Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials

Finite element (FE) calculations of a cylindrical cell containing a spherical hole have been performed under large strain conditions for varying triaxiality with three different constitutive models for the matrix material, i.e. rate independent plastic material with isotropic hardening, visco-plastic material under both isothermal and adiabatic conditions, and porous plastic material with a second population of voids nucleating strain controlled. The “mesoscopic” stress-strain and void growth responses of the cell are compared with predictions of the modified Gurson model in order to study the effects of varying triaxiality and strain rate on the critical void volume fraction. The interaction of two different sizes of voids was modelled by changing the strain level for nucleation and the stress triaxiality. The study confirms that the void volume fraction at void coalescence does not depend significantly on the triaxiality if the initial volume fraction of the primary voids is small and if there are no secondary voids. The strain rate does not affect f_c either. The results also indicate that a single internal variable, f, is not sufficient to characterize the fracture processes in materials containing two different size-scales of void nucleating particles.     

Endochronic analysis for finite elasto-plastic deformation and application to metal tube under torsion and metal rectangular block under biaxial compression

The ordinary differential constitutive equations of endochronic theory are extended to simulate elasto-plastic deformation in the range of finite strain using the concept of corotational rate. Different corotational stress rates (Jaumann, Cotter-Rivlin, Truesdell, Dienes and Mandel) are incorporated into the theory. In addition, a new formulation of the plastic spin, which can be used in the Mandel stress rate, is derived. Theoretical simulations of the axial effects for various materials subjected to simple and pure torsional loading cases are discussed in this study. It is shown that the endochronic theory incorporated with the Mandel stress rate yields the most satisfactory result, as indicated from comparison with the experimental data found in literature.

Finally, theoretical investigation of the deformation subjected to finite proportional and non-proportional biaxial compression is presented. The true relationship between stress and strain can be converted to a nominal stress-strain relationship for biaxial loading through the explicit transformation equations derived in this paper. Experimental data tested by Khan and Wang [1990] (“An Experimental study of Large finite Plastic Deformation in Annealed 1100 Aluminum During Proportional and Non-proportional Biaxial Compression” Int. J. Plasticity, 6, 485) are suitably described by the theory demonstrated from a comparison with the theoretical prediction according to rigid-plastic and elastic-plastic models employed by Huang and Khan [1991]. “An Analysis of Finite Elastic-Plastic Deformation under Biaxial Compression”, Int. J. Plasticity, 7, 219).     

Experimental and numerical determinations of the initial surface of phase transformation under biaxial loading in some polycrystalline shape-memory alloys

Biaxial proportional loading such as tension (compression)–internal pressure and bi-compression tests are performed on a Cu-Zn-Al and Cu-Al-Be shape memory polycrystals. These tests lead to the experimental determination of the initial surface of phase transformation (austenite→martensite) in the principal stress space (σ₁,σ₂). A first “micro–macro” modeling is performed as follows. Lattice measurements of the cubic austenite and the monoclinic martensite cells are used to determine the “nature” of the phase transformation, i.e. an exact interface between the parent phase and an untwinned martensite variant. The yield surface is obtained by a simple (Sachs constant stress) averaging procedure assuming random texture. A second modeling, performed in the context of the thermodynamics of irreversible processes, consists of a phenomenological approach at the scale of the polycrystal. These two models fit the experimental phase transformation surface well.     

Anisotropic hardening in single crystals and the plasticity of polycrystals

Based on the dislocation structures developed during plastic deformation, an anisotropic hardening law is developed to describe the latent hardening behavior of slip systems under multislip. This theory incorporates the concept of isotropic hardening, kinematic hardening, and the two-parameter representation; it automatically includes the strength differential between the forward and reversed slips and between the acute and obtuse cross slips. The self-hardening modulus of a slip system is found to be “associated” with the latent hardening law involved, and, based on some experimental evidence, two specific sets of self-hardening modulus are suggested. An important feature of this associated modulus is that the slip system with a soft latent hardening (e.g., the reversed system with a Bauschinge effect) will have an enhanced self-hardening modulus. This newly developed hardening law, together with its associated latent hardening moduli, is then applied to examine the strain-hardening behavior of a polycrystal. Although crystals with a stronger latent hardening will, in general, also lead to a stranger strain-hardening for the polycrystal, the stress-strain behavior of the polycrystal using the kinematic hardening law of single crystals is found to be not necessarily softer than that using the isotropic hardening law. Within the range of experimentally measured latent hardening ratio of slip systems, the anisotropic theory is also used to calculate the motion of yield surface of a polycrystal. The general results, employing four selected types of anisotropic hardening, all show the essential features of experimental observations by Phillips and his co-workers. The application is highlighted with a reasonably successful quantitative modeling of initial and subsequent yield surfaces of an aluminum.     

