Elastoplastic Matrix Research Articles

The plastic behavior of matrix and its effect on the interface stress transfer in fiber-reinforced composites

ABSTRACT Existing shear-lag models for fiber-reinforced elasto-plastic composites always assume a rectangular-shaped plastic zone in the matrix, which is not consistent with numerical simulations. In this paper, a shear-lag model considering a more realistic parabolic shape of the plastic zone in an elasto-plastic matrix is proposed, in order to investigate the effect of matrix plasticity on the interface stress transfer. A closed-form solution to the profile of parabolic shape related to the applied stress is obtained so that the plastic zone evolution can be analytically characterized. Compared with the case with a purely elastic matrix, a plastic matrix could lead to a decrease in the interfacial shear stress (ISS) and an increase in the fiber stress. The smaller the plastic tangent modulus of the matrix, the lower the ISS is and the larger the fiber stress becomes, and the ISS and fiber stress keep being on same magnitudes of yielding strength and Young’s modulus of matrix, respectively. The present model with a parabolic plastic shape is more accurate to describe the mechanical behaviors of plasticity in matrix, which is of guiding value for the design of elasto-plastic composites used in vehicle accessories, medical devices, civil engineering and so on.

Analytical probabilistic progressive damage modeling of single composite filaments of material extrusion

This research focuses on investigating the elastic and damage characteristics of individual composite beads utilized in additively manufactured (3D printed) composites. A novel analytical probabilistic progressive damage model (PPDM) is introduced to comprehensively capture the elastic and damage attributes of these single 3D printed beads, which constitute the foundational elements of 3D printed composites. The PPDM is formulated based on assumptions of a perfectly elastic, perfectly plastic, and elasto-plastic matrix. For the elasto-plastic matrix, a novel formulation of the damage is developed to offer a more realistic view of the internal mechanics. Furthermore, a novel method of measuring the Young’s modulus of 3D printed composites is assessed. To this end, the Sentmanat extensional rheometer (SER) fixture is used to conduct a series of tensile tests on Kevlar and glass fiber-reinforced nylon beads. The experimental data of the SER are compared with those from an Instron tensile machine. Finally, the results of the PPDM and the experiments are compared and reveal a strong agreement in both the elastic region and the ultimate strength and strain.

Fracture spacings of fiber inclusions in a ductile geological matrix and development of microboudins: 3D numerical modeling

Tensile fractures and resultant microboudinage structures of brittle fiber inclusions (e.g., tourmaline, piedmontite and amphibole) in the soft matrix of deforming minerals are of great significance for determining the finite strains and paleostresses of naturally deformed rocks. Using statistical strength theory, damage mechanics, and continuum mechanics, we have reproduced in a series of numerical models the sequential fractures of either homogeneous or heterogeneous fiber inclusions under axial tension in an elastoplastic matrix. The results clarify that: (1) The spacing between fractures in fibers is inversely proportional to the applied strain. As the applied strain increases, the fracture spacing systematically decreases as sequential fractures fill in until fracture saturation is reached. (2) As fiber length increases, the critical tensile strain for fracture saturation rises. For the same fiber diameter, saturation fracture spacing increases slightly with rising fiber length. For the same fiber length, however, saturation fracture spacing decreases significantly with lessening fiber diameter. Hence, fracture spacing at the saturation state depends on the volume fraction of fiber. (3) The rupture mode of fibers strongly depends on the non-uniform distribution of mechanical properties, which provides an effective approach for estimating the inhomogeneity of fibers by analyzing the formation of fractures. Furthermore, due to material heterogeneity, new fractures are unlikely to occur in the middle of existing adjacent fractures.

