Orthotropic Plasticity Research Articles

A covariant formulation for finite strain modelling of orthotropic elasticity and orthotropic plasticity with plasticity-induced evolution of orthotropy: Application to natural fibres

We introduce a rate-independent model for orthotropic elastic and orthotropic plastic material behaviour in a hyper-elasto-plastic framework at finite strains. The model is based on the postulate of covariance and does not rely on a multiplicative decomposition of the deformation gradient. Furthermore, a plastic-deformation-induced evolution of orthotropy is considered, similar to the notion of plastic spin. We propose that the orthotropic strain energy function and the orthotropic yield criterion are guided by identical structural tensors that evolve with plasticity. The modelled material behaviour is significant for natural fibres such as flax, hemp, or pulp fibres. Our formulation has three findings. Firstly, the covariant formulation of plasticity provides rate equations for the plastic variables and the structural tensors suitable for reproducing stress–strain diagrams of natural fibres. Secondly, the introduction of plastic-deformation-induced evolution of orthotropy in the proposed covariant setting results in a non-associative plasticity algorithm. Thirdly, the covariant setting allows the incorporation of suitable constitutive equations for the structural tensors to evolve orthotropy. The latter successfully models the stiffness increase in stress–strain diagrams of cyclic tensile tests of natural fibres.

Evaluation of ductile fracture criteria in combination with a homogenous polynomial yield function for edge splitting damage of DP steels

Abstract This study investigates the formability characteristics of dual phase steels (DP600 and DP800) under flange stretching conditions through hole expansion tests. The hole-splitting initiation was numerically predicted using ductile damage functions coupled with an orthotropic plasticity model. Therefore, a polynomial yield criterion coupled with three damage criteria, namely generalized plastic work, void growth model, and a shear ductile fracture model, is implemented into Marc software by the user defined material subroutine. Thus, the fracture stroke, hole expansion ratio, and fracture initiation location were estimated for both steels. The polynomial yield criterion could capture the anisotropic features of the dual phase steels. Furthermore, the stress triaxiality-based criteria were reasonably accurate in stretching limit predictions of both steels’ grades. Nevertheless, plastic work predicted the fracture strokes and hole expansion ratios noticeably lower than the experimental outcomes for both steels. In addition, all the numerical results captured the exact fracture initiation location for DP600, while a slight discrepancy was observed for DP800. All ductile fracture models pointed out the identical crack location, which shows the cruciality of the yield criterion for locating the fracture initiation in hole expansion test. Consequently, both void growth model and shear ductile fracture model showed accurate performances conforming to the stress triaxiality was found to be more dominant than the Lode parameter.

An orthotropic plasticity model at finite strains with plasticity-induced evolution of orthotropy based on a covariant formulation

We present a rate-independent model for isotropic elastic–orthotropic plastic material behaviour in a hyper-elasto-plastic setting at finite strains, which is based on a covariant formulation that includes plastic-deformation-induced evolution of orthotropy. The model relies on a treatment by Lu and Papadopoulos, who made use of the postulate of covariance for an anisotropic elasto-plastic solid and derived constitutive equations of evolving anisotropies at finite strains. The latter is tantamount to the notion of plastic spin. This treatment does not rely on a multiplicative decomposition of the deformation gradient. We test our model on in-plane sheet-metal forming processes, which are governed by the evolution of pre-existing preferred material orientations. Hence, we advocate an orthotropic yield criterion directed by evolving structural tensors to describe this material behaviour. Our formulation yields two key findings. Firstly, the covariant formulation of plasticity yields suitable evolution equations for the structural tensors characterising the symmetry group of the orthotropic yield function. Secondly, the constitutive equations for the plastic variables and the structural tensors, which are both symmetric second-order tensors, give results that are in good agreement with experimental and numerical findings from in-plane sheet forming processes.

Finite element analyses of crack extensions in curved compact tension specimens of irradiated Zr-2.5Nb pressure tube materials by nodal release method and cohesive zone modeling approach

Crack extensions in thin CCT specimens of irradiated and hydrided irradiated Zr-2.5Nb pressure tube materials are simulated using the nodal release method and cohesive zone modeling (CZM) approach. The load vs displacement curves from the 3-D FE analyses with consideration of plastic orthotropy using the nodal release method match with the experimental data. The changing separation work rates of the 3-D FE analyses using the nodal release method and the experimental J-R curves are adopted as references for the changing cohesive energies in 2-D plane strain and plane stress FE analyses with the CZM approach. The simulation results with the CZM approach match well with the experimental data.

