Algorithmic Tangent Moduli Research Articles

On a finite strain viscoplastic theory based on a new internal dissipation inequality

This work is focused on the theoretical development and numerical implementation of a viscoplastic law. According to the second law of thermodynamics a dissipation inequality described in the rotated material coordinate system is developed. Based on this dissipation inequality and the principle of maximum dissipation a finite strain viscoplastic model described also in the rotated material coordinate system is formulated. The evolution equations are expressed in terms of the material time derivatives of the rotated elastic logarithmic strain, the accumulated plastic strain and the strain-like tensor conjugate to the rotated back stress. The mathematical structure of this theory is concise and similar to that of the infinitesimal viscoplastic theory. These characteristics make the numerical implementation of this theory easy. The stress integration algorithm and the algorithmic tangent moduli for the infinitesimal theory can be applied to the numerical implementation of the present finite strain theory with a little reformulation. The complicated algorithmic formulations for most of other finite plastic laws can be therefore circumvented. In order to check the effectivity of the present finite strain theory a set of numerical examples under strict deformation conditions are presented. These numerical examples prove the excellent performance of the present viscoplastic material law at describing the finite strain elastoplastic and viscoplastic problems.

Large strain finite element analysis of sand: model, algorithm and application to numerical simulation of tire–sand interaction

Based on a review of several existing models for the elastic responses of geomaterials, a new nonlinear elastic law is suggested. A fully implicit return mapping algorithm for non-associative elastoplastic models of the geomaterials is presented within the regime of large strains. This algorithm is formulated in the principal axes. Algorithmic tangent moduli are derived. A modified critical state model in conjunction with the new nonlinear elastic law is implemented into the general purpose finite element software MARC, using the proposed algorithm. Furthermore, the tire–sand interaction is numerically simulated, employing the modified model and the algorithm.

A re-formulation of the exponential algorithm for finite strain plasticity in terms of cauchy stresses

The role of the stress measure to be chosen as the argument in the definition of yield functions is discussed in the context of finite strain plasticity theory. Motivated by physical arguments, the exponential algorithm for multiplicative finite strain plasticity is revisited such that Cauchy stresses are adopted as arguments in the yield function. Using logarithmic strain measures, the return map algorithm is formulated in principal axes. The algorithmic tangent moduli are obtained in a slightly modified, unsymmetric format compared to the standard formulation in terms of Kirchhoff stresses. However, the global structure of the exponential algorithm is unchanged. The algorithm is applied to the re-formulation of the Cam—Clay model in terms of Cauchy stresses. The typical calibration procedure of the Cam—Clay model based on Cauchy stresses is demonstrated. As an alternative, a modification of the Cam—Clay model, which allows re-calibration of Cauchy stress-based test data to be used within the framework of a Kirchhoff-based finite strain model is also discussed. The relevance of the adequate choice of the stress measure is illustrated by means of selected numerical analyses.

CONSIDERATION OF AGING OF SHOTCRETE IN THE CONTEXT OF A 3-D VISCOPLASTIC MATERIAL MODEL

The structure of the classical theory of viscoplasticity and its algorithmic formulation in the context of an operator split scheme is extended to account for the effect of aging. Within this framework, a 3-D multisurface constitutive model for shotcrete is proposed. A rotating-crack model based on the Rankine criterion is adopted in conjunction with a Drucker–Prager yield criterion for describing the mechanical behaviour of young concrete in the compressive regime. The model is suitable for the representation of the time-dependent behaviour of shotcrete including creep and the time dependence of strength and stiffness. A computationally effective return mapping scheme is achieved by formulating the material model in terms of the principal directions and transforming the obtained relations to global axes. Linearization of the return mapping algorithm yields the algorithmic tangent moduli.

Large Strain Finite-Element Analysis of Snow

The modified Cam-clay model and the classical Drucker-Prager plasticity model are used for the numerical representation of snow. Both models are recast within an algorithmic framework for multiplicative finite deformation plasticity. A fully implicit return project algorithm for the Cam-clay model in conjunction with a modification of the elastic stress-strain law and the computation of the exact algorithmic tangent moduli provide an efficient implementation of this model. Both material models are calibrated according to the experimental data obtained from hydrostatic and shear-box tests of snow and verified by reanalyses of these tests. Numerical results obtained from finite-element analyses of the mechanical interaction between a single tread rubber block from an automobile tire moving over a snow-covered surface are presented.

On the numerical treatment and analysis of finite deformation ductile single crystal plasticity

This contribution is concerned with the numerical analysis and simulation of instability phenomena such as shearband localization in plastic flow processes within the framework of ductile single (fcc) crystal viscoplasticity. To this end, a family of implicit constitutive algorithms and their implementation are considered in detail. Issues of accuracy and conservation of plastic incompressibility are examined. Moreover, the implementation within an overall Newton-Raphson finite element solution strategy relies on the correct evaluation of the algorithmic tangent moduli which result from linearizing the stress update integrator. Likewise, these algorithmic tangent moduli are employed within the numerical analysis of the spatial localization tensor. Examples demonstrating the performance of the proposed strategy at the local constitutive level as well as at the global structural level are given.

A new model of generalized plasticity and its numerical implementation

A previously proposed simple model of generalized plasticity is modified so that it allows behavior that is asymptotically perfectly plastic or strain-softening. Initially presented in uniaxial form, the model is then generalized to multiaxial stress. Numerical implementation is developed first through direct integration of the rate equations, including special cases in which the solution may be obtained analytically, and then by means of a return-map algorithm, which is particularly well suited to the finite-element method; consistent algorithmic tangent moduli are derived as well. Numerical examples are presented to illustrate the various approaches.