Creep,plasticity, and fatigue of single crystal superalloy

Single crystal components in gas turbine engines are subject to such extreme temperatures and stresses that life prediction becomes highly inaccurate resulting in components that can only be shown to meet their requirements through experience. Reliable life prediction methodologies are required both for design and life management. In order to address this issue we have developed a thermo-viscoplastic constitutive model for single crystal materials. Our incremental large strain formulation additively decomposes the inelastic strain rate into components along the octahedral and cubic slip planes. We have developed a crystallographic-based creep constitutive model able to predict sigmoidal creep behavior of Ni base superalloys. Inelastic shear rate along each slip system is expressed as a sum of a time dependent creep component and a rate independent plastic component. We develop a new robust, computationally efficient rate-independent crystal plasticity approach and combined it with creep flow rule calibrated for Ni-based superalloys. The transient variation of each of the inelastic components includes a back stress for kinematic hardening and latent hardening parameters to account for the stress evolution with inelastic strain as well as the evolution for dislocation densities. The complete formulation accurately predicts both monotonic and cyclic tests at different crystallographic orientations for constant and variable temperature conditions (low cycle fatigue (LCF) and thermo-mechanical fatigue (TMF) tests). Based on the test and modeling results we formulate a new life prediction criterion suitable for both LCF and TMF conditions.     

A viscoplastic constitutive model for nickel-base superalloy,part 2: modeling under anisothermal conditions

In Part 2 of this study, extensive deformation tests were carried out on the nickel-base polycrystalline superalloy IN738LC under isothermal and anisothermal conditions between 450 and 950 °C. Under the isothermal conditions, the material showed almost no rate/time-dependency below 700 °C, while it showed distinct rate/time-dependency above 800 °C. Regarding the cyclic deformation, slight cyclic hardening behavior was observed when the temperature was below 700 °C and the imposed strain rate was fast, whereas in the case of the temperature above 800 °C or under slower strain rate conditions, the cyclic hardening behavior was scarcely observed. Unique inelastic behavior was observed under in-phase and out-of-phase anisothermal conditions: with an increase in the number of cycles, the stress at higher temperatures became smaller and the stress at lower temperatures became larger in absolute value although the stress range was approximately constant during the cyclic loading. In other words, the mean stress continues to evolve cycle-by-cycle in the direction of the stress at lower temperatures. Based on the experimental results, it was assumed that evolution of the variable Y that had been incorporated into a kinematic hardening rule in Part 1 of this study is active under higher temperatures and is negligible under lower temperatures. The material constants used in the constitutive equations were determined with the isothermal data, and were expressed as functions of temperature empirically. The extended viscoplastic constitutive equations were applied to the anisothermal cyclic loading as well as the monotonic tension, stress relaxation, creep and cyclic loading under the isothermal conditions. It was demonstrated that the present viscoplastic constitutive model was successful in describing the inelastic behavior of the material adequately, including the anomalous inelastic behavior observed under the anisothermal conditions, owing to the consideration of the variable Y.     

Constitutive modeling of textured body-centered-cubic (bcc) polycrystals

A new latent hardening model for body-centered-cubic (bcc) single crystals motivated by the inapplicability of the Schmid law (Critical Resolved Shear Stress Criterion) is presented. This model is based on the asymmetry of shearing resistance of the {112} slip planes depending on the shearing direction in the sense of ‘twin’ and ‘anti-twin’. For the interpretation of deformation of polycrystalline aggregates depending upon initial texture, a constitutive law for bcc single crystals is developed. This law is based on a rigorous constitutive theory for crystallographic slip that accounts for the effects of strain hardening, rate-sensitivity and thermal softening. The deformation response of textured polycrystal is investigated by means of a Taylor type averaging scheme and an established numerical procedure. Results for textured tungsten polycrystals at low and high strain rates for two different textures [001] and [011] are presented and compared with experimental results. The predictions compare well with experimental observations for the [001] texture. In the [011] texture, due to the reduced symmetry of deformation, lateral tensile stresses develop even under uniaxial compression. These lateral tensile stresses are responsible for observed lack of ductility and transgranular failure in the [011] texture.     

Strain gradient crystal plasticity analysis of a single crystal containing a cylindrical void

The effects of void size and hardening in a hexagonal close-packed single crystal containing a cylindrical void loaded by a far-field equibiaxial tensile stress under plane strain conditions are studied. The crystal has three in-plane slip systems oriented at the angle 60° with respect to one another. Finite element simulations are performed using a strain gradient crystal plasticity formulation with an intrinsic length scale parameter in a non-local strain gradient constitutive framework. For a vanishing length scale parameter the non-local formulation reduces to a local crystal plasticity formulation. The stress and deformation fields obtained with a local non-hardening constitutive formulation are compared to those obtained from a local hardening formulation and to those from a non-local formulation. Compared to the case of the non-hardening local constitutive formulation, it is shown that a local theory with hardening has only minor effects on the deformation field around the void, whereas a significant difference is obtained with the non-local constitutive relation. Finally, it is shown that the applied stress state required to activate plastic deformation at the void is up to three times higher for smaller void sizes than for larger void sizes in the non-local material.     

