Green-Naghdi Rate Research Articles

Numerical implementation of the hypoplastic model for SPH analysis of soil structure development in extremely large deformation

The extremely large deformation of soil is a significant challenge due to the dynamic changes in soil structure, but it is poorly understood owing to inadequate effective numerical methods. This paper presents a numerical method for analysing the responses of sand-like granular soil undergoing extremely large deformation. The proposed method utilizes the Smoothed Particle Hydrodynamics (SPH) technique, which is a meshfree particle-based approach capable of circumventing the computational difficulties associated with mesh distortion. The Gudehus-Bauer (G-B) hypoplastic model, which can capture the pressure- and density- dependent mechanical behaviour of granular soil, is implemented into the in-house developed SPH code. Concerning the characteristics of large deformation, the Green-Naghdi rate is applied in this study to achieve objective stress integration. An adaptive explicit stress integration scheme, termed Modified Euler automatic Substepping with Error Control (M-E-SEC) is utilized to handle the time integration of the SPH equation system. Additionally, a stability correction method is introduced to suppress the breakdown problem that occurs at low-stress states. The effectiveness of the proposed method is validated by experimental data. Furthermore, several numerical examples are presented to elucidate the evolution of shear band structures in granular soils subject to extremely large deformation levels.

Consistent numerical implementation of hypoelastic constitutive models

In hypoelastic constitutive models, an objective stress rate is related to the rate of deformation through an elasticity tensor. The Truesdell, Jaumann, and Green–Naghdi rates of the Cauchy and Kirchhoff stress tensors are examples of the objective stress rates. The finite element analysis software ABAQUS uses a co-rotational frame which is based on the Jaumann rate for solid elements and on the Green–Naghdi rate for shell and membrane elements. The user subroutine UMAT is the platform to implement a general constitutive model into ABAQUS, but, in order to update the Jacobian matrix DDSDDE in UMAT, the model must be expressed in terms of the Jaumann rate of the Kirchhoff stress tensor. This study aims to formulate and implement various hypoelastic constitutive models into the ABAQUS UMAT subroutine. The developed UMAT subroutine codes are validated using available solutions, and the consequence of using wrong Jacobian matrices is elucidated. The UMAT subroutine codes are provided in the “Electronic Supplementary Material” repository for the user’s consideration.

On instability of constitutive models for isotropic elastic-perfectly plastic material behaviour at finite deformations

The main goal of this work is to test the possibility of a newly introduced constitutive law to model the behaviour of the isotropic elastic-perfectly plastic material which is exposed to large elastoplastic deformations. The proposed constitutive relation is based on the hypo-elastic relation and the inelastic INTERATOM model. The verification of the model is done by its implementation into the commercial software ABAQUS/Standard via the user subroutine UMAT. For that purpose, the large simple shear problem is studied where selected objective corotational rates, i.e. the logarithmic rate, the Jaumann rate and the Green-Naghdi rate, are individually implemented in the aforementioned constitutive relations. The obtained results are compared mutually and with the relevant literature. The proposed constitutive model is also used to test the behaviour of the part of a real engineering structure, i.e. a seismic isolator, in order to obtain the correct input data for further analysis of superstructure behaviour due to seismic excitation.

Numerical analysis of finite hypo-elastic cyclic deformation with large rotations

Constitutive relations which describe engineering materials behaviour during the finite elastoplastic deformations are usually presented in the form of rates of stresses and strains. One of the possible approaches in the constitutive relations formulation is the additive decomposition of the total deformation rate into its elastic part and its plastic part. The elastic deformation rate contributes to any elastoplastic deformation at any stage. Hence, its exact and well-considered formulation is of particular importance and it has to be properly predicted by the corresponding material law. This is of great importance in particular when deformation cyclic processes are considered, in which case small errors may accumulate, even if the total deformation is small. The implementation of the most frequently used corotational rates, i.e. the Jaumann rate and the Green-Naghdi rate, in the hypo-elastic constitutive relations regarding small and moderate rotations gives accurate results for low number of repeated deformation cycles. With increased number of cycles, however, the implementation of these rates results in different and physically non-admissible material responses. This instability in results is particularly observable during the cyclic deformations with large rotations, which is the main subject of this work. In contrast to the aforementioned objective rates, the results of the logarithmic rate implementation into the hypo-elastic constitutive relations for the case of pure elastic deformation describe a physically stable process.

