Strain Space Research Articles

A strain acceleration method to identify the onset of diffuse necking

This paper presents an innovative ‘strain acceleration method’ for determining the onset of diffuse necking in sheet forming tests using data obtained from digital image correlation (DIC). The method identifies the onset time of diffuse necking and provides the corresponding in-plane principal strain values by detecting a local extreme in the second derivative of the minor principal in-plane strain with respect to time at the edges of the sheet surface region where diffuse necking occurs. Results obtained from applying the method to tensile testing on two different materials and comparisons with available methods based on force-time or principal strain rate evolutions confirm its accuracy and validity. The new method was implemented in a computer software to be used for research and education that also enables determination of localized necking and fracture and plotting the strain loading paths in principal strain space.

Forming limit and failure characterization of as-received and bi-axial prestrained Cu-Cr-Zr-Ti alloy sheets in the context of multistage forming

The characterization of forming limit utilizing Cu-Cr-Zr-Ti alloy sheets during the multistage forming process is imperative for successfully fabricating thrust chamber liners used in satellite launch vehicles. Therefore, the present work studies the effect of prestrain on the forming limit and failure behaviour of this alloy under different deformation paths. A stretch-forming setup was utilized to induce 8% equi-biaxial prestrain (EBP), and subsequently, these prestrained specimens were deformed till the onset of failure using the same setup. After prestraining, a shift in the failure strains was observed in the principal strain space, whereas it was found to be absent in the polar effective plastic strain, principal stress and effective plastic strain-stress triaxiality loci. The cup height along different deformation paths was marginally decreased in the prestrained specimens because of the reduction in failure strains, and the fracture occurred in the prestrained region. Further, failure analysis revealed intergranular cracks in the specimens deformed along plane strain and uniaxial deformation paths due to the presence of micron-size Cr-rich precipitates along grain boundaries. However, intergranular and transgranular cracks were present in the prestrained specimens, which were deformed equi-biaxially similar to the as-received specimens. The presence of nano-size Cr-rich precipitates inside the grains acted as nucleation sites for voids during equi-biaxial stretching, leading to transgranular cracks. It was also observed that failure in the prestrained and as-received specimens occurred at a similar effective plastic strain value due to the combined effect of void density and void size. Based on experimental observations, it was inferred that the forming limit of Cu-Cr-Zr-Ti sheets under a two-stage forming process was primarily affected by the presence of micro and nano-size Cr-rich precipitates.

Quantized plastic deformation

In engineering crystal plasticity inelastic mechanisms correspond to tensorial zero-energy valleys in the space of macroscopic strains. The flat nature of such valleys is in contradiction with the fact that plastic slips, mimicking lattice-invariant shears, are inherently discrete. A reconciliation has recently been achieved in the mesoscopic tensorial model (MTM) of crystal plasticity, which introduces periodically modulated energy valleys while also capturing in a geometrically exact way the crystallographically-specific aspects of plastic slips. In this paper, we extend the MTM framework, which in its original form had the appearance of a discretized nonlinear elasticity theory, by explicitly introducing the concept of plastic deformation. The ensuing model contains a novel matrix-valued spin variable, representing the quantized plastic distortion, whose rate-independent evolution can be described by a discrete (quasi-)automaton. The proposed reformulation of the MTM leads to a considerable computational speedup associated with the use of a robust and efficient hybrid Gauss-Newton–Cauchy energy minimization algorithm. To illustrate the effectiveness of the new approach, we present a detailed case-study focusing on the aspects of crystal plasticity that are beyond reach for the classical continuum theory. Thus, we provide compelling evidence that the re-formulated MTM is fully adequate to deal with the intermittency of plastic response under quasi-static loading. In particular, our numerical experiments show that the statistics of dislocational avalanches, associated with plastic yield in 2D square crystals, exhibits a power-law tail with a critical exponent matching the value predicted by general theoretical considerations and also independently observed in discrete-dislocation-dynamics (DDD) simulations.

