Micromechanical Considerations Research Articles

Yield Surfaces and Plastic Potentials for Metals, with Analysis of Plastic Dilatation and Strength Asymmetry in BCC Crystals

Since the 1980s, constitutive modeling has steadily migrated from phenomenological descriptions toward approaches that are based on micromechanics considerations. Despite significant efforts, crystal plasticity remains an open field of research. Among the unresolved issues are the anomalous behavior of metals at low temperatures and the stress upturn at extreme dynamics. This work is focused on the low-temperature responses of body-centered-cubic (bcc) metals, among them, molybdenum (Mo). At these conditions, the plastic flow strength is governed by the motion of screw dislocations. The resultant non-planarity of core structures and slip causes the following: the shear stress includes non-glide components, the Schmid law is violated, there is a tension-compression asymmetry, and the yield surface and plastic potential are clearly decoupled. We find that the behavioral complexities can be explained by atomistically resolved friction coefficients in macroscopic yield and flow. The plastic flow mechanisms establish the departure point into the follow-up analysis of yield surfaces. For example, we know that while the von Mises stress is explained based on energy considerations, we will also show that the stress has a clear geometric interpretation. Moreover, the von Mises stress is just one case within a much broader class of equivalent stresses. Possible correlations among non-Schmid effects (as represented macroscopically by friction coefficients), volume change (i.e., residual elastic dilatation) from dislocation lines, and elastic anisotropy are investigated. Extensions to the shock regime are also established.

Compression of Fibrous Assemblies: Revisiting the Stress–Density Relation

Abstract Many engineering materials are made from fibers, and fibrous assemblies are often compacted during the fabrication process. Compression leads to the formation of contacts between fibers, and this causes stiffening. The relation between the uniaxial stress, S, and the volume fraction of fibers, φ, is of power law form. The derivation of this relation based on micromechanics considerations takes as input the structural evolution represented by the dependence of the mean segment length of the network, lc, on the current density, ρ (ρ is defined as the total length of fiber per unit volume of the network). In this work, we revisit this problem while considering that the mean segment length should be defined exclusively by fiber contacts that transmit load. We use numerical simulations of the compression of crimped fiber assemblies to show that, when using this definition, ρ∼1/lc2 at large enough strains. Purely geometric considerations require that ρ∼1/lc, and we observe that this applies in the early stages of compaction. In pre-stressed networks, the density–mean segment length scaling is of the form ρ∼1/lc2 at all strains. This has implications for the relation between stress and the fiber volume fraction. For both ρ versus lc scalings, S∼(φn−φ0n), where φ0 is the initial or reference fiber volume fraction; however, n = 3 when ρ∼1/lc and n = 2 for ρ∼1/lc2. These predictions are compared with experimental data from the literature.

Hydrogen assisted cracking paths in cold drawn pearlitic steels

This paper analyses hydrogen assisted cracking (HAC) paths in cold drawn pearlitic steels in which progressive cold drawing produces a preferential orientation of the pearlitic microstructure in the matter of colonies and lamellae, thereby inducing strength anisotropy in the steel, and thus the resistance to HAC is a directional property that depends on the angle in relation to the drawing direction. Therefore, an initial transverse crack changes its propagation direction to approach that of the wire axis, thus producing mixed mode propagation, the deflection angle being an increasing function of the cold drawing degree. This experimental result may be explained by micro-mechanical considerations on the basis of the lamellar microstructure of the steels. A relationship is established between the microstructural angles and the deflection angles of the macroscopic HAC path, thus providing a materials science type relationship between the microstructure and the macroscopic crack.

Stress corrosion cracking paths in cold drawn pearlitic steels

This paper analyses stress corrosion cracking (SCC) paths in cold drawn pearlitic steels in which progressive cold drawing produces a preferential orientation of the pearlitic microstructure in the matter of colonies and lamellae, thereby inducing strength anisotropy in the steel, and thus the resistance to SCC is a directional property that depends on the angle in relation to the drawing direction. Therefore, an initial transverse crack changes its propagation direction to approach that of the wire axis, thus producing mixed mode propagation, the deflection angle being an increasing function of the cold drawing degree. This experimental result may be explained by micro-mechanical considerations on the basis of the lamellar microstructure of the steels. A relationship is established between the microstructural angles and the deflection angles of the macroscopic SCC path, thus providing a materials science type relationship between the microstructure and the macroscopic crack.

