Macroscopic Yield Surface Research Articles

Establishing the relationship between generalized crystallographic texture and macroscopic yield surfaces using partial input convex neural networks

In this study, we present a methodology to predict the macroscopic yield surface of metals and metallic alloys with general crystallographic textures. In previous work, we have established the use of partially input convex neural networks (pICNN) as macroscopic yield functions of crystal plasticity simulations. However, this work was performed with an over-abundance of data, and on limited crystallographic textures. Here, we extend this study to approach more realistic material states (i.e., complex crystallographic textures), and consider data-availability as a major driver for our approach. We present our modified framework capable of handling generalized material states and demonstrate its effectiveness on samples with multi-modal textures deformed under plane stress conditions. We further describe an adaptive algorithm for the generation of training data as informed by the shape of yield surfaces to reduce the time for both the generation of training data as well as pICNN training. Finally, we will discuss errors in both training and test datasets, limitations, and future extensibility.

Yield criteria for voided materials with anisotropic matrix behavior

This paper presents a theoretical and micromechanical procedure for determining the macroscopic behavior of voided materials with anisotropic matrix constitutive behavior. Within the Gurson approach (Gurson, 1977), it combines the IPE (Isotropy Plasticity Equivalent) concept proposed by Karafillis and Boyce (1993) to the procedure proposed by Benallal (2017) to build macroscopic behavior of voided materials with general isotropic matrix behavior. For sake of clarity, the simple Gurson kinematically admissible velocity field is used and can be replaced by any more suitable trial field. The matrix may have any anisotropic behavior represented by the linear transformation of the IPE setting, which is traceless. A general parametric constitutive relation is obtained for the six-dimensional yield surface of the porous solid. A numerical algorithm is then proposed to evaluate and visualize arbitrary sections of this full six-dimensional macroscopic yield surface and various illustrations are provided for different matrix plastic constitutive behavior including Lode angle effects and materials symmetries. Some qualitative aspects of the macroscopic yield locus are summarized and discussed.

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method

Evaluation of macroscopic yield surfaces of periodic porous microstructures employing transformation field analysis and high order finite element method

The in-plane elastic-plastic response of an incompressible, filled hexagonal honeycomb

Exact solutions are derived for the small-strain, in-plane, elasto-plastic response of a hexagonal honeycomb using slender beam theory; incompressibility of the honeycomb is enforced by filling its voids with an incompressible, inviscid fluid. The honeycomb has sides of equal length, but its inclined struts subtend an angle that can deviate from 120° with respect to the vertical side walls. The relative density is sufficiently small that the struts are slender and can be treated as Euler-Bernoulli beams. Exact solutions are obtained for the elastic moduli and macroscopic yield surface of the rigid, ideally plastic lattice under general in-plane loading: the solutions satisfy equilibrium, compatibility and the constitutive response of each elastic, ideally plastic beam. Prior to conducting an elastic analysis, and a rigid, ideally plastic analysis, initial insight is gained by exploring the vector space of inextensional collapse mechanisms of the pin-jointed, compressible version of the hexagonal truss. Two inextensional collapse mechanisms of the compressible honeycomb are identified from the null space of the kinematic matrix. The presence of an incompressible, inviscid fluid in the voids of the honeycomb locks-up one mechanism but the other mechanism survives and generates macroscopic shear strain. Consequently, the incompressible hexagonal honeycomb with rigid joints has a high shear compliance and a low shear strength, with values equal to that of the unfilled, compressible honeycomb. In contrast, macroscopic tensile straining of the incompressible honeycomb requires the stretching of bars in addition to bar-bending, and the tensile modulus and strength of the incompressible honeycomb are thereby elevated. Explicit analytical formulae are derived for the macroscopic tensile modulus and strength of the incompressible honeycomb.

