Secant Formulation Research Articles

Combined mean-field and full-field homogenization of porous elasto-plastic materials and composites under arbitrary stress triaxialities

A predictive micromechanical approach is proposed for porous materials where the unreinforced or reinforced matrix phase is elasto-plastic with hardening. The cavities can be spheres, long or short cylinders. The approach is based on an alternative microstructure made of elasto-plastic inhomogeneities embedded in a homogenized porous matrix phase, and the volume fractions are determined from a maximum packing argument. The effective properties of single hollow solids are computed with an energy-based approach coupled with full-field finite element (FE) analyses. Next, the alternative microstructures are homogenized with mean-field (MF) models. For reinforced porous materials, a two-level method is adopted, where the proposed approach is used at the lower level to obtain a fictitious homogenized matrix, in which reinforcements are embedded at the upper level. The present work is restricted to monotonic and proportional loadings and to the secant formulation of isotropic or transversely isotropic elasto-plasticity. However, no constitutive models are supposed or identified. The predictions were verified against reference full-field FE results on the actual microstructures in 3D and 2D plane strain or stress, for arbitrary stress triaxialities, and good agreement was found in all cases.

A time-incremental homogenization method for elasto-viscoplastic particulate composites based on a modified secant formulation

Starting from the exact solution of the heterogeneous linear viscoelastic Eshelby ellipsoidal inclusion problem obtained in the time domain for an ellipsoidal inclusion embedded in an isotropic matrix (Berbenni et al., 2015), a direct time-incremental homogenization Mori–Tanaka scheme for two-phase non linear elasto-viscoplastic materials is proposed. The extension to non-linear behavior is based on a “modified secant” formulation to obtain the linear comparison composite (LCC) properties. In contrast with more classic homogenization method based on the first moment of stresses (i.e. classic secant formulation), the present approach is based on the second-order moment of stresses making use of the Hill’s lemma. Hence, a modified secant viscoplastic compliance is considered for the matrix phase. The effective behavior as well as the evolution laws of the averaged strains and stresses per phase are directly solved in the time domain, without the need of Laplace-Carson transforms. The estimates provided by the present homogenization model are compared to full-field calculations from the literature for two-phase non linear particulate composites constituted of an isotropic elasto-viscoplastic matrix phase and isotropic elastic particles, and, for radial and non-radial loadings.

Micro-mechanical model for predicting the elasto-plastic behavior of composites based on secant formulation method

To predict the Elasto-Plastic Behaviors (EPBs) of aligned inclusions reinforced composites, this paper develops an interpolative Mori-Tanaka/Double-Inclusion (MT-DI) homogenization model with the secant formulation, and gives the numerical implementation algorithms of the developed MT-DI model with the secant formulation. The Finite Element (FE) homogenization method is implemented to provide the “exact” EPBs of the composites and thus validate the MT-DI model with the secant formulation. The MT-DI model with the 2nd-order secant formulation is validated to provide the more accurate predictions, while the MT-DI model with the 1st-order secant formulation always gives the stiffer predictions. The results show that using the macro-stress and macro-strain as the inputs, the MT-DI model with the secant formulation gives the identical predictions. The predictions of the MT-DI model with the secant formulation vary between those of the MT and DI models with the secant formulation. Meanwhile, the MT-DI model with the secant formulation does not predict the accurate EPBs for the phases of the composites.

A modified secant formulation to predict the overall behavior of elasto-viscoplastic particulate composites

To model the mechanical response of rate-dependent elasto-viscoplastic (or equivalently nonlinear viscoelastic) particulate composites, the proposed approach consists in linearizing the (nonlinear) relation between the viscous strain rate and the stress. To obtain a tractable problem, the linearized properties (viscous modulus, stress-free strain rate) are chosen uniform per phase: in each phase, they are computed for reference stresses per phase. Thus, we obtain a fictitious linear viscoelastic solid whose microstructure is the same as the previous one and submitted to additional uniform stress-free strains per phase. For two-phases isotropic particulate composites, this general formulation associated to Hashin–Shtrikman estimates of the above-mentioned fictitious linear viscoelastic solid yields closed-form expressions of the time evolution of the effective behavior as well as phase-averaged mechanical fields expressed as a set of nonlinear first-order differential equations. Since the linearized properties in the matrix phase are based on the second-order moment of the stress field, these estimates coincides with the upper bound (Ponte Castañeda, 1992) in the purely viscoplastic regime. These estimates are also consistent with the one given for non aging linear viscoelastic composites (correspondence principle). Finally, full-field computations are used to asses the quality of these estimates for various loading conditions varying from a pure shear to a pure differential swelling loading for moderate to highly contrasted behaviors and including non monotonic and non radial effects.