Auxetic orthotropic materials: Numerical determination of a phenomenological spline-based stored density energy and its implementation for finite element analysis

Auxetic materials, which have negative Poisson’s ratio, show potential to be used in many interesting applications. Finite element analysis (FEA) is an important phase in implementing auxetic materials, but may become computationally expensive because simulation often needs microscale details and a fine mesh. It is also necessary to check that topological aspects of the microscale reflects not only micro but macromechanical behavior. This work presents a phenomenological approach to the problem using data-driven spline-based techniques to properly characterize orthotropic auxetic material requiring neither analytical constraints nor micromechanics, expanding on previous methods for isotropic materials. Hyperelastic energies of auxetic orthotropic material are determined from experimental data by solving the equilibrium differential functional equations directly, so no fitting or analytical estimation is necessary. This offers two advantages; (i) it allows the FEA study of orthotropic auxetic materials without requiring micromechanics considerations, reducing modeling and computational time costs by two to three orders of magnitude; (ii) it adapts the hyperelastic energies to the nature of the material with precision, which could be critical in scenarios where accuracy is essential (e.g. robotic surgery).

Non-standard contact conditions in generalized continua: microblock contact model for a Cosserat body

Generalized continuum theories involve non-standard boundary conditions that are associated with the additional kinematic variables introduced in those theories, e.g., higher gradients of the displacement field or additional kinematic degrees of freedom. Accordingly, formulation of a contact problem for such a continuum necessarily requires that adequate contact conditions are formulated for the additional kinematic variables and/or for the respective generalized tractions. In this paper, we address several related open problems, namely, how to enhance the classic contact conditions to include the effects of the additional kinematic variables, how to link the enhanced contact model to the underlying microstructure of the solid, and how to do it in a consistent manner. As a first step towards a new class of contact models for generalized continua, a microblock contact model is derived for a Cosserat solid based on simple micromechanical considerations. To illustrate the non-trivial effects introduced by the non-standard boundary conditions, the problem of compression of an infinite strip with nonaligned microblocks is considered, and the analytical solution is derived for the corresponding boundary layers. A Hertz-like contact problem is also solved numerically with the focus on non-standard features of the solution and on the related size effects.

Understanding the complete loss of uniform plastic deformation of some ultrafine-grained metallic materials in tensile straining

Reducing the grain size ranges among the most powerful methods to manage both high strength and high toughness of metallic structural materials. However, below certain grain sizes (often around 1 μm) some metallic materials lose most of their tensile ductility. In uniaxial tensile straining and related tensile forming processes, necking is then initiated immediately after first yielding. This in turn leads to a complete loss of uniform plastic deformation, a severe drawback in the use of these materials. Here, a new macromechanical approach in combination with micromechanical considerations is used to explain this long-standing riddle in materials science and engineering. The approach is based on the yield point phenomenon and the stress state that necessarily develops at the Lüders fronts. Using a simple new material model and a novel strain hardening law, it is clearly shown, that below a critical grain size the necessary nominal stress to spread out a Lüders band may be higher than the tensile strength. This invariably leads to a special form of plastic instability and the complete loss of uniform plastic deformation. Micro- and macromechanical considerations, experiments, analytical calculations and extensive finite element calculations are combined in this comprehensive study and fit together well. Furthermore, many features of the yield point phenomenon (the upper and lower yield strength, the spreading out of simple and complex Lüders bands, the stress fluctuations within the Lüders region) and the Hall-Petch relationship may now be understood in a new consistent way.

Homogenized Gurson-type behavior equations for strain rate sensitive materials

In this paper, the classical Gurson model for ductile porous media is extended for strain-rate-dependent materials. Based on micromechanical considerations, approximate closed-form macroscopic behavior equations are derived to describe the viscous response of a ductile metallic material. To this end, the analysis of the expansion of a long cylindrical void in an ideally plastic solid introduced by McClintock (J Appl Mech 35:363, 1968) is revisited. The classical Gurson yield locus has been modified to explicitly take into account the strain rate sensitivity parameter for strain rate power-law solids. Two macroscopic approaches are proposed in this work. Both models use the first term of a Taylor series expansion to approximate integrals to polynomial functions. The first proposed closed-form approach is analytically more tractable than the second one. The second approach is more accurate. In order to compare the proposed approximate Gurson-type macroscopic functions with the behavior of the original Gurson yield locus, numerical finite element analyses for cylindrical cells have been conducted for a wide range of porosities, triaxialities, and strain rate sensitivity parameters. The results presented evidence that, for large values of the rate sensitivity parameters, the proposed extended Gurson-type models have the important quality to better predict the behavior of rate sensitive materials than the classical one. They also provide simpler and accurate alternatives to more traditional viscoplastic models.