On rate sensitivity of f.c.c. metals,instantaneous rate sensitivity and rate sensitivity of strain hardening

An up-to-date approach to the phenomenon of rate sensitivity observed in f.c.c. metals is discussed. It is shown that the rate sensitivity of strain hardening, which so far has been neglected, plays an equal or even dominant role in an estimation of the total rate sensitivity. It is suggested that the presence of the rate sensitivity of strain hardening is developed by the athermal generation of structural defects (dislocations), while at the same time collision and partial annihilation of dislocations occurs with the assistance of thermal activation. These micromechanisms of plastic deformation are capable of developing so-called strain rate and temperature history effects. Some experimental evidence of strain rate history effects, for both polycrystals and monocrystals, are provided in this paper; they are discussed within the framework of instantaneous rate sensitivity versus rate sensitivity of strain hardening. Both rate sensitivities are reviewed within the framework of thermal activation strain rate analysis. Experimental data are provided for aluminium, copper and lead. They conclusively demonstrate the importance of rate sensitivity of strain hardening which is developed by dynamic recovery (annihilation of defects during plastic deformation).Some fundamentals of how to construct constitutive relations have been discussed on the basis that the total flow stress τ is the sum of the effective stress τ1 and the internal stress τ_μ. General relations for structural evolution have been analysed, which are able to describe strain rate and temperature history effects. On the basis of earlier results a general relationship for structural evolution has been proposed using the concept of the effective dislocation multiplication coefficient M_eff. It is shown that an evolutionary relationship should be of differential type with generation and annihilation terms.Finally, some recommendations are provided as to constitutive modeling and future studies of both instantaneous rate sensitivity and rate sensitivity of strain hardening.     

On constitutive modelling of porous neo-Hookean composites

In this paper a hyperelastic constitutive model is developed for neo-Hookean composites with aligned continuous cylindrical pores in the finite elasticity regime. Although the matrix is incompressible, the composite itself is compressible because of the existence of voids. For this compressible transversely isotropic material, the deformation gradient can be decomposed multiplicatively into three parts: an isochoric uniaxial deformation along the preferred direction of the material (which is identical to the direction of the cylindrical pores here); an equi-biaxial deformation on the transverse plane (the plane perpendicular to the preferred direction); and subsequent shear deformation (which includes “along-fibre” shear and transverse shear). Compared to the multiplicative decomposition used in our previous model for incompressible fibre reinforced composites [Guo, Z., Peng, X.Q., Moran, B., 2006, A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus. J. Mech. Phys. Solids 54(9), 1952–1971], the equi-biaxial deformation is introduced to achieve the desired volume change. To estimate the strain energy function for this composite, a cylindrical composite element model is developed. Analytically exact strain distributions in the composite element model are derived for the isochoric uniaxial deformation along the preferred direction, the equi-biaxial deformation on the transverse plane, as well as the “along-fibre” shear deformation. The effective shear modulus from conventional composites theory based on the infinitesimal strain linear elasticity is extended to the present finite deformation regime to estimate the strain energy related to the transverse shear deformation, which leads to an explicit formula for the strain energy function of the composite under a general finite deformation state.     

Simply approximate estimation of local failure position of a free-free slender shell

Dynamic plastic failure characteristics of a space free-free slender shell subjected to intense dynamic loading of suddenly applied pressure unsymmetrical triangle distributed along its span was studied. Both rigid perfectly plastic (r-p-p) analytical method and finite element method based elastic perfectly plastic (e-p-p) material idealization and shell element model were adopted to predict the local failure position in the structure. It was shown that both r-p-p and e-p-p model could estimate a plastic “kink” taking place in the slender shell, which reflects the strain localization of deformation. The comparison for the position of “kink” predicted by using r-p-p and e-p-p methods is found to be reasonable good.     

