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 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.

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.

Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation—Part II: Finite Strain Elastoplasticity

An Eulerian rate formulation of finite strain elastoplasticity is developed based on a fully integrable rate form of hyperelasticity proposed in Part I of this work. A flow rule is proposed in the Eulerian framework, based on the principle of maximum plastic dissipation in six-dimensional stress space for the case of J2 isotropic plasticity. The proposed flow rule bypasses the need for additional evolution laws and/or simplifying assumptions for the skew-symmetric part of the plastic velocity gradient, known as the material plastic spin. Kinematic hardening is modeled with an evolution equation for the backstress tensor considering Prager’s yielding-stationarity criterion. Nonlinear evolution equations for the backstress and flow stress are proposed for an extension of the model to mixed nonlinear hardening. Furthermore, exact deviatoric/volumetric decoupled forms for kinematic and kinetic variables are obtained. The proposed model is implemented with the Zaremba–Jaumann rate and is used to solve the problem of rectilinear shear for a perfectly plastic and for a linear kinematic hardening material. Neither solution produces oscillatory stress or backstress components. The model is then used to predict the nonlinear hardening behavior of SUS 304 stainless steel under fixed-end finite torsion. Results obtained are in good agreement with reported experimental data. The Swift effect under finite torsion is well predicted by the proposed model.

Eulerian Framework for Inelasticity Based on the Jaumann Rate and a Hyperelastic Constitutive Relation—Part I: Rate-Form Hyperelasticity

An integrable Eulerian rate formulation of finite deformation elasticity is developed, which relates the Jaumann or other objective corotational rate of the Kirchhoff stress with material spin to the same rate of the left Cauchy–Green deformation measure through a deformation dependent constitutive tensor. The proposed constitutive relationship can be written in terms of the rate of deformation tensor in the form of a hypoelastic material model. Integrability conditions, under which the proposed formulation yields (a) a Cauchy elastic and (b) a Green elastic material model are derived for the isotropic case. These determine the deformation dependent instantaneous elasticity tensor of the material. In particular, when the Cauchy integrability criterion is applied to the stress-strain relationship of a hyperelastic material model, an Eulerian rate formulation of hyperelasticity is obtained. This formulation proves crucial for the Eulerian finite strain elastoplastic model developed in part II of this work. The proposed model is formulated and integrated in the fixed background and extends the notion of an integrable hypoelastic model to arbitrary corotational objective rates and coordinates. Integrability was previously shown for the grade-zero hypoelastic model with use of the logarithmic (D) rate, the spin of which is formulated in principal coordinates. Uniform deformation examples of rectilinear shear, closed path four-step loading, and cyclic elliptical loading are presented. Contrary to classical grade-zero hypoelasticity, no shear oscillation, elastic dissipation, or ratcheting under cyclic load is observed when the simple Zaremba–Jaumann rate of stress is employed.

Determination of Hardening Coefficient of Large Strain Constitutive Model Based on Torsion Tests

Uni-tension test and torsion test with specimens of A3 steel are completed, and deformation rate, the spin of Euler configuration to Lagrange configuration and generalized Jaumann rate of Kirchhoff stress in torsion problem are given. Large strain constitutive relationship with generalized Jaumann stress rate is studied based on torsion test. The simple shear problem is discussed by using the constitutive model given in this paper. The result indicates that the hardening coefficient of constitutive model with generalized Jaumann rate may be given by the same expression as that of constitutive model with Jaumann objective rate when determining by using torsion tests and the shear stress response curve in simple shear problem obtained by using the constitutive model in this paper is similar to the shear stress-strain curve given in solid circular shaft torsion tests.

Errors Caused by Non-Work-Conjugate Stress and Strain Measures and Necessary Corrections in Finite Element Programs

Many finite element programs including standard commercial software such as ABAQUS use an incremental finite strain formulation that is not fully work-conjugate, i.e., the work of stress increments on the strain increments does not give a second-order accurate expression for work. In particular, the stress increments based on the Jaumann rate of Kirchhoff stress are work-conjugate with the increments of the Hencky (logarithmic) strain tensor but are paired in many finite element programs with the increments of Green’s Lagrangian strain tensor. Although this problem was pointed out as early 1971, a demonstration of its significance in realistic situations has been lacking. Here it is shown that, in buckling of compressed highly orthotropic columns or sandwich columns that are very “soft” in shear, the use of such nonconjugate stress and strain increments can cause large errors, as high as 100% of the critical load, even if the strains are small. A similar situation may arise when severe damage such as distributed cracking leads to a highly anisotropic tangential stiffness matrix, or when axial cracks between fibers severely weaken a uniaxial fiber composite or wood. A revision of these finite element programs is advisable, and will in fact be easy—it will suffice to replace the Jaumann rate with the Truesdell rate. Alternatively, the Green’s Lagrangian strain could be replaced with the Hencky strain.