Size-effects on yield surfaces for micro reinforced composites

Size effects in heterogeneous materials are studied using a rate independent higher order strain gradient plasticity theory, where strain gradient effects are incorporated in the free energy of the material. Numerical studies are carried out using a finite element method, where the components of the plastic strain tensor appear as free variables in addition to the displacement variables. Non-conventional boundary conditions are applied at material interfaces to model a constraint on plastic flow due to dislocation blocking. Unit cell calculations are carried out under generalized plane strain conditions. The homogenized response of a material with cylindrical reinforcing fibers is analyzed for different values of the internal material length scale and homogenized yield surfaces are presented. While the main focus is on initial yield surfaces, subsequent yield surfaces are also presented. The center of the yield surface is tracked under uniaxial loading both in the transverse and longitudinal directions and an anisotropic Bauschinger effect is shown to depend on the size of the fibers. Results are compared to conventional predictions, and size-effects on the kinematic hardening are accentuated.     

A plasticity theory without drucker's postulate,suitable for granular materials

A plasticity theory is introduced which starts with a dilatancy rule and a function of plastic strain rates which represents the energy dissipated during plastic deformation. Yield surfaces and flow rules are then derived using energy conservation and the theory of envelopes. This method allows valid plasticity theories to be derived for frictional materials, but gives results for non-frictional materials which are identical to those of the classical theories.A dissipation function which includes deformation by granule rearrangement and granule distortion is presented and used to obtain a range of yield surfaces and flow rules, which are similar to those used in the critical state theory of soil mechanics. The microstructural features which may control the governing parameters of the dissipation functions are discussed.     

Mechanism-based strain gradient Drucker–Prager elastoplasticity for pressure-sensitive materials

Modeling strain gradient plasticity effects has achieved considerable success in recent years. However, incorporating the full mechanisms of the pressure-sensitive yielding and the size dependence of plastic deformation still remains an open challenge. In this work, a mechanism-based stain gradient (MSG) plasticity theory for pressure sensitive materials with a variable material length-scale parameter is presented. The flow theory of MSG plasticity based on the Drucker–Prager yield function is established following the same hierarchical framework of MSG plasticity proposed by Gao et al., 1999, Huang et al., 2000 and Qiu et al. (2003) in order to link the strain gradient plasticity theory on the mesoscale to the Taylor dislocation model on the micro-scale. The incremental constitutive relation based on the associated flow rule is derived for the Drucker–Prager yield function on the micro-scale, including the higher-order stress introduced as the thermodynamic conjugate of strain gradient at the mesoscale. The proposed theory successfully predicts the experimental results. The numerical results show that when the pressure-sensitivity index defined by the Drucker–Prager yield function takes different values, the material response curves are different and the material strength increases with the increase of pressure-sensitivity index. It proves that this procedure is able to represent the material behavior of pressure-sensitive materials such as geomaterials, polymeric materials, metallic foams and metallic glass.     

Mechanism-based strain gradient crystal plasticity—I. Theory

We have been developing the theory of mechanism-based strain gradient plasticity (MSG) to model size-dependent plastic deformation at micron and submicron length scales. The core idea has been to incorporate the concept of geometrically necessary dislocations into the continuum plastic constitutive laws via the Taylor hardening relation. Here we extend this effort to develop a mechanism-based strain gradient theory of crystal plasticity. In this theory, an effective density of geometrically necessary dislocations for a specific slip plane is introduced via a continuum analog of the Peach-Koehler force in dislocation theory and is incorporated into the plastic constitutive laws via the Taylor relation.     

A study on subsequent yield surface based on the distributed-element model

This study considers extension of the distributed-element model to account for the deformation-induced anisotropy demonstrated in subsequent yield surfaces. By modifying the shape parameter of the strength-distribution function in the model as a function of the accumulated plastic deformation and the preloading direction, the DEM is able to account appropriately for major features of the subsequent yield surfaces demonstrated by real materials under proportional loading. The proposed modeling technique is confirmed by performing numerical simulations and comparing with experimental results available in the literature.     