3D numerical modelling of twisting cracks under bending and torsion of skew notched beams

The testing of mode III and mixed mode failure is every so often encountered in the dedicated literature of mechanical characterization of brittle and quasi-brittle materials. In this work, the application of the mixed strain displacement ε-u finite element formulation to three examples involving skew notched beams is presented. The use of this FE technology is effective in problems involving localization of strains in softening materials.The objectives of the paper are: (i) to test the mixed formulation in mode III and mixed mode failure and (ii) to present an enhancement in terms of computational time given by the kinematic compatibility between irreducible displacement-based and the mixed strain-displacement elements.Three tests of skew-notched beams are presented: firstly, a three point bending test of a PolyMethyl MethaAcrylate beam; secondly, a torsion test of a plain concrete prismatic beam with square base; finally, a torsion test of a cylindrical beam made of plain concrete as well. To describe the mechanical behavior of the material in the inelastic range, Rankine and Drucker-Prager failure criteria are used in both plasticity and isotropic continuum damage formats.The proposed mixed formulation is capable of yielding results close to the experimental ones in terms of fracture surface, peak load and global loss of carrying capability. In addition, the symmetric secant formulation and the compatibility condition between the standard irreducible method and the strain-displacement one is exploited, resulting in a significant speedup of the computational procedure.

An adaptive model order reduction with Quasi‐Newton method for nonlinear dynamical problems

SummaryModel Order Reduction (MOR) methods are extremely useful to reduce processing time, even nowadays, when parallel processing is possible in any personal computer. This work describes a method that combines Proper Orthogonal Decomposition (POD) and Ritz vectors to achieve an efficient Galerkin projection, which changes during nonlinear solving (online analysis). It is supported by a new adaptive strategy, which analyzes the error and the convergence rate for nonlinear dynamical problems. This model order reduction is assisted by a secant formulation which is updated by the Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) formula to accelerate convergence in the reduced space, and a tangent formulation when correction of the reduced space is needed. Furthermore, this research shows that this adaptive strategy permits correction of the reduced model at low cost and small error. Copyright © 2015 John Wiley & Sons, Ltd.

A set of enhanced formulations for existing nonlinear homogenization schemes and their evaluation

Abstract On the basis of non-biased comparative evaluations of various linearization procedures used in nonlinear homogenization, performed both at the global and local scales for power-law composites (Rekik et al., 2005, 2007, 2012), we propose in this paper six ad hoc enhancements of some of the linearization procedures considered in Rekik et al. (2007). Both “stress–strain” approaches (the secant and affine formulations) or “variational formulations” (the tangent second-order method (Ponte Castaneda, 1996)) are considered. The main idea consists in proposing alternatives for the usual reference strains used by the secant, affine and tangent second-order procedures. The new linear comparison composites generated by the linearization step around the chosen alternative descriptors of the strain field statistics explicitly account for either intraphase strain fluctuations or both inter- and intraphase strain fluctuations. As a first illustration, the relevance and limitations of the enhanced linearization procedures are tested for rigidly-reinforced and porous power-law composites. For isochoric loadings, it is shown that two variants of the enhanced tangent second-order formulation lead to accurate estimates of the exact effective response which are in good agreement with the efficient second-order scheme of Ponte Castaneda (2002a). Further, the modified secant formulation provides good results for strongly nonlinear rigidly-reinforced composites away from low particulate volume fraction and the percolation threshold; however some new inherent limitations of secant formulations are also established. At last, a very discriminant situation is tested: it consists of a porous medium submitted to a pure hydrostatic loading at low pore concentrations. It is shown that one variant of the proposed enhanced second-order formulations leads to accurate estimates alike the efficient and more sophisticated formulations proposed in Bilger et al. (2002); Danas et al. (2008).

Experiments and nonlinear homogenization sustaining mean-field theories for refractory mortarless masonry: The classical secant procedure and its improved variants

Abstract In order to support the optimization of mortarless brick linings of steel producing furnaces it is proposed in this paper to investigate only the compressibility of dry refractory joints. To this end, optical measurements based on the Digital Image Correlation method were carried out during compression of Magnesia-Carbon brick samples with dry joints. The second main objective is to assess the accuracy of the secant linearization schemes (the classical secant procedure and its modified extension) to reproduce the reference local and global behaviour of refractory mortarless linings accounting for the identified inelastic convex power-law behaviour of the Magnesia-Carbon dry joint. The reference nonlinear solution is obtained by means of finite elements method. Under normal compressive loading, unlike for usual (concave) power-law viscoplastic composites for which the secant schemes are known to provide too stiff results, it was found that the modified secant scheme leads to good overall predictions. The classical secant procedure underestimates the reference local and overall behaviour. To improve the latter result, an empirical improved secant formulation was proposed and implemented. It leads to better estimates at local and global levels.