Basis-free expressions for families of objective strain tensors, their rates, and conjugate stress tensors

The objective (Lagrangian and Eulerian) strain tensors, their rates, and conjugate stress tensors used in continuum mechanics equations are considered. These tensors are represented in the eigenprojections of the right and left stretch tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The novelty of the research lies in the simultaneous derivation of expressions for Lagrangian versions of tensors and their Eulerian counterparts with decomposition of the obtained expressions into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . We consider the Lagrangian and Eulerian Doyle–Ericksen families of strain tensors, which, in turn, are subfamilies of the well-known Hill family of strain tensors. Basis-free expressions are obtained for the material rates of Lagrangian tensors from the previously introduced family of generalized strain tensors, whose subfamilies are the Lagrangian and Eulerian Hill families of strain tensors, and similar expressions are obtained for the Green–Naghdi rates of Eulerian strain tensors, which are Eulerian counterparts of the material rates of Lagrangian strain tensors. Basis-free expressions for stress tensors are derived from the classical definitions of the conjugacy of stress and strain tensors. Expressions are found for the Lagrangian and Eulerian symmetric Atluri stress tensors that do not have conjugate strain tensors from the Hill family, and the obtained expressions are decomposed into components coaxial and orthogonal to the tensors $$\mathbf {U}$$ and $$\mathbf {V}$$ . The ranges of allowed ratios of different principal stretches at which the Lagrangian and Eulerian Hencky and Atluri stress tensors approximate the rotated/standard Kirchhoff stress tensors are determined. It is shown that the range of allowed ratios for the Hencky stress tensors is wider than that for the Atluri stress tensors.

Is an additive decomposition of a rate of deformation and objective stress rates passé?

The manuscript critically examines a half a century old idea of decomposing the rate of deformation into elastic and plastic parts. We show that even the most recent additive variant based on the logarithmic rate is inconsistent with the notion of elasticity in so-called unloading stress ratchetting obstacle test while the earlier corotational variants based on the Jaumann and Green–Naghdi rates are well known not to be integrable. We then propose a stress-dependent so-called kinetic logarithmic stress rate that is both integrable and passes the unloading stress ratchetting obstacle test. We also demonstrate that the additive hypo-elasto-plasticity model based on the proposed kinetic logarithmic rate is weakly invariant under isochoric reference change for simple materials in the sense of Noll.

Numerical Approximation of Elasticity Tensor Associated With Green-Naghdi Rate.

Objective stress rates are often used in commercial finite element (FE) programs. However, deriving a consistent tangent modulus tensor (also known as elasticity tensor or material Jacobian) associated with the objective stress rates is challenging when complex material models are utilized. In this paper, an approximation method for the tangent modulus tensor associated with the Green-Naghdi rate of the Kirchhoff stress is employed to simplify the evaluation process. The effectiveness of the approach is demonstrated through the implementation of two user-defined fiber-reinforced hyperelastic material models. Comparisons between the approximation method and the closed-form analytical method demonstrate that the former can simplify the material Jacobian evaluation with satisfactory accuracy while retaining its computational efficiency. Moreover, since the approximation method is independent of material models, it can facilitate the implementation of complex material models in FE analysis using shell/membrane elements in abaqus.

Unified Description of the Mechanical Properties of Typical Marine Soil and Its Application

This study employed a modified elastoplastic constitutive model that can systematically describe the monotonic and cyclic mechanical behaviors of typical marine soils combining the subloading, normal, and superloading yield surfaces, in the seismic response analysis of three-dimensional (3D) marine site. New evolution equations for stress-induced anisotropy development and the change in the overconsolidation of soils were proposed. This model can describe the unified behaviour of unsaturated soil and saturated soil using independent state variables and can uniquely describe the multiple mechanical properties of soils under general stress states, without changing the parameter values using the transform stress method. An effective stress-based, fully coupled, explicit finite element–finite difference method was established based on this model and three-phase field theory. A finite deformation analysis was presented by introducing the Green-Naghdi rate tensor. The simulation and analysis indicated that the proposed method was sufficient for simulating the seismic disaster process of 3D marine sites. The results suggested that the ground motion intensity would increase due to the local uneven complex topography and site effect and also provided the temporal and spatial distribution of landslide and collapse at the specific location of the marine site.

