Material Point Level Research Articles

Hydromechanical modelling of the influence of water saturation on the penetration performance of concrete target

The objective of the present paper is to study numerically the hydro-mechanical behaviour of concrete targets subjected to a rigid projectile, with special attention to the influence of water saturation. An elastoplastic model, encompassing two primary plastic mechanisms (shear and pore collapse), is improved to capture the influence of water saturation. The adopted model is implemented in the finite element code Abaqus/Explicit and validated by simulating two dynamic compression tests at the material point level and two penetration tests at the structure level. Not only a strong influence of water saturation on both volumetric and deviatoric behaviours of concrete is observed at material point level, but also the time-evolutions of projectile velocity, projectile deceleration and penetration depth in concrete targets are also satisfactorily reproduced by the numerical simulations. After that, a series of parametric studies are performed to understand the influence of water saturation on the penetration performance of concrete. It is found that with the presence of pore water, penetration resistance of concrete increases with decreasing water saturation, indicating a higher projectile deceleration and lower penetration depth obtained in saturated targets. Numerical modelling and analysis will enhance our understanding of the vulnerability of concrete infrastructures subjected to near-field detonations or impacts.

Enhancing damage prediction in bulk metal forming through machine learning-assisted parameter identification

The open-source parameter identification tool ADAPT (A diversely applicable parameter identification Tool) is integrated with a machine learning-based approach for start value prediction in order to calibrate a Gurson–Tvergaard–Needleman (GTN) and a Lemaitre damage model. As representative example case-hardened steel 16MnCrS5 is elaborated. An artificial neural network (ANN) is initially trained by using load–displacement curves derived from simulations of a boundary value problem—instead of using data generated for homogeneous states of deformation at material point or one-element level—with varying material parameter combinations. The ANN is then employed so as to predict sets of material parameters that already provide close solutions to the experiment. These predicted parameter sets serve as starting values for a subsequent multi-objective parameter identification by using ADAPT. ADAPT allows for the consideration of input data from multiple scales, including integral data such as load–displacement curves, full-field data such as displacement and strain fields, and high-resolution experimental void data at the micro-scale. The influence of each data set on prediction quality is analyzed. Using various types of input data introduces additional information, enhancing prediction accuracy. The validation is carried out with respect to experimental void measurements of forward rod extruded parts. The results demonstrate, by incorporating void measurements in the optimization process, that it is possible to improve the quantitative prediction of ductile damage in the sense of void area fractions by factor 28 in forward rod extrusion.

Neural integration for constitutive equations using small data

Data-driven models based on deep learning algorithms intend to overcome the limitations of traditional constitutive modelling by directly learning from data. However, the need for extensive data that collate the full state of the material is hindered by traditional experimental observations, which typically provide only small data—sparse and partial material state observations. To address this issue, we develop a novel deep learning algorithm referred to as Neural Integration for Constitutive Equations to discover constitutive models at the material point level from scarce and incomplete observations. It builds upon the solution of the initial value problem describing the time evolution of the material state, unlike the majority of data-driven approaches for constitutive modelling that require large data of increments of state variables. Numerical benchmarks demonstrate that the method can learn accurate, consistent, and robust constitutive models from incomplete, sparse, and noisy data collecting simple conventional experimental protocols.

A phase‐field model for hydrogen‐promoted fracture based on a mixed rate‐type variational setting

AbstractCertain metals experience a substantial deterioration in mechanical properties when exposed to a hydrogen environment, an effect termed hydrogen embrittlement. To understand, predict, and counteract this hydrogen‐assisted material degradation, sufficiently accurate material models are needed. According to the current hypothesis, hydrogen diffusion is driven by gradients of concentration and hydrostatic stress. To capture this, a phase‐field model is formulated as a multi‐field problem coupling deformation, crack propagation, and diffusion to analyze hydrogen‐promoted fracture. Here, the displacements, a fracture‐related phase‐field, the hydrogen lattice occupancy, and the chemical potential are considered as primary field variables. Approaches proposed in the literature often use an extrapolation of the hydrostatic stress calculated at the material point level onto the nodes and later use the B‐matrix to compute the gradient of hydrostatic stress. In order to circumvent this potentially inaccurate extrapolation, the model is recast into a mixed rate‐type variational setting, where the chemical potential—whose gradient governs the hydrogen flux—is obtained from the numerical solution of a saddle point problem. A representative boundary value problem is presented to demonstrate the applicability of the developed numerical framework.