A continuum damage mechanics model for fatigue degradation and failure of short fiber reinforced composites

The present study is concerned with the development and implementation of a continuum damage mechanics model describing the degradation and failure of short fiber reinforced composites subjected to static and fatigue loads. Towards a numerically efficient formulation, the model is defined entirely on the macroscopic scale, although being motivated by microstructural considerations. Based on experimental observations, an anisotropic linear elastic formulation is adopted as a base formulation. The elastic model is coupled with a damage formulation assuming that damage is driven by the approach of a state point in strain space to a Tsai-Hill type failure envelope. The model is implemented as a user defined subroutine into a finite element formulation. As a numerically highly efficient alternative, an enhanced Wöhler-Miner type post-processing procedure is proposed. Both numerical approaches are found in good agreement with experimental data obtained in coupon experiments as well as in experiments on specimens with more complex geometry and loading conditions.

Finite strain modelling of Shape Memory Alloys in the logarithmic strain space: A comparative study with other finite strain approaches

Finite strain models for Shape Memory Alloys (SMAs) offer significant advantages over small strain models in that they excel in their ability to characterize SMA behaviour under large strain and finite rotations while eliminating the inherent assumptions associated with the small strain framework. The paper at hand presents a numerical model for SMAs at finite strain, employing the logarithmic strain measure to obtain an additive split of the total strain into its elastic and inelastic (transformation) parts. Notably, the model maintains the thermodynamic consistency validated using the Clausius–Duhem inequality. Additionally, we compare this model with two other finite strain SMA models employing logarithmic strain measures: a hypoelastic model, which splits the rate of deformation tensor additively, and a hyperelastic model, which decomposes the deformation gradient multiplicatively. The numerical formulation for the three finite strain models is specifically tailored to address superelasticity in SMAs. It is derived from the established small strain model and seamlessly extended into the domain of finite strain framework. Several numerical examples are carried out to showcase the logarithmic strain space approach applied to SMAs and to compare its response with that of the other aforementioned finite strain models in SMAs. The meticulous examination reveals that the logarithmic strain space approach aligns well with the existing finite strain models, demonstrating its effectiveness and compatibility.

Deformation of Cylindrical Shell Made of 9X2 Steel Under Complex Loading

The development of the construction industry in terms of the design and manufacture of shell structures of non-standard architectural forms made of materials with complex mechanical properties requires using modern integrated computer-aided design systems with step-by-step modeling of deformation of structural elements under operating conditions, as well as taking into account their subsequent behavior after accumulation of residual strains during plastic deformation. The purpose of the study is to simulate the process of plastic deformation of a thin-walled cylindrical shell made of 9X2 GOST 5950-2000 (Interstate Standard) steel under compression and torsion with theoretical calculations based on the general theory of elastoplastic processes by A.A. Ilyushin. The constitutive equations of the theory of elastoplastic processes by A.A. Ilyushin for complex loading path and deformation of materials in the deviatoric strain space are presented. Based on the presented solutions, according to the strain path of the 9X2 steel shell implemented in the model, the graphs showing the relation of the vector and scalar properties of the material to the arc length of the strain path are constructed. A conclusion is made about the degree of hardening of the material in question and its dependence on the magnitude of the angle of convergence at the kink point of the complex path. The graphs of changes in the constitutive plasticity functions with respect to the increments of the arc length of the strain path are presented.

A multiscale bifurcation analysis using micromechanical-based constitutive tensor for granular material

The current study presents a multiscale approach that investigates material instability and localization phenomena in plastic granular materials and the discrete-continuum duality. A bifurcation and stability analysis in continuum mechanics usually requires the material’s tangent (stiffness) operator, the computation of which in a micromechanical approach such as Discrete Element Modeling (DEM) requires specific treatment. To bridge the discrete and continuum worlds, a new computational approach incorporating strain probing is proposed to reconstruct the elastoplastic constitutive tensor and its spectral characteristics from DEM simulations. The probing technique permits the computation of the tangent operator that inherits microstructural information from the discrete world to analyze bifurcation in elastoplasticity at the macro level. An incrementally linear constitutive tensor is computed, distinct for each of the ensemble of probing directions belonging to a particular tensorial zone or sector of incremental stress or strain space, thus making it directionally non-linear. Following such an approach, material instability can be evaluated from the spectral characteristics of the tangent constitutive tensor deduced from DEM probing calculations belonging to an identified tensorial zone. A meso-scale analysis is finally offered to detect shear band localization through the well-known Rice’s criterion as a continuum-based concept extended to a micromechanical discrete modeling framework. These new numerical results show that the multiscale proposed approach, which allows access to microstructural information, is consistent with a continuum one such as when predicting the localization angle during shear banding in a granular specimen in DEM.

