Multiscale modeling of irradiated polycrystalline FCC metals

We propose a set of models for the post-irradiation deformation response of polycrystalline FCC metals. First, a defect- and dislocation-density based evolution model is developed to capture the features of irradiation-induced hardening as well as intra-granular softening. The proposed hardening model is incorporated within a rate-independent single crystal plasticity model. The result is a non-homogeneous deformation model that accounts for defect absorption on the active slip planes during plastic loading. The macroscopic non-linear constitutive response of the polycrystalline aggregate of the single crystal grains is then obtained using a micro–macro transition scheme, which is realized within a Jacobian-free multiscale method (JFMM). The Jacobian-free approach circumvents explicit computation of the tangent matrix at the macroscale by using a Newton–Krylov process. This has a major advantage in terms of storage requirements and computational cost over existing approaches based on homogenized material coefficients in which explicit Jacobian computation is required at every Newton step. The mechanical response of neutron-irradiated single and polycrystalline OFHC copper is studied and it is shown to capture experimentally observed grain-level phenomena.     

Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

In order to model the effects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic-plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations reflect the macroscopic experimentally observed tensile stress-strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress-strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress-strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension.     

A new thermo-elasto-plasticity constitutive theory for polycrystalline metals

In this study, the behavior of polycrystalline metals at different temperatures is investigated by a new thermo-elasto-plasticity constitutive theory. Based on solid mechanical and interatomic potential, the constitutive equation is established using a new decomposition of the deformation gradient. For polycrystalline copper and magnesium,the stress–strain curves from 77 to 764 K(copper), and 77 to 870 K(magnesium) under quasi-static uniaxial loading are calculated, and then the calculated results are compared with the experiment results. Also, it is determined that the present model has the capacity to describe the decrease of the elastic modulus and yield stress with the increasing temperature, as well as the change of hardening behaviors of the polycrystalline metals. The calculation process is simple and explicit,which makes it easy to implement into the applications.     

Inelastic behaviour of polycrystalline metals under complex loading condition

In this paper, a simple mechanical model of polycrystal is suggested on the basis of the crystal plasticity. This model incorporates interactions among grains in polycrystal and interactions among slip systems in a crystal grain component. Especially, an equation for a slip strain of slip system is based on a theory of thermo-activation motion of dislocation. In this way, the model can take into account rate (creep) properties and temperature effects as well as an induced plastic anisotropy depending on deformation path, under multi-axial condition.Some computations are done by using the model for typical sets of strain paths and compared with corresponding experimental results of aluminum alloy at 200°C. The computational results reproduce accurately typical features observed in the experiments.     

Intergranular strain evolution near fatigue crack tips in polycrystalline metals

The deformation field near a steady fatigue crack includes a plastic zone in front of the crack tip and a plastic wake behind it, and the magnitude, distribution, and history of the residual strain along the crack path depend on the stress multiaxiality, material properties, and history of stress intensity factor and crack growth rate. An in situ, full-field, non-destructive measurement of lattice strain (which relies on the intergranular interactions of the inhomogeneous deformation fields in neighboring grains) by neutron diffraction techniques has been performed for the fatigue test of a Ni-based superalloy compact tension specimen. These microscopic grain level measurements provided unprecedented information on the fatigue growth mechanisms. A two-scale model is developed to predict the lattice strain evolution near fatigue crack tips in polycrystalline materials. An irreversible, hysteretic cohesive interface model is adopted to simulate a steady fatigue crack, which allows us to generate the stress/strain distribution and history near the fatigue crack tip. The continuum deformation history is used as inputs for the micromechanical analysis of lattice strain evolution using the slip-based crystal plasticity model, thus making a mechanistic connection between macro- and micro-strains. Predictions from perfect grain-boundary simulations exhibit the same lattice strain distributions as in neutron diffraction measurements, except for discrepancies near the crack tip within about one-tenth of the plastic zone size. By considering the intergranular damage, which leads to vanishing intergranular strains as damage proceeds, we find a significantly improved agreement between predicted and measured lattice strains inside the fatigue process zone. Consequently, the intergranular damage near fatigue crack tip is concluded to be responsible for fatigue crack growth.     

