J2 Deformation Theory Research Articles

Experimental and numerical studies on the collapse of Titanium alloy ring-stiffened cylinder

This paper investigates the collapse of ring-stiffened Titanium alloy cylinders used for subsea resource exploration under hydrostatic pressure through experimental and numerical methods. Extensive material tests were first conducted on Titanium alloy specimens to obtain the fundamental mechanical properties and variation characteristics. Then, a 4236 mm-long test cylinder was fabricated with an inner radius of 650 mm and a thickness (ts) of 18.85 mm. The initial geometric imperfection was measured at evenly-spaced positions in the axial direction and 48 locations around the circumference by dial gauges. Afterward, the test cylinder was transferred into a custom hyperbaric pressure vessel and pressurized to collapse. As to the numerical analysis, a user-defined material subroutine implementing the incremental J2 deformation theory was developed to predict plastic bifurcation pressure. Moreover, a nonlinear finite element (FE) model, which incorporated the measured geometric imperfection and material nonlinearity, was used to reproduce the experiment. The numerical results were found to exhibit reasonably good agreement with the test data. In addition, parametric studies were conducted regarding material properties, geometric parameters, and imperfection sizes on the load-carrying capacity of Titanium alloy ring-stiffened cylinders.

Necking of thin-walled cylinders via bifurcation of incompressible nonlinear elastic solids.

Necking localization under quasi-static uniaxial tension is experimentally observed in ductile thin-walled cylindrical tubes, made of soft polypropylene. Necking nucleates at multiple locations along the tube and spreads throughout, involving the occurrence of higher-order modes, evidencing trefoil and fourth-foiled (but rarely even fifth-foiled) shaped cross-sections. No evidence of such a complicated necking occurrence and growth was found in other ductile materials for thin-walled cylinders under quasi-static loading. With the aim of modelling this phenomenon, as well as all other possible bifurcations, a two-dimensional formulation is introduced, in which only the mean surface of the tube is considered, paralleling the celebrated Flügge 's treatment of axially-compressed cylindrical shells. This treatment is extended to include tension and a broad class of nonlinear-hyperelastic constitutive law for the material, which is also assumed to be incompressible. The theoretical framework leads to a number of new results, not only for tensile axial force (where necking is modelled and, as a particular case, the classic Considère formula is shown to represent the limit of very thin tubes), but also for compressive force, providing closed-form formulae for wrinkling (showing that a direct application of the Flügge equation can be incorrect) and for Euler buckling. It is shown that the J2-deformation theory of plasticity (the simplest constitutive assumption to mimic through nonlinear elasticity the plastic branch of a material) captures multiple necking and occurrence of higher-order modes, so that experiments are explained. The presented results are important for several applications, ranging from aerospace and automotive engineering to the vascular mechanobiology, where a thin-walled tube (for instance an artery, or a catheter, or a stent) may become unstable not only in compression, but also in tension.

Elastic–plastic bifurcation analysis of cylindrical shells with axisymmetric variable thickness

This study focuses on the elastic–plastic bifurcation analysis of axial compressed thin-walled cylindrical shells with variable thickness. The general asymptotic formula of the elastic–plastic critical buckling load, which is a function of the Coefficient of the Thickness Variation (CTV) and the buckling load of shells with constant thickness, is derived based on the hybrid perturbation-weighted residual method. The general formula can be reduced to the classical elastic buckling solution in Koiter (1994) when considering the elastic case with the first-order term. Furthermore, a finite element bifurcation analysis procedure is also presented by adopting the incremental J2 deformation theory of plasticity through a custom user-defined subroutine appended to the nonlinear finite element software ABAQUS. The robustness of this numerical scheme and the theoretical formula is confirmed by the decent agreement with the available experimental results in the literature. In the end, the influence of the thickness variation parameter and D/t on the elastic–plastic buckling load of shells with varying thickness is also discussed.

