A comparative study of stress update algorithms for rate‐independent and rate‐dependent crystal plasticity

The paper presents a comparative discussion of stress update algorithms for single‐crystal plasticity at small strains. The key result is a new unified fully implicit multisurface‐type return algorithm for both the rate‐independent and the rate‐dependent setting, endowed with three alternative approaches to the regularization of possible redundant slip activities. The fundamental problem of the rate‐independent theory is the possible ill condition due to linear‐dependent active slip systems. We discuss three possible algorithmic approaches to deal with this problem. This includes the use of alternative generalized inverses of the Jacobian of the currently active yield criterion functions as well as a new diagonal shift regularization technique, motivated by a limit of the rate‐dependent theory. Analytical investigations and numerical experiments show that all three approaches result in similar physically acceptable predictions of the active slip of rate‐independent single‐crystal plasticity, while the new proposed diagonal shift method is the most simple and efficient concept. Copyright © 2001 John Wiley & Sons, Ltd.     

Implicit boundary method for analysis using uniform B‐spline basis and structured grid

Several analysis techniques such as extended finite element method (X‐FEM) have been developed recently, which use structured grid for the analysis. Implicit boundary method uses implicit equations of the boundary to apply boundary conditions in X‐FEM framework using structured grids. Solution structures for test and trial functions are constructed using implicit equations such that the boundary conditions are satisfied even if there are no nodes on the boundary. In this paper, this method is applied for analysis using uniform B‐spline basis defined over a structured grid. Solution structures that are C¹ or C² continuous throughout the analysis domain can be constructed using B‐spline basis functions. As a structured grid does not conform to the geometry of the analysis domain, the boundaries of the analysis domain are defined independently using equations of the boundary curves/surfaces. Compared with conforming mesh, it is easier to generate structured grids that overlap the geometry and the elements in the grid are regular shaped and undistorted. Numerical examples are presented to demonstrate the performance of these B‐spline elements. The results are compared with analytical solutions as well as with traditional finite element solutions. Convergence studies for several examples show that B‐spline elements provide accurate solutions with fewer elements and nodes compared with traditional FEM. They also provide continuous stress and strain in the analysis domain, thus eliminating the need for smoothing stress/strain results. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical implementation of a J_2- and J_3-dependent plasticity model based on a spectral decomposition of the stress deviator

A new plasticity model with a yield criterion that depends on the second and third invariants of the stress deviator is proposed. The model is intended to bridge the gap between von Mises’ and Tresca’s yield criteria. An associative flow rule is employed. The proposed model contains one new non-dimensional key material parameter, that quantifies the relative difference in yield strength between uniaxial tension and pure shear. The yield surface is smooth and convex. Material strain hardening can be ascertained by a standard uniaxial tensile test, whereas the new material parameter can be determined by a test in pure shear. A fully implicit backward Euler method is developed and presented for the integration of stresses with a tangent operator consistent with the stress updating scheme. The stress updating method utilizes a spectral decomposition of the deviatoric stress tensor, which leads to a stable and robust updating scheme for a yield surface that exhibits strong and rapidly changing curvature in the synoptic plane. The proposed constitutive theory is implemented in a finite element program, and the influence of the new material parameter is demonstrated in two numerical examples.     

A consistent formulation for the integration of combined plasticity and creep

Within the context of the consistent tangent update, this paper outlines a stress update algorithm for combined creep and plasticity. The algorithm is implicit, providing unconditional stability, and utilizes local Newton iteration to solve SCALAR forms of the coupled constitutive equations for the creep and plastic strain increments. The tangent for the local iteration is obtained accurately providing quadratic convergence at the Gauss point level. Quadratic convergence of the global iteration procedure is also maintained using an explicitly derived consistent tangent for combined plasticity and creep. Further, combination with an automatic time-stepping scheme provides an efficient, stable, accurate and robust computational algorithm. The algorithm has been implemented in the general purpose FE package LUSAS¹.     

A large strain plasticity model for implicit finite element analyses

The theoretical basis and numerical implementation of a plasticity model suitable for finite strains and rotations are described. The constitutive equations governing J ₂ flow theory are formulated using strains-stresses and their rates defined on the unrotated frame of reference. Unlike models based on the classical Jaumann (or corotational) stress rate, the present model predicts physically acceptable responses for homogeneous deformations of exceedingly large magnitude. The associated numerical algorithms accommodate the large strain increments that arise in finite-element formulations employing an implicit solution of the global equilibrium equations. The resulting computational framework divorces the finite rotation effects on strain-stress rates from integration of the rates to update the material response over a load (time) step. Consequently, all of the numerical refinements developed previously for small-strain plasticity (radial return with subincrementation, plane stress modifications, kinematic hardening, consistent tangent operators) are utilized without modification. Details of the numerical algorithms are provided including the necessary transformation matrices and additional techniques required for finite deformations in plane stress. Several numerical examples are presented to illustrate the realistic responses predicted by the model and the robustness of the numerical procedures.     

