimental results under the true-triaxial condition. Triaxial compression test data in general is readily available for rocks, but in practice true-triaxial facilities are seldom used. This creates a requirement for 3D failure criterion, which can be used to predict the true-triaxial strength using simple triaxial compression test data. We have proposed three criteria by altering the deviatoric function of modified Mohr–Coulomb criterion (Singh et al. in Int J Rock Mech Min Sci 48(4):546–555, 2011). The proposed criteria represent the behaviour of rock from lower to very high confining pressures. The criteria have unique yield surfaces. The yield surface changes its shape with the change in the mean stress. It is observed beyond certain value of the mean stress; the yield surfaces converge to von-Mises yield surface which means the shape of the yield surface is independent of pressure beyond the certain value of mean normal stress. The criteria require three parameters that can be easily calculated from the triaxial compression test results. The proposed criteria also satisfy the Drucker stability postulate, hence could be adopted as a plastic potential function. For the assessment of the criteria, true-triaxial data from twelve rocks are considered. Results of the criteria are compared with the modified Mohr–Coulomb criteria both 2D and 3D proposed by Singh et al. (Int J Rock Mech Min Sci 48(4):546–555, 2011). Performance of the new failure criteria for the considered rocks are observed to be better than the modified Mohr–Coulomb criteria (2D and 3D).

Study on Mohr–Coulomb-based three-dimensional strength criteria and its application in the stability analysis of vertical borehole

In the present study, the six extensions for the Mohr–Coulomb criterion into three-dimensional stress state are discussed in detail. The prediction of strength from these criteria are compared with the experimental data of true-triaxial tests from 16 rocks. These extended criteria are Mogi–Coulomb, Modified Mohr–Coulomb A and B, and Mohr–Coulomb criterion with three different deviatoric functions (elliptical, hyperbolic, and spatial mobilized plane functions). The strength parameters of these criteria are calculated using the triaxial compression points from the true-triaxial tests data. Such consideration is taken because in general triaxial compression tests, data are only available for the majority of rocks. It is observed from the comparison that on average Mohr–Coulomb criterion with elliptical smoothening (MC-ED) performed better than other criteria (considered in the study). Hence, MC-ED is considered for vertical borehole stability analysis. MC-ED criterion also satisfies the Drucker stability postulate. To demonstrate the application of MC-ED criterion in the borehole stability analysis, four cases are considered. Results from the MC-ED criterion are compared with the Mohr–Coulomb criterion. The Mohr–Coulomb analysis calculates the higher minimum overbalance pressure required as compared to the MC-ED and lower fracture pressure. It is observed that MC-ED results are relatively closer to the actual values used in the case study considered in this paper.

A compressible hyper-viscoelastic material constitutive model for human brain tissue and the identification of its parameters

In this paper, we have introduced a compressible hyper-viscoelastic constitutive model for human brain tissue. The model is calibrated with the reported experimental data from different regions of the brain. The parameters of the model are determined in a simultaneous calibration for tension, compression, shear, and compression–relaxation tests data. They are obtained in an iterative procedure in conjunction with a finite elements (FE) modeling of the tissue, as well as, with the Nelder–Mead Simplex optimization procedure. In the calibration procedure, the compressibility of the material is taken into account, and the respective time-dependent volumetric parameter is also determined. Additionally, the Drucker stability condition is enforced to assess the physical meaning of the extracted constitutive parameters. This proposed model provides an improved prediction of the experimental data and tissue response under various loading conditions. The results show that, under inhomogeneous deformation, the suggested approach will lead to a better material calibration of brain tissue compared to the simple mathematical model fitting.

Modeling and experimental verification of nonlinear behavior of cellulose nanocrystals reinforced poly(lactic acid) composites

In this paper, a constitutive modeling framework was introduced to characterize the time-dependent behavior in poly(lactic acid) reinforced by cellulose nanocrystals (CNCs). The mechanical properties of PLA and its corresponding nanocomposites were analyzed at three different extension rates (1, 5, 10 mm/min) using a stress relaxation test. Six hyperelastic strain energy functions were employed to describe the instantaneous mechanical behavior. The deviatoric and volumetric material constants were verified using Drucker stability criterion, and the instability limits of each model were reported. Furthermore, a hyper-viscoelastic model consisting of Neo-Hookean strain energy function and time-dependent Prony series was used to study the viscoelastic behavior of PLA and the corresponding nanocomposites. The model constants were determined using finite element (FE) simulations through an iterative process for the whole loading history of the relaxation test. Nelder–Mead Simplex method was applied to optimize the results of the FE simulations using the experimental data. The mechanical response observed for PLA and nanocomposites from model exhibited a good agreement with both short and long-term experimental data, suggesting the applicability of the estimated material parameters by this technique.

Uncertainty quantification for constitutive model calibration of brain tissue.