Thermomechanically coupled theory in the context of the multiphase-field method

The modeling and simulation of microstructure evolution is usually subject to the multiphase-field method. In this context, thermomechanical coupling is often neglected, even when non-isothermal phase transformations are considered. Using a simplified example, the present work shows that this assumption is not justified for small strains and small strain rates with respect to a non-vanishing coefficient of thermal expansion. To this end, both a thermomechanically coupled and a thermomechanically weakly coupled theory are briefly revisited. The difference between the coupled and the weakly coupled theory regarding the growth of an inclusion is discussed. The considered elastoplastic inclusion, subjected to eigenstrains, is embedded in an elastoplastic matrix under load. It is shown, that the weakly coupled theory overestimates the growth of the inclusion, and, thus, the volume concentration, compared to the coupled theory. Moreover, only the application of the coupled theory reflects a load-induced anisotropic growth of the inclusion, which exhibits an isotropic material behavior, due to non-vanishing uniaxial Neumann boundary conditions. In addition, it is shown that the smaller the heat conduction coefficient, the more pronounced the anisotropic growth of the inclusion.

Physically recurrent neural networks for path-dependent heterogeneous materials: Embedding constitutive models in a data-driven surrogate

Driven by the need to accelerate numerical simulations, the use of machine learning techniques is rapidly growing in the field of computational solid mechanics. Their application is especially advantageous in concurrent multiscale finite element analysis (FE2) due to the exceedingly high computational costs often associated with it and the high number of similar micromechanical analyses involved. To tackle the issue, using surrogate models to approximate the microscopic behavior and accelerate the simulations is a promising and increasingly popular strategy. However, several challenges related to their data-driven nature compromise the reliability of surrogate models in material modeling. The alternative explored in this work is to reintroduce some of the physics-based knowledge of classical constitutive modeling into a neural network by employing the actual material models used in the full-order micromodel to introduce non-linearity. Thus, path-dependency arises naturally since every material model in the layer keeps track of its own internal variables. For the numerical examples, a composite Representative Volume Element with elastic fibers and elasto-plastic matrix material is used as the microscopic model. The network is tested in a series of challenging scenarios and its performance is compared to that of a state-of-the-art Recurrent Neural Network (RNN). A remarkable outcome of the novel framework is the ability to naturally predict unloading/reloading behavior without ever seeing it during training, a stark contrast with popular but data-hungry models such as RNNs. Finally, the proposed network is applied to FE2 examples to assess its robustness for application in nonlinear finite element analysis.

Ductile fracture of high entropy alloys: From the design of an experimental campaign to the development of a micromechanics-based modeling framework

Cantor-type high entropy alloys form a new family of metallic alloys characterized by a combination of high strength and high fracture toughness. An experimental study on the CoCrNi alloy is first performed to determine the damage and fracture mechanisms under various stress states. A micromechanics-based ductile fracture model is identified and validated using these experimental data. The model corresponds to a hyperelastic finite strain multi-yield surface constitutive description coupled with multiple nonlocal variables. The yield surfaces consist of three distinct nonlocal solutions corresponding to three different modes of void expansion within an elastoplastic matrix: a void growth mode governed by a Gurson-based yield surface corrected for shear effects, an internal necking-driven coalescence mode governed by an extension of the Thomason yield surface based on the maximum principal stress, and a shear-driven coalescence mode governed by the maximum shear stress. This advanced formulation embedded in a large strain finite element setup captures the effects not only of the stress triaxiality but also of the Lode variable. In particular, the analysis shows that a failure model accounting for these two invariants of the stress tensor captures the fracture in high-entropy alloys over a wide range of conditions.

Precomputation of Critical State Soil Plastic Models

In this paper, a simple precomputing procedure is proposed to improve the numerical performance of the technological application of critical state soil models. In these models, if associated plasticity is assumed, the normalization of the stress space allows both the yield surface and the plastic components of the elastoplastic matrix to be defined as a function of a single variable. This approach facilitates their parameterization and precomputation, preventing the repetition of calculations when the boundary value problems appear at the yield surface with the calculation of plastic strain. To illustrate the scope of the procedure, its application on a modified Cam Clay model is analysed, which shows that the method allows a significant reduction of about 50% (as compared with the conventional explicit integration algorithm) in the computational time without reducing the precision. Although it is intended for critical state models in soils, the approach can be applied to other materials and types of constitutive models provided that parameterization is possible. It is therefore a methodology of practical interest, especially when a large volume of calculations is required, for example when studying large-scale engineering systems, performing sensitivity analysis, or solving optimization problems.