Kinematic hardening rules with critical state of dynamic recovery,part I: formulation and basic features for ratchetting behavior

Kinematic hardening rules formulated in a hardening/dynamic recovery format are examined for simulating rachetting behavior. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be activated fully. Discussing their basic features, the authors show that they can predict much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. Comparisons with multilayer and multisurface models are made also, resulting in a finding that the simple one in the present rules is similar to the multilayer model with total strain rate replaced by inelastic (or plastic) strain rate. Part II of this work deals with applications to experiments.     

Implicit finite element formulations for multi-phase transformation in high carbon steel

An anomalous plastic deformation observed during the phase transformation of steels was implemented into the finite element modeling. The constitutive equations for the transformation plasticity originally proposed by Greenwood and Johnson [Greenwood, G.W., Johnson, R.H., 1965. The deformation of metals under small stresses during phase transformation. Proc. Roy. Soc. A 283, 403] and further extended by Leblond et al. [Leblond, J.B., Mottet, G., Devaux, J.C., 1986a. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, I. Derivation of general relations. J. Mech. Phys. Solids 34, 395–409; Leblond, J.B., Mottet, G., Devaux, J.C., 1986b. A theoretical and numerical approach to the plastic behavior of steels during phase transformations, II. Study of classical plasticity for ideal-plastic phases. J. Mech. Phys. Solids 34, 411–432; Leblond, J.B., Devaux, J., Devaux, J.C., 1989a. Mathematical modeling of transformation plasticity in steels, I: case of ideal-plastic phases. Int. J. Plasticity 5, 511–572; Leblond, J.B., 1989b. Mathematical modeling of transformation plasticity in steels, II: coupling with strain hardening phenomena. Int. J. Plasticity 5, 573–591] were modified to consider the thermo-mechanical response of generalized multi-phase steel during phase transformations from austenite at high temperature. An implicit numerical solution procedure to calculate the plastic deformation of each constituent phase was newly proposed and implemented into the general purpose implicit finite element program via user material subroutine. The new algorithms include efficient calculation of consistent tangent modulus for the transformation plasticity and application of general anisotropic yield functions without limitation to the isotropic yield function. Besides the thermo-elastic–plastic constitutive equations, non-isothermal transformation kinetics was characterized by the Johnson–Mehl–Avrami–Kolmogorov (JMAK) equation and additivity relationship for the diffusional transformation, while the model proposed by Koistinen and Marburger was used for the diffusionless transformation. Numerical verifications for the continuous cooling experiments under various loading conditions were conducted to demonstrate the applicability of the developed numerical algorithms to the high carbon steel SK5.     

Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence

We applied a technique that defines and extracts “structures” from a DNS dataset of a turbulence variable in a way that allows concurrent quantitative and visual analysis. Local topological and statistical measures of enstrophy and strain-rate structures were compared with global statistics to determine the role of mean shear in the dynamical interactions between fluctuating vorticity and strain-rate during transition from isotropic to shear-dominated turbulence. We find that mean shear adjusts the alignment of fluctuating vorticity, fluctuating strain-rate in principal axes, and mean strain-rate in a way that (1) enhances both global and local alignments between vorticity and the second eigenvector of fluctuating strain-rate, (2) two-dimensionalizes fluctuating strain-rate, and (3) aligns the compressional components of fluctuating and mean strain-rate. Shear causes amalgamation of enstrophy and strain-rate structures, and suppresses the existence of strain-rate structures in low-vorticity regions between enstrophy structures. A primary effect of shear is to enhance “passive” strain-rate fluctuations, strain-rate kinematically induced by local vorticity concentrations with negligible enstrophy production, relative to “active,” or vorticity-generating strain-rate fluctuations. Enstrophy structures separate into “active” and “passive” based on the level of the second eigenvalue of fluctuating strain-rate. We embedded the structure-extraction algorithm into an interactive visualization-based analysis system from which the time evolution of a shear-induced hairpin enstrophy structure was visually and quantitatively analyzed. The structure originated in the initial isotropic state as a vortex sheet, evolved into a vortex tube during a transitional period, and developed into a well-defined horseshoe vortex in the shear-dominated asymptotic state.     