On the energy decomposition in variational phase-field models for brittle fracture under multi-axial stress states

Phase-field models of brittle fracture are typically endowed with a decomposition of the elastic strain energy density in order to realistically describe fracture under multi-axial stress states. In this contribution, we identify the essential requirements for this decomposition to correctly describe both nucleation and propagation of cracks. Discussing the evolution of the elastic domains in the strain and stress spaces as damage evolves, we highlight the links between the nucleation and propagation conditions and the modulation of the elastic energy with the phase-field variable. In light of the identified requirements, we review some of the existing energy decompositions, showcasing their merits and limitations, and conclude that none of them is able to fulfil all requirements. As a partial remedy to this outcome, we propose a new energy decomposition, denoted as star-convex model, which involves a minimal modification of the volumetric-deviatoric decomposition. Predictions of the star-convex model are compared with those of the existing models with different numerical tests encompassing both nucleation and propagation.

Agar dehydration: a simple method for long-term storage of Acanthamoeba spp. collection at room temperature.

This study describes dehydration of agar containing cysts as a novel and inexpensive method for long-term storage of Acanthamoeba spp. collections at room temperature. Five hundred microliters of axenically cultured Acanthamoeba spp. trophozoites (106 cells/mL) in PYG media or 150µl of amoeba suspension (106 cells or cysts/mL) from monoxenic plate culture was spread onto the surface of non-nutritive agar (NNA, 2-3-mm thick) without or with a layer of heat-inactivated Escherichia coli, respectively. The plates were sealed and incubated at 30°C. After the encystment, the Parafilm® was removed, and the plates were kept at the same temperature until the NNA was completely dehydrated. The dehydrated cyst-containing NNA was cut in rectangles and stored in airtight tubes at room temperature for up to 3years. Cyst viability was assessed by inoculating them in fresh NNA with a layer of E. coli and in PYG followed by incubation at 30°C. One hundred percent of samples from all specimens (19) stored over the 3years allowed new cultures to be re-established; however, two strains showed reduced viability, at 66.7% and 62.5%, after 2years of room temperature storage. One hundred percent of the cyst samples produced axenically and maintained in dry NNA allowed the re-establishment of axenic cultures through direct incubation in PYG, with excystment occurring within 24 or 48h. For the first time, we report the dehydration of cyst-containing agar as an economical and effective method for the long-term storage of Acanthamoeba spp. collections at room temperature. It enables the creation of large collections using reduced space and economical transport of Acanthamoeba strains, in addition to allowing better organization of the collection.

Anisotropic ductile fracture of a stainless steel under biaxial loading: Experiments and predictions

The anisotropic ductile fracture behavior of a stainless steel is investigated using a combination of experiments and analysis. The material is SS-304L, an austenitic, low-carbon stainless steel, received in the form of tubes of 2.38 mm dia. and 0.15 mm thickness. The tubes are inflated under volume control in a custom apparatus. At the same time, the axial force on the tubes is kept proportional to the pressure induced by the inflation. This leads to quasi-proportional stress paths in the meridional-hoop engineering stress space. A total of 15 discrete paths are successfully tested. Stereo-type Digital Image Correlation is used to measure the strains. In every case, the tubes develop a series of instabilities before bursting. The failure is oriented along the meridional or the hoop direction of the tube, depending on which stress is greater. A mild plastic anisotropy is detected in these experiments. Hence these results are then used for calibrating the anisotropic yield criterion Yld2004-3D. This is introduced in a finite element model of the experiments, which includes a thickness imperfection designed to capture the two failure orientations observed in the experiments. The numerical model using the Yld2004-3D criterion reproduces the experiments well, e.g., it captures the experimental stress-strain and induced strain paths better than von Mises. It is then used to probe the conditions at the onset of fracture (hybrid method). It is found that most paths lead to essentially proportional loading during deformation. A significant anisotropy in the fracture behavior is detected, with the meridional-stress-dominated paths being able to develop much higher strains than the hoop-dominated ones. These results are then captured by the DF2016 ductile fracture criterion, modified to use the anisotropic yield criterion Yld91. The proposed criterion is flexible enough to represent the fracture anisotropy very well, without being unnecessarily complex. The fracture forming limit curve (i.e., the fracture envelope in strain space) predicted by the DF2016/Yld91 model is also found to be very close to the experiments. The results and findings of this work help establish a framework to reliably design components and processes when significant fracture anisotropy is expected.