Incremental variational homogenization of elastoplastic composites with isotropic and Armstrong-Frederick type nonlinear kinematic hardening

In order to investigate the behavior of elastoplastic composites exhibiting both isotropic and nonlinear kinematic hardening, we extend the Double Incremental Variational (DIV) formulation of Lucchetta et al. (2019), based on both the incremental variational principles introduced by Lahellec and Suquet (2007) and the formulation proposed by Agoras et al. (2016). However, the Armstrong-Frederick model (Armstrong and Frederick), which is very often used to describe nonlinear kinematic hardening and refers to the framework of non-associated plasticity (Chaboche, 1977), cannot be handled within the framework of generalized standard materials as required by the incremental variational principles on which the DIV formulation relies. That is why we work with an approximation of this model, namely the modified Chaboche model (Chaboche, 1983). As the dissipation potential associated with this model depends on internal variables, we have to extend the incremental variational principles of Lahellec and Suquet to such a situation. Then, we apply twice the variational procedure of Ponte Castañeda (1991), first to linearize the local behavior and then to deal with the intraphase heterogeneity of the thermoelastic Linear Comparison Composite (LCC) induced by the linearisation step. The resulting thermoelastic LCC with per-phase homogeneous properties is homogenized by classical linear schemes. We develop and implement this new incremental variational procedure for two-phase matrix-inclusions composites with an isotropic elastoplastic matrix exhibiting combined isotropic and nonlinear kinematic hardening. For various cyclic loadings, the predictions of the proposed DIV formulation compare favorably with Finite Element simulations based on the Armstrong-Frederick model.

Development of a two-scale damage model for incorporating the fatigue crack nucleation from surface inclusions

In high cycle fatigue, damage and plasticity occur at the microscale, a scale smaller than the representative volume element (mesoscale). Therefore, the two-scale damage model is used for fatigue life assessment. In this model, a spherical inclusion is assumed to be fully embedded in the matrix. However, experimental observations show that the microplasticity and microcracks nucleate more easily in grains on the free surface rather than in the bulk. Hence, the aim of this work is to extend the two-scale damage model to take into account the nucleation of fatigue cracks on free surfaces. In order to do this, a surface scale transition law is derived to link the micro and mesoscales, considering a hemispherical damaged inclusion at the surface of an elasto-plastic matrix. Furthermore, a numerical scheme is proposed by developing an in-house Python code in conjunction with Abaqus software to compute the damage. The material parameters required in model are identified using monotonic and fatigue test results of smooth specimens. As a validation, for the butt-welded specimens subjected to fatigue loading, the obtained results of the proposed surface law are compared with the results of the pre-existing non-surface laws and the experimental fatigue data. It is confirmed that the use of surface localization law leads to a noticeable reduction in the number of cycles to crack initiation of about 25%.

A new robust theoretical prediction model for flange wrinkling in conventional spinning

Flange wrinkling is one of the most common defects affecting the forming quality of workpieces in conventional spinning. To this end, the flange wrinkling mechanism is analyzed and a new robust theoretical prediction model for the flange wrinkling is developed in this work. It is found that the initiation of flange wrinkling is closely related to the excessive circumferential compressive stress in the zone affected by the roller action rather than the whole flange. Then, a theoretical model for the critical circumferential stress producing flange wrinkling is developed based on the generalized variational principle of limit analysis. By comparing the instant maximum circumferential stress in the zone affected by the roller action with the calculated critical stress, the flange wrinkling can be predicted. Compared with the prediction model based on energy method, the main improvements of the new model can be summarized as follows. First of all, the incremental displacement amplitude of deflection mode in the effective compressive region, which dramatically affects the critical stress but overlooked in the model based on energy method, is considered in the new model. Secondly, the sheet blank is regarded as a rigid plastic material in the new model, so the calculation of elastoplastic matrix which is very sensitive to the mechanical states does not need to be considered, thereby avoiding the adverse effect of the extraction deviation of mechanical states on prediction accuracy. Thus, compared with the prediction model based on energy method, the new model has higher prediction accuracy and better robustness under the extraction deviation of mechanical states. In addition to realizing the prediction of flange wrinkling, the new model can also be used to analyze the influence of sheet blank geometric parameters and material parameters on the initiation of flange wrinkling under various forming conditions. In view of this, the new model is applied to reveal the influence of material parameters (material strengthening coefficient K, hardening index n and elastic modulus E) on the initiation of flange wrinkling by combining response surface methodology and central composite design (CCD). The results show that K has the greatest influence on the initiation of flange wrinkling, followed by E and n. Considering the interaction between the parameters, the interaction between K and E also has a greater effect on the initiation of flange wrinkling. The smaller E and larger K are helpful to reduce the risk of flange wrinkling. Changing the value of n has no significant effect on delaying the occurrence of flange wrinkling. The results can deepen the understanding on the prediction and rules of flange wrinkling in conventional spinning.

