Creep damage modeling of a polycrystalline material

A polycrystalline material is investigated under creep conditions within the framework of continuum micromechanics. Geometrical 3D model of a polycrystalline microstructure is represented as a unit cell with grains of random crystallographical orientation and shape. Thickness of the plains, separating neighboring grains in the unit cell, can have non-zero value. Obtained geometry assigns a special zone in the vicinity of grain boundaries, possessing unordered crystalline structure. A mechanical behavior of this zone should allow sliding of the adjacent grains. Within the grain interior crystalline structure is ordered, what prescribes cubic symmetry of a material. The anisotropic material model with the orthotropic symmetry is implemented in ABAQUS and used to assign elastic and creep behavior of both the grain interior and grain boundary material. Appropriate parameters set allows transition from the orthotropy to the cubic symmetry for the grain interior. Material parameters for the grain interior are identified from creep tests for single crystal copper. Model parameters for the grain boundary are set from the physical considerations and numerical model validation according to the experimental data of the grain boundary sliding in a polycrystalline copper [2]. As the result of analysis representative number of grains and grain boundary thickness in the unit cell are recommended. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Method to Estimate the Inelastic Response in Notched Structures of Anisotropic Materials with Elastic ‐ Perfectly Plastic Behavior

The fatigue strength of engineering structures is often limited by notches which arise from constructive features or manufacturing defects. The constitutive behavior in notch regions is then characterized by small plastic zones which are contained in an elastic region. To avoid costly plastic calculations, approximate methods have been developed to estimate the inelastic stress‐strain response at the notch tip. One of the first and best known approximate models is that of Neuber which, over the last 40 years, has received considerable attention particularly in connection with fatigue life prediction. Numerous studies have been conducted to the verification and the generalization of the Neuber approach which respect to multiaxiality, cyclic loading and creep conditions. Recently an extension of the Neuber method to anisotropic materials has been proposed in [1] and applied to directionally solidified and single crystal Nickel based superalloys as they are used in high temperature material applications. In this short notice we modify the approach in [1] for the special case of an elastic ‐ perfectly plastic anisotropic material.     

Creep of Single Crystals – Modelling and Numerical Aspects

A number of constitutive models, utilizing both microstructural and/or phenomenological considerations, have been developed for the simulation of the creep behaviour of nickel-base single crystal superalloys at elevated temperatures. In this work, emphasis is placed on the rate-dependent single crystal plasticity model [1]. A strategy for the identification of the material parameters of the model to fit the results from experiments has been implemented. The parameter fitting methodology rests upon a two-membered evolution strategy. In addition, a proposal is made for the extension of the Cailletaud model [1] by means of an evolution equation for a damage variable which enables the modelling of the tertiary creep stage. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlocal damage-viscoplastic model for high temperature creep of single crystal superalloys

A new nonlocal damage-viscoplastic model for high temperature creep of single crystal superalloys is developed. It is based on the variational formulation consisting of free energy, plastic and damage dissipation potentials. Evolution equations for plastic strain and damage variables are derived from the minimum principle for dissipation potentials [1]. The model is capable of describing different stages of creep in a unified way. The evolution of dislocation densities of gamma and gamma prime phases in superalloys incorporates plastic deformation. It results in the time-dependence of the creep rate in primary and secondary creep. Tertiary creep is taken into account by introducing local and nonlocal damage variables. Herein the nonlocal one is considered as numerical treatment to remove mesh-dependence. Numerical results and comparisons with experimental data of the single crystal superalloy LEK94 are shown. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Rank-One Convexification Approach for the Modeling of Magnetic Shape Memory Response

We present an incremental energy minimization model for magnetic shape memory alloys (MSMAs) whose derivation departs from the constrained theory of magnetoelasticity [1], but additionally accounts for elastic deformations, magnetization rotation, and dissipative mechanisms. The minimization of the proposed incremental energy yields the evolution of the internal state variables. In this sense, the presented modeling concept clearly distinguishes itself from standard phenomenological approaches to MSMA modeling [4]. The extended model is applied to simulate the response of single crystalline Ni₂MnGa. It is shown to accurately capture the nonlinear, anisotropic, hysteretic, and highly stress level-dependent features of MSMA behavior, based on just a few fundamental material parameters, which is validated by comparison to experimental data. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalized constitutive equation for creep of polymers at multiaxial loading

Summary This paper introduced a unified formulation for generalized deformation models including load dependent effects (2nd order effects). It is given in more detail for stationary creep of isotropic, orthotropic, and anisotropic material behavior. A further generalization of the introduced 6-parameter constitutive equation is possible by coupling creep and damage. These generalizations include the classical theory of creep damage [13]. The proof of the proposed theory is given in [20–22] for special cases with a reduced number of material parameters. The results of calculations show a good agreement with results from multiaxial tests.Dedicated to our colleague and friend Vitauts Tamus on the occasion of his sixtieth birthdayPublished in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6, pp. 723–733, November–December, 1995.     