Advanced Geomechanical Model to Predict the Impact of CO2-Induced Microstructural Alterations on the Cohesive-Frictional Behavior of Mt. Simon Sandstone

We investigated the influence of CO2-induced geochemical reactions on the cohesive-frictional properties of host rock within the context of CO2 storage in a saline aquifer and focused on the Mt. Simon sandstone. The research objective was to model geo-mechanical changes due to host rock exposure to CO2-saturated brine while accounting for heterogeneity, double-scale porosity, and granular structure. We formulated a three-level multi-scale model for host rocks. We conducted scanning electron microscopy analyses to probe the microstructure and grid nanoindentation to measure the mechanical response. We derived new nonlinear strength upscaling solutions to correlate the effective strength characteristics and the macroscopic yield surface to the micro-structure at the nano-, micro-, and meso-scales. Specifically, our theoretical model links CO2-induced microstructural alterations to a reduction in the size of the yield surface, and a drop in the value of the friction coefficient. In turn, regarding the Illinois Basin Decatur Project, the CO2-induced drop in friction coefficient is linked to an increase in the risk of fault slip and a higher probability of induced microseismicity during and after the end of CO2 underground injection operations. The theoretical model presented is essential for the geo-mechanical modeling of CO2 underground injection operations at multiple length-scales.

Effective behaviour of porous ductile solids with a non-quadratic isotropic matrix yield surface

In this study, we examine the macroscopic yielding of isotropic porous ductile solids having a matrix yield function dependent on the second and third deviatoric stress invariants. Numerical limit analyses using a three-dimensional finite element model of a hollow sphere with a Hershey-Hosford matrix yield function are conducted for different shapes of the matrix yield surface and porosity levels. These numerical results are then used to elucidate first-order effects of the third deviatoric stress invariant on the macroscopic yielding and further used as reference data to assess the performance of two porous plasticity models that incorporate effects of the third deviatoric stress invariant using the isotropic non-quadratic Hershey-Hosford yield function. The first model is derived from an upper-bound limit analysis of the hollow sphere representative volume element using the Gurson-Rice trial velocity field, but with a rather general isotropic matrix yield function. The second model is a simple, heuristic extension of the Gurson model incorporating the equivalent stress measure of the Hershey-Hosford yield function.From the numerical limit analyses, it is found that the contours of the macroscopic yield surface in the deviatoric plane transform from the hexagonal shape of the underlying matrix yield surface to a rounded triangular shape that converges to the circular shape of the Gurson model as the macroscopic stress triaxiality ratio increases. This shape transformation is dependent upon the porosity level. The upper-bound model was found to be in very good agreement with the numerical data for all stress states, shapes of the matrix yield surface, and porosity levels. The heuristic model provides good predictions for low and moderate levels of porosity pertinent to ductile fracture, but the predictions deteriorate when the stress triaxiality ratio and the porosity level increase.We also address the issue of how representative the spherical unit cell is for the description of real porous solids. To that end, we make comparisons between a space-filling representative volume element in the form of a cubic unit cell with a centric spherical void and the hollow sphere model. These results show that the hollow sphere model generally provides slightly higher values for the yield limits. The shape of the yield loci is similar for the two models in the case of non-quadratic matrix yield surfaces, while the cubic model gives a different shape of the yield loci for low and intermediate stress triaxiality ratios when the von Mises yield function is used for the matrix.

Construction of macroscale yield surfaces for ductile composites based on a virtual testing strategy

The paper describes an approach, based on computational homogenization, for constructing the macroscopic yield surface pertinent to a meso-heterogeneous ductile composite, whereby it is assumed that each constituent is elastic-plastic with (possible) hardening; however, the strategy does not depend on the particular choice of plasticity model (standard, crystal) on the mesoscale. The aim is to compute the macroscale yield properties with bounds that are defined with a priori given confidence, in terms of an “outer yield surface” and an “inner yield surface”. However, these bounds will inevitably be approximate in practice due to the difficulty to achieve guaranteed bounds for general incremental plasticity.The following ingredients are essential to the approach: (1) A “virtual testing” strategy is proposed, whereby a sufficiently large number of realizations of the presumed random meso-structure, determined by the chosen probability, are utilized as Statistical Volume Elements (SVE). (2) In order to obtain the pertinent bounds on the “directional yield stress”, SVE-computations are carried out with Dirichlet and Neumann boundary conditions. (3) A stress-driven, rather than the more traditional strain-driven, format of the SVE-problem is adopted in order to cover the macroscopic stress space in a systematic fashion.Illustrative numerical examples in 2D (particle composites) and 3D (polycrystals of Duplex Stainless Steel) conclude the paper.