An anisotropic damage model for finite strains with full and reduced regularization of the damage tensor

AbstractThe modeling of damage as an anisotropic phenomenon enables the consideration of arbitrarily oriented microcracks at the material point level. Yet, the incorporation of material softening into structural simulations still requires a regularization of for example, the degrading variable. There exist different possibilities for a regularization in case of anisotropic damage with varying numbers of nonlocal degrees of freedom corresponding to for example, the symmetric integrity tensor , the principal traces of the damage tensor or a scalar damage hardening variable. Here, we propose a finite strain formulation with a symmetric second order damage tensor of which all six independent components are regularized with a corresponding nonlocal degree of freedom. Due to the significant increase in computational cost caused by the full regularization of the damage tensor, alternative approaches for a reduced regularization with fewer nonlocal degrees of freedom are discussed. Thereafter, the results of a numerical example using the model with full regularization are presented.

Numerical Aspects of a Continuum Sintering Model Formulated in the Standard Dissipative Framework

Robust and computationally efficient numeric algorithms are required to simulate the sintering process of complex ceramic components by means of the finite element method. This work focuses on a thermodynamically consistent sintering model capturing the effects of both, viscosity and elasticity, within the standard dissipative framework. In particular, the temporal integration of the model by means of several implicit first and second order accurate one step time integration methods is discussed. It is shown in terms of numerical experiments on the material point level that the first order schemes exhibit poor performance when compared to second order schemes. Further numerical experiments indicate that the results translate directly to finite element simulations.

Finite element modelling of unsaturated soils using a modified version of the Barcelona Basic Model

In this paper, unsaturated soil behaviour is numerically simulated using a modified version of the Barcelona Basic Model. The model has been implemented as a User-Defined Soil Model (UDSM) in the finite element code PLAXIS. Unlike the original formulation based on net stress ([1]), this model isformulated based on the well-known Bishop’s effective stress, with suction as an additional stress variable. Furthermore, a non-associated flow rule and the dependency on Lode’s angle proposed by [2] are incorporated. The Soil Water Characteristic Curve (SWCC) follows the Van Genuchten model ([3]). First, the simulation of laboratory tests along different stress paths and levels of suction are compared againstexperimental data. Then, an unsaturated soil embankment subject to a time-dependent precipitation is simulated by means of a fully coupled flow-deformation analysis. The results show the capability of the model to reproduce the main features of unsaturated soil behaviour both at material point level and inboundary value problems.

Two-scale off-and online approaches to geometrically exact elastoplastic rods

This work compares two different computational approaches to geometrically exact elastoplastic rods. The first approach applies an elastoplastic constitutive model in terms of stress resultants, i.e. forces and moments. It requires knowledge of the rod’s elasticity and yield-criterion in terms of stress resultants. Furthermore a resultant-type hardening expression must be formulated. These are obtained by integrating elastoplastic stress and hardening measures from three-dimensional continuum mechanics over the rod’s deformed cross-section, which is performed in an offline stage. The second approach applies an FE ^2 approach as established in computational homogenization. Therein, the macro-scale describing the geometrically exact rod is coupled to the micro-scale, i.e., the cross-section of the rod. A novelty of the presented work is the determination of a hardening tensor for use in the stress resultant approach. The mechanical response of both approaches is first compared on the material point level, a single cross-section of a uniformly strained rod. Later, also the mechanical response and the deformation of finitely and non-uniformly strained rods are investigated.

Energetically motivated crack orientation vector for phase-field fracture with a directional split

The realistic approximation of structural behavior in a post fracture state by the phase-field method requires information about the spatial orientation of the crack surface at the material point level. For the directional phase-field split, this orientation is specified by the crack orientation vector, that is defined perpendicular to the crack surface. An alternative approach to the determination of the orientation based on standard fracture mechanical arguments, i.e. in alignment with the direction of the largest principle tensile strain or stress, is investigated by considering the amount of dissipated strain energy density during crack evolution. In contrast to the application of gradient methods, the analytical approach enables the determination of all local maxima of strain energy density dissipation and, in consequence, the identification of the global maximum, that is assumed to govern the orientation of an evolving crack. Furthermore, the evaluation of the local maxima provides a novel aspect in the discussion of the phenomenon of crack branching. As the directional split differentiates into crack driving contributions of tension and shear stresses on the crack surface, a consistent relation to Mode I and Mode II fracture is available and a mode dependent fracture toughness can be considered. Consequently, the realistic simulation of rock-like fracture is demonstrated. In addition, a numerical investigation of Gamma -convergence for an AT-2 type crack surface density is presented in a two-dimensional setup. For the directional split, also the issues internal locking as well as lateral phase-field evolution are addressed.