Micromechanical modeling of stress-induced phase transformations. Part 1. Thermodynamics and kinetics of coupled interface propagation and reorientation

The universal (i.e. independent of the constitutive equations) thermodynamic driving force for coherent interface reorientation during first-order phase transformations in solids is derived for small and finite strains. The derivation is performed for a representative volume with plane interfaces, homogeneous stresses and strains in phases and macroscopically homogeneous boundary conditions. Dissipation function for coupled interface (or multiple parallel interfaces) reorientation and propagation is derived for combined athermal and drag interface friction. The relation between the rates of single and multiple interface reorientation and propagation and the corresponding driving forces are derived using extremum principles of irreversible thermodynamics. They are used to derive complete system of equations for evolution of martensitic microstructure (consisting of austenite and a fine mixture of two martensitic variants) in a representative volume under complex thermomechanical loading. Viscous dissipation at the interface level introduces size dependence in the kinetic equation for the rate of volume fraction. General relationships for a representative volume with moving interfaces under piece-wise homogeneous boundary conditions are derived. It was found that the driving force for interface reorientation appears when macroscopically homogeneous stress or strain are prescribed, which corresponds to experiments. Boundary conditions are satisfied in an averaged way. In Part 2 of the paper [Levitas, V.I., Ozsoy, I.B., 2008. Micromechanical modeling of stress-induced phase transformations. Part 2. Computational algorithms and examples. Int. J. Plasticity (2008)], the developed theory is applied to the numerical modeling of the evolution of martensitic microstructure under three-dimensional thermomechanical loading during cubic-tetragonal and tetragonal-orthorhombic phase transformations.     

Weakly nonlocal envelope solitary waves: numerical calculations for the Klein-Gordon (φ) equation

“Weakly nonlocal” solitary waves differ from ordinary solitary waves by possessing small amplitude, oscillatory “wings” that extend indefinitely from the large amplitude “core”. Such generalized solitary waves have been discovered in capillarygravity water waves, particle physics models, and geophysical Rossby waves. In this work, we present explicit calculations of weakly nonlocal envelope solitary waves. Each is a sine wave modulated by a slowly-varying “envelope” that itself propagates at the group velocity. Our example is the cubically nonlinear Klein-Gordon equation, which is a model in particle physics (φ⁴ theory) and in electrical engineering (with a different sign). Both cases have weakly nonlocal“breather” solitons. Via the Lorentz invariance, each breather generates a one-parameter family of nonlocal envelope solitary waves. The φ⁴ breather was described and calculated in earlier work. This generates envelope solitons which have “wings” that are (mostly) proportional to the second harmonic of the sinusoidal factor. In this article, we calculate breathers and envelope solitary waves for the second, electrical engineering case. Since these, unlike the φ⁴ waves, contain only odd harmonics, the envelope solitary waves are nonlocal only via the third harmonic.     

A theoretical and numerical approach to the plastic behaviour of steels during phase transformations—II. Study of classical plasticity for ideal-plastic phases

The response of phase-transforming steels to variations of the applied stress (i.e. the ∑-term of the classical plastic strain rate Ė^cp defined in Part I) is studied both theoretically and numerically for ideal-plastic individual phases. It is found theoretically that though the stress-strain curve contains no elastic portion, it is nevertheless initially tangent to the elastic line with slope equal to Young's modulus. Moreover an explicit formula for the beginning of the curve is derived for medium or high proportions of the harder phase, and a simple upper bound is given for the ultimate stress (maximum Von Mises stress). The finite element simulation confirms and completes these results, especially concerning the ultimate stress whose discrepancy with the theoretical upper bound is found to be maximum for low proportions of the harder phase. Based on these results, a complete model is proposed for the ∑-term of the classical plastic strain rate Ė^cp in the case of ideal-plastic phases.     