Serpentine locomotion through elastic energy release.

A model for serpentine locomotion is derived from a novel perspective based on concepts from configurational mechanics. The motion is realized through the release of the elastic energy of a deformable rod, sliding inside a frictionless channel, which represents a snake moving against lateral restraints. A new formulation is presented, correcting previous results and including situations never analysed so far, as in the cases when the serpent's body lies only partially inside the restraining channel or when the body has a muscle relaxation localized in a small zone. Micromechanical considerations show that propulsion is the result of reactions tangential to the frictionless constraint and acting on the snake's body, a counter-intuitive feature in mechanics. It is also experimentally demonstrated that the propulsive force driving serpentine motion can be directly measured on a designed apparatus in which flexible bars sweep a frictionless channel. Experiments fully confirm the theoretical modelling, so that the presented results open the way to exploration of effects, such as variability in the bending stiffness or channel geometry or friction, on the propulsive force of snake models made up of elastic rods.

Stress Corrosion Cracking of Progressively Cold-Drawn Pearlitic Steels: From Tintoretto to Picasso

Progressive cold drawing in eutectoid steels produces a preferential orientation of the pearlitic microstructure in the wire axis or drawing direction. This affects the posterior behaviour of the steels under conditions of stress corrosion cracking (SCC). The experimental results show that cold drawing induces strength anisotropy in the steel, and thus the resistance to SCC is a directional property that depends on the angle in relation to the drawing direction. Therefore, an initial transverse crack changes its propagation direction to approach that of the wire axis, thus producing mixed mode propagation, the deflection angle being an increasing function of the cold drawing degree. This experimental result may be explained by micro-mechanical considerations on the basis of the lamellar microstructure of the steels. A relationship is established between the microstructural angles and the deflection angles of the macroscopic SCC crack, thus providing a materials science type relationship between the microstructure and the macroscopic crack paths with regard to hydrogen assisted cracking (HAC) associated with the cathodic regime and in the matter of localised anodic dissolution (LAD) linked to the anodic regime. Whereas in hot rolled (not cold drawn at all) and slightly drawn steels the phenomenon of SCC develops in mode I, i.e., the SCC crack is a straight line linked with the classical linear perspective painting by Tintoretto, in the case of heavily drawn steels, the SCC deflected crack is a polygonal line associated with the multi-perspective cubist painting by Picasso, these conclusions being valid for both HAC and LAD.

Continuum damage model for ferroelectric materials and its application to multilayer actuators

In this paper a micromechanical continuum damage model for ferroelectric materials is presented. As a constitutive law it is implemented into a finite element (FE) code. The model is based on micromechanical considerations of domain switching and its interaction with microcrack growth and coalescence. A FE analysis of a multilayer actuator is performed, showing the initiation of damage zones at the electrode tips during the poling process. Further, the influence of mechanical pre-stressing on damage evolution and actuating properties is investigated. The results provided in this work give useful information on the damage of advanced piezoelectric devices and their optimization.

Experimental investigation and constitutive modelling of creep-damage behaviours in monzogranite

In this study, the creep-damage behaviour of monzogranite is first investigated through triaxial creep tests under different confining pressures. Creep deformation and its rate in the whole creep process are compared and analysed on the basis of the experimental results. It is seen that the rate of creep strains is closely related to applied stress and can be described by an exponential function in terms of deviatoric stress. A discrete viscoplastic damage constitutive model is then formulated in the discrete thermodynamic framework and with micro-mechanical consideration. Comparisons between experimental data and numerical prediction show that the model can well describe the typical phenomena of creep behaviour.

Determination of the mode I, mode II, and mixed-mode I–II delamination toughness of a graphite/polyimide composite at room and elevated temperatures

A study is described in which the mode I, mode II, and mixed-mode delamination toughnesses of a woven fabric graphite/polyimide composite are determined at room temperature and at 300℃. It is shown that, with some modifications, commonly used delamination toughness test methods can be extended for use into this relatively high-temperature regime. At room temperature, the mode II toughness was found to be significantly higher than that obtained in mode I. Relative to their fracture toughnesses at room temperature, the mode I and mixed-mode toughnesses increased at elevated temperature, while the mode II toughness decreased. Thus, at 300℃, the mode I toughness was larger than that obtained for mode II. The temperature dependence in all three modes was found to be consistent with what is predicted based on micromechanical considerations for the effects of temperature on the resin. In addition, the local tow orientations influenced the fracture path and produced toughening mechanisms that were both temperature and mode mix dependent.