Integration of self-consistent polycrystal plasticity with dislocation density based hardening laws within an implicit finite element framework: Application to low-symmetry metals

We present an implementation of the viscoplastic self-consistent (VPSC) polycrystalline model in an implicit finite element (FE) framework, which accounts for a dislocation-based hardening law for multiple slip and twinning modes at the micro-scale grain level. The model is applied to simulate the macro-scale mechanical response of a highly anisotropic low-symmetry (orthorhombic) crystal structure. In this approach, a finite element integration point represents a polycrystalline material point and the meso-scale mechanical response is obtained by the mean-field VPSC homogenization scheme. We demonstrate the accuracy of the FE-VPSC model by analyzing the mechanical response and microstructure evolution of α-uranium samples under simple compression/tension and four-point bending tests. Predictions of the FE-VPSC simulations compare favorably with experimental measurements of geometrical changes and microstructure evolution. Specifically, the model captures accurately the tension–compression asymmetry of the material associated with twinning, as well as the rigidity of the material response along the hard-to-deform crystallographic orientations.     

On the evolution of lattice deformation in austenitic stainless steels—The role of work hardening at finite strains

In this work, a three dimensional crystal plasticity-based finite element model is presented to examine the micromechanical behaviour of austenitic stainless steels. The model accounts for realistic polycrystal micromorphology, the kinematics of crystallographic slip, lattice rotation, slip interaction (latent hardening) and geometric distortion at finite deformation. We utilise the model to predict the microscopic lattice strain evolution of austenitic stainless steels during uniaxial tension at ambient temperature with validation through in situ neutron diffraction measurements. Overall, the predicted lattice strains are in very good agreement with those measured in both longitudinal and transverse directions (parallel and perpendicular to the tensile loading axis, respectively). The information provided by the model suggests that the observed nonlinear response in the transverse {200} grain family is associated with a competitive bimodal evolution of strain during inelastic deformation. The results associated with latent hardening effects at the microscale also indicate that in situ neutron diffraction measurements in conjunction with macroscopic uniaxial tensile data may be used to calibrate crystal plasticity models for the prediction of the inelastic material deformation response.     

A crystallographic model for the study of local deformation processes in polycrystals

Most engineering materials possess a polycrystalline structure. Under load the anisotropy of the constituent grains leads to strong inhomogeneities of stresses and strains on the grain level. In order to investigate the local deformation processes, a new crystallographic model for pure fcc metals in the low temperature range has been developed. It is based on the framework of crystal plasticity and uses the finite element method (FEM). The rate dependent constitutive equations consider isotropic as well as kinematic hardening, whereby the mutual interactions of dislocation processes on the different slip systems are taken into account. Comprehensive calculations show that the essential features of both single crystals—which serve as a test object for the constitutive equations—and polycrystals are reproduced correctly. Moreover the simulations allow a deeper understanding of the mechanisms that control the local deformation behaviour of metals, especially of the mutual interactions of slip system activity, local hardening and resulting local strain. Furthermore, the model may serve as a physically motivated base for a later inclusion of damage terms which allow investigations of damage and fatigue on the local scale.     

Modeling of polycrystals using a gradient crystal plasticity theory that includes dissipative micro-stresses

This study investigates thermodynamically consistent dissipative hardening in gradient crystal plasticity in a large-deformation context. A viscoplastic model which accounts for constitutive dependence on the slip, the slip gradient as well as the slip rate gradient is presented. The model is an extension of that due to Gurtin (Gurtin, M. E., J. Mech. Phys. Solids, 52 (2004) 2545–2568 and Gurtin, M. E., J. Mech. Phys. Solids, 56 (2008) 640–662)), and is guided by the viscoplastic model and algorithm of Ekh et al. (Ekh, M., Grymer, M., Runesson, K. and Svedberg, T., Int. J. Numer. Meths Engng, 72 (2007) 197–220) whose governing equations are equivalent to those of Gurtin for the purely energetic case. In contrast to the Gurtin formulation and in line with that due to Ekh et al., viscoplasticity in the present model is accounted for through a Perzyna-type regularization. The resulting theory includes three different types of hardening: standard isotropic hardening is incorporated as well as energetic hardening driven by the slip gradient. In addition, as a third type, dissipative hardening associated with plastic strain rate gradients is included. Numerical computations are carried out and discussed for the large strain, viscoplastic model with non-zero dissipative backstress.     