An Improved Heat Equation to Model Ductile-to-Brittle Failure Mode Transition at High Strain Rates Using Fully Coupled Thermal-Structural Finite Element Analysis

In this paper a universal heat equation for fully coupled thermal structural finite element analysis of deformable solids capable of predicting ductile-to-brittle failure mode transition at high strain rates is presented. In the problem mathematical formulation appropriate strain measures describing the onset and the growth of ductile and total damage and heat generation rate per unit volume to model dissipation-induced heating have been employed, which were extended with the heat equation. The model was implemented into a finite element code utilizing an improved weak form for updated Lagrangian formulation, an extended NoIHKH material model for cyclic plasticity of metals applicable in wide range of strain rates and the Jaumann rate in the form of the Green-Naghdi rate in the co-rotational Cauchy’s stress objective integration. The model verification showed excellent agreement with the modelled experiment at low strain rates. Plastic bending of a cantilever has been studied at higher strain rates. A few selected analysis results are presented and briefly discussed.

Green-Naghdi rate of the Kirchhoff stress and deformation rate: the elasticity tensor

The elasticity tensor providing the power-conjugation of the Green-Naghdi rate of the Kirchhoff stress and the deformation rate is required, e.g. by the commercially available Finite Element package ABAQUS/Standard for the material user subroutine UMAT, used to input material behaviours other than those included in the libraries of the package. This elasticity tensor had been studied in the literature, but its symmetries have only been briefly discussed, and only its component form in Cartesian coordinates was known. In this work, we derived a covariant, component-free expression of this elasticity tensor and thoroughly studied its symmetries. We found that, although symmetry on both pair of feet (indices) has been deemed to be desirable in the literature, the expression of the tensor available to-date in fact possesses only symmetry on the first pair of feet (indices), whereas the second pair lacks symmetry, and therefore carries a skew-symmetric contribution. This contribution is unnecessary, as it is automatically filtered in the contraction of the elasticity tensor with the symmetric deformation rate tensor. In order to avoid carrying this unnecessary skew-symmetric contribution in the computations, we employ a tensor identity that naturally symmetrises the second pair of feet of the elasticity tensor. We demonstrated the validity and robustness of the implementation of the user-defined material based on this tensor representation by simulating a benchmark problem consisting in biaxial tests of porcine and human atrial tissue, with material properties taken from previously performed experiments. We compared the results obtained by means of our user-defined material and those obtained through an equivalent built-in material, and obtained identical results.

Energy-Conservation Error Due to Use of Green–Naghdi Objective Stress Rate in Commercial Finite-Element Codes and Its Compensation

The objective stress rates used in most commercial finite element programs are the Jaumann rate of Kirchhoff stress, Jaumann rates of Cauchy stress, or Green–Naghdi rate. The last two were long ago shown not to be associated by work with any finite strain tensor, and the first has often been combined with tangential moduli not associated by work. The error in energy conservation was thought to be negligible, but recently, several papers presented examples of structures with high volume compressibility or a high degree of orthotropy in which the use of commercial software with the Jaumann rate of Cauchy or Kirchhoff stress leads to major errors in energy conservation, on the order of 25–100%. The present paper focuses on the Green–Naghdi rate, which is used in the explicit nonlinear algorithms of commercial software, e.g., in subroutine VUMAT of ABAQUS. This rate can also lead to major violations of energy conservation (or work conjugacy)—not only because of high compressibility or pronounced orthotropy but also because of large material rotations. This fact is first demonstrated analytically. Then an example of a notched steel cylinder made of steel and undergoing compression with the formation of a plastic shear band is simulated numerically by subroutine VUMAT in ABAQUS. It is found that the energy conservation error of the Green–Naghdi rate exceeds 5% or 30% when the specimen shortens by 26% or 38%, respectively. Revisions in commercial software are needed but, even in their absence, correct results can be obtained with the existing software. To this end, the appropriate transformation of tangential moduli, to be implemented in the user's material subroutine, is derived.

Review of energy conservation errors in finite element softwares caused by using energy-inconsistent objective stress rates

The paper briefly summarizes the theoretical derivation of the objective stress rates that are work-conjugate to various finite strain tensors, and then briefly reviews several practical examples demonstrating large errors that can be used by energy inconsistent stress rates. It is concluded that the software makers should switch to the Truesdell objective stress rate, which is work-conjugate to Green’s Lagrangian finite strain tensor. The Jaumann rate of Cauchy stress and the Green-Naghdi rate, currently used in most software, should be abandoned since they are not work-conjugate to any finite strain tensor. The Jaumann rate of Kirchhoff stress is work-conjugate to the Hencky logarithmic strain tensor but, because of an energy inconsistency in the work of initial stresses, can lead to severe errors in the cases of high natural orthotropy or strain-induced incremental orthotropy due to material damage. If the commercial softwares are not revised, the user still can make in the user’s implicit or explicit material subroutines (such as UMAT and VUMAT in ABAQUS) a simple transformation of the incremental constitutive relation to the Truesdell rate, and the commercial software then delivers energy consistent results.