An elastic-plastic analysis of an asymmetric cracked specimen subjected to longitudinal shear

Some recent elastic-plastic analyses of cracked specimens subjected to symmetric mode III loading are extended to include asymmetric loading and geometry. Solutions are given for arbitrary work hardening behaviour in any specimen that is amenable to a linear elastic analysis. It is shown that asymmetry has a major influence on the shape of the plastic zone, but does not affect the J-integral unil the loading is well into the large scale yielding range. In particular the “plastic zone corrected” estimate of J, obtained by elastically solving a problem for a crack longer than the actual one, is shown to remain a valid two-term asymptotic expansion in the presence of asymmetry. The general results are applied to a crack at an angle to a uniform stress field in a power law hardening material. The growth of the plastic zone is displayed graphically for various hardening exponents and crack orientations. No other asymmetric solution is available, but values of J are compared with those obtained from a fully plastic analysis in the symmetric case.     

An elastoplastic framework for granular materials becoming cohesive through mechanical densification. Part II – the formulation of elastoplastic coupling at large strain

The two key phenomena occurring in the process of ceramic powder compaction are the progressive gain in cohesion and the increase of elastic stiffness, both related to the development of plastic deformation. The latter effect is an example of ‘elastoplastic coupling’, in which the plastic flow affects the elastic properties of the material, and has been so far considered only within the framework of small strain assumption (mainly to describe elastic degradation in rock-like materials), so that it remains completely unexplored for large strain. Therefore, a new finite strain generalization of elastoplastic coupling theory is given to describe the mechanical behaviour of materials evolving from a granular to a dense state.The correct account of elastoplastic coupling and of the specific characteristics of materials evolving from a loose to a dense state (for instance, nonlinear – or linear – dependence of the elastic part of the deformation on the forming pressure in the granular – or dense – state) makes the use of existing large strain formulations awkward, if even possible. Therefore, first, we have resorted to a very general setting allowing general transformations between work-conjugate stress and strain measures; second, we have introduced the multiplicative decomposition of the deformation gradient and, third, employing isotropy and hyperelasticity of elastic response, we have obtained a relation between the Biot stress and its ‘total’ and ‘plastic’ work-conjugate strain measure. This is a key result, since it allows an immediate achievement of the rate elastoplastic constitutive equations. Knowing the general form of these equations, all the specific laws governing the behaviour of ceramic powders are finally introduced as generalizations of the small strain counterparts given in Part I of this paper.     

Constitutive equations for hot-working of metals

Elevated temperature deformation processing - “hot-working,” is an important step during the manufacturing of most metal products. Central to any successful analysis of a hot-working process is the use of appropriate rate and temperature-dependent constitutive equations for large, interrupted inelastic deformations, which can faithfully account for strain-hardening, the restoration processes of recovery and recrystallization and strain rate and temperature history effects. In this paper we develop a set of phenomenological, internal variable type constitutive equations describing the elevated temperature deformation of metals. We use a scalar and a symmetric, traceless, second-order tensor as internal variables which, in an average sense, represent an isotropic and an anisotropic resistance to plastic flow offered by the internal state of the material. In this theory, we consider small elastic stretches but large plastic deformations (within the limits of texturing) of isotropic materials. Special cases (within the constitutive framework developed here) which should be suitable for analyzing hot-working processes are indicated.     

Spring-back evaluation of automotive sheets based on isotropic-kinematic hardening laws and non-quadratic anisotropic yield functions: Part I: theory and formulation

In order to improve the prediction capability of spring-back in automotive sheet forming processes, the modified Chaboche type combined isotropic-kinematic hardening law was formulated based on the modified equivalent plastic work principle to account for the Bauschinger effect and transient behavior. As for the yield stress function, the non-quadratic anisotropic yield potential, Yld2000-2d, was utilized under the plane stress condition. Besides the theoretical aspect of the constitutive law including the general plastic work principle for monotonously proportional loading, the method to determine hardening parameters as well as numerical formulations to update stresses were developed based on the incremental deformation theory and the consistency requirement as summarized in Part I, while the characterization of material properties and verifications with experiments are discussed in Part II and III, respectively.