Suitability of yield functions for the approximation of subsequent yield surfaces

Many consitutive models in the plasticity of metals are based on the existence of a yield function, which is not only used to mark the elastic limit but also as a potential function for the plastic strain rate. The construction of this function therefore deserves the utmost interest. Measurements of the elastic limit show the essential features of how the geometry of the yield function contour lines should change with further straining. Against these typical geometric forms of the yield surface existing proposals for invariant formulations of the yield function taking into account isotropic, kinematic and formative hardening are tested. Even if no evolution equations for the constitutive variables contained in the yield functions are specified, best approximations of measured yield surfaces can be computed by optimisation of a quality function. It can be shown that most of the representations are not even able to describe the experimental results qualitatively. The numeric results show further that the yield function is essentially of grade three in the deviatoric stresses. The evolution of internal variables can be deduced from the approximations of the measurements.     

The bimodal theory of plasticity: A form-invariant generalisation

The bimodal plasticity model of fibre-reinforced materials is currently available and applicable only in association with thin-walled fibrous composites containing a family of straight fibres which are conveniently assumed parallel with the x₁-axis of an appropriately chosen Cartesian co-ordinate system. Based on reliable experimental evidence, the model suggests that plastic slip in the composite operates in two distinct modes; the so-called matrix dominated mode (MDM) which depends on a matrix yield stress, and the fibre dominated mode (FDM) which depends also on the fibre yield stress. Each mode is activated by different states of applied stress, has its own yield surface (or surfaces) in the stress space and has its own segment on the overall yield surface of the composite. This paper employs theory of tensor representations and produces a form-invariant generalisation of both modes of the model. This generalisation furnishes the model with direct applicability to relevant plasticity problems, regardless of the shape of the fibres or the orientation of the co-ordinate system. It thus provides a proper mathematical foundation that underpins important physical concepts associated with the model while it also elucidates several technical relevant issues. A most interesting of those issues is the revelation that activation of the MDM plastic regime is possible only if the applied stress state allows the fibres to act like they are practically inextensible. Moreover, activation of the more dominant, between the two MDM plastic slip branches is possible only if conditions of material incompressibility hold, in addition to the implied condition of fibre inextensibility. A direct mathematical connection is thus achieved between basic, experimentally verified concepts of the bimodal plasticity model and a relevant mathematical model originated earlier from the theory of ideal fibre-reinforced materials. An additional issue of discussion involves the number of independent yield stress parameters that the bimodal theory needs to take into consideration. Moreover, an analytical expression is provided of a relatively simple mathematical surface that possesses all known features of the FDM yield surface; currently captured with the aid of both experimental and computational means. The present study is guided by the existing relevant experimental evidence which, however, is principally associated with the plastic behaviour of solids reinforced by strong fibres. Nevertheless, several of the outlined developments are expected to be applicable to composite materials containing a single family of more compliant or even weak fibres.     

Research note on a reexamination of neutral loading experiments

In the examination of the published results from neutral loading experiments, the question as to whether plastic deformation occurs is found to depend on both the material and initial loading strain. Provided that initial loading is elastic, then a subsequent stress path that follows the boundary of the initial yield surface for a hardening material is truly neutral with a wholly elastic response. However, when initial loading is elastic-plastic, then further plastic deformation is produced from a subsequent stress path that follows an isotropic expansion of the initial yield surface. These results enable the appropriateness of the kinematic hardening rule and more recent developments in plasticity theory to be appraised. Neutral loading of a non-hardening material produces plastic flow. Whether the absence of hardening is inherent or induced by plastic prestrain, it is shown that the Prandtl-Reuss theory then represents the observed behavior. In general, the purely elastic and nonhardening solutions provide respectively lower and upper bounds on the deformation.     

An endochronic theory accounted for deformation induced anisotropy

An anisotropic quadratic form of plastic strain increment is used to define the intrinsic time in the endochronic theory of plasticity. Based on this new definition, a yield function can be derived. This new version of endochronic theory can describe the expansion, translation, rotation, and distortion of the yield surface. While the initial yielding is in the form of the Mises yield criterion, the distortion of subsequent yield surfaces is expressed by the compression or stretching of the Mises yield surface. The effect of sharp front and blunt rear of the yield surface is considered to be of secondary importance and neglected in the interest of keeping the equations simple. This idealization will not much affect the prediction power of the model, because the plastic strain increment is in the radial direction emanating from the center of the current yield surface and is not normal to the current yield surface. In this theory, the plastic deformation is thus not sensitive to the exact shape of the yield surface. It has been shown that the proposed theory is capable of describing the experimental results of three different metals considered. The test series investigated include several different paths of prestress.     