Periodic necking of misfit hyperelastic filaments embedded in a soft matrix

The necking instability is a precursor to tensile failure and rupture of materials. A quasistatically loaded free-standing uniaxial specimen typically exhibits necking at a single location, thus corresponding to a long wavelength bifurcation mode. If confined to a substrate or embedded in a matrix the same filament can exhibit periodic necking and fragmentation thus creating segments of finite length. While such periodic instabilities have been extensively studied in ductile metal filaments and thin sheets, less is known about necking in hyperelastic materials. Nonetheless, in recent years, there has been a renewed interest in the role of necking in novel materials, for the advancement of fabrication processes and to explain fragmentation phenomena observed in 3D printed active biological matter. In both cases materials are not well described by the existing frameworks that employ J2 deformation theory of plasticity, and existing studies do not account for the significant role of misfit stretches that may emerge in these systems through chemical, or biological contraction. To address these limitations, in this paper we begin by experimentally demonstrating the role of the surrounding matrix on the necking and fragmentation of a compliant filament embedded in a tunable rubber matrix. Using a generalized hyperelastic model with strain softening, our analytical bifurcation analysis explains the experimental observations and is shown to agree with numerical predictions. The analysis reveals three distinct bifurcation modes: the long wavelength necking, thus recovering the Considère criterion; the periodic necking observed in our experiments; and a short wavelength mode that is characterized by localization along the center cord of the filament and is independent of the film-to-matrix stiffness ratio. We find that the softening coefficient and the filament misfit stretch can significantly influence the stability threshold and observed wavelength, respectively. Our results can guide the design and fabrication of novel composite materials and can explain the fragmentation processes observed in active biological materials.

Interactive Plastic Local Buckling of Box-shaped Aluminium Members under Uniform Compression

A theoretical approach for predicting the ultimate resistance of aluminium alloy members subjected to local buckling under uniform compression is presented focusing the attention on box sections. Starting from the J2 deformation theory of plasticity, the theory of plastic buckling of plates is presented including, as an original contribution, also the variability of the Poisson’s ratio depending on the stress levels. The differential equation of the plates at the onset of buckling is developed and the corresponding solution is provided. Starting from the obtained closed-form solution, the interactive buckling either in the elastic or in the plastic range is analysed. The interaction between the plate elements constituting the member section is explicitly accounted for using a completely theoretical approach where the evaluation of the critical stress leading to local buckling in the plastic range is derived as the value corresponding to the existence of a non-trivial solution for which the determinant of the matrix of the equation system is equal to zero. The accuracy of the plastic buckling theory is pointed out by comparing the buckling resistance obtained by the theoretical approach with the values of the ultimate resistance provided by stub column tests.Finally, some advances concerning the use of the effective thickness approach, currently adopted by Eurocode 9, are also proposed and herein presented for the first time and compared with test results.

Mechanical behavior and failure of glass/carbon fiber hybrid composites: Multiscale computational predictions validated by experiments

A comprehensive, novel and computationally low cost multiscale model based on finite element analysis is proposed which includes repeating unit cell micromechanics, material nonlinearity, and progressive damage analysis to predict the mechanical behavior and failure of glass/carbon fiber hybrid composites subjected to various loading conditions. The computational micromechanics is used to predict the homogenized properties of the composite from the mechanical properties of its constituents (fiber, and matrix), together with the volume fraction and spatial distribution of the fibers within it. Hashin’s failure criteria at the meso-scale is implemented to determine failure of lamina while nonlinearity of epoxy matrix is introduced to model through J2 deformation theory of plasticity. It is shown that the introduced multiscale model can be used for 3D macroscale structural analysis through a user defined material (UMAT) subroutine developed at the finite-element software ABAQUS/Implicit. The results of 3-point bending and tensile tests accompanied by acoustic emission, and in-plane shear tests with digital image correlation analysis, are used to validate the proposed multiscale model. It is shown that damage progress and strain distribution under various loading conditions can be predicted successfully, thus a reasonable correlation between the model and collected experimental data is achieved.

Topology optimization considering the Drucker–Prager criterion with a surrogate nonlinear elastic constitutive model

We address material nonlinear topology optimization problems considering the Drucker–Prager strength criterion by means of a surrogate nonlinear elastic model. The nonlinear material model is based on a generalized J2 deformation theory of plasticity. From an algorithmic viewpoint, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. The objective function of the optimization problem consists of maximizing the strain energy of the system in equilibrium subjected to a volume constraint. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved through direct minimization of the structural strain energy using Newton’s method with an inexact line search strategy. Four numerical examples demonstrate features of the proposed nonlinear topology optimization framework considering the Drucker–Prager strength criterion.