Large strain deformation of polycrystalline metals at low homologous temperatures

Large strain compression data (true strains to about ?3.0) are presented for polycrystalline α U and α Fe at room temperature. The results, together with other published data at low homologous temperatures (≈0.2 T_m), where T_m is the absolute melting temperature, suggest that a steady-state flow stress σ_s is approached after extensive strain-hardening, α U exhibits a very high strain-hardening rate, with σ_s ≈ 2900 MPa (420 ksi) indicating that cold-working is a very potent method of strengthening this metal. All the data evaluated can be fit by the stress-strain relation σ = σ_s? exp (?(Nε)^p)(σ_s? σ_y), where σ_y is the yield stess, p is a constant equal to a for the metals analyzed, N is a constant associated with the strain-hardening characteristics of a material, σ is true stress, and ε is true strain.     

A multiscale approach for modeling scale-dependent yield stress in polycrystalline metals

Modeling of scale-dependent characteristics of mechanical properties of metal polycrystals is studied using both discrete dislocation dynamics and continuum crystal plasticity. The initial movements of dislocation arc emitted from a Frank-Read type dislocation source and bounded by surrounding grain boundaries are examined by dislocation dynamics analyses system and we find the minimum resolved shear stress for the FR source to emit at least one closed loop. When the grain size is large enough compared to the size of FR source, the minimum resolved shear stress levels off to a certain value, but when the grain size is close to the size of the FR source, the minimum resolved shear stress shows a sharp increase. These results are modeled into the expression of the critical resolved shear stress of slip systems and continuum mechanics based crystal plasticity analyses of six-grained polycrystal models are made. Results of the crystal plasticity analyses show a distinct increase of macro- and microscopic yield stress for specimens with smaller mean grain diameter. Scale-dependent characteristics of the yield stress and its relation to some control parameters are discussed.     

A two-dimensional dislocation dynamics model of the plastic deformation of polycrystalline metals

Two-dimensional dislocation dynamics (2D-DD) simulations under fully periodic boundary conditions are employed to study the relation between microstructure and strength of a material. The material is modeled as an elastic continuum that contains a defect microstructure consisting of a preexisting dislocation population, dislocation sources, and grain boundaries. The mechanical response of such a material is tested by uniaxially loading it up to a certain stress and allowing it to relax until the strain rate falls below a threshold. The total plastic strain obtained for a certain stress level yields the quasi-static stress-strain curve of the material. Besides assuming Frank-Read-like dislocation sources, we also investigate the influence of a pre-existing dislocation density on the flow stress of the model material. Our results show that - despite its inherent simplifications - the 2D-DD model yields material behavior that is consistent with the classical theories of Taylor and Hall-Petch. Consequently, if set up in a proper way, these models are suited to study plastic deformation of polycrystalline materials.     

Anisotropic yield criterion for polycrystalline metals using texture and crystal symmetries

An anisotropic yield criterion for polycrystalline metals which uses texture data and takes advantage of crystal symmetries is presented. A linear transformation is developed to map an anisotropic yield surface for a polycrystal to an appropriate isotropic yield surface. The transformation developed reflects the symmetry of the material being modeled. First, the transformation is determined. Then, information regarding the orientation distribution (texture) of the crystals in a polycrystalline aggregate is used to determine, via averaging, the transformation for the polycrystal. The transformation, along with appropriate isotropic yield surface, provides a phenomenological approach to modeling yield, yet accounts for microstructural texture. The approach reduces to the Hill (1950) anisotropic plasticity theory under certain conditions. The yield surfaces and R-values for various face-centered-cubic ( fcc) polycrystalline textures are computed by this method. Results compare favorably with those given by other theories, and with experiment. The method proves to have the computational efficiency of phenomenological approaches to modeling yield, while effectively incorporating the physics of more complex crystallographic approaches.     

Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis

The objective of this contribution is to develop an elastic-plastic-damage constitutive model for crystal grain and to incorporate it with two-scale finite element analyses based on mathematical homogenization method, in order to characterize the macroscopic tensile strength of polycrystalline metals. More specifically, the constitutive model for single crystal is obtained by combining hyperelasticity, a rate-independent single crystal plasticity and a continuum damage model. The evolution equations, stress update algorithm and consistent tangent are derived within the framework of standard elastoplasticity at finite strain. By employing two-scale finite element analysis, the ductile behaviour of polycrystalline metals and corresponding tensile strength are evaluated. The importance of finite element formulation is examined by comparing performance of several finite elements and their convergence behaviour is assessed with mesh refinement. Finally, the grain size effect on yield and tensile strength is analysed in order to illustrate the versatility of the proposed two-scale model.     

On the modeling and simulation of induced anisotropy in polycrystalline metals with application to springback

The presence of initial, and the development of induced, anisotropic elastic and inelastic material behavior in polycrystalline metals, can be traced back to the influence of texture and dislocation substructural development on this behavior. As it turns out, via homogenization or other means, one can formulate effective models for such structure and its effect on the macroscopic material behavior with the help of the concept of evolving structure tensors. From the constitutive point of view, these quantities determine the material symmetry properties. Most importantly, all dependent constitutive fields (e.g., stress) are by definition isotropic functions of the independent constitutive variables, which include these evolving structure tensors. The evolution of these tensors during loading results in an evolution of the anisotropy of the material. From an algorithmic point of view, the current approach leads to constitutive models which are quite amenable to numerical implementation. To demonstrate the applicability of the resulting constitutive formulation, we apply it to the case of metal plasticity with combined hardening involving both deformation- and permanently induced anisotropy. Comparison of simulation results based on this model for the bending tension of aluminum-alloy sheet-metal strips with corresponding experimental ones show good agreement.     

The influence of a limiting dislocation flux on the mechanical response of polycrystalline metals

Certain physical results from the study of dislocation motions lead to the concept of ideal viscoplasticity. In particular the existence of a limiting dislocation velocity, coupled with an upper limit on dislocation density, provides an upper bound on the dislocation flux. Thus, the plastic strain rate in polycrystalline metals must also be bounded in many cases of interest. This physical situation can be idealized by postulating transition functions from zero to maximum flux in as simple as possible a manner consistent with the problem under investigation and the solution features to be examined. A drastic transition function is given here which leads to multiaxial stress, strain, strain-rate relations of reasonable simplicity and these are illustrated by application to several example problems. A common feature of the solutions of the examples treated is that the material response is partly rate dependent and partly rate independent. This indicates that the corresponding physical situations are characterized by large dislocation fluxes during part of the time and very small fluxes at other times.     

An anisotropic elastic-plastic constitutive model for single and polycrystalline metals. I-theoretical developments

An anisotropic elastic-plastic constitutive model for single and polycrystalline metals is proposed. The anisotropic hardening of single crystals, at first, is discussed with the viewpoint of yield surface and a new formulation of it is proposed. Then, a model for the anisotropic hardening of polycrystals is suggested by increasing the number of slip systems and incorporating the interaction of all slip systems. The interaction of grains through grain boundaries is shown to be similar to, and incorporated into, the interaction of slip systems in grains. The numerical predictions and their comparisons with experiments will follow in Part II of this paper.     

On the mathematical description of the tensile stress-strain curves of polycrystalline face centered cubic metals

The stress-strain curves of polycrystalline face centered cubic metals (aluminium, copper, gold, silver and nickel) were measured by an Instron tensile test machine controlled by a computer registering 20 points per second along the load-elongation curve. The large number of measured points makes it possible to perform a detailed analysis of the mathematical structure of the stress-strain curves. General conditions are introduced which must be fulfilled by the functions which approximately describe the whole stress-strain curves. It is shown that the limiting value of a special set of suitable functions in practice describes the stress-strain curves exactly.     