Topology and material orientation optimization based on evolution equations

Many modern high-performance materials have inherent anisotropic elastic properties and its local material orientation can be considered to be an additional design variable for the topology optimization [1–3]. We extend our previous model for topology optimization with variational controlled growth [4–6] for linear elastic anisotropic materials, for which the material orientation is introduced as an additional design variable. We solve the optimization problem purely with the principles of thermodynamics by minimizing the Gibbs energy. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An axisymmetric problem of an embedded mixed-mode crack in a functionally graded magnetoelectroelastic infinite medium

This paper investigates the problem of an axisymmetric penny shaped crack embedded in an infinite functionally graded magneto electro elastic medium. The loading consists of magnetoelectromechanical loads applied on the crack surfaces assumed to be magneto electrically impermeable. The material’s gradient is parallel to the axisymmetric direction and is perpendicular to the crack plane. An anisotropic constitutive law is adopted to model the material behavior. The governing equations are converted analytically using Hankel transform into coupled singular integral equations, which are solved numerically to yield the crack tip stress, electric displacement and magnetic induction intensity factors. A similar problem but with a different crack morphology, that is a plane crack embedded in an infinite functionally graded magneto electro elastic medium, was considered by the authors in a previous work (Rekik et al., 2012) [25]. While the overall solution schemes look similar, the axisymmetric problem resulted in more mathematical complexities and let to different conclusions with respect to the influence of coupling between elastic, electric and magnetic effects. The main focus of this paper is to study the effect of material non-homogeneity on the fields’ intensity factors to understand further the behavior of graded magnetoelectroelastic materials containing penny shaped cracks and to inspect the effect of varying the crack geometry.     

Micro-mechanical analysis of creep behavior in a multipass weld

A method of evaluating creep response of the multipass weld based on the micro-macro mechanics approach is introduced. Multipass weld microstructure that consists from columnar, coarse and fine grained zones is considered. Materials of these constituents assumed to be isotropic. Weld metal properties of inelastic behavior have general type of symmetry and are described by the anisotropic creep constitutive model. To model the microstructure of the multipass weld metal the representative volume element (RVE) is created for CAE Abaqus. Material properties of weld metal grain type zones are taken from the experiments. Numerical tests on uniform loading of the RVE are performed. Creep material properties for equivalent weld material are found for welds with different number of passes. The symmetry type of the creep material properties of multi-pass weld are evaluated for the equivalent weld material. As an example of macro model analysis of the welding, the creep calculation of the cylindrical shell with the welding under the uniform inner pressure is performed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear micromechanical model of filament-wound composites considering fiber undulation

Flexible-matrix composites with highly anisotropic properties have successfully been used in numerous fields to improve the performance of conventional structures or to facilitate new innovations. Many of them are designed on the basis of tubes which are produced efficiently by the filament winding process. To predict the elastic behavior of filament-wound flexible-matrix composites, aspects of the nonlinear behavior of the flexible material have to be considered, as well as the features of the distinct fiber undulation geometry inherent to the filament winding process. The present study considers these characteristics in the micromechanical modeling of the elastic behavior by including a nonlinear material model to represent the strain-dependent moduli and manufacturing-dependent geometries. The structure is characterized by a unit cell and subcells, analyzed separately and combined based on different sets of isostress and isostrain assumptions that depend on the winding angle. On the basis of experimentally obtained nonlinear lamina properties, an iterative method of solution is chosen to calculate the axial stress–strain behavior of tubes with various winding parameters. The resulting predictions are validated by testing tubes in tension and compression. The model shows good agreement with the experiments. Predictions made using the model show a strong influence of filament winding parameters on the axial modulus of flexible-matrix composite tubes.     

Effect of random irregularities of the creep of reinforced layered plastics

The authors investigate the creep of inhomogeneous materials consisting of a large number of stiff orthotropic elastic layers alternating with layers of linear isotropic viscoelastic material. The elastic layers are assumed to be almost plane; the functions describing the irregularities (curvature) form a random field. The averaged characteristics of the medium are found together with the variation of the averaged displacements and strains in time. An analogous problem was previously considered in [1, 6] on the assumption that the binder layers are elastic. The present paper is based on the equations of [1] and the elastic-viscoelastic correspondence principle [4]. When the correlation scales of the irregularities are small as compared with the dimensions of the body and the characteristic distances over which the averaged parameters of the stress-strain state vary appreciably is considered in detail. A relation is established between the creep functions for simple cases of the state of stress and the parameters characterizing the properties of the components, the properties of the random field of initial irregularities, etc. The development of perturbations with different wave numbers is investigated. The theory is used to describe the creep of reinforced layered plastics.Mekhanika Polimerov, Vol. 2, No. 5, pp. 755–762, 1966     

Parameter–identification of macroscopic material models based on virtual testing of given material mesostructures