Theoretical predictions of dynamic necking formability of ductile metallic sheets with evolving plastic anisotropy and tension-compression asymmetry

In this paper, we have investigated necking formability of anisotropic and tension-compression asymmetric metallic sheets subjected to in-plane loading paths ranging from plane strain tension to near equibiaxial tension. For that purpose, we have used three different approaches: a linear stability analysis, a nonlinear two-zone model and unit-cell finite element calculations. We have considered three materials –AZ31-Mg alloy, high purity α-titanium and OFHC copper– whose mechanical behavior is described with an elastic-plastic constitutive model with yielding defined by the CPB06 criterion (15) which includes specific features to account for the evolution of plastic orthotropy and strength differential effect with accumulated plastic deformation (48). From a methodological standpoint, the main novelty of this paper with respect to the recent work of N’souglo et al. (42) –which investigated materials with yielding described by the orthotropic criterion of Hill (24)– is the extension of both stability analysis and nonlinear two-zone model to consider anisotropic and tension-compression asymmetric materials with distortional hardening. The results obtained with the stability analysis and the nonlinear two-zone model show reasonable qualitative and quantitative agreement with forming limit diagrams calculated with the finite element simulations, for the three materials considered, and for a wide range of loading rates varying from quasi-static loading up to 40000 s− 1, which makes apparent the capacity of the theoretical models to capture the mechanisms which control necking formability of metallic materials with complex plastic behavior. Special mention deserves the nonlinear two-zone model, as it does not need prior calibration –unlike the stability analysis– and it yields accurate predictions that rarely deviate more than 10% from the results obtained with the unit-cell calculations.

Calibration of orthotropic plasticity- and damage models for micro-sandwich materials

Sandwich structures are commonly used to increase bending-stiffness without significantly increasing weight. In particular, micro-sandwich materials have been developed with the automotive industry in mind, being thin and formable. In the present work, it is investigated if micro-sandwich materials may be modeled using commercially available material models, accounting for both elasto-plasticity and fracture. A methodology for calibration of both the constitutive- and the damage model of micro-sandwich materials is presented. To validate the models, an experimental T-peel test is performed on the micro-sandwich material and compared with the numerical models. The models are found to be in agreement with the experimental data, being able to recreate the force response as well as the fracture of the micro-sandwich core.

Evaluation of Hoffman and Xia plasticity models against bi-axial tension experiments of planar fiber network materials

The anisotropic properties and pressure sensitivity are intrinsic features of the constitutive response of fiber network materials. Although advanced models have been developed to simulate the complex response of fibrous materials, the lack of comparative studies may lead to a dubiety regarding the selection of a suitable method. In this study, the pressure-sensitive Hoffman yield criterion and the Xia model are implemented for the plane stress case to simulate the mechanical response under a bi-axial loading state. The performance of both models is experimentally assessed by comparison to bi-axial tests on cruciform-shaped specimens loaded in different directions with respect to the material principal directions. The comparison with the experimentally measured forces shows the ability of the Hoffman model as well as the Xia model with shape parameter k≤2 to adequately predict the material response. However, this study demonstrates that the Xia model consistently presents a stiffer bi-axial response when k≥3 compared to the Hoffman model. This result highlights the importance of calibrating the shape parameter k for the Xia model using a bi-axial test, which can be a cumbersome task. Also, for the same tension-compression response, the Hill criterion as a special case of the Hoffman model presents a good ability to simulate the mechanical response of the material for bi-axial conditions. Furthermore, in terms of stability criteria, the Xia model is unconditionally convex while the convexity of the Hoffman model is a function of the orthotropic plastic matrix. This study not only assesses the prediction capabilities of the two models, but also gives an insight into the selection of an appropriate constitutive model for material characterization and simulation of fibrous materials. The UMAT implementations of both models which are not available in commercial software and the calibration tool of the Xia model are shared with open-source along with this work.

Experimental and numerical determination of the screw deformation in laminated beech veneer lumber as a result of moisture‐induced swelling

The strong hygroscopic behavior of wood can lead to severe moisture-induced swelling deformations, especially perpendicular to the grain direction. Fully threaded screws arranged in this direction can be used as reinforcement and to constrain those deformations. Depending on the wood moisture content increase, tensile stresses in the screws and compression stresses in the surrounding solid wood evolve. A sufficiently accurate prediction of the maximum stresses occurring is important for reliable structural designs. For this reason, moisture-induced swelling experiments for two different configurations of screw diameter and specimen dimensions were performed within this work. Due to its homogeneity and its high swelling rate perpendicular to the grain, beech laminated veneer lumber (LVL) was used. A moisture content increase from oven-dry to about 7% led to screw tensile stresses in the magnitude of the yield strength and to highly nonlinear effects in the surrounding wood material. By means of a two-dimensional finite-element-based numerical simulation model, taking into account orthotropic plasticity and moisture-dependent material properties, the experiments could be simulated with high accuracy. The numerical results agree well with experimental observations. Based on a small parameter study, new insights into the complex interaction behavior between beech LVL and fully threaded screws could be gained.