슬관절 전방 십자 인대의 반복 변형하에서의 역학적 거동에 관한 수치적 연구

Anterior cruciate ligament(ACL) of human body experiences a large deformation. May during everyday when large deformation is repeated by various activities such as outdoor activity, ACL easily get damaged. In order to acknowledge the effect of the cyclic large deformation to ACL, the constitutive equations for ACL are derived from experiment data. The concept of the objective stress rate plays a important role wherever large deformation occurs. In order to obtain the objective stress rates the eigenprojection technique is used. A comparison is made for four different cases: Jaumann rate, Green-Naghdi rate, logarithmic rate and twirl tensor of Eulerian triad rate for an isotropic material subject to cyclic deformation, such as simple shear motion. Four different materials are studied to compare the behavior of the materials for ACL using different objective rates. Finally, more complicated model with fibers for soft tissues is used to calculate the behavior subjected to cyclic large deformation.

An Eulerian multiplicative constitutive model of finite elastoplasticity

An Eulerian rate-independent constitutive model for isotropic materials undergoing finite elastoplastic deformation is formulated. Entirely fulfilling the multiplicative decomposition of the deformation gradient , a constitutive equation and the coupled elastoplastic spin of the objective corotational rate therein are explicitly derived. For the purely elastic deformation , the model degenerates into a hypoelastic-type equation with the Green–Naghdi rate. For the small elastic- and rigid-plastic deformations, the model converges to the widely-used additive model where the Jaumann rate is used. Finally, as an illustration, using a combined exponential isotropic-nonlinear kinematic hardening pattern, the finite simple shear deformation is analyzed and a comparison is made with the experimental findings in the literature.

Objective stress rates, cyclic deformation paths, and residual stress accumulation

As one of the basic constitutive ingredients in finite elastoplasticity, Truesdell's hypoelastic equation of grade zero has been widely used to present an objective Eulerian rate formulation of the behaviour. Although it has been known that a stress rate employed in this basic rate equation would actually result in path-dependence behaviour inconsistent with the notion of hyperelasticity, sometimes it has been believed that the definition of the objective stress rate in this basic rate equation might be insubstantial, on account of the fact that the elastic may be small for metals and alloys, etc. Towards clarifying this issue, here we present a unified, direct study on the path-dependence properties of Truesdell equation for all possible deformation paths. Towards this goal, explicit closed-form solutions for the stress response for any given deformation path are derived for classical stress rates including Zaremba-Jaumann rate, Green-Naghdi rate, Oldroyd rate, Cotter-Rivlin rate, and Truesdell rate, etc. From these solutions in terms of explicit path integrations, it may become straightforward to understand how the stress response relies on deformation paths. In particular, cyclic deformation paths composed of any number of deformation cycles are taken into consideration. Explicit exact expressions in simple form are established between the residual stress and the cycle number. These results clearly suggest that the monotone accumulation phenomena of residual stresses with increasing cycle number, found in previous numerical examples, may actually be general facts for all possible cyclic deformation paths. These phenomena imply that, even for a cyclic deformation path undergoing small deformation, the final residual stress would reach such a noticeable level that in any event it could not be regarded to be negligible. Examples are given in the cases of the simple shearing and smooth cyclic plane deformation paths.

Objective stress rates, path-dependence properties and non-integrability problems

In 1984, Simo and Pister [1] demonstrated that the widely-used hypoelastic equation fails to be exactly integrable to deliver finite elastic relations for several well-known objective stress rates, including Jaumann rate, Green-Naghdi rate, Truesdell rate and Lie derivative (upper Oldroyd rate), etc. Successfully treating complicated systems of nonlinear partial differential equations in the Cauchy stress, they proved that Bernstein’s integrability conditions for hypoelastic rate equations could not be fulfilled for each of the foregoing classical stress rates.

Study on dynamic constitutive relations for concrete with finite deformation

The differences between finite deformation and infinitesimal deformation are discussed. They are exercised on elasto-viscoplastic constitutive relations of concrete. Then, a rate-dependent mechanics model was presented on the basis of Ottosen's four-parameter yield criterion, where different loading surface transferring laws were taken into account, when material was in hardening stage or in softening stage, respectively. The model is well established, so that it can be applied to simulate the response of concrete subject to impact loading. Green-Naghdi stress rate was introduced as objective stress rate. Appropriate hypothesis was postulated in accordance with many experimental results, which could reflect the mechanical behaviour of concrete with large deformation. Available thoughts as well as effective methods are also provided for the research on related engineering problems.