Stress‐controlled weakly periodic boundary conditions: Axial stress under varying orientations

SummaryThe accuracy of multiscale modeling approaches for the analysis of heterogeneous materials hinges on the representativeness of the micromodel. One of the issues that affects this representativeness is the application of appropriate boundary conditions. Periodic boundary conditions are the most common choice. However, when localization takes place, periodic boundary conditions tend to overconstrain the microscopic problem. Weakly periodic boundary conditions have been proposed to overcome this effect. In this study, the effectiveness of weakly periodic boundary conditions in restoring transverse isotropy of representative volume elements (RVE) for a fiber‐reinforced composite with elastoplastic matrix is investigated. The formulation of weakly periodic boundary conditions is extended to allow for force‐controlled simulations where a uniaxial stress can be applied. A series of simulations is performed where the orientation of applied stress is gradually varied and the influence of this orientation on the averaged response is examined. An original method is presented to test the correlation between the ultimate principal stress and average localization angle of shear bands within an RVE. It is concluded that weakly periodic boundary conditions alleviate anisotropy in the RVE response but do not remove it.

Real time imaging of strain fields induced by the ferrite-to-austenite transformation in high purity iron

During the ferrite-to-austenite transformation, the accommodation of the volume misfit between ferrite and austenite induces strain localization in both phases. A detailed understanding of this mechanism is a necessary step towards the improvement of thermomechanical treatments of iron alloys. Full-field measurements of displacement during heating of high purity iron samples are thoroughly post-processed to track the transformation in situ. They allow a fine characterization of the interaction between local plastic events and microstructure evolutions during the transformation. The deformation of austenite grains bears strong signatures in terms of orientation that can be used to monitor their individual growth in temperature. The parent phase acts as an elastoplastic matrix embedding an inclusion everywhere except at grain boundaries where additional localizations induced by crystal rotations precede the transformation interface.

A nonlocal approach of ductile failure incorporating void growth, internal necking, and shear dominated coalescence mechanisms

An advanced modeling framework is developed for predicting the failure of ductile materials relying on micromechanics, physical ingredients, and robust numerical methods. The approach is based on a hyperelastic finite strain multi-surface constitutive model with multiple nonlocal variables. The three distinct nonlocal solutions for the expansion of voids embedded in an elastoplastic matrix are considered: a void growth phase governed by the Gurson–Tvergaard–Needleman yield surface, a void necking coalescence phase governed by a heuristic extension of the Thomason yield surface based on the maximum principal stress, and a competing void shearing coalescence phase triggered by the maximum shear stress. The first solution considers the diffused plastic deformation around the voids while the last two solutions correspond to a state of plastic localization between neighboring voids. This combination captures the Lode variable and shear effects, which play important roles in dictating the damage evolution rates. The implicit nonlocal formulation with multiple nonlocal variables, including the volumetric and deviatoric parts of the plastic strain, and the mean equivalent plastic strain of the matrix, regularizes the problem of the loss of solution uniqueness when material softening occurs whatever the localization mechanism. The predictive capability of the proposed model is demonstrated through different numerical simulations in which complex failure patterns such as slant and cup-cone of respectively plane strain and axisymmetric samples under tensile loading conditions develop.