On the effect of grains interface parameters on the macroscopic properties of polycrystalline materials

In this paper, the influence of microscopic parameters on the macroscopic behaviour of polycrystalline materials under different loading configuration is investigated. Linear elastic grains with zero-thickness cohesive interfaces are considered at the microscale with in depth introduction of effective parameters. A multiscale method based on homogenisation technique is employed to bridge the scales. In order to minimize the homogenisation error, a representative volume element (RVE) of the microscopic structure is statistically determined to be used in the numerical analysis. For each loading condition of the RVE, several numerical examinations are conducted to illustrate the relationship between the microscopic parameters. Finally, the effects of microscopic critical fracture energies, maximum tensile and shear strengths of grain interfaces on the mechanical properties, i.e. stress-strain curve and yield surface at the macroscale are discussed in details. It is shown that macroscopic yield surface and stress-strain curves can be used to characterise the microscopic properties.

Stiffness and yield strength of architectured foams based on the Schwarz Primitive triply periodic minimal surface

Architectured foams based on the mathematically-known non-self-intersecting Schwarz Primitive triply periodic minimal surfaces (TPMS) might have superior mechanical properties as compared to stochastic or lattice foams. In this paper, the elastic properties (uniaxial, shear and bulk moduli, Zener anisotropy index, and Poisson's ratio) and yield strengths under several different stress states of the Schwarz Primitive TPMS sheet-based foam (referred to as P-foam) is investigated by finite element analyses of unit cells under periodic boundary conditions. The P-foam has a cubic symmetry with three independent elastic constants and yield strengths and is interconnected and intertwined in the three-dimensional space. The relative density (i.e., the density of the foam with respect to the density of the base solid material) of the P-foam can be varied by changing the thickness of the Schwarz Primitive TPMS-based sheet solid. It is found that the shear modulus and shear yield strength as well as the bulk modulus and hydrostatic yield strength scale linearly with the P-foam relative density indicating a stretching-dominated behavior. On the other hand, the uniaxial modulus and yield strength scale with a power law indicating a coupled stretching- and bending-dominated deformation. The macroscopic yield surface of the P-foam for various stress states is adequately and equally described by the extension of Hill's anisotropic yield function.

Nonlinear homogenisation of trabecular bone: Effect of solid phase constitutive model.

Micro-finite element models have been extensively employed to evaluate the elastic properties of trabecular bone and, to a limited extent, its yield behaviour. The macroscopic stiffness tensor and yield surface are of special interest since they are essential in the prediction of bone strength and stability of implants at the whole bone level. While macroscopic elastic properties are now well understood, yield and post-yield properties are not. The aim of this study is to shed some light on what the effect of the solid phase yield criterion is on the macroscopic yield of trabecular bone for samples with different microstructure. Three samples with very different density were subjected to a large set of apparent load cases (which is important since physiological loading is complex and can have multiple components in stress or strain space) with two different solid phase yield criteria: Drucker-Prager and eccentric-ellipsoid. The study found that these two criteria led to small differences in the macroscopic yield strains for most load cases except for those that were compression-dominated; in these load cases, the yield strains for the Drucker-Prager criterion were significantly higher. Higher density samples resulted in higher differences between the two criteria. This work provides a comprehensive assessment of the effect of two different solid phase yield criteria on the macroscopic yield strains of trabecular bone, for a wide range of load cases, and for samples with different morphology.

Capturing yield surface evolution with a multilinear anisotropic kinematic hardening model

Many authors have observed experimentally that the macroscopic yield surface changes substantially its shape during plastic flow, specially in metals which suffer significant work hardening. The evolution is frequently characterized by a corner effect in the stress direction of loading, and a flatter shape in the opposite direction. In order to incorporate this effect many constitutive models for yield surface evolution have been proposed in the literature. In this work we perform some numerical predictions for experiments similar to the ones performed in the literature using a multilayer kinematic hardening model which employs the associative Prager’s translation rule. Using this model we prescribe offsets of probing plastic strain, so apparent yield surfaces can be determined in a similar way as it is performed in the actual experiments. We show that similar shapes to those reported in experiments are obtained. From the simulations we can conclude that a relevant part of the apparent yield surface evolution may be related to the anisotropic kinematic hardening field.