The results of a study comparing model calibration techniques for Ogden's constitutive model that describes the hyperelastic behavior of brain tissue are presented. One and two-term Ogden models are fit to two different sets of stress-strain experimental data for brain tissue using both least squares optimization and Bayesian estimation. For the Bayesian estimation, the joint posterior distribution of the constitutive parameters is calculated by employing Hamiltonian Monte Carlo (HMC) sampling, a type of Markov Chain Monte Carlo method. The HMC method is enriched in this work to intrinsically enforce the Drucker stability criterion by formulating a nonlinear parameter constraint function, which ensures the constitutive model produces physically meaningful results. Through application of the nested sampling technique, 95% confidence bounds on the constitutive model parameters are identified, and these bounds are then propagated through the constitutive model to produce the resultant bounds on the stress-strain response. The behavior of the model calibration procedures and the effect of the characteristics of the experimental data are extensively evaluated. It is demonstrated that increasing model complexity (i.e., adding an additional term in the Ogden model) improves the accuracy of the best-fit set of parameters while also increasing the uncertainty via the widening of the confidence bounds of the calibrated parameters. Despite some similarity between the two data sets, the resulting distributions are noticeably different, highlighting the sensitivity of the calibration procedures to the characteristics of the data. For example, the amount of uncertainty reported on the experimental data plays an essential role in how data points are weighted during the calibration, and this significantly affects how the parameters are calibrated when combining experimental data sets from disparate sources.

Determination of elastomeric foam parameters for simulations of complex loading

Background: Finite element (FE) analysis has shown promise for the evaluation of elastomeric foam personal protection devices. Although appropriate representation of foam materials is necessary in order to obtain realistic simulation results, material definitions used in the literature vary widely and often fail to account for the multi-mode loading experienced by these devices. This study aims to provide a library of elastomeric foam material parameters that can be used in FE simulations of complex loading scenarios. Method of Approach: Twelve foam materials used in footwear were tested in uni-axial compression, simple shear and volumetric compression. For each material, parameters for a common compressible hyperelastic material model used in FE analysis were determined using: (a) compression; (b) compression and shear data; and (c) data from all three tests. Results: Material parameters and Drucker stability limits for the best fits are provided with their associated errors. The material model was able to reproduce deformation modes for which data was provided during parameter determination but was unable to predict behavior in other deformation modes. Conclusions: Simulation results were found to be highly dependent on the extent of the test data used to determine the parameters in the material definition. This finding calls into question the many published results of simulations of complex loading that use foam material parameters obtained from a single mode of testing. The library of foam parameters developed here presents associated errors in three deformation modes that should provide for a more informed selection of material parameters.

HOLONOMIC SOFTENING: MODELS AND ANALYSIS1*

This paper deals with a special class of holonomic (path-independent) structural analysis problems involving nonlinear or piecewise linear softening. In particular, the formulation takes the form of a complementarity problem, an important class of mathematical problems characterized by the orthogonality of two sign-constrained vectors. A feature and difficulty associated with softening, which violates Drucker's stability postulate, is multiplicity of solutions. The main aims of this paper are to give a precise mathematical description of a wide class of softening models. This is achieved via a theoretically and computationally advantageous complementarity format. Second, key ideas underlying a recently developed complementarity solver, PATH, which has the potential of capturing any multiplicity of solutions or to show that none exists, are outlined. Two examples concerning discretized truss structures—a prototype of other more advanced finite element based structural models—are given for illustrative purposes.

The drucker stability of a material

An invariant formulation of the criterion for the stability of a material, which enables one to investigate the local stability under arbitrary stress conditions, is obtained on the basis of Drucker's postulate [1]. It is shown that the criterion obtained has an explicit physical meaning at large plastic deformations. A relation is established between the stability parameter and other fundamental parameters of the stressed state.

Irreversibility, Stability and Maximum Dissipation in Elastoplastic Damage

The relation between the maximum dissipation principle, the irreversibility condition of Ilyushin and Drucker's stability postulate is studied in the present paper. For simple elastoplastic material models, it is well known that these three postulates require orthogonality between the plastic strain rate and the yield surface. The objective of this work is to investigate this relation for more general material models containing internal variables coupled with the elastic strains. It can be shown that the irreversibility and stability conditions pose the same restriction on the directions of dissipative fluxes, but do not determine uniquely this direction. The postulate of maximum dissipation is a stronger restriction to the evolution equations than Ilyushin's condition. The results are applied to damage mechanics models. For models describing elasticity and damage, a normality law in damage force space is obtained. In the case of plasticity with damage, the postulates imply orthogonality neither in the stress space nor in the damage force space. Usual non-associated flow rules for elastoplasticity with damage violate Drucker's stability postulate. The only convenient way to formulate a model, which fulfills the postulates of Ilyushin and Drucker a priori, is the use of an associated flow rule.