Energetically stable curve fitting to hyperelastic models based on uniaxial and biaxial tensile tests

Hyperelastic constitutive laws in biomechanics are used to model soft tissues, and material model parameters are often determined by performing curve fitting on data from uniaxial or biaxial tensile tests. The strain energy function of the applied constitutive law must to be energetically stable; however, this condition is not inherently provided by most currently available models. This study provides a procedure to determine stable strain energy functions in a biaxial strain space based on either uniaxial or biaxial tensile tests. Instead of conservative, strain-independent conditions, a stability region is defined in the strain space based on the sample’s tensile tests, thus allowing optimisation within a wider parameter space, resulting in better approximations. An extension of the Levenberg–Marquardt algorithm incorporating user-defined stability constraints is proposed, and the constrained optimisation algorithm is applied to isotropic and anisotropic models. The uniqueness of solutions of the Fung model is also discussed. The material model parameters of stable solutions for soft tissue measurements from various literature sources are determined to demonstrate the proposed procedure. Applying appropriate constraints in the optimisation algorithm resulted in stable and physically permissible constrained solutions for the strain energy function, in contrast to the results of most unconstrained optimisation cases.

A closed-form expression of a ductile Fracture Limit Surface (FLS) for general plane stress deformation paths

Ductile fracture in metals is a complex phenomenon caused by the nucleation, growth, and aggregation of micron-sized voids. Existing theoretical models for fracture use experimentally determined Fracture Limit Curves (FLCs) in two-dimensional strain space, which require multiple curves to account for anisotropic effects. This paper proposes a practical closed-form analytical function for a Fracture Limit Surface (FLS) which has a conical shape in the Elastic Distortional Deformation Space (EDDS). The EDDS is a space generated by the components of the elastic distortional metric of the microstructural vectors that characterize elastic distortional deformations and the orientations of anisotropy. The FLS is determined by simulating only four experimental FLCs using a large deformation Eulerian formulation of the microstructural vectors and it depends only on the current state of elastic distortional deformation. It is shown that the failure limit points predicted by the FLS agreed very well with those predicted using a popular evolution equation for damage-based ductile fracture. Path-independence of the FLS is also examined using experimental data. In addition, the example of a B-pillar forming process is presented to show that the FLS can be applied to practical engineering problems with material points that experience general loading paths.

A forming limit framework accounting for various failure mechanisms: Localization, ductile and cleavage fracture

The forming limits and failure properties of three distinct advanced high-strength steels (AHSS) have been investigated under various stress states in tensile tests with optimized specimen geometries. In addition to the commonly observed failure patterns governed by localized necking and ductile fracture for two of the AHSS, after substantial plastic deformation at room temperature, cleavage fracture occurs for a large range of stress states in a laboratory quenching and partitioning steel with superior tensile properties. The competition between failure patterns, encompassing ductile and cleavage fractures with and without necking, is governed by the mechanical properties of materials and the stress states, as a transition of failure mechanisms occurs with increasing triaxiality. The forming limit framework is, therefore, further extended to seamlessly integrate cleavage fracture in this study, where the competition between various failure mechanisms is demonstrated using three AHSS in the space of critical strain and principal stress. These findings shed light on the importance of considering cleavage fracture strength as a parameter besides the strength-ductility synergy in advanced high-strength metallic materials, and the proposed framework also gives a more comprehensive guide in designing and conducting the sheet metal forming processes.