Buckling analysis of elastic–plastic nanoplates resting on a Winkler–Pasternak foundation based on nonlocal third-order plate theory

The paper analyzes the elastic–plastic buckling behavior of thick, rectangular nanoplates embedded in a Winkler–Pasternak foundation, adopting the Reddy third-order plate theory in nonlocal elasticity. Elasto-plasticity is accounted for by considering two alternative plasticity theories, namely the J2 flow incremental and the J2 deformation theory, with material properties defined by a Ramberg–Osgood relation. An iterative procedure is proposed to obtain the critical load, and the corresponding critical mode, of plates simply supported on two opposite edges under applied uniaxial and biaxial loading conditions. Extensive analysis investigates the effects of geometrical, constitutive, and nonlocal parameters on the critical behavior of plates with different boundary conditions. To the best of the authors’ knowledge, there are no findings about elastoplastic buckling of nanoplates in the existing literature. It is therefore hoped that the results obtained may provide a helpful basis for comparison for future investigations.

Mixed mode fracture in power law hardening materials for plane stress

The classic nonlinear fracture problem of a fully yielded, mixed mode stationary crack in a power law hardening material for conditions of plane stress under small-scale yielding is reconsidered. It has been determined that two different asymptotic solutions are required to represent the full range of mixed mode loading. Neither asymptotic solution has the double root of the linear elastic counterpart, i.e., the nonlinear plane stress problem does not have a mixed mode asymptotic solution. The mode II dominant asymptotic solution consists of two terms. The leading term is the pure mode II HRR term, while the second term is symmetric with an eigenvalue slightly weaker than the HRR eigenvalue. This two-term solution applies to a relatively large range of mixed mode loading. The mode I dominant asymptotic solution also consists of a symmetric and an antisymmetric term with different eigenvalues, and has a limited range of applicability near mode I. The pure mode I HRR term is the symmetric term. Contrary to expected behavior based on energy considerations and experience with higher order solutions, the antisymmetric term has an eigenvalue that is stronger than the HRR eigenvalue. This antisymmetric asymptotic solution, which cannot exist without the presence of the mode I HRR term, depends on two small parameters: the distance from the crack tip, r, and the ratio of mode II to mode I loading, K2/K1. The interpretation is that this two-term asymptotic solution exists for small r in the limit as K2/K1 approaches zero. An unusual feature of the second term is that it does not exist in the limit as r approaches zero, and therefore from a mathematical point of view this term does not cause the J-integral to be infinite. The asymptotic results are confirmed with full-field finite element analysis by using the J2 deformation theory of plasticity using a computational domain that covers eleven decades of radial detail. This validates the asymptotic solutions and shows that a two-parameter fracture theory can be used very near mode I and near mode II. The transition from one asymptotic solution to the other, which is demonstrated to occur near mode I, gives rise to a loss of dominance of these two-term asymptotic solutions. The hardening exponent, “n”, plays an important role in the ranges of validity of the two asymptotic solutions. Finally, the asymptotic solutions are shown to agree with solutions from the non-hardening limit, and the comparisons are consistent with those of the full-field results.

Prediction of Wrinkling of a Beverage Can Subjected to the Redrawing Process by J2 Deformation Theory

Wrinkling of beverage cans is one of the problems faced by can manufacturers and aluminum suppliers. The bottom of an aluminum can is wrinkled by compression during the forming process. In this study, to predict the occurrence of wrinkles during the redrawing process of AA3104 (t = 0.265 mm), which is the material used to fabricate aluminum cans, the classical plasticity J2 deformation theory (J2D) and flow theory (J2F) were considered. J2F considers only the deformation perpendicular to the yield locus, whereas J2D considers the deformation perpendicular to the yield locus and that tangential to the yield locus. Wrinkles are predicted using finite element (FE) analyses based on J2D and J2F, and the results are compared. J2F could not predict the number and amplitude of wrinkles. By contrast, the wrinkles predicted using J2D exhibited good agreement with sample data obtained for a real can. To find the difference between the results obtained using J2F and J2D, evolutions of stress path in a wrinkled element are compared. It was confirmed that compressive stress is more dominant in the J2D case than in the J2F case. Moreover, the measured effective strain of the element is small, under 0.04. In conclusion, J2D is more suitable for predicting the wrinkling behavior of aluminum cans than J2F. In addition, ANOVA and ANOM analysis are performed to evaluate the influence of the design parameters, namely friction coefficient, thickness, and outer profile angle, and the parameters are optimized to reduce wrinkles by combining the Taguchi method with FE simulation based on the J2D theory.