Variational modeling of temperature induced and cooling-rate dependent phase transformations in polycrystalline steel

Phase transformations in steel have a high impact on the material properties. The material is modified by a changed stiffness as well as residual stresses which are a result of transformation strains and plastic deformations during the phase transition. In the current work, we propose a variational material model which is able to display the phase transformations which occur during thermal processes like induction hardening and grinding. Depending on the cooling rate, the material model is able to show the transformation between austenite and martensite, austenite and ferrite as well as a mixture of both. The polycrystalline structure of the steel is taken into account by use of different oriented grains which are considered by a combined Voigt–Reuss bound for the energy of the phase and grain mixture. The material model is derived using the principle of the minimum of the dissipation potential for non-isothermal problems. Compared to classical approaches in material science, the model is thermomechanically coupled. Thus, the phase transformation is also influenced by the stress state and individual martensitic variants are favored during the transformation process, which is an important property for applications to process chains. The model’s ability to show the complex material behavior is proven by calculations on the material point level as well as by finite element simulations.

Cylindrical microplane model for compressive kink band failures and combined friction/inelasticity in fiber composites II: Analyses

This two-part paper presents a novel 3D cylindrical microplane model for the longitudinal compressive failure of fiber reinforced composites by kink band formation. The model formulation was described in part I, and here in part II, its implementation and numerical analyses are described. Through both material point and structure level analyses, the model is shown to predict well the geometric features of the failure, including damage localization, in-plane and out of plane kink band formation, and local fiber rotation. Further, the model is able to capture the entire load displacement curve including the peak load, the prolonged post-peak load plateau, and most importantly, the quasi-brittle fracture mechanics-based strength size effect, in agreement with experiments. The model is also able to capture the effect of the misalignment angle on the peak and post-peak loads. The validated model is used to investigate and quantify the effect of friction on the failure behavior. Results suggest that friction is a major effect, contributing to a higher peak load and post-peak load plateau. Additionally, a simpler and efficient alternative for the combined friction/damage is proposed, which involves a simple modification of the microplane level damage law.

Strain localization criteria for viscoplastic geomaterials

AbstractThis work presents a viscoplastic localization criterion to detect quasi‐instantaneous (i.e., load‐induced) and delayed (creep‐induced) strain localization in rate‐dependent solids. The study is based on the theory of controllability and a viscoplastic description of the mechanical response. Analytical precursors of unstable states are defined through systems of ordinary differential equations (OEDs). The use of the proposed criteria is illustrated at the material point level through a set of strain localization analyses simulating active strain localization of a porous rock. In addition, full‐field finite element simulations of compression tests conducted under various pressures are reported to demonstrate the role of local unstable viscoplasticity in the spontaneous propagation of deformation bands under stationary boundary conditions. The study shows that the viscoplastic localization criterion maintains a negative sign as long as the behavior is unstable, that is, the rate of deformation is accelerating. The sign switch coincides with the transition to decelerating deformation. The analyses revealed that pulses of overstress always emerge in correspondence with the growth of unstable behavior, and the peak matches the transition to stable behavior. The local responses recovered from full‐field analyses were consistent with those observed in analyses at material point level and the predictions of the presented theory.

Effects of spatial autocorrelation structure for friction angle on the runout distance in heterogeneous sand collapse

This paper proposes a stochastic method for analyzing the runout distance of sand collapse considering the spatial variability of shear strength, in which random field theory and generalized interpolation material point method are integrated into a Monte-Carlo simulation basis. The random field is generated by Cholesky matrix decomposition method and implemented into the material point level, hence heterogeneity and large deformations are simultaneously considered in the modeling process. A sand collapse case is simulated with both homogeneous and heterogeneous condition assumptions by the proposed method. The effect of five theoretical autocorrelation functions (ACFs) on the runout distance of the collapse is highlighted since the ACFs are commonly adopted to characterize the spatial variability of soil properties due to sparse site observation data. It is shown that the deterministic analysis may underestimate the runout distance, while the heterogeneous model provides realistic results. Moreover, five ACFs and different coefficients of variation of friction angle (COVφ) are compared to investigate their influences on the runout distance modeling. The results show that the uncertainty of runout distance increases with the increase in COVφ. Meanwhile, the variances of the runout distance also become larger with COVφ increasing. Based on the proportion of the runout distance which exceeds the deterministic value, the results indicate that the deterministic analysis notably underestimates the risk induced by large runout distances in real heterogeneous granular flows (e.g., landslide, debris-avalanches).