Mechanism-based strain gradient crystal plasticity—II. Analysis

In part I of this series (Mechanism-based strain gradient crystal plasticity—I. Theory. J. Mech. Phys. Sol. (2005), accepted for publication), we have proposed a theory of mechanism-based strain gradient crystal plasticity (MSG-CP) to model the effect of inherent anisotropy of a crystal lattice on size-dependent non-uniform plastic deformation at micron and submicron length scales. In the present paper, several example problems are investigated to show how crystal anisotropy is reflected by the MSG-CP theory.     

An analysis of exhaustion hardening in micron-scale plasticity

The rate-dependent behavior of micron-scale model planar crystals is investigated using the framework of mechanism-based discrete dislocation plasticity. Long-range interactions between dislocations are accounted for through elasticity. Mechanism-based constitutive rules are used to represent the short-range interactions between dislocations, including dislocation multiplication and dislocation escape at free surfaces. Emphasis is laid on circumstances where the deformed samples are not statistically homogeneous. The calculations show that dimensional constraints selectively set the operating dislocation mechanisms, thus giving rise to the phenomenon of exhaustion hardening whereby the applied strain rate is predominantly accommodated by elastic deformation. When conditions are met for this type of hardening to take place, the calculations reproduce some interesting qualitative features of plastic deformation in microcrystals, such as flow intermittency over coarse time-scales and large values of the flow stress with no significant accumulation of dislocation density. In addition, the applied strain rate is varied down to 0.1 s⁻¹ and is found to affect the rate of exhaustion hardening.     

Advances in experiments on metal sheets and tubes in support of constitutive modeling and forming simulations

This work is a review of experimental methods for observing and modeling the anisotropic plastic behavior of metal sheets and tubes under a variety of loading paths, such as biaxial compression tests; biaxial tension tests on metal sheets and tubes using closed-loop electrohydraulic testing machines; the abrupt strain path change method for detecting a yield vertex and subsequent yield loci without unloading; in-plane stress reversal tests on metal sheets; and multistage tension tests. Observed material responses are compared with the predictions of phenomenological plasticity models. Special attention is paid to the plastic deformation behavior of materials commonly used in industry, and to verifying the validity of conventional anisotropic yield criteria for those materials and associated flow rules at large plastic strains. The effects of using appropriate anisotropic yield criteria on the accuracy of simulations of forming defects, such as large springback and fracture, are also presented to highlight the importance of accurate material testing and modeling.     

Directional distortional hardening in metal plasticity within thermodynamics

This paper presents a complete theory for metal plasticity that includes isotropic, kinematic, and directional distortional hardening, within the framework of thermodynamics. Directional distortion is defined here as the formation of a region of high curvature on the yield surface, approximately in the direction of loading, and a region of flattening approximately in the opposite direction, as observed in experiments on various types of metals. The distinguishing features of this theory are the introduction of a fourth order tensor-valued internal variable, whose evolution in conjunction with a directional scalar multiplier describes the evolving directional distortion, and the fact that the hardening laws for all internal variables are derived on the basis of sufficient conditions to satisfy the thermodynamic requirement of positive dissipation. The applicability of the theory is illustrated by fitting experimental data on distorted yield surfaces in the course of plastic deformation.     

An endochronic theory accounting for deformation induced anisotropy of metals under biaxial load

Using the experimental results of yield surfaces obtained by Wu and Yeh [1991] (Int. J. Plasticity, 7, 803) for 304 stainless steel, this work provides a verification of the endochronic theory of plasticity accounting for deformation induced anisotropy. The experiments were performed under proportional loading conditions. The main difference between this paper and other papers that attempt to describe the distortion of a yield surface is that, in addition to distortion, motion of yield surface (kinematic hardening) has also been addressed by this paper. The result has shown that the theory predicts the experimental data with substantial accuracy. However, since in this theory the plastic strain increment, although normal to the initial yield surface, is in the radial direction emanating from the center of the subsequent yield surface, validity of the present model must be further studied for the case involving nonproportional loading conditions.     