Damage modelling strategies for unidirectional laminates subjected to impact using CZM and orthotropic plasticity law

In the present paper, the crashworthiness of a thick unidirectional carbon fibre reinforced polymer is investigated. This material is manufactured via compression moulding process. A Low-Velocity Impact (LVI) test is implemented on a quasi-isotropic laminate for the experimental evaluation of the energy absorption due to impact, while the internal failure mechanism is detected using Computerized Tomography (CT). Two different Finite Element (FE) models are applied to model the damage onset and propagation: firstly, a shell-based model and, secondly, a solid-based model using Cohesive Zone Method (CZM). In the CZM, the analytical modelling of the cohesive element properties is adopted, and the critical force evaluated experimentally is related to the energy release rate of mode II, and to the equivalent elastic properties of the laminate. The strength and weakness of the proposed approach in mimicking the impact behaviour and the actual failure mechanism, are discussed, and validated versus numerical simulation. The models are in good agreement with the experimental results; in fact, the relative error of the maximum force is about 4 per cent, and it occurs in the shell-based model.

Inverse parameter identification of an anisotropic plasticity model for sheet metal

The increasing economic and ecological demands on the mobility sector require efforts to reduce resource consumption in both the production and utilization phases. The use of lightweight construction technologies can save material and increase energy efficiency during operation. Multi-material systems consisting of different materials and geometries are used to achieve weight reduction. Since conventional joining processes reach their limits in the connection of these components, new methods and technologies are necessary in order to be able to react versatilely to varying process and disturbance variables. For fundamental investigations of new possibilities in joining technology, numerical investigations are helpful to identify process parameters. To generate valid results, robust and efficient material models are developed which are adapted to the requirements of versatile joining technologies, for instance to the high plastic strains associated with self-piercing riveting. To describe the inherent strain-induced plastic orthotropy of sheet metal an anisotropic Hill-plasticity model is formulated. Tensile tests for different sheet orientations are conducted both experimentally and numerically to adjust the anisotropic material parameters by inverse parameter identification for aluminium EN AW-6014 and steel HCT590X. Then, the layer compression test is used to validate the model and the previously identified parameters.

Effect of high strain rate on tensile response and failure analysis of titanium/glass fiber reinforced polymer composites

The high strain rate tensile response of titanium-based fiber metal laminates (FMLs), consisting of layers of titanium Ti-6Al-4V alloy sheet and glass fiber reinforced composites, is examined. A hand layup method is used to fabricate four different layups of FMLs, exhibiting the same thickness of the total metal layer. A split Hopkinson tensile bar apparatus is used to load titanium and composite under a high strain rate to obtain baseline data. High-speed digital image correlation is used to measure the strain directly on the specimen gage region. The elastic-plastic response of FMLs up to maximum stress is predicted by the classical laminated plate model and orthotropic plasticity model. This is followed by a behavior considering the mechanics of delamination. The results show that the layup sequence of titanium-based FMLs considerably affects the failure behavior of composites following ultimate strength. This strength increases at high strain rates and seems higher for titanium-based FMLs than aluminium-based FMLs. This is primarily caused by the rate-dependent response of the titanium and composite. The failure strain of glass fiber reinforced epoxy (GFRP) constituent, failure strain, and toughness of FMLs are affected by isolating composite layers by metallic layers within FMLs and are found to be rate sensitive. Isolation of composite layers from one another by metallic layers results in more progressive failure of FMLs. The proposed models are validated with experiments of aluminium-based FMLs available in the literature.

Strain Localization of Orthotropic Elasto-Plastic Cohesive-Frictional Materials: Analytical Results and Numerical Verification.

Strain localization analysis for orthotropic-associated plasticity in cohesive–frictional materials is addressed in this work. Specifically, the localization condition is derived from Maxwell’s kinematics, the plastic flow rule and the boundedness of stress rates. The analysis is applicable to strong and regularized discontinuity settings. Expanding on previous works, the quadratic orthotropic Hoffman and Tsai–Wu models are investigated and compared to pressure insensitive and sensitive models such as von Mises, Hill and Drucker–Prager. Analytical localization angles are obtained in uniaxial tension and compression under plane stress and plane strain conditions. These are only dependent on the plastic potential adopted; ensuing, a geometrical interpretation in the stress space is offered. The analytical results are then validated by independent numerical simulations. The B-bar finite element is used to deal with the limiting incompressibility in the purely isochoric plastic flow. For a strip under vertical stretching in plane stress and plane strain as well as Prandtl’s problem of indentation by a flat rigid die in plane strain, numerical results are presented for both isotropic and orthotropic plasticity models with or without tilting angle between the material axes and the applied loading. The influence of frictional behavior is studied. In all the investigated cases, the numerical results provide compelling support to the analytical prognosis.