Rate-independent crystalline and polycrystalline plasticity,application to FCC materials

This paper deals with the simulation of the mechanical response and texture evolution of cubic crystals and polycrystals for a rate-independent elastic–plastic constitutive law. No viscous effects are considered. An algorithm is introduced to treat the difficult case of multi-surface plasticity. This algorithm allows the computation of the mechanical response of a single crystal. The corresponding yield surface is made of the intersection of several hyper-planes in the stress space. The problem of the multiplicity of the slip systems is solved thanks to a pseudo-inversion method. Self and latent hardening are taken into account. In order to compute the response of a polycrystal, a Taylor homogenization scheme is used. The stress–strain response of single crystals and polycrystals is computed for various loading cases. The texture evolution predicted for compression, plane strain compression and simple shear are compared with the results given by a visco-plastic polycrystalline model.     

Time-dependent ratchetting experiments of SS304 stainless steel

The time-dependent strain cyclic characteristics and ratchetting behaviours of SS304 stainless steel were investigated by uniaxial/multiaxial cyclic loading tests at room and elevated temperatures (350 and 700 °C). The effects of loading rate, peak/valley strain or stress holds, ambient temperature and non-proportional loading path on the cyclic softening/hardening and ratchetting behaviours of the material were discussed. It is shown that: the cyclic deformation of the material presents remarkable time-dependence at room temperature and 700 °C; the cyclic hardening feature and ratchetting strain depend significantly on straining or stressing rate, hold-time, ambient temperature and the non-proportionality of loading path; the time-dependent ratchetting is resulted from the slight opening of hysteresis loop and visco-plasticity together, and the viscosity is a dominating factor at 700 °C; at 350 °C, abnormal rate-dependence and quick shakedown of ratchetting are observed due to the dynamic strain aging of the material at this temperature. Some significant conclusions are obtained, which are useful to construct a constitutive model to describe the time-dependent ratchetting behaviour of the material. It is also stated that the unified visco-plastic constitutive model discussed here cannot provide reasonable simulation to the time-dependent ratchetting at 700 °C, especially to that with certain peak/valley stress hold, since the effect of the high viscosity on time-dependent ratchetting cannot be properly described by using a unified visco-plastic flow rule.     

Strain hardening in 2D discrete dislocation dynamics simulations: A new ‘2.5D’ algorithm

The two-dimensional discrete dislocation dynamics (2D DD) method, consisting of parallel straight edge dislocations gliding on independent slip systems in a plane strain model of a crystal, is often used to study complicated boundary value problems in crystal plasticity. However, the absence of truly three dimensional mechanisms such as junction formation means that forest hardening cannot be modeled, unless additional so-called ‘2.5D’ constitutive rules are prescribed for short-range dislocation interactions. Here, results from three dimensional dislocation dynamics (3D DD) simulations in an FCC material are used to define new constitutive rules for short-range interactions and junction formation between dislocations on intersecting slip systems in 2D. The mutual strengthening effect of junctions on preexisting obstacles, such as precipitates or grain boundaries, is also accounted for in the model. The new ‘2.5D’ DD model, with no arbitrary adjustable parameters beyond those obtained from lower scale simulation methods, is shown to predict athermal hardening rates, differences in flow behavior for single and multiple slip, and latent hardening ratios. All these phenomena are well-established in the plasticity of crystals and quantitative results predicted by the model are in good agreement with experimental observations.     

Analysis of dynamic shear bands in an FCC single crystal

We study plane strain dynamic thermomechanical deformations of an fcc single crystal compressed along the crystallographic direction [010] at an average strain rate of 1000 sec⁻¹. Two cases are studied; one in which the plane of deformation is parallel tothe plane (001) of the single crystal, and another one with deformation occuring in the plane (101̄) of the single crystal. In each case, the 12 slip systems are aligned symmetrically about the two centroidal axes. We assume that the elastic and plastic deformations of the crystal are symmetrical about these two axes. The crystal material is presumed to exhibit strain hardening, strain-rate hardening, and thermal softening. A simple combined isotropic-kinematic hardening expression for the critical resolved shear stress, proposed by Weng, is modified to account for the affine thermal softening of the material. When the deformation is in the plane (001) of the single crystal, four slip systems (111)[11̄0], (111̄)[11̄0], (11̄;1̄;)[110], and (11̄1)[110] are active in the sense that significant plastic deformations occur along these slip systems. However, when the plane of deformation is parallel to the plane (101̄;) of the single crystal, slip systems (11̄;1)[110], (11̄1)[011], (111)[11̄0], and (111)[01̄1] are more active than the other eight slip systems. At an average strain of 0.0108, the maximum angle of rotation of a slip system within a shear band, about an axis perpendicular to the plane of deformation, is found to be 20.3° in the former case, and 22.9° in the latter.     