An enhanced algorithmic scheme for relaxed incremental variational damage formulations at finite strains

AbstractRelaxed damage formulations enable to overcome the mesh dependency induced by a loss of ellipticity. However, such approaches require the construction of a convex hull of the associated energy potential. Within this contribution, an improved one‐dimensional, relaxed damage formulation of [1] with a successive, computationally optimal convexification scheme, see [2], is presented. Instead of solving a non‐convex optimization problem at the material point level, we iterate over a discrete grid and build consecutively linear Taylor expansions to construct the convex hull. Due to the reduced computational effort, integration of the one‐dimensional mechanical model over orientation distributions to obtain the three‐dimensional response as proposed in [3] becomes more feasible.

Consistent Implicit Time Integration for Viscoplastic Modelingof Subsidence above Hydrocarbon Reservoirs

The viscoplastic model proposed by Vermeer and Neher in 1999 is still currently used in the oil and gas industry for subsidence modeling, to predict the deformation of the ground surface induced by hydrocarbon withdrawal from underground reservoirs. Even though several different implementations of this model have been proposed in the literature, also very recently, a consistent fully implicit implementation is still missing, probably due to the technical difficulties involved in the rigorous derivation of the analytical tangent matrix. To fill this gap and to provide an effective tool to the engineering community, a fully implicit backward Euler integration is proposed and validated in this work. The consistent expression of the tangent stiffness matrix is also derived analytically, and its validation strategy is described in detail. The model was implemented in a commercial finite element code through a user-defined material subroutine. The advantages of the proposed implicit formulation in terms of stability with respect to an explicit formulation were assessed and validated. The examples include studies at material point level and at field scale for a case study of subsidence in a synthetic reservoir.

Numerically Efficient Three-Dimensional Model for Non-Linear Finite Element Analysis of Reinforced Concrete Structures.

The paper concerns the non-linear finite element analysis (NLFEA) of Reinforced Concrete (RC) structures for engineering applications. The required level of complexity of constitutive models for such analysis was discussed and non-linear elastic models combined with the smeared cracking approach proved to be efficient. A new constitutive hypoelastic-brittle model of concrete based on these assumptions was proposed. Moreover, a method including the tension stiffening effect (TS) was developed. This phenomenon is connected with the bond properties between concrete and steel and, in some situations, has significant influence on the deflections of RC structures. It is often neglected by or included in the constitutive model of concrete. In the paper, an alternative approach was presented, in which this phenomenon is taken into account by generalising the material model of reinforcing steel. This approach is consistent with modern design standards and has solid physical foundations. The proposed models were implemented in the Abaqus code via UMAT user’s procedure coded in FORTRAN. Model verification and validation were presented in four case studies, concerning: a Willam’s test (examination on material point level), a beam with bending failure, and two beams with shear failure (with and without stirrups). The obtained results were compared with experimental outcomes and numerical results obtained by other researchers. The presented approach enables the accurate prediction not only of load capacity but of structural deformability, due to the precise description of TS. Thus, it promises to be a useful engineering tool.

Micromorphic approach to phase-field modeling of multivariant martensitic transformation with rate-independent dissipation effects

A micromorphic formulation of the phase-field model of martensitic transformation is developed within the incremental energy minimization framework. In contrast to the conventional phase-field formulation, the order parameters are viewed as local variables and the corresponding evolution equations are solved at the material-point level, i.e. at the Gauss points in the finite-element setting. From a computational standpoint, such a treatment is advantageous for complex evolution laws that may lead to computational difficulties if treated globally, as in the conventional phase-field formulation. In the micromorphic formulation, each order parameter is coupled to its micromorphic counterpart governed by a global Helmholtz-type PDE. This coupling ensures that the interfacial energy and related size effects are correctly captured by the model. In this work, the micromorphic approach is applied to a finite-strain multivariant phase-field model that incorporates rate-independent dissipation. The augmented Lagrangian technique is then used to transform the resulting non-smooth incremental minimization problem to a smooth and unconstrained saddle-point problem. Microstructure evolution under nano-indentation is studied to illustrate the approach.