Homogenization of periodic materials with viscoelastic phases using the generalized FVDAM theory

In the last decades, new generations of advanced materials have been designed and manufactured for specific applications. The micromechanics plays an important role in the development of heterogeneous materials, enabling efficient analyses of composite materials with complex geometries, circumventing the traditional trial-and-error approach, producing substantial cost savings. The unit cell problem to the analysis of periodic heterogeneous media can be solved by the well-established 0th order version of the finite-volume theory, named finite-volume direct averaging micromechanics (FVDAM) theory. This standard version of the FVDAM theory employs an incomplete second-order displacement field within individual subvolumes of a discretized analysis domain together with a surface-averaging framework, which does not enforce displacement or traction continuity in a point-wise manner. This, in turn, produces interfacial interpenetrations and non-traction stress discontinuities, thereby demanding very refined meshes in order to produce good interfacial conformability and pointwise stress continuity between adjacent subvolumes. To overcome these shortcomings, a generalized FVDAM theory has been proposed to enable analysis of periodic heterogeneous materials in the finite-deformation domain. The generalization is based on a higher-order displacement field representation and on the definition of elasticity-based surface-averaged kinematic and static variables related through a local stiffness matrix. Herein, we specialize the generalized FVDAM theory to the infinitesimal analysis of periodic materials with viscoelastic phases, where a total or secant formulation is employed, with the viscoelastic strains evaluated incrementally using an algorithm based on the concept of state variables. The generalized or 2nd order version considerably improves interfacial conformability and pointwise traction and non-traction stress continuity between adjacent subvolumes in comparison with the 0th order version, but with a higher computational cost. Furthermore, for the same mesh discretization, these two versions provide comparable macroscopic response. Considering these features, the 0th order version is recommended to evaluate the effective elastic properties and the homogenized creep and relaxation functions, while the 2nd order version is more efficient in the evaluation of the microscopic displacement and stress fields.

Formulation of consistent nonlinear restoring stiffness for dynamic analysis of tension leg platform and its influence on response

Shortcomings of the traditionally used nonlinear restoring stiffness of TLPs, i.e. unrealistically high stiffness of horizontal motions, their uncoupling and secant formulation are pointed out. Therefore, new consistent restoring stiffness is derived. The platform is considered as a rigid body moored by flexible pretensioned tendons. Global horizontal low frequency motions (surge, sway and yaw) with large amplitudes as a result of dominant second order wave excitation and small stiffness, and vertical local motions (heave, roll and pitch) of higher frequency and small amplitudes excited by the first order wave forces, are distinguished. Hence, horizontal displacements represent position parameters in analysis of vertical motions. First, the linear restoring stiffness, which consists of the tendon conventional axial stiffness, the tendon geometric stiffness and the platform hydrostatic stiffness, is established. Then it is extended to large displacements resulting in new secant restoring stiffness. It depends on surge, sway and yaw displacements and is the same in any horizontal direction. Also, the tangent stiffness, which gives more accurate results, is derived. Heave is defined as vertical projection of axial tendon vibrations and platform tangential oscillations, which are analyzed in their natural moving coordinate system. Inertia force due to setdown, as a slave d.o.f. of the master horizontal motions, is taken into account in the dynamic equilibrium equations. As a result the complete tangential stiffness matrix of horizontal and vertical motions includes 7 d.o.f. The known secant restoring stiffness matrices are compared with the new one and noticed differences are discussed. All theoretical contributions are illustrated by relatively simple numerical example.

On the formulation of anisotropic elastic degradation. I. Theory based on a pseudo-logarithmic damage tensor rate

In spite of its appeal, anisotropic damage is being introduced in the constitutive equations of engineering materials at a slow pace. One of the main reasons is the difficulty of establishing general evolution laws. This originates from the lack of physical meaning of the thermodynamic forces conjugate to the damage variables, which finally constitute the space in which loading functions and `damage rules' are defined. In this article, the authors propose a new `pseudo-logarithmic' rate of damage, which has the advantage of exhibiting a simple and meaningful conjugate force with very convenient properties. A main advantage is the physical interpretation of the corresponding “damage rule”, which clearly separates the effects of its volumetric part, responsible for isotropic degradation, from its deviatoric part, responsible for anisotropic effects. This new concept is applied to a second-order tensor secant formulation, which is developed using traditional concepts of continuum damage mechanics within the general theoretical framework of elastic degradation and damage recently proposed by the authors. A first example of anisotropic damage formulation based on these concepts, the `generalized pseudo-Rankine' model, is presented and verified with analytical and numerical examples in a companion `Part II' paper.

Bracing Requirements of Plane Frames

Secant formulation for the elastic‐plastic limit analysis of planar frames with rigid and semirigid connections is presented. A series of one‐, two‐, three‐, and 14‐story braced frames are considered. The results of limit analysis of these frames for different bracing areas are included. In evaluating these results, both strength and serviceability criteria are considered. Based on these studies, semiempirical formulas are developed to calculate the required bracing areas for frames with type‐FR and type‐PR connections. The effect of connection flexibility in semirigid frames is accounted for by introducing a correction factor to the formula developed for rigid frames. The performance of the proposed formulas are evaluated by comparing with the results of limit analysis, and good correlation is observed. A step‐by‐step scheme is presented for easy calculation of the required bracing area. The limitations of the scheme are also identified. In the case of semirigid frames, if the connections do not have enough moment capacity to allow plastic hinges to be formed in the members, the connections must be treated as hinged.