On the homogenization of metal matrix composites using strain gradient plasticity

The homogenized response of metal matrix composites（MMC） is studied using strain gradient plasticity.The material model employed is a rate independent formulation of energetic strain gradient plasticity at the micro scale and conventional rate independent plasticity at the macro scale. Free energy inside the micro structure is included due to the elastic strains and plastic strain gradients. A unit cell containing a circular elastic fiber is analyzed under macroscopic simple shear in addition to transverse and longitudinal loading. The analyses are carried out under generalized plane strain condition. Micro-macro homogenization is performed observing the Hill-Mandel energy condition,and overall loading is considered such that the homogenized higher order terms vanish. The results highlight the intrinsic size-effects as well as the effect of fiber volume fraction on the overall response curves, plastic strain distributions and homogenized yield surfaces under different loading conditions. It is concluded that composites with smaller reinforcement size have larger initial yield surfaces and furthermore,they exhibit more kinematic hardening.     

Shakedown theory for elastic plastic kinematic hardening bodies

Shakedown static and kinematic theorems for elastic–plastic (generally nonlinear) kinematic hardening solids are derived in classical (path-independence) spirit with new constructions. The generally plastic-deformation-history-dependent hardening curve is assumed to be limited by the initial yield stress and ultimate yield strength, and to obey a positive hysteresis postulate for closed plastic cycles, but else can be arbitrary and unspecified. The theorems reveal that the shakedown of structures is not affected by the particular form of the hardening curve, but just by the initial and ultimate yield stresses. While the ultimate yield strength is clearly defined macroscopically and attached to the incremental collapse mode with unbounded plastic deformations, the initial yield stress, which is responsible for the bounded cyclic plasticity collapse mode, should not be taken as the convenient one at a fixed amount of plastic deformation (0.2%), but is suggested to be taken as low as the fatigue limit to preserve the classical load-history-independence spirit of the shakedown theorems. Otherwise, for our pragmatic application purpose, it may be given empirical values between the low fatigue limit and high ultimate yield stresses according to particular loading processes considered, which may range anywhere between the high-cycle and low-cycle ones. The theorems appear as simple as those of Melan and Koiter for perfect plasticity but applied to the much larger class of more realistic kinematic hardening materials.     

Viscoplasticity with a saturation stress: distinguishing features of the model

Viscoplastic models including a saturation stress are considered. The existence of the saturation stress significantly changes the mathematical structure of solutions near maximum friction surfaces (surfaces where the friction stress is equal to the local shear yield stress). The main features of solutions based on such theories are: (a) sliding must occur at the maximum friction surfaces under certain conditions, (b) the velocity field may be singular in the vicinity of maximum friction surfaces. The objective of the present paper is to study these features of solutions. The mathematical structure obtained is considered to be advantageous for a class of materials and may lead to a convergence of viscoplastic solutions to the corresponding rigid perfectly plastic solutions. It seems that the latter is of importance for the construction of a unified theory that could describe the material behavior in the range from rate-independent plasticity to viscoplasticity. In the present paper, the study of the main features of the model is based on the exact closed-form solution to the problem of flow between two coaxial rotating cylinders. In the case of sliding, in addition to the aforementioned features, the asymptotic behavior of the velocity field in the vicinity of the maximum friction surface is found for a class of constitutive laws.     

Mechanism-based strain gradient plasticity in C0 axisymmetric element

Non-uniform plastic deformation of materials exhibits a strong size dependence when the material and deformation length scales are of the same order at micro- and nano-metre levels. Recent progresses in testing equipment and computational facilities enhancing further the study on material characterization at these levels confirmed the size effect phenomenon. It has been shown that at this length scale, the material constitutive condition involves not only the state of strain but also the strain gradient plasticity. In this study, C⁰ axisymmetric element incorporating the mechanism-based strain gradient plasticity is developed. Classical continuum plasticity approach taking into consideration Taylor dislocation model is adopted. As the length scale and strain gradient affect only the constitutive relation, it is unnecessary to introduce either additional model variables or higher order stress components. This results in the ease and convenience in the implementation. Additional computational efforts and resources required of the proposed approach as compared with conventional finite element analyses are minimal. Numerical results on indentation tests at micron and submicron levels confirm the necessity of including the mechanism-based strain gradient plasticity with appropriate inherent material length scale. It is also interesting to note that the material is hardened under Berkovich compared to conical indenters when plastic strain gradient is considered but softened otherwise.