Shrinkage-induced deformations and creep of structural concrete: 1-year measurements and numerical prediction

The material models for creep and shrinkage operating on the material point level in FEM are usually intended for challenging complex applications and structures, where the average cross-sectional approach does not suffice. The identification of the growing number of material parameters induced by increasing model capabilities relies on very specific and narrow-oriented yet interconnected experiments which are scarce. The presented comprehensive experiments aim to provide a clearer image of the complicated interaction among the basic phenomena: drying, shrinkage, creep, and microcracking. The cornerstone of this ongoing research is a unique set of 30 partially-sealed unreinforced concrete beams with span 1.75–3.0 m subjected to drying.To minimize material variation, all specimens in this study were cast from a single batch of ordinary strength structural concrete with slag-blended binder. The resulting experimental database will be suitable both for validation and development of the constitutive models.

Effect of the shape of the softening damage law on the predicted tensile fracturing and energy dissipation in textile composites

The fracturing behavior of fiber-reinforced composites is often modeled using continuum damage mechanics-based approaches which commonly assume a linear softening damage law, i.e. a linearly decreasing stress with increasing strain. The objective of this paper is to assess the suitability of this assumption for composites and to systematically analyze the effect of the assumed shape of the softening damage law at material point, on the predicted fracturing behavior of the structure. The material considered is an epoxy/carbon-fiber twill woven composite. Numerical evaluations are conducted by predicting the strength size effect and progressive accumulation of damage in a structure and comparing with experimental measurements. Damage models with a wide range of shapes of softening laws are considered. These include linear, bilinear, trilinear, exponential, power law and sigmoidal. The effect of introducing a sudden drop in the damage law is also evaluated. All modeling is conducted via the crack band model (CBM), to ensure the mesh objectivity of results. It is found that despite using the right strength and fracture energy as input at the material point level, not all softening law shapes can accurately predict the strength size effect and post-peak softening. Especially the sudden drop feature leads to severe underprediction of strength. Bilinear softening shows the best combination of simplicity and accuracy, especially if formulated via the R-curve approach. Further insights are obtained by assessing the predicted fracture process zone size and fracture energy via the size effect method. Finally structure level predictions are made by considering plug initiated axial crushing of a cylindrical crushcan. The close ties between the shape of the material softening law and predicted process zone, and structural fracturing are revealed, and shown to be an important consideration in formulating damage models.

Thermodynamics-based Artificial Neural Networks for constitutive modeling

Machine Learning methods and, in particular, Artificial Neural Networks (ANNs) have demonstrated promising capabilities in material constitutive modeling. One of the main drawbacks of such approaches is the lack of a rigorous frame based on the laws of physics. This may render physically inconsistent the predictions of a trained network, which can be even dangerous for real applications.Here we propose a new class of data-driven, physics-based, neural networks for constitutive modeling of strain rate independent processes at the material point level, which we define as Thermodynamics-based Artificial Neural Networks (TANNs). The two basic principles of thermodynamics are encoded in the network’s architecture by taking advantage of automatic differentiation to compute the numerical derivatives of a network with respect to its inputs. In this way, derivatives of the free-energy, the dissipation rate and their relation with the stress and internal state variables are hardwired in the architecture of TANNs. Consequently, our approach does not have to identify the underlying pattern of thermodynamic laws during training, reducing the need of large data-sets. Moreover the training is more efficient and robust, and the predictions more accurate. Finally and more important, the predictions remain thermodynamically consistent, even for unseen data. Based on these features, TANNs are a starting point for data-driven, physics-based constitutive modeling with neural networks.We demonstrate the wide applicability of TANNs for modeling elasto-plastic materials, using both hyper- and hypo-plasticity models. Strain hardening and softening are also considered for the hyper-plastic scenario. Detailed comparisons show that the predictions of TANNs outperform those of standard ANNs. Finally, we demonstrate that the implementation of the laws of thermodynamics confers to TANNs high robustness in the presence of noise in the training data, compared to standard approaches.TANNs’ architecture is general, enabling applications to materials with different or more complex behavior, without any modification.