Nonlinear reduced order homogenization of materials including cohesive interfaces

The mechanical response of composite materials is strongly influenced by the nonlinear behavior of the interface between the constituents. In order to make reliable yet computationally efficient predictions for such materials, a reduced order model is developed. Conceptual ideas of the NTFA (Michel and Suquet, Int J Solids Struct 40:6937---6955, 2003, Comput Methods Appl Mech Eng 193:5477---5502, 2004) and of the pRBMOR (Fritzen, Hodapp and Leuschner Comput Methods Appl Mech Eng 260:143---154, 2013, Fritzen et al., Comput Methods Appl Mech Eng 278:186---217, 2014) are adopted. The key idea is to parameterize the displacement jumps on the cohesive interfaces by a reduced basis of global ansatz functions. Micromechanical considerations and the potential structure of the constitutive models lead to a variational formulation and reduced equilibrium conditions. The effect of the preanalysis phase on the accuracy is investigated using geometrically optimal training directions. The reduced model is tested for three-dimensional microstructures. Besides the effective stress response, the tension---compression asymmetry and the distribution of the separation of the interface are investigated. Memory savings on the order of $$10^5$$105 are realized. The computing time is reduced considerably.

Quantitative intravoxel analysis of microCT-scanned resorbing ceramic biomaterials – Perspectives for computer-aided biomaterial design

Abstract

Damage model for ferroelectric materials and its applications to multilayer actuators

AbstractIn this paper a damage model for ferroelectric materials is presented. It is implemented in terms of a user element in the commercial FEM‐code Abaqus. The model is based on micromechanical considerations of domain switching and its interaction with microcrack growth and coalescence. Finite element analysis of a multilayer actuator is performed, showing principal stresses leading to crack initiation and damage of the actuator. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Macroscopic stress and strain in a doubly periodic array of dislocation dipoles.

It is known that in two-dimensional periodic arrays of dislocations the summation of the periodic image fields is conditionally convergent. This is due to the long-range character of the elastic fields of dislocations. As a result, the stress field obtained for a doubly periodic array of dislocation dipoles may contain a spurious constant stress that depends on the adopted summation scheme. In the present work, we provide, based on micromechanical considerations, a simple physical explanation of the origin of the conditional convergence of lattice sums of image interactions. In this context, the spurious stresses are found in a closed form for an arbitrary elastic anisotropy, and this is achieved without using the stress field of an individual dislocation. An alternative procedure is also developed where the macroscopic spurious stresses are determined using the solution of the Eshelby's inclusion problem.

An Experimental Study and Constitutive Modeling of Saturated Porous Rocks

This paper is devoted to the experimental characterization and constitutive modeling of saturated porous rocks. A typical porous chalk is investigated. Drained hydrostatic and triaxial compression tests are first performed to characterize the basic mechanical behavior of chalk. Drained triaxial tests with constant interstitial pressure are then carried out to study the effects of interstitial pressure on the plastic deformation and failure criterion. Finally, undrained triaxial compression tests are performed to investigate poromechanical coupling in saturated conditions. Based on the experimental data and some relevant micromechanical considerations, a micromechanics-based plastic model is proposed and extended to poroplastic coupling using the effective stress concept. The proposed model is verified through comparisons between the numerical results and experimental data for both drained and undrained tests.

A discrete viscoplastic damage model for time-dependent behaviour of quasi-brittle rocks

A discrete viscoplastic damage model is proposed for the description of time-dependent behaviour of quasi-brittle rocks. This model is inspired by micromechanical considerations. Rocks are considered as quasi-brittle materials containing a spatial distribution of microcracks. Viscoplastic strains of rocks are generated by the time-dependent frictional sliding of microcracks surfaces while the damage of rocks is related to the growth of microcracks. The two inelastic mechanisms are coupled. The model is formulated in the framework of discrete thermodynamics of irreversible processes. The local strains and stresses applied to microcracks are related to the macroscopic counterparts through appropriate kinematic and static conditions. Specific viscoplastic flow rule as well as damage evolution law is defined for each family of microcracks. The validity of proposed model is verified through comparisons between numerical results and experimental data in triaxial compression tests and creep tests for sandstone.

Nonlinear Constitutive Model and Finite Element Analysis of Damage Processes in Ferroelectrics

In this paper a damage model for ferroelectric materials is presented. It is implemented in terms of a user element in a commercial FEM-code Abaqus. The model is based on micromechanical considerations of domain switching and its interaction with microc- rack growth and coalescence. The influence of damage evolution on the effective material properties is demonstrated. Further, a finite element analysis of a multilayer actuator is performed, showing damage and crack patterns.