Multiscale micromechanical model of duplex low density steel with B2 ordered precipitates

In this study we reproduced the experimentally established micro-structure of a, Fe-15Mn-10Al-0.8C-5Ni, low density steel into a representative volume element (RVE), i.e. the smallest representative feature on which measurement can be made to yield a value representative of the whole structure. The pertinent structural parts were split into, an austenite (γ) phase, micro-meter sized B2 precipitates located inside γ and a composite ferrite(α)+nano-sized B2 “phase”. We simulated the mechanical elastoplastic response of each of these parts through J2 deformation theory of plasticity. The equivalent inclusion method and mean field approach were employed to describe the behaviour or precipitates. The simulated composite mechanical response was in good agreement with those measured through tensile tests. The simulated responses suggests that the type of chosen boundary condition affects the development of shear strain localization. It was found that in those γ grains located between any closely spaced composite α+B2 the strain localization is pronounced. In addition, inhomogeneity between phases led to stress concentration which in turn intensified the localization.

The dynamics of a shear band

A shear band of finite length, formed inside a ductile material at a certain stage of a continued homogeneous strain, provides a dynamic perturbation to an incident wave field, which strongly influences the dynamics of the material and affects its path to failure. The investigation of this perturbation is presented for a ductile metal, with reference to the incremental mechanics of a material obeying the J2–deformation theory of plasticity (a special form of prestressed, elastic, anisotropic, and incompressible solid). The treatment originates from the derivation of integral representations relating the incremental mechanical fields at every point of the medium to the incremental displacement jump across the shear band faces, generated by an impinging wave. The boundary integral equations (under the plane strain assumption) are numerically approached through a collocation technique, which keeps into account the singularity at the shear band tips and permits the analysis of an incident wave impinging a shear band. It is shown that the presence of the shear band induces a resonance, visible in the incremental displacement field and in the stress intensity factor at the shear band tips, which promotes shear band growth. Moreover, the waves scattered by the shear band are shown to generate a fine texture of vibrations, parallel to the shear band line and propagating at a long distance from it, but leaving a sort of conical shadow zone, which emanates from the tips of the shear band.

A path-dependent necking instability analysis of the thin substrate composite plates considering nonlinear reinforced layer effects

In this paper, an analytical path-dependent instability model is proposed for the thin substrate composite plates, considering the nonlinear reinforced layer effects. The model is introduced by extending a modified maximum force criterion (MMFC) and the vertex criteria to predict diffuse and localized necking, respectively, for the thin substrate-supported plates. The MMFC considers the strain hardening on the diffuse necking as well as the loading conditions. The vertex criteria presented by St $$ \ddot{\mathrm{o}}\mathrm{re} $$ n and Rice are usually based on the J 2 deformation theory of classical plasticity, which explores the localized necking through the rate discontinuity assumption at the necking band. Both models will be combined with the strain path effect through a linear adoption of an equivalent strain. It will be investigated by applying a pre-strain in the major and minor directions. Moreover, a dependent to yield criterion (DYC) angle is used for prediction of the necking band angle in the vertex theory. Also, a nonlinear neo-Hookean reinforced layer is considered to make a sensible delay at plastic deformation. Since the reinforcing layers are the materials with high stretchability, the effects of material properties and thickness values will be studied for supporting metal layers. Also, it is assumed that the incompressibility condition remains for corresponded elastomer and metal layer strains. Finally, the quadratic Hill criterion is used to investigate the anisotropy effect. The model is verified by experimental and other theoretical results.