SOME NOVEL DEVELOPMENTS IN FINITE ELEMENT PROCEDURES FOR GRADIENT-DEPENDENT PLASTICITY

Improved algorithms are proposed for a gradient plasticity theory in which the Laplacian of an invariant plastic strain measure enters the yield function. Particular attention is given to the type of finite elements that can be used within the format of gradient-dependent plasticity. Assuming a weak satisfaction of the yield function, mixed finite elements are developed, in which the invariant plastic strain measure and the displacements are discretized. Two families of finite elements are developed: one in which the invariant plastic strain measure is interpolated using C¹-continuous polynomials, and one in which penalty-enhanced C⁰-continuous interpolants are used. The performance of both families of finite elements is assessed numerically in one-dimensional and two-dimensional boundary value problems. The regularizing effect of the used gradient enhancement in computations of elastoplastic solids is demonstrated, both for mesh refinement and for the directional bias of the grid lines.     

Study of the effects of stress and strain on martensite transformation: Kinetics and transformation plasticity*

In this paper, the effects of stress and strain on the kinetics and plasticity during martensitic transformation are studied. The mathematical models of transformation kinetics and plasticity under stress are developed. According to experimental results, the transformation plasticity parameter k is concluded not to be a constant, but it varies with the stresses.     

A micro‐mechanical damage model based on gradient plasticity: algorithms and applications

As soon as material failure dominates a deformation process, the material increasingly displays strain softening and the finite element computation is significantly affected by the element size. Without remedying this effect in the constitutive model one cannot hope for a reliable prediction of the ductile material failure process. In the present paper, a micro‐mechanical damage model coupled to gradient‐dependent plasticity theory is presented and its finite element algorithm is discussed. By incorporating the Laplacian of plastic strain into the damage constitutive relationship, the known mesh‐dependence is overcome and computational results are uniquely correlated with the given material parameters. The implicit C¹ shape function is used and can be transformed to arbitrary quadrilateral elements. The introduced intrinsic material length parameter is able to predict size effects in material failure. Copyright © 2002 John Wiley & Sons, Ltd.     

An Edge-Based Smoothed Discrete Shear Gap Method Using the C0-Type Higher-Order Shear Deformation Theory for Analysis of Laminated Composite Plates

The edge-based smoothing discrete shear gap method (ES-DSG3) using three-node triangular elements is combined with a C⁰-type higher-order shear deformation theory (HSDT) to give a new linear triangular plate element for static, free vibration, and buckling analyses of laminated composite plates. In the ES-DSG3, only the linear approximation is necessary, and the discrete shear gap method (DSG) for triangular plate elements is used to avoid the shear locking and spurious zero energy modes. In addition, the stiffness matrices are calculated relying on smoothing domains associated with the edges of the triangular elements through an edge-based strain smoothing technique. Using the C⁰-type HSDT, the shear correction factors in the original ES-DSG3 can be removed and replaced by two additional degrees of freedom at each node. The numerical examples demonstrated that the ES-DSG3 show remarkably excellent performance compared to several other published elements in the literature.     

The geometric nature of plasticity laws

This paper concerns the plasticity constitutive laws in small strain. In the thermodynamic approach developed here, the key concept is that of internal variables. The differential nature of plasticity law has been pointed out for a long time. If we unite the invariance condition of these laws in a state variable transformation, this involves, ultimately, that the natural mathematic frame of plasticity theory is Differential Geometry. The system state is defined as a point of a differentiable manifold. The state variable are the local coordinates of this point in a chart. The internal stresses are the components of a covariant vector of the cotangent bundle to internal state manifold and the elastic domain is a convex part of cotangent vector space. The plastic yield criteria such as von Mises condition define a Riemannian structure over the manifold. The metric element is identified with the internal dissipation element. Constitutive laws link the covariant derivatives of the thermodynamic stress with the state variable. Hardening modulus splits up in two parts, kinematic hardening and metric hardening. This last is defined by Christoffel connection coefficients. Applied to von Mises isotropic yield condition, the metric hardening is identified with isotropic hardening. The Baltov-Sawczuk model is also analysed. The use of appropriate polar coordinates simplifies significantly the computations. Generalization to a significant category of non-differentiable yield criteria, such as Tresca condition, is considered by introducting a metric tensor family. The adaptation of Drucker's postulate to the proposed model requires the introduction of parallel transport of the internal stress covector. Generally, this transport is different over distinctive paths joining two points. This fact expresses internal state manifold curvature. The Riemann-Christoffel tensor is computed for von Mises, Baltov-Sawczuk and Tresca models.     