Phonon stability boundary and deep elastic strain engineering of lattice thermal conductivity

Recent studies have reported the experimental discovery that nanoscale specimens of even a natural material, such as diamond, can be deformed elastically to as much as 10% tensile elastic strain at room temperature without the onset of permanent damage or fracture. Computational work combining ab initio calculations and machine learning (ML) algorithms has further demonstrated that the bandgap of diamond can be altered significantly purely by reversible elastic straining. These findings open up unprecedented possibilities for designing materials and devices with extreme physical properties and performance characteristics for a variety of technological applications. However, a general scientific framework to guide the design of engineering materials through such elastic strain engineering (ESE) has not yet been developed. By combining first-principles calculations with ML, we present here a general approach to map out the entire phonon stability boundary in six-dimensional strain space, which can guide the ESE of a material without phase transitions. We focus on ESE of vibrational properties, including harmonic phonon dispersions, nonlinear phonon scattering, and thermal conductivity. While the framework presented here can be applied to any material, we show as an example demonstration that the room-temperature lattice thermal conductivity of diamond can be increased by more than 100% or reduced by more than 95% purely by ESE, without triggering phonon instabilities. Such a framework opens the door for tailoring of thermal-barrier, thermoelectric, and electro-optical properties of materials and devices through the purposeful design of homogeneous or inhomogeneous strains.

Damage in a comprehensive model for shape memory alloys in logarithmic strain space

Shape memory alloys (SMAs) possess considerable complexity in response to thermomechanical load due to their strong dependence on strain, stress, temperature and history. This high complexity is a limiting factor in the fatigue analysis of SMA structures. With this in mind, a comprehensive phenomenological model is developed to address nearly all of the features in SMAs with a particular focus on the development of damage. Due to the fact that many SMA structures experience large deformations, the model is developed in finite strains using the logarithmic strain space approach. Besides the balance of linear momentum, the model includes the heat conduction equation and two micromorphic nonlocal balance laws. The latter are introduced to regularize the softening associated with the damage and transformation. The material model incorporates a damage evolution law that depends on the integration of a simplified dissipation term. The model is applied to a few examples to observe the interaction of the features and to illustrate the applicability of the model for rupture and fatigue analysis. The interaction of the features is shown to produce a reasonable representation of the cyclic loading of wires at different loading rates. Rupture is shown to occur due to a large amount of plastic strain and can be performed due to the nonlocal approach for the damage. The fatigue analysis shows some promising accuracy but only if the dissipated area is captured properly.

Electrohydraulic System Analysis of Variable Recruitment Fluidic Artificial Muscle Bundles With Interaction Effects

Abstract This study investigates the system-level performance of variable recruitment (VR) fluidic artificial muscle (FAM) actuator bundles using a model that incorporates FAM interaction effects. A VR bundle combines multiple FAMs to act as one actuator in which the FAMs are sequentially recruited to increase overall efficiency. In a VR bundle, inactive/low-pressure FAMs are compressed beyond their free strains, exerting resistive forces opposing that of active FAMs. A recent model that captures this behavior is used to simulate sinusoidal contraction of a VR bundle with a hanging mass load. The implications of inter-FAM effects on the force–strain space of a VR bundle are discussed and a method of recruitment state transition required to track a sinusoid is proposed. The dynamics of the electrohydraulic subsystems are presented and used to evaluate its system efficiency and bandwidth limits. Three different electrohydraulic configurations are considered: (1) continuous motor operation with constant pump displacement, (2) intermittent motor operation with constant pump displacement, and (3) continuous motor operation with variable pump displacement. Simulation results show the superior bandwidth capabilities of VR bundles by demonstrating its ability to track sinusoids with amplitudes up to 16% strain at frequencies greater than 0.5 Hz, compared to that of a single equivalent cross section area motor unit (SEMU). In addition to increased bandwidth limit, system efficiencies averaged over a range of amplitudes show up to 170% increase when comparing a VR bundle using variable pump displacement to a SEMU using constant pump displacement.