Modeling of large simple shear using a viscoplastic overstress model and classical plasticity model with different objective stress rates

Large simple shear deformation, which usually has been used to test the applicability of various models, is studied by applying the classical plasticity model and the viscoplasticity theory based on overstress (VBO) with the logarithmic stress rate. Hypo-elastic, elastic-perfectly plastic, elastic-plastic linear kinematic and isotropic hardening models are employed. In addition to the logarithmic rate, the influences of the convected Truesdell and co-rotational Jaumann and Green-Naghdi rates on stress-strain behavior in simple shear are investigated. It is observed that unlike the Jaumann rate the logarithmic rate does not exhibit any oscillatory response. The responses of the logarithmic rate to every type of model discussed here are acceptable except for the elastic-perfectly plastic case. The finite viscoplasticity theory based on overstress, which has been developed by Krempl and his co-workers, with zero isotropic stress rate and nonzero hardening modulus and elastic-plastic kinematic hardening model gives the same results at the rate of 10−5 1/s. It is shown that elastic-perfectly plastic, elastic-plastic kinematic hardening can be modeled with FVBO.

Numerical study of consistency of rate constitutive equations with elasticity at finite deformation

AbstractThe present work is concerned with the numerical study of the elasticity consistency of the spatial rate equations using the conventional Oldroyd, Truesdell, Cotter–Rivlin, Jaumann and Green–Naghdi rates and the three novel co‐rotational ΩE‐ and Ω¯L‐based, logarithmic rates, and of the rotated material rate equation describing the relationship between the material time derivatives of the rotated Kirchhoff stress and material logarithmic strain. To this end, three integration procedures for updating stress are presented. The stress responses of several typical deformation processes are simulated. According to the numerical results we know that among the spatial rate equations only the logarithmic rate equation is consistent with elasticity under constant material parameters. Integrating the other spatial rate equations will provide path‐dependent stress response. These numerical conclusions support the arguments in H. Xiao et al. (Acta Mechanica 1999; 138:31–50). The reasons leading to elasticity inconsistency of spatial rate equations are analysed. If the material parameters are assumed to be strain‐dependent, the logarithmic rate equation loses also its elasticity‐consistent property. The numerical results prove also that the spatial logarithmic and rotated material rate equations are equivalent to each other. Copyright © 2002 John Wiley & Sons, Ltd.

The choice of objective rates in finite elastoplasticity: general results on the uniqueness of the logarithmic rate

A Eulerian rate formulation of finite elastoplasticity is a composite one composed of a rate equation for elastic behaviour and a flow rule for plastic behaviour as well as an evolution equation for hardening behaviour, in which objective Eulerian tensor rates are used. Among a large variety of objective rates (actually infinitely many), how to choose suitable ones has been one of the crucial points in finite elastoplasticity. It is realized that the foregoing composite formulation of elastoplasticity should fulfil certain consistency criteria in order to avoid inconsistencies or contradictions. These criteria narrow the choice of objective rates. These authors (Bruhns and co–workers and Xiao and co–workers) have recently introduced the self–consistency criterion : in a composite formulation of elastoplasticity, the rate equation intended for characterization of elastic behaviour should be exactly integrable to deliver an elastic relation. It has been demonstrated in the work of the aforementioned authors that if a composite formulation of elastoplasticity with objective rates is required to fulfil the just–stated self–consistency criterion, as it should be, then the newly discovered logarithmic rate is the only possible choice among all objective corotational rates, including the Zaremba–Jaumann rate and the Green–Naghdi rate, etc. In the afore–mentioned result, non–corotational rates, including Oldroyd rates, Cotter–Rivlin rate and Truesdell rate, etc., are not taken into consideration in general. It is the main goal of this paper to further establish the uniqueness of the logarithmic rate among all corotational and non–corotational objective rates. Essential to the attainment of this goal is the use of the yielding–stationarity criterion : the vanishing of the stress rate implies that the yield function is stationary. It is shown that the just–stated criterion is necessary for a composite Eulerian rate formulation of finite elastoplasticity to be consistent and means that the stress rate must be a corotational rate. The main goal of this article is, thus, attained by combining the just–stated result and the established result stated before.