Inclusion of primary creep in the estimation of the C t parameter

The problem of time-dependent fracture under transient creep conditions is investigated via finite element analyses of fracture specimens with stationary cracks. The constitutive models consist of linear elasticity with combinations of power-law secondary creep and two primary creep laws. Two proposed parameters are studied. One is a contour integral, C(t), which characterizes the crack tip singularity strength. The other one, C _t, is evaluated based on the load line deflection rate and has been used successfully in correlating experimental creep crack growth data.It is evident that accurate constitutive modeling is essential to good agreement with experimental data. The inclusion of primary creep resolves earlier discrepancies between the experimental and analytical load line deflection rates which are used to calculate the respective values of C _t. The loading boundary condition is also an important factor that has been addressed. A more general formulation of C _twhich includes primary creep is presented. In small scale and transition creep, the C _tparameter does not characterize the crack tip stress singularity but rather is related to the crack tip creep zone growth rate. At times past transition time, C _tand C(t) both approach a path-independent integral, C ^*(t), which characterizes the stationary crack tip stress field. The relationship between C _tand C(t) is discussed. The interpretation and estimation of the C _tparameter are given based on the numerical results and analytical manipulations.     

Characterization of strain hardening for a simple pressure-sensitive plasticity model

Summary Paralleling the development of strain hardening for the pressure-independent von Mises criterion, a simple plasticity model in strain space was presented to characterize strain hardening for pressure-sensitive compressible materials. Two hardening moduli,H _T andH _C , which emerged from the constitutive equations and can becalculated from uniaxial stress-strain curves in tension and compression, were used to characterize the strainhardening responses forgeneral and special stress systems. The results indicated the implications and restrictions of the yield function on the hardening responses. It was also shown that strain softening, under general stress systems, can be a natural consequence of pressure-sensitive yielding. Consequently, a strain-space formulation is recommended for most (if not all) pressure-sensitive plasticity models. Preliminary application to the yielding of polymers under hydrostatic pressure gave reasonable results for polyethylene at moderate pressure and small strains; the results for polycarbonate were generally poor. Finally, the advantages and limitations of the present approach were discussed.With 6 Figures     

Comparison among estimation methods of C* parameter for axially oriented external surface crack in cylinders

Abstract

C^* is usually used to describe the creep crack growth. ASTM E1457 allows C^* to be calculated from creep load line displacement rate. However in components it is difficult or impossible to measure load line displacement rate. Therefore for the components C^* must be determined by finite element methods or reference stress concepts. Estimates of C^* obtained by reference stress methods will depend on the collapse mechanism adopted and therefore several estimations are proposed. This paper presents a numerical study of non-linear fracture mechanics parameter predictions under elevated temperature for axially oriented external surface crack in cylinder. Comparison of C^* calculated from FE analysis and different reference methods is conducted. The values of C^* obtained from the API579 net section solution are also found to be slightly conservative and give the closest agreement to the F.E. contour integral C^*. In addition, the comparison between C^* of homogeneous material and TYPE IV cracking is conducted. The difference between homogeneous material and TYPE IV cracking is almost negligible and therefore the reference stress solutions for homogenous material could be applied to estimate C^* for TYPE IV cracking.     

Procedures to investigate injectivity of polynomial maps and to compute the inverse

In this paper a polynomial map from C ⁿ to C ^m is studied in order to investigate if it is injective out of a set of measure zero. We propose a procedure, based on truncated Gröbner basis computations, which when successful, allows to reduce the problem to an easier map, and so gives a speed-up of the general algorithms using Gröbner basis techniques. Moreover, for the special case of a polynomial map from C ⁿ to C ⁿ where the polynomials are at most quadratic, we propose two criteria for non-injectivity based on the structure of the Jacobian matrix and requiring only basic symbolic computations.     

A novel four‐node quadrilateral smoothing element for stress enhancement and error estimation

A four‐node, quadrilateral smoothing element is developed based upon a penalized‐discrete‐least‐squares variational formulation. The smoothing methodology recovers C¹‐continuous stresses, thus enabling effective a posteriori error estimation and automatic adaptive mesh refinement. The element formulation is originated with a five‐node macro‐element configuration consisting of four triangular anisoparametric smoothing elements in a cross‐diagonal pattern. This element pattern enables a convenient closed‐form solution for the degrees of freedom of the interior node, resulting from enforcing explicitly a set of natural edge‐wise penalty constraints. The degree‐of‐freedom reduction scheme leads to a very efficient formulation of a four‐node quadrilateral smoothing element without any compromise in robustness and accuracy of the smoothing analysis. The application examples include stress recovery and error estimation in adaptive mesh refinement solutions for an elasticity problem and an aerospace structural component. Copyright © 1999 John Wiley & Sons, Ltd.     