Inelastic Buckling of FGM Cylindrical Shells Subjected to Combined Axial and Torsional Loads

A semi-analytical procedure is presented to solve the elastoplastic buckling problem of cylindrical shells made of functionally groded materials (FGMs) under combined axial and torsional loads. The elastoplastic properties are assumed to vary smoothly according to the power law distribution rule, introduced in the J2 deformation theory for formulation of the constitutive relation of FGMs in the framework of the Tamura–Tomota–Ozawa (TTO) model, which is a volume fraction-based material model. The critical condition is deduced by the Ritz method. Assuming the uniform prebuckling strain, a biaxial stress state analysis is conducted to determine the analytical position of the elastoplastic interface, which is used in the integration of the elastoplastic internal force and all the material-related structural parameters. Finally, an iterative procedure is adopted to find the exact elastoplastic critical load. Numerical results indicate the effects of the inhomogeneous parameter and the elastoplastic material properties of their constituents on the stability region and plastic flow region of the materials.

An efficient model for diffuse to localized necking transition in rate-dependent bifurcation analysis of metallic sheets

A plastic instability model is extended to predict evolution of a bounded deformation in necking phenomenon. There are several studies to predict the plastic instability such as the vertex model proposed by Stören and Rice, which considers strain and stress rate discontinuities in the necking band. Here, the model is established on the J2 deformation theory of classical plasticity. Effect of stress triaxiality is investigated on the localization in bifurcation analysis. Also, a modified maximum force criterion is applied to predict diffuse necking considering loading conditions. Although considering the effect of strain rate hardening plays an important role to give accurate results, usually imposing it leads to emerging relatively the main complex constitutive equations. Therefore, a delicate bridge between the diffuse and localized models is made using the maximum force assumption to overcome on the complexity of the problem. Also, by investigating the strain rate behavior on the plastic instability, the forming limit diagrams are obtained illustrating more accurate results than other existing models. The anisotropy effect is studied by application of a quadratic Hill's criteria. The necking band angle will be investigated per different conditions through extending the vertex model coupled with the angle-dependent yield criterion.

Buckling and postbuckling of elastoplastic FGM plates under inplane loads

Elastoplastic buckling behaviors of rectangular plates made from functionally graded materials (FGMs) are investigated in this paper. The elastoplastic material properties are assumed to vary smoothly through the thickness of the plates. The three dimensional material constitutive relation of FGMs is found by introducing the material homogenization method, named Tamura-Tomota-Ozawa(TTO) model, into J2 deformation theory or J2 flow theory. The uniform strain hypothesis helps to simplify the prebuckling state and derive the analytical expression of the position of the material elastoplastic interface. The buckling governing equations and the buckling critical condition of the structures are formulated under the framework of the classical plate theory. An iterative algorithm is designed to obtain the elastoplastic buckling critical load, a converging result between the prebuckling and the buckling critical internal forces. ABAQUS simulation well verifies the present theoretical predictions from J2 flow theory, and is resorted to investigate the postbuckling behaviors of FGM plates. Discussions are addressed for the effects of the constituent distribution, the material plastic flow, the preloaded states of the plates, and the regions of buckling types are plotted as well.

Stability of hydrostatic-pressured FGM thick rings with material nonlinearity

Buckling behaviors of elastoplastic ceramic/metallic functionally graded material (FGM) rings are investigated by using the first order shear deformation theory. The hydrostatic-pressured rings are assumed to be in both the plane-stress case and the plane-strain case, which lead respectively to a uniaxial and a biaxial elastoplastic stress states in prebuckling stage. A uniform strain hypothesis helps to deal with the elastoplastic stress states. By introducing in the graded material properties, the constitutive model of FGMs is formulated under the framework of J2 deformation theory. By considering the kinetic relations of von-Kárman type and employing the principle of virtual displacement, the equilibrium equations and the buckling governing equations of FGM circular rings are formulated, and the analytical solution of the anisotropic rings is obtained. Finally, the elastoplastic buckling problem is numerically solved through a semi-analytical method, which is proposed to seek the real circumferential strain of FGM rings at the buckling point and determinate the elastoplastic buckling critical hydrostatic pressure. The effects of the inhomogeneous and geometrical parameters on the buckling critical load and the position of the elastoplastic interface are discussed. Results show that, in both the plane-stress and the plane-strain cases, the elastoplastic critical loads are generally lower than their elastic counterparts due to material flow, and the plane-strain critical load is generally larger than the plane-stress one. The elastoplastic critical load does not always decrease monotonously with the increase of the inhomogeneous parameters, which is quite different from their elastic counterparts.