Experimental Investigations and Numerical Modeling of Incompressible Elastomers during Non-Homogeneous Deformations

Abstract Three hyperelastic constitutive models, the Arruda and Boyce model, the Yeoh model, and the Ogden model, are used to simulate the response of vinyl elastomers in large, non-homogeneous deformation states. A single uniaxial compression test is used to characterize the elastomers to obtain model constants. Drucker stability is enforced by restriction of the coefficients for the Yeoh and Ogden models to produce physically feasible uniaxial results. The ability of the models to predict the global force versus deformation responses and deformed shapes for large deformation shear, tensile deformation of a long bar with fixed grips, and inflation of a thin disk is examined. The experiments used in this study are designed to provide known boundary conditions to eliminate ambiguity in the modeling. We demonstrate for the first time that each of these models may be sufficiently characterized via a simple, homogeneous compression test to allow accurate predictions of large, non-homogeneous deformations involving rotations of the principal stretch directions.

A simple constitutive relationship for isotropic solids with stress-state type dependent properties

The general form of the relationships between two symmetric coaxial tensors of second rank in an isotropic medium is considered. The condition under which the simple tensor linear form of these relationships, which includes the Lode's angle, can be used is determined. The range of validity of the proposed relationships is studied by means of Drucker's stability postulate and the existence and uniqueness conditions of the generalised solution of the problems. The applications that are built on the proposed form of the constitutive relationships, that cover creep, plastic, and behaviour are given briefly.

Parallel Elastoplastic Models of Inelastic Material Behavior

Parallel mathematical models are presented and analyzed as a basis for representing the mechanical behavior of inelastic materials. These models are developed within the framework of incremental elastoplasticity theory. The models are shown to simply represent strain hardening through the adjustment of internal stresses within internal elements of the model, to capture Bauschinger's effect during stress reversal, and to provide hysteretic loops in closed stress cycles. The models are shown to provide stable, nonassociated plastic flow for frictional materials such as geologic materials, i.e., to obey Drucker's stability postulate. A simple parallel model for rock is presented that gives the same response as a conventional elastoplastic material model when used to fit idealized uniaxial strain and triaxial compression test data. However, the two models give much different results in triaxial extension after initial hardening occurred on the compression side first. It is believed that this modeling concept holds significant promise for improving mathematical models of the inelastic mechanical behavior of a variety of engineering materials.

Large Strain Viscoelastic Constitutive Models for Rubber, Part II: Determination of Material Constants

Abstract A method for determining material constants in large strain viscoelastic materials was demonstrated for a highly saturated nitrile rubber. Material constant selection was based on viscoelastic stress relaxation data at small and large strains, under both tension and compression, and was constrained to assure Drucker stability. Assuming that the viscoelastic strain energy function was both time and strain separable, a Prony series was constructed for the time dependent material constants. For comparison, four different Prony series were developed from collocation methods and a nonlinear regression analysis, each separately based on either large or small tensile strain relaxation data. In addition, a final Prony series was constructed from dynamic data. These Prony series were included in this comparison to judge their ability to predict both large and small strain material behavior. Finite element analyses of large and small step-strain relaxation tests and a single cycle hysteresis loop at large deformations were performed for each set of Prony series. The results were then compared to experimental behavior. The Prony series based on the constrained method accurately predicted step-strain relaxation behavior at all strain levels, for both tension and compression. The finite element results for the other Prony series show that large strain material behavior was best predicted by those Prony series based on large strain material behavior. Similar findings were found for small strain material behavior. The constrained Prony series and the two large strain based Prony series best modeled the experimental hysteresis loop.

Large Strain Viscoelastic Constitutive Models for Rubber, Part I: Formulations

Abstract Constitutive models for large strain isothermal viscoelastic deformations of rubber are reviewed. The models discussed are for materials which have separable long and short term stresses, and for which the short term stresses have separable time and strain effects. They include the history integral model, an internal stress variable model, an internal stretch variable model, and an internal solid model. The internal stretch variable model and the internal solid model were motivated by molecular descriptions of rubber viscoelasticity. The material tests required to determine the constants for these large strain models are discussed. The classical problem of determining a rubber energy density function, used to describe both the long term and short term energy in these models, is reviewed. A method to assure Drucker stability, a commonly overlooked issue, is presented for experimentally determined Rivlin energy density function expansions. Also, an improved method for determining the coefficients of a Prony series, which defines a material's relaxation time spectrum, from experimental relaxation data is presented.

Stability of Frictional Materials

The stability of frictional materials that display a nonassociated flow rule is discussed. Particular emphasis is put on an analysis of recent experiments conducted by Lade et al. (1987), where stable behavior was observed, when the material was expected to be ``unstable'' in the sense of Drucker's stability postulate. It was therefore concluded that Drucker's postulate regarding material stability is not valid for frictional materials. In the present paper, the writers present an analysis and describe features in the experiments by Lade et al. that show that the reported stability may be less surprising.