General time‐dependent C(t) and J(t) estimation equations for elastic‐plastic‐creep fracture mechanics analysis

This paper presents equations for estimating the crack tip characterizing parameters C(t) and J(t), for general elastic‐plastic‐creep conditions where the power‐law creep and plasticity stress exponents differ, by modifying the plasticity correction term in published equations. The plasticity correction term in the newly proposed equations is given in terms of the initial elastic‐plastic and steady‐state creep stress fields. The predicted C(t) and J(t) results are validated by comparison with systematic elastic‐plastic‐creep FE results. Good agreement with the FE results is found.     

A general framework for generating convex yield surfaces for anisotropic metals

Summary The aim of this paper is to develop a general procedure to create yield surfaces (both isotropic and anisotropic) for elastic plastic metals with particular reference to sheet metal forming operations. Due to the fact that the forming limit of sheet metals requires a very accurate prediction of the yield limits and their normals, it is desirable to create yield functions that offer independent control of the yield points and the normal to the yield surface at a particular stress state without affecting its global convexity properties and the yield points at other states of stress. We achieve this by creating a class of yield surfaces that are obtained by the intersection of a number of elementary convex surfaces (such as planes and cylinders); the edges and corners that are a result of the intersection are smoothened by aL ^p-norm smoothing technique to give rounded corners. The yield surfaces generated by this procedure are centrosymmetric in nature and are thus limited in their applicability.We show that many of the standard yield surfaces can be recast into this form. We obtain a new yield surface that fits the experiments obtained for the 6022-T4 aluminum alloy as reported by Barlat et al. [1]. The same yield surface is also used to fit the results of the Taylor-Bishop-Hill polycrystal theory applied to a material with copper texture. The data were obtained from Choi et al. [2].     

Non-linear fracture mechanics analyses of part circumferential surface cracked pipes

This paper provides engineering estimates of non-linear fracture mechanics parameters for pipes with part circumferential inner surface cracks, subject to internal pressure and global bending. Solutions are given in the form of two different approaches, the GE/EPRI approach and the reference stress approach. For the GE/EPRI approach, the plastic influence functions for fully plastic J solutions are tabulated based on extensive 3-D FE calculations using deformation plasticity, covering a wide range of pipe and crack geometries. The developed GE/EPRI-type fully plastic J estimation equations are then re-formulated using the concept of the reference stress approach for wider applications. The proposed reference stress based estimates are validated against detailed 3-D elastic-plastic and elastic-creep FE results. For a total of 26 cases considered in this paper, agreement between the proposed reference stress based J and C ^* estimates and the FE results is excellent. An important aspect of the proposed estimates is that they not only are simple and accurate but also can be used to estimate J and C ^* at an arbitrary point along the crack front.     

Experimental estimation of the probability distribution of fatigue crack growth lives

This paper deals with a method to estimate numerically the reliability of fatigue sensitive structures with respect to fatigue crack growth. A method is proposed to experimentally determine the probability distribution functions of material parameters of the Paris law, da/dN = C(ΔK/K₀)^m, using stress intensity factor controlled tests. The auto-correlation function of the resistance to fatigue crack growth, 1/C, is also estimated from the experimental data. The results of a high tensile strength steel show that the distribution of the parameter, m, is approximately normal and that of 1/C is a 3-parameter Weibull. The merit of the proposed method is that only a small number of tests are required to determine these functions. The probability distribution of the fatigue crack length after a given number of load cycles or the number of load cycles for a crack to reach a given length can be estimated by simulations of non-Gaussian random processes having these functions.     

Crack growth kinetics of 7050-T73651 aluminium alloy under constant load at 150°C

Crack growth tests at 150°C under constant load conditions were performed on compact tension specimens of 7050-T73651 aluminium alloy. The d.c. potential drop method was employed to monitor crack lengths throughout the tests. Fracture mechanics parameters such as the stress intensity factor (K) and energy-rate line integral (C ^*) were used to establish correlation with the crack propagation rates. As a result of experiments it was found that crack growth rates (da/dt) versus K, over discrete ranges of rates, can be described as a power-law equation in the form da/dt = AK ⁿ. The parameter C ^* appears to correlate well with the crack propagation rates through a single power law over a wide range of rates.