Non-associative [formula omitted] plasticity model for finite element buckling analysis of shells in the inelastic range

The development and numerical implementation of a special-purpose constitutive model is described for investigating the structural stability of cylindrical metal shells under axial compression and bending, which buckle in the inelastic range. The model employs von Mises yield surface (J2 plasticity) and the rate form of J2 deformation theory, leading to a non-associated flow rule. Special emphasis is given on plastic flow continuity. The numerical implementation is conducted through both the classical Euler-backward and Euler-forward substitution numerical schemes, where stress and strain tensors are described in curvilinear coordinates, with the extra constraint of zero normal stress through the shell thickness. The numerical results will be compared with available experimental data. The model is implemented within an in-house finite element technique for the nonlinear analysis of relatively thick cylindrical metal shells that uses a “tube-element” discretization, and it is employed for the solution of some benchmark problems.

Progressive damage and failure response of hybrid 3D textile composites subjected to flexural loading, part II: Mechanics based multiscale computational modeling of progressive damage and failure

A mechanics based multiscale computational model is presented to predict the deformation, damage and failure response of hybrid 3D textile composites (H3DTCs) subjected to three-point bending. The geometry of the textile architecture was incorporated in a mesoscale finite element (FE) model, while the H3DTC was homogenized at the macroscale. The mesoscale model is a collection of repeat unit cells (RUCs) that are composed of different types of fiber tows embedded in a surrounding matrix. Matrix microdamage was modeled by a (pre-peak) nonlinear stress versus strain response, using a modified J2 deformation theory of plasticity incorporating a secant-modulus approach. Fiber tow pre-peak nonlinear response was computed using a novel, two-scale model, in which the subscale micromechanical analysis was carried out in closed-form based upon a unit cell of a fiber–matrix concentric cylinder. Consequently, the influence of matrix microdamage developing at the microscale manifests as the progressive degradation of fiber tow stiffness at the mesoscale. The smeared crack approach (SCA) was employed to model the post-peak softening of the constituents due to failure, including matrix macro-cracking, tow kinking, and tow breaking. This method offers a mesh objective result by relating the post-peak softening response to a traction–separation law that is associated with each failure mechanism through a characteristic length. Thus, the total energy release rate during failure in a continuum element is related to the fracture toughness of the material.The load–deflection responses, along with the progressive damage and failure events, including fiber tow kinking and rupture, are successfully predicted through the proposed computational model. In addition, the textile architecture-dependent effect, observed in the asymmetric H3DTCs, is also captured, demonstrating the predictive capability of the proposed modeling scheme. Since all the inputs are from the constituent level, the model is useful in understanding how the macroscopic response of H3DTCs is influenced by textile architecture and constituent properties. The experimental studies are presented in Part I of this two-part sequence (Zhang et al., 2015).

Effect of Crack Front Curvature on Experimental Evaluation of J-Integral for Single-Edge Bend Specimens

Abstract This paper presents three-dimensional (3D) finite element analyses of plane-sided single-edge bend specimens to investigate the effect of the crack front curvature on the accuracy of the average J-integral (J) evaluated over the crack front using the plastic eta-factor-based approach specified in ASTM E1820-11. Specimens with average crack lengths of 0.3, 0.5, and 0.7 and thickness-to-width ratios of 1, 0.5, and 0.25 were analyzed. The analysis was based on the J2 deformation theory of plasticity and assumed stationary cracks with sharp crack tips. The curved crack front was characterized by a power-law expression proposed by Nikishikov et al. that was validated using crack front data collected in this study. The plastic eta factors specified in ASTM E1820-11 and proposed by Zhu et al. were employed. The analysis results indicate that the errors in the plastic eta-factor-based J were generally between −7 % (i.e., underestimation) and 6 % (i.e., overestimation) for the specimens that had curved crack fronts with crack front curvature equal to the maximum allowable value specified in ASTM E1820-11. Based on the analysis results, crack front straightness criteria that are in most cases less stringent than those specified in ASTM E1820-11 are recommended. The suggested criteria could potentially lead to reduced specimen rejection rates and cost savings.