Yield surfaces of heterogeneous media with debonded inclusions

Purpose – The purpose of this paper is to present knowledge in estimating yield surfaces of heterogeneous media by use of homogenization, especially where the macroscopic behaviour is driven by weak interfaces between phase constituents. Design/methodology/approach – A computational homogenization procedure is used to determine the yield surface of a Representative Volume Element (RVE) that contains a fully debonded inclusion embedded within ideally plastic matrix, whereby the interface is modelled by a Coulomb type friction law. Findings – The macroscopic behaviour of the RVE is shown to coincide an RVE with a hole for expanding loads, whereas for compressive loads, it was shown to approach an RVE with a fully bonded inclusion. Originality/value – The present paper builds on Gurson’s work in estimating macroscopic yield surfaces of heterogeneous materials. The work is novel in the sense that there had been no previous publications discussing influence of weak interfaces on yield surfaces.

Macroscopic multiaxial yield and failure surfaces for light closed-cell foams

This article deals with the development of a multiaxial “plasticity” criterion for the simulation of the macroscopic behavior of light closed-cell foams. The paper proposes a numerical methodology for the determination of the mechanical macroscopic properties given the geometry of the cells (whose walls are assumed to be thin) and the properties of the bulk material. The focus is on foams with relative densities ranging from 0.0526 to 0.1525. The bulk material is assumed to be isotropic with perfectly plastic behavior. We consider tetrakaidecahedron cells having three planes of symmetry in three orthogonal directions. As expected for a material having a cubic symmetry, one observes that the elastic homogenized material is fully described by three parameters (E∗,ν∗,G∗) and not only two as for the isotropic bulk material (E,ν). We propose a formula which, starting from only a knowledge of the two elastic properties and density of the bulk material, leads to the three macroscopic elastic properties of the thin-wall regular tetrakaidecahedral foam. Their yield and failure strengths in tension, compression and shear are also calculated. Two types of macroscopic behavior can be clearly identified.The first type characterizes foams made of a high-yield-strength bulk material: the foam’s macroscopic behavior is determined mainly by buckling of the cell walls.The second type applies when the bulk material has a low yield strength. In that case, yielding of the cell walls is the main failure mechanism. A loading surface of the Wang and Pan form is proposed for all these cases, both in the early nonlinear stage and at ultimate failure. Lastly the effect of irregularity of the geometries on the load surface is presented for the two previous extreme cases. It is observed that these irregularities play a crucial role in tensile case for the lightest foams having the highest yield strength.It is interesting to note that such a loading surface also leads to a good approximation of the “yield” strength obtained experimentally for both open-cell foams Combaz et al. (2010, 2011) and closed-cell foams Wang and Pan (2006).

Computational assessment of the microstructure-dependent plasticity of lamellar gray cast iron – Part II: initial yield surfaces and directions

In this paper, the initial multiaxial yield behavior of three different gray cast iron materials with lamellar shaped graphite inclusions is numerically investigated by means of the finite-element method. Therefore, volume elements including the real microstructure of the materials are loaded bi- and triaxially beyond macroscopic yield. The shape of the obtained yield surfaces are compared to the surfaces of four continuum models which, amongst others, are proposed in literature to describe the inelastic behavior of gray cast iron with lamellar shaped graphite inclusions. It is found that the presented continuum models and the macroscopic yield surfaces obtained with microstructure-based finite-element models deviate. Furthermore, the initial inelastic flow direction is computed at the onset of macroscopic yielding. The analysis show that the inelastic flow is normal to the yield surface.

Generalized distortional hardening in a thermomechanically coupled framework

AbstractIn many metals like aluminum the evolution of the microstructure leads to an evolution of the macroscopic yield surface. Clearly, isotropic and kinematic hardening models cannot capture this effect realistically. For that purpose, a new generalized distortional hardening model is proposed. This novel approach belongs to the class of so‐called generalized standard materials and is thermodynamically consistent. Although the approach is relatively simple, it shows several advantages like yield surface convexity and yield limit saturation. This novel model is embedded into a thermomechanically coupled finite strain framework. Several numerical simulations show the predictive capabilities. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The stiffness and strength of the gyroid lattice