Constitutive modelling of anisotropic creep deformation in single crystal blade alloys SRR99 and CMSX-4

A damage mechanics based model has been developed to model stress rupture and creep behaviour of the first and second generation single crystal superalloys SRR99 and CMSX-4. In this article the creep behaviour of CMSX-4 in several different orientations at 950°C is simulated using finite elements, these simulations are compared with the results of creep tests. In order that the effects of rotation and specimen bending can be accounted for in the analysis the entire creep specimen is modelled. The FE program ABAQUS has been used and the slip system model is written using a User MATerial subroutine (UMAT). EBSD (electron back scattered diffraction) measurements of the lattice rotations occurring during creep indicate that the active slip systems at 950°C are the <101>{111} and <112>{111} systems, our simulations show that the creep results can be explained by activating these two families of slip system. There is strong microstructural evidence that the significant components of the hardening matrix should be those causing self and latent hardening of the <101>{111} systems and latent hardening by the <101>{111} systems on the <112>{111} systems.     

Crystal plasticity-based forming limit prediction for non-cubic metals: Application to Mg alloy AZ31B

A viscoplastic crystal plasticity model is incorporated within the Marciniak–Kuczynski (M–K) approach for forming limit curve prediction. The approach allows for the incorporation of crystallographic texture-induced anisotropy and the evolution of the same. The effects of mechanical twinning on the plastic response and texture evolution are also incorporated. Grain-level constitutive parameters describing the temperature dependent behavior of hexagonal close packed Mg alloy, AZ31B, sheets at discrete temperatures are used as a first application of the model. A trade-off between significant strain hardening behavior at lower temperatures (∼150 °C), and significant strain rate hardening at higher temperatures (∼200 °C) lead to similarities in the predicted forming limits. The actual formability of this alloy depends strongly on temperature within this range, and this distinction with the current modeling is related to more localized instability-based failure mechanisms at the lower temperatures than is assumed in the M–K approach. It is shown that the strain path dependence in the strain hardening response is significant and that it influences the forming limits in a predictable way. For broader applicability, a means of incorporating dynamic recrystallization into the crystal plasticity model is required.     

On the modelling of anisotropic material behaviour in viscoplasticity

A uniaxial viscoplastic deformation is motivated as a discrete sequence of stable and unstable equilibrium states and approximated by a smooth family of stable states of equilibrium depending on the history of the mechanical process. Three-dimensional crystal viscoplasticity starts from the assumption that inelastic shearings take place on slip systems, which are known from the particular geometric structure of the crystal. A constitutive model for the behaviour of a single crystal is developed, based on a free energy, which decomposes into an elastic and an inelastic part. The elastic part, the isothermal strain energy, depends on the elastic Green strain and allows for the initial anisotropy, known from the special type of the crystal lattice. Additionally, the strain energy function contains an orthogonal tensor-valued internal variable representing the orientation of the anisotropy axes. This orientation develops according to an evolution equation, which satisfies the postulate of full invariance in the sense that it is an observer-invariant relation. The inelastic part of the free energy is a quadratic function of the integrated shear rates and corresponding internal variables being equivalent to backstresses in order to consider kinematic hardening phenomena on the slip system level. The evolution equations for the shears, backstresses and crystallographic orientations are thermomechanically consistent in the sense that they are compatible with the entropy inequality. While the general theory applies to all types of lattices, specific test calculations refer to cubic symmetry (fcc) and small elastic strains. The simulations of simple tension and compression processes of a single crystal illustrates the development of the crystallographic axes according to the proposed evolution equation. In order to simulate the behaviour of a polycrystal the initial orientations of the anisotropy axes are assumed to be space-dependent but piecewise constant, where each region of a constant orientation corresponds to a grain. The results of the calculation show that the initially isotropic distribution of the orientation changes in a physically reasonable manner and that the intensity of this process-induced texture depends on the specific choice of the material constants.     

Modeling transients in the mechanical response of copper due to strain path changes

When copper is deformed to large strains its texture and microstructure change drastically, leading to plastic anisotropy and extended transients when it is reloaded along a different strain path. For predicting these transients, we develop a constitutive model for polycrystalline metals that incorporates texture and grain microstructure. The directional anisotropy in the single crystals is considered to be induced by variable latent hardening associated with cross-slip, cut-through of planar dislocation walls, and dislocation-based reversal mechanisms. These effects are introduced in a crystallographic hardening model which is, in turn, implemented into a polycrystal model. This approach successfully explains the flow response of OFHC Cu pre-loaded in tension (compression) and reloaded in tension (compression), and the response of OFHC Cu severely strained in shear by equal channel angular extrusion and subsequently compressed in each of the three orthogonal directions. This new theoretical framework applies to arbitrary strain path changes, and is fully anisotropic.     

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.