Assessment of crystal plasticity based calculation of the lattice spin of polycrystalline metals for FE implementation

When texture is incorporated in the finite element simulation of a metal forming process, much computer time can be saved by replacing continuous texture and corresponding yield locus updates by intermittent updates after strain intervals of e.g. 20%. The hypothesis that the evolution of the anisotropic properties of a polycrystalline material during such finite interval of plastic deformation can be modelled by just rotating the initial texture instead of continuously updating it by means of a polycrystal deformation model is tested in this work. Two spins for rotating the frame have been assessed: the classical rigid body spin and a crystal plasticity based “Mandel spin” (calculated from the rotated initial texture) which is the average of the spins of all the crystal lattices of the polycrystal. Each of these methods was used to study the evolution of the yield locus and the r-value distribution during the 20% strain interval. The results were compared to those obtained by simulating the texture evolution continuously using a polycrystal deformation model. When the texture was not updated during deformation, it was found that for most initial textures the Mandel spin does not perform better than the rigid body spin, except for some special initial textures for which the Mandel spin is much better. The latter ones are textures which are almost stable for the corresponding strain mode. When the texture was updated after each strain interval of e.g. 20% the Mandel spin performed much better than the rigid body spin.     

Characterization of yielding behavior of polycrystalline metals with single crystal plasticity based on representative characteristic length

Single crystal plasticity based on a representative characteristic length is proposed and introduced into a homogenization approach based on finite element analyses, which are applied to characterization of distinctive yielding behaviors of polycrystalline metals, yield-point elongation, and grain size strengthening. The computational manner for an implicit stress update is derived with the framework of a standard multi-surface plasticity at finite strain, where the evolution of the characteristic lengths are numerically converted from the accumulated slips of all of slip systems by exploiting the mathematical feature of the characteristic length as the intermediate function of the plastic internal variables. Furthermore, a constitutive model for a single crystal reproduces the stress–strain curve divided into three parts. Using two-scale finite element analysis, the macroscopic stress–strain response with yield-point elongation under a situation of low dislocation density is reproduced. Finally, the grain size effect on the yield strength is analyzed with modeling of the grain boundary in the context of the proposed constitutive model and is discussed from both macroscopic and microscopic views.     

Characterizing texture in polycrystalline materials by ultrasonic waves

A method for characterizing texture from measurements of ultrasonic wave velocities is proposed. In polycrystalline aggregates, ultrasonic wave velocities are strongly affected by orientation distribution coefficients (ODCs), which are usually used to describe the degree of preferred grain orientation in textured materials. In this work, velocities of longitudinal and transverse waves propagating into aluminum alloy 6061 were measured under pure shear, simple shear and uniaxial tension. From the measured ultrasonic wave velocities, the ODCs W₄₀₀ and W₄₂₀ were calculated to infer the deformation-induced texture. The predicted pole figures, obtained using ultrasonic velocities, were in good qualitative agreement with the finite element polycrystal model analyzed pole figures.     

Inelastic constitutive equations of polycrystalline metals subjected to finite deformation part I: Effect of grain rotation

In this article, a set of inelastic constitutive equations of polycrystalline metals is derived by combining a finite deformation kinematics of single crystal component, and a shear stress-shear strain relation of slip system based on a thermoactivated motion of dislocation. Interactions among grains are incorporated by “constant deformation gradient assumption.” The forms of these equations are rather simple internal variable theory types. By using these equations, some fundamental effects of grain rotations on inelastic behaviors of polycrystalline metals in a finite deformation range under complex loading and elevated temperature conditions are demonstrated. Some comments are given on a problem of plastic spin tensor.     

Scattering by an anisotropic circle

The scattering by a circle is considered when the outside medium is isotropic and the inside medium is anisotropic (orthotropic). The problem is a scalar one and is phrased as a scattering problem for elastic waves with polarization out of the plane of the circle (SH wave), but the solution is with minor modifications valid also for scattering of electromagnetic waves. The equation inside the circle is first transformed to polar coordinates and it then explicitly contains the azimuthal angle through trigonometric functions. Making an expansion in a trigonometric series in the azimuthal coordinate then gives a coupled system of ordinary differential equations in the radial coordinate that is solved by power series expansions. With the solution inside the circle complete the scattering problem is solved essentially as in the classical case. Some numerical examples are given showing the influence of anisotropy, and it is noted that the effects of anisotropy are generally strong except at low frequencies where the dominating scattering only depends on the mean stiffness and not on the degree of anisotropy.