Recently, a nanoscale lattice material, based upon the gyroid topology has been self-assembled by phase separation techniques (Scherer et al., 2012) and prototyped in thin film applications. The mechanical properties of the gyroid are reported here. It is a cubic lattice, with a connectivity of three struts per joint, and is bending-dominated in its elasto-plastic response to all loading states except for hydrostatic: under a hydrostatic stress it exhibits stretching-dominated behaviour. The three independent elastic constants of the lattice are determined through a unit cell analysis using the finite element method; it is found that the elastic and shear modulus scale quadratically with the relative density of the lattice, whereas the bulk modulus scales linearly. The plastic collapse response of a rigid, ideally plastic gyroid lattice is explored using the upper bound method, and is validated by finite element calculations for an elastic-ideally plastic lattice. The effect of geometrical imperfections, in the form of random perturbations to the joint positions, is investigated for both stiffness and strength. It is demonstrated that the hydrostatic modulus and strength are imperfection sensitive, in contrast to the deviatoric response. The macroscopic yield surface of the imperfect lattice is adequately described by a modified version of Hill’s anisotropic yield criterion. The article ends with a case study on the stress induced within a gyroid thin film, when the film and its substrate are subjected to a thermal expansion mismatch.

Numerical assessment of the macroscopic strength criterion of reinforced soils using semidefinite programming

SUMMARYThe macroscopic strength properties of reinforced soils, regarded as periodic composite materials, are investigated by means of a FEM‐based formulation of both static and kinematic approaches of yield design applied to the reinforced soil's unit cell. Because the reinforced soil's individual constituents obey a 3D Mohr–Coulomb strength condition, such a numerical problem can be treated through an optimization procedure using semidefinite programming. The whole numerical procedure is applied to the derivation of both lower bound and upper bound estimates to the macroscopic yield surface of a soil reinforced either by columnar inclusions (stone columns) or a double array of trenches (cross trench reinforcement). The so‐obtained results highlight the efficiency of the proposed numerical method. Copyright © 2014 John Wiley & Sons, Ltd.

Thermodynamically consistent modeling of yield surface distortion

AbstractFrom a macroscopic point of view, the deformation of most metals results in an evolution of the symmetry groups characterizing the isotropy of the considered materials. With respect to plastic deformation for instance, the shape of the macroscopic yield surface evolves during deformation. In the present paper, a novel constitutive framework capturing this evolution is proposed. This framework is based on the fundamentals of thermodynamics. Furthermore, it also shows a variational structure such that all state variables follow jointly from minimizing an incrementally defined energy functional. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical homogenization of nonlinear composites with a polarization-based FFT iterative scheme

This paper presents an extension of the polarization based iterative scheme in the context of nonlinear composites. The polarization based algorithm, used for computing the effective properties of composite materials, converges independently of the value of the contrast between the phases. However the application of this scheme in the initial form is restricted to the case of linear elasticity. In this paper we propose some modifications of this iterative scheme in order to deal with the most general case of nonlinear constituents. Two applications are proposed to show the capability of the method. The first concerns a composite made up of a power-law type nonlinear matrix with rigid inclusions while in the second illustration we deals with the determination of the macroscopic yield surface of porous plastic materials.

A new macroscopically anisotropic pressure dependent yield function for metal matrix composite based on strain gradient plasticity for the microstructure

Metal matrix composites with long aligned elastic fibers are studied using an energetic rate independent strain gradient plasticity theory with an isotropic pressure independent yield function at the microscale. The material response is homogenized to obtain a conventional macroscopic model that exhibits anisotropic yield properties with a pressure dependence. At the microscale free energy includes both elastic strains and plastic strain gradients, and the theory demands higher order boundary conditions in terms of plastic strain or work conjugate higher order tractions. The mechanical response is investigated numerically using a unit cell model with periodic boundary conditions containing a single fiber deformed under generalized plane strain conditions. The homogenized response can be modeled by conventional plasticity with an anisotropic yield surface and a free energy depending on plastic strain in addition to the elastic strain. Hill's classical anisotropic yield criterion is extended to cover the composite such that hydrostatic pressure dependency, Bauschinger stress and size-effects are considered. It is found that depending on the fiber volume fraction, the anisotropic yield surface of the composite is inclined compared to a standard pressure independent yield surfaces. The evolution of the macroscopic yield surface is investigated by quantifying both anisotropic hardening (expansion) and kinematic hardening (translation), where the coefficients of anisotropy and the Bauschinger stress are extracted.