Nonlinear dynamics of earthquake-resistant structures using shape memory alloy composites

Earthquake-resistant structures have been widely investigated in order to produce safe buildings designed to resist seismic activities. The remarkable properties of shape memory alloys, especially pseudoelastic effect, can be exploited in order to promote the essential energy dissipation necessary for earthquake-resistant structures. In this regard, shape memory alloy composite is an idea that can make this application feasible, using shape memory alloy fibers embedded in a matrix. This article investigates the use of shape memory alloy composites in a one-story frame structure subjected to earthquakes. Different kinds of composites are analyzed, comparing the influence of matrix type. Both linear elastic matrix and elastoplastic matrix with isotropic and kinematic hardening are investigated. Results indicate the great energy dissipation capability of shape memory alloy composites. A parametric analysis allows one to conclude that the maximum shape memory alloy volume fraction is not the optimum design condition for none of the cases studied, highlighting the necessity of a proper composite design. Despite the elastoplastic behavior of matrix also dissipates a considerable amount of energy, the associated residual strains are not desirable, showing the advantage of the use of shape memory alloys.

Numerical homogenization method in the modeling of gas hydrate bearing sediments

The study of the mechanical behavior of gas hydrate bearing soils represents a major interest. The hydrate inclusions in sediments change their microstructure and their mechanical properties with it. We developed a numerical homogenization code in order to simulate the macro-mechanical response of a periodic unit-cell using local elastoplastic laws and complex geometries to define the microstructural components of the material. We applied it to a real image of fine-grained sediments containing gas hydrate veins. This technique allowed us to study the effect of an elastoplastic soil matrix and of different volume fractions of gas hydrates with complex shapes on the overall response of the material.

An inverse micro-mechanical analysis toward the stochastic homogenization of nonlinear random composites

An inverse Mean-Field Homogenization (MFH) process is developed to improve the computational efficiency of non-linear stochastic multiscale analyzes by relying on a micro-mechanics model. First full-field simulations of composite Stochastic Volume Element (SVE) realizations are performed to characterize the homogenized stochastic behavior. The uncertainties observed in the non-linear homogenized response, which result from the uncertainties of their micro-structures, are then translated to an incremental-secant MFH formulation by defining the MFH input parameters as random effective properties. These effective input parameters, which correspond to the micro-structure geometrical information and to the material phases model parameters, are identified by conducting an inverse analysis from the full-field homogenized responses. Compared to the direct finite element analyzes on SVEs, the resulting stochastic MFH process reduces not only the computational cost, but also the order of uncertain parameters in the composite micro-structures, leading to a stochastic Mean-Field Reduced Order Model (MF-ROM). A data-driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic MF-ROM embedded in a Stochastic Finite Element Method (SFEM). The two cases of elastic Unidirectional (UD) fibers embedded in an elasto-plastic matrix and of elastic UD fibers embedded in a damage-enhanced elasto-plastic matrix are successively considered. In order to illustrate the capabilities of the method, the stochastic response of a ply is studied under transverse loading condition.

A simple and flexible model order reduction method for FFT-based homogenization problems using a sparse sampling technique

This work is concerned with the development of a novel model order reduction technique for FFT solvers. The underlying concept is a compressed sensing technique which allows the reconstruction of highly incomplete data using non-linear recovery algorithms based on convex optimization, provided the data is sparse or has a sparse representation in a transformed basis. In the context of FFT solvers, this concept is utilized to identify a reduced set of frequencies on a sampling pattern in the frequency domain based on which the Lippmann–Schwinger equation is discretized. Classical fixed-point iterations are performed to solve the local problem. Compared to the unreduced solution, a significant speed-up in CPU times at a negligibly small loss of accuracy in the overall constitutive response is observed. The generation of highly resolved local fields is easily possible in a post-processing step using reconstruction algorithms which are available as open source routines. The developed reduction technique does not require any time-consuming offline computations, e.g. for the generation of snapshots, is not restricted to any kinematic or constitutive assumptions and its implementation is straightforward. Composites consisting of elastic inclusions embedded in an i) elastic and ii) elastoplastic matrix are investigated as representative simulation examples.