Improving failure sub-models in an orthotropic plasticity-based material model

Theoretical details of two failure criteria implemented in an orthotropic plasticity model are presented. Improvements to the well-known Puck Failure criterion and a recently developed Generalized Tabulated Failure criterion are used to illustrate how to link a failure sub-model to existing deformation and damage sub-models in the context of explicit finite element analysis. These models are implemented in LS-DYNA, a commercial transient dynamic finite element code. Two validation tests are used to evaluate the failure sub-model implementation and improvements - a stacked-ply test carried out at room temperature under quasi-static tensile and compressive loadings, and a high-speed, projectile impact test where there is significant damage and material failure of the impacted panel. Results indicate that developed procedures and improvements provide the analyst with a reasonable and systematic approach to building predictive impact simulation models.

A uniquely defined multiplicative elasto-plasticity model with orthotropic yield function and plastic spin

A rate-independent model for isotropic elastic–orthotropic plastic material behaviour including the plastic spin is presented in this paper. The plastic spin, as introduced by Dafalias, is the spin of the continuum relative to the material substructure. The model is based on a specific multiplicative decomposition of the deformation gradient tensor, which introduces a uniquely defined intermediate configuration as motivated by Casey. We focus our attention on metal sheets in forming processes, in which pre-existing preferred orientations govern the orthotropic plastic behaviour. As a result, we advocate a Hill-type yield criterion enriched by the notion of plastic spin to describe this material behaviour. Our formulation yields three key findings: firstly, the uniquely defined intermediate configuration, namely a plastically stretched intermediate configuration, allows for a neat implementation of the plastic spin; secondly, the algorithmic formulation is straightforward and shows no additional difficulties in the implementation; and thirdly, a good agreement of our numerical model with experimental and numerical results from in-plane sheet forming processes reported in the literature is achieved.

Influence of the Replacement of the Actual Plastic Orthotropy with Various Approximations of Normal Anisotropy on Residual Stresses and Strains in a Thin Disk Subjected to External Pressure

Plastic anisotropy is very common to metallic materials. This property may significantly affect the performance of structures. However, the actual orthotropic yield criterion is often replaced with a criterion based on the assumption of normal anisotropy. The present paper aims to reveal the influence of this replacement on the distribution of strains and residual strains in a thin hollow disk under plane stress conditions. The boundary-value problem is intentionally formulated such that it is possible to obtain an exact semi-analytical solution without relaxing the boundary conditions. It is assumed that the disk is loaded by external pressure, followed by elastic unloading. The comparative analysis of the distributions of residual strains shows a significant deviation of the distribution resulting from the solutions based on the assumption of normal anisotropy from the distribution found using the actual orthotropic yield criterion. This finding shows that replacing the actual orthotropic yield criterion with the assumption of normal anisotropy may result in very inaccurate predictions. The type of anisotropy accepted is of practical importance because it usually results from such processes as drawing end extrusion with an axis of symmetry.

A three-pronged approach to predict the effect of plastic orthotropy on the formability of thin sheets subjected to dynamic biaxial stretching

In this paper, we have investigated the effect of material orthotropy on the formability of metallic sheets subjected to dynamic biaxial stretching. For that purpose, we have devised an original three-pronged methodology which includes a linear stability analysis, a nonlinear two-zone model and finite element calculations. We have studied 5 different materials whose mechanical behavior is described with an elastic isotropic, plastic anisotropic constitutive model with yielding based on Hill (1948) criterion. The linear stability analysis and the nonlinear two-zone model are extensions of the formulations developed by Zaera et al. (2015) and Jacques (2020), respectively, to consider Hill (1948) plasticity. The finite element calculations are performed with ABAQUS/Explicit (2016) using the unit-cell model developed by Rodríguez-Martínez et al. (2017), which includes a sinusoidal spatial imperfection to favor necking localization. The predictions of the stability analysis and the two-zone model are systematically compared against the finite element results – which are considered as the reference approach to validate the theoretical models – for loading paths ranging from plane strain stretching to equibiaxial stretching, and for different strain rates ranging from 100s−1 to 50000s−1. The stability analysis and the two-zone model yield the same overall trends obtained with the finite element simulations for the 5 materials investigated, and for most of the strain rates and loading paths the agreement for the necking strains is also quantitative. Notably, the differences between the finite element results and the two-zone model rarely go beyond 5%. Altogether, the results presented in this work provide new insights into the mechanisms which control dynamic formability of anisotropic metallic sheets.