Statistically compatible hyper-reduction for computational homogenization

The computational bottleneck of reduced order models (ROMs) in nonlinear homogenization is usually given by the local material laws, which need to be evaluated in a large number of microscopic integration points. Hyper-reduction methods use only a small subset of the integration points and reach tremendous speed-ups at high accuracy. However, the underintegration breaks the overall compatibility of the microscopic strain field and is in this sense disrespecting the microscopic boundary value problem. Here, a new type of generalized integration points is introduced in strain space in order to remedy this shortcoming. Being inspired by results from nonlinear homogenization theory, the concept of statistical compatibility is developed and forms the theoretical basis for the new integration points, which respect the compatibility of the microscopic strain field in a statistical sense. The statistically compatible integration points can be derived offline and replace the conventional ones in a Galerkin-projection based setting with global modes identified via proper orthogonal decomposition (POD). The method is tested for various reinforced composites, indicating that 10-20 integration points are often sufficient to reach errors smaller than 3%, with CPU-times in the μs-range (per time step). A possible extension of the method for problems with higher nonlinearity and stronger field fluctuations is discussed within the context of a porous microstructure.

Interplay between yielding, 'recovery', and strength of yield stress fluids for direct ink writing: new insights from oscillatory rheology.

Formulation design and rheology are critical for successful manufacturing via direct ink writing (DIW), thus linking rheology and printability is a growing area of research amongst the DIW and rheology communities. This work provides an extensive rheological investigation into the material strength, yielding and 'recovery' properties of graphite (Gr)-hydrogel based formulations. Using state-of-the-art Large Amplitude Oscillatory Shear (LAOS) techniques, Fourier Transform (FT) rheology and sequence of physical process (SPP) analysis, and 3-step 'recovery' tests we provide new insights on the yielding phenomenon, energy transitions and microstructural changes that the formulations undergo. The insights from the rheology experiments are combined with in situ and continuous monitoring during the printing process. From these analyses, we select rheological metrics or descriptors to quantify flowability, recoverability, and material strength. There is a threshold concentration of Gr powders (30 wt%) at which there is a shift in the yielding process. Below this threshold (for the F127 hydrogel and mixtures with low Gr content), perfect plastic dissipation ratio (ϕ) values are close to 0 in the LVR and then steeply increase to close to 1 after the cross-over in a narrow strain (and stress) space. As Gr concentration increases, and print quality gets worse, ϕ values consistently increase in the LVR and at any given γ0, evidencing an increased energy dissipation throughout the flow transition region. Lissajous-Bowditch curves and SPP Cole-Cole plots illustrate these trends. The extent of the 'recovery' (quantified by the mutation time, λI, and the storage modulus 'recovered' after large deformations ) is also directly related to Gr content, with higher loading resulting in lesser recovery. Our findings provide a comprehensive set of metrics to characterise complex (yield stress) fluids for DIW using three property maps, one for each stage: flowability or yielding process, recoverability and material strength. The results demonstrate that considering these three maps holistically provides insightful trends to guide formulation design and assess performance in DIW.

Effective stress analysis of soil-structure systems in practice through strain space multiple mechanism model: validation through 15 case histories

Effective stress analysis of soil-structure systems in practice through strain space multiple mechanism model: validation through 15 case histories

Effect of steady-state strength of medium to dense sand on the performance of sheet pile wall as identified through centrifuge tests and Type-B and C simulations

This study focuses on five centrifuge experiments of a sheet pile wall performed by three institutions, which cover relative densities ranging from 55 % to 66 %. A strain space multiple mechanism model incorporated in the computer program FLIP is used for the simulation. In the Type-B simulation performed before the centrifuge test data release, the steady-state strength Sus is not considered, i.e., Sus is assumed to be infinity (Sus=∞), and only cyclic resistance is considered. In the Type-C simulation performed after the release of the centrifuge test data, Sus and cyclic resistance are considered. When Sus=∞, the computed displacements of the sheet pile wall are consistent with or unconservative by a factor of one-half of the measured displacements. However, when Sus ranges from 1500 to 7500 kPa evaluated for clean sands, the computed displacements are generally consistent with the measured displacements, or slightly conservative. When Sus ranges from 100 to 400 kPa evaluated for sands with fines content of 5 %, or when Sus ranges from 20 to 50 kPa inferred from cyclic laboratory tests, the computed displacements are conservative by a factor of one and a half or two of the measured displacements. The steady-state strength of medium to dense sand is identified as a significant factor influencing the performance of a sheet pile wall.