Micromechanical estimation of the effective wear of elastoplastic fiber-reinforced composites

A micromechanical approach recently elaborated for estimating the effective wear of linearly elastic heterogeneous materials is, in the present work, extended to elastoplastic fibrous composites consisting of an elastoplastic matrix reinforced by elastic unidirectional fibers. The local wear behavior of both the fiber and matrix phases is assumed to be governed by the well-known Archard law. It is shown that: (i) when the normal force supported by an elastoplastic fibrous composite is smaller than a critical value, the effective wear of the composite complies still with Archard law; (ii) once the normal force applied to the composite is larger than the critical value, the effective wear of the composite depends nonlinearly on the normal force so that Archard law is no longer valid. Using the composite cylinder assemblage (CCA) model and adopting the Tresca yield criterion for the matrix phase, the effective wear of elastoplastic fiber-reinforced composites is analytically and explicitly estimated while distinguishing different elastic and elastoplastic stages. These results contribute to developing a micromechanical approach of theoretical and practical interest for understanding and designing the tribological properties of nonlinear inhomogeneous materials.

A double incremental variational procedure for elastoplastic composites with combined isotropic and linear kinematic hardening

We investigate the nonlinear behavior of elasto-plastic composites with isotropic and linear kinematic hardening. We first rely on the incremental variational principles introduced by Lahellec and Suquet (2007a). We also take advantage of an alternative formulation, recently proposed by Agoras et al. (2016) for visco-plastic composites without hardening, which consists in a double application of the variational procedure of Ponte-Castañeda. We extend in this paper this approach to elasto-plastic composites with combined linear kinematic and isotropic work-hardening. The first application of the variational procedure linearizes the local behavior, including hardening, and leads to a thermo-elastic Linear Comparison Composite (LCC) with a heterogeneous polarization field inside the phases. The second one deals with the heterogeneity of the polarization and results in a new thermo-elastic LCC with a per-phase homogeneous polarization field, which effective behavior can then be estimated by classical linear homogenization schemes. We develop and implement this new incremental variational procedure for composites comprised of linear elastic spherical particles isotropically distributed in an elasto-plastic matrix. The predictions of the model are compared with results available in the literature for cyclic proportional and non-proportional loadings. New results for elasto-plastic composites with combined isotropic and kinematic hardening are also provided. They are in good agreement with the numerical computations we carried out, at both local and macroscopic scales.

High Performance Reduced Order Modeling Techniques Based on Optimal Energy Quadrature: Application to Geometrically Non-linear Multiscale Inelastic Material Modeling

A High-Performance Reduced-Order Model (HPROM) technique, previously presented by the authors in the context of hierarchical multiscale models for non linear-materials undergoing infinitesimal strains, is generalized to deal with large deformation elasto-plastic problems. The proposed HPROM technique uses a Proper Orthogonal Decomposition procedure to build a reduced basis of the primary kinematical variable of the micro-scale problem, defined in terms of the micro-deformation gradient fluctuations. Then a Galerkin-projection, onto this reduced basis, is utilized to reduce the dimensionality of the micro-force balance equation, the stress homogenization equation and the effective macro-constitutive tangent tensor equation. Finally, a reduced goal-oriented quadrature rule is introduced to compute the non-affine terms of these equations. Main importance in this paper is given to the numerical assessment of the developed HPROM technique. The numerical experiments are performed on a micro-cell simulating a randomly distributed set of elastic inclusions embedded into an elasto-plastic matrix. This micro-structure is representative of a typical ductile metallic alloy. The HPROM technique applied to this type of problem displays high computational speed-ups, increasing with the complexity of the finite element model. From these results, we conclude that the proposed HPROM technique is an effective computational tool for modeling, with very large speed-ups and acceptable accuracy levels with respect to the high-fidelity case, the multiscale behavior of heterogeneous materials subjected to large deformations involving two well-separated scales of length.