For many heterogeneous materials such as composites and polycrystals, the material modeling for the constituents on a representative mesoscale can be considered as known, including concrete values of their inherent material parameters. Typical examples are isotropic elastic–plastic models for the constituents of composites or anisotropic crystal–plasticity models for the grains of polycrystals. This knowledge can be exploited with regard to the modeling of the homogenized macroscopic response. In particular, parameters in macroscopic models may be identified by virtual experiments provided by a computational deformation–driving of representative mesostructures. This paper outlines the general concept for the parameter–identification of macroscopic materialmodels based on the virtual testing of given material mesostructures. The virtual test data are obtained in the form of multi–dimensional stress–strain paths by applying different deformation gradients to a given mesostructure. After specifying a corresponding macroscopic material model covering the observed effects on the macroscale, the material parameters are identified by a least–square–type optimization procedure that optimizes the macroscopic material parameters. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Anisotropic creep modeling based on elastic projection operators

The paper presents a unified approach for creep modeling of anisotropic materials, and is specified in more detail to the cases of isotropy, cubic symmetry and transversal isotropy. Thereby an additive decomposition of the elastic and inelastic strain tensors into dilational and isochoric Kelvin modes is assumed. Each of these modes is obtained from fourth order projection operators, resulting from solution of the eigenvalue problem for the fourth order anisotropic elasticity tensor. For simplicity the amount of strain rate for each mode is determined with a Norton type ansatz in terms of an equivalent stress, and the experimental phenomenon of primary creep is taken into account by a back stress tensor of Armstrong‐Frederick type, which is also decompose into Kelvin modes. Two numerical creep simulations investigate the crystal orientation for a compact tension specimen made out of CMSX‐4 superalloy.     

Numerical Calculation of Fiber Orientation in Three-Dimensional Arterial Walls

In this contribution an approach for the fiber reorientation in three-dimensional arterial walls is presented. In detail the load-bearing capacity of the tissue is increased by re orienting the fibers with respect to the principal stresses, cf. [1]. The improved fiber reorientation algorithm is combined with the polyconvex nonlinear anisotropic material model presented in [3]. The results of a three-dimensional finite element simulation, where the reorientation approach is applied to a short segment of a patient-specific arterial geometry, are presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic constants for a material reinforced in two directions in the planar case

A simplified model is proposed for an elastic material (glass laminate) reinforced in two perpendicular directions. On the basis of an analysis of the cooperation of the reinforcement and the binder in the assumed model, approximate formulas are obtained for the reduced elastic constants of the reinforced material treated as an anisotropic and macroscopically homogeneous body. A slightly different approach to the solution of a similar problem is described in [2, 4]. The formulas obtained are in agreement with the experimental data and the recommendations of other authors.Mekhanika Polimerov, Vol. 3, No. 2, pp. 266–272, 1967     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

A model of weak viscoelastic nematodynamics

The paper develops a continuum theory of weak viscoelastic nematodynamics of Maxwell type. It can describe the molecular elasticity effects in mono-domain flows of liquid crystalline polymers as well as the viscoelastic effects in suspensions of uniaxially symmetric particles in polymer fluids. Along with viscoelastic and nematic kinematics, the theory employs a general form of weakly elastic thermodynamic potential and the Leslie–Ericksen–Parodi type constitutive equations for viscous nematic liquids, while ignoring inertia effects and the Frank (orientation) elasticity in liquid crystal polymers. In general case, even the simplest Maxwell model has many basic parameters. Nevertheless, recently discovered algebraic properties of nematic operations reveal a general structure of the theory and present it in a simple form. It is shown that the evolution equation for director is also viscoelastic. An example of magnetization exemplifies the action of non-symmetric stresses. When the magnetic field is absent, the theory is reduced to the symmetric, fluid mechanical case with relaxation properties for both the stress and director. Our recent analyses of elastic and viscous soft deformation modes are also extended to the viscoelastic case. The occurrence of possible soft modes minimizes both the free energy and dissipation, and also significantly decreases the number of material parameters. In symmetric linear case, the theory is explicitly presented in terms of anisotropic linear memory functionals. Several analytical results demonstrate a rich behavior predicted by the developed model for steady and unsteady flows in simple shearing and simple elongation.     

Nonlinear creep in high-density polyethylene under time-dependent stresses

The authors have performed experiments on the creep of specimens of high-density poly-ethylene in states of plane stress which vary linearly with time. The theoretical curves are calculated from equations of two types, namely the cubic and principal cubic theories of viscoelasticity [3]. The parameters of the material are taken from a previous paper [1].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 796–803, September–October, 1973.     

Solution-precipitation creep – micromechanical modelling and numerical results

Our aim is to present a continuum mechanical model for solution-precipitation creep as well as to compare the numerical results based on that model with experimental observations. The formulation of the problem is based on the minimization of a Lagrangian consisting of elastic power and dissipation. Elastic energy is chosen to be in a standard form but dissipation is strongly adapted to the solution-precipitation process by introducing two new quantities: the velocity of material transport within the crystallite-interfaces and the normal velocity of precipitation or solution respectively. The model enables one to give an analytical solution for the case of a single crystal and numerical solution based on a finite element method for more complex, polycrystalline materials. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)