On the New Shear Constraint for Plane-Stress Orthotropic Plasticity Modeling of Sheet Metals

A shear constraint was very recently proposed by Abedini et al. (Int. J. Solids and Structures 151: 118–134 2018) to evaluate and calibrate advanced non-quadratic anisotropic yield criteria and to eliminate what they called non-physical numerical artifacts in those criteria. This investigation points out that such a shear constraint is in fact unnecessary for plane-stress orthotropic plasticity in general. Using the well-known Hill’s 1948 quadratic and Gotoh’s 1977 quartic yield functions for orthotropic sheet metals in plane stress, it is shown analytically that pure shear stressing and pure shear straining loading conditions are not equivalent except for very special cases. By conducting a series of shearing experiments on an aluminum sheet metal, the actual test results are shown not to provide any unequivocal supporting evidence at all to the newly proposed shear constraint. The so-called non-physical numerical artifacts of the non-equivalence in pure shear stressing and pure shear straining of a sheet metal are in fact the intrinsic features of an anisotropic material in general. The newly proposed shear constraint should thus not be accepted to be universally applicable at all for anisotropic plasticity modeling of sheet metals. Such a proposed constraint itself shall be regarded as a provisional simplifying assumption of reduced anisotropy only for some particular sheet metals under consideration.

Uni-axial tensile response and failure of glass fiber reinforced titanium laminates

Abstract Layers of thin metallic sheets and fiber-reinforced composite layers bonded together is known as fiber metal laminates (FMLs). In the current study, titanium Ti–6Al–4V sheets are used with thicknesses 0.2 mm, 0.4 mm and, 0.6 mm along with layers of unidirectional glass-fiber reinforced composite to produce FMLs and the tensile response of these laminates are examined. Four different stacking sequences of FMLs have been considered exhibiting the same thickness for the total metal layer and the hand layup process is used to prepare these laminates. An orthotropic plasticity theory of macro mechanical type and classical laminated plate theory are considered to model the elastic-plastic type of behavior of titanium-based FMLs. A plastic potential function based on three parameters is used to the model the orthotropic plasticity. Behavior with linearly elastic and orthotropic elastic-plastic types is assumed for the layers of glass composite and titanium, respectively in the laminated plate theory. The results show that the initial modulus of FMLs has not been influenced by the sequence of the layup, while this sequence of layup does considerably affect the response of FMLs following ultimate strength. The properties of FMLs such as failure strain and toughness which are important parameters from a design point of view of FMLs can be altered to absorb energy at dissimilar rates, i.e., by insulating the composite layers from each other by metallic layers in case of FMLs 3/2–0.4, 4/3–0.2(O) and 4/3–0.4(O). The model with orthotropic plasticity is found to be precise up to the total strain level of 2.1%, i.e., close to the failure of composite layers within FMLs, when compared with results measured from experiments. The behavior with stress-strain predicted by laminated plate theory also finds realistic up to the total strain of 2.1% to describe the corresponding behavior obtained from experiments.

A constitutive model for a rate and temperature-dependent, plastically anisotropic titanium alloy

Aircraft engine fan blades are notably designed to withstand impact loading involving large deformation, high strain rate, non-proportional loading paths and self-heating. Due to their high strength-to-weight ratio and good toughness, Ti–6Al–4V titanium alloys are promising candidates for the blades leading edge. An extensive experimental campaign on a Ti–6Al–4V titanium alloy provided in the form of cold rolled plates has been carried out. The thermo-mechanical characterization consisted in tension, compression and shear tests performed at various strain rates and temperatures, and under monotonic as well as alternate loading paths. A constitutive model has been accordingly developed accounting for the combined effect of plastic orthotropy and tension/compression asymmetry, nonlinear isotropic and kinematic strain hardening, strain rate hardening, and thermal softening. The constitutive model has been implemented as a user material subroutine into the commercial finite element computation code LS-DYNA. The performances of the model have been estimated by conducting numerical simulations considering a volume element under various loading paths as well as the specimens used for the experimental campaign.

Investigation of Burst Failure of a Thick-Walled Spherical Vessel with Plastic Orthotropy under Internal and External Pressures

Investigation of Burst Failure of a Thick-Walled Spherical Vessel with Plastic Orthotropy under Internal and External Pressures