Nonlocal Continuum Damage Mechanics Research Articles

Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

We present a new Integrated Finite Element Neural Network framework (I-FENN), with the objective to accelerate the numerical solution of nonlinear computational mechanics problems. We leverage the swift predictive capability of neural networks (NNs) and we embed them inside the finite element stiffness function, to compute element-level state variables and their derivatives within a nonlinear, iterative numerical solution. This process is conducted jointly with conventional finite element methods that involve shape functions: the NN receives input data that resembles the material point deformation and its output is used to construct element-level field variables such as the element Jacobian matrix and residual vector. Here we introduce I-FENN to the continuum damage analysis of quasi-brittle materials, and we establish a new non-local gradient-based damage framework which operates at the cost of a local damage approach. First, we develop a physics informed neural network (PINN) to resemble the non-local gradient model and then we train the neural network offline. The network learns to predict the non-local equivalent strain at each material point, as well as its derivative with respect to the local strain. Then, the PINN is integrated in the element stiffness definition and conducts the local to non-local strain transformation, whereas the two PINN outputs are used to construct the element Jacobian matrix and residual vector. This process is carried out within the nonlinear solver, until numerical convergence is achieved. The resulting method bears the computational cost of the conventional local damage approach, but ensures mesh-independent results and a diffused non-local strain and damage profile. As a result, the proposed method tackles the vital drawbacks of both the local and non-local gradient method, respectively being the mesh-dependence and additional computational cost. We showcase through a series of numerical examples the computational efficiency and generalization capability of I-FENN in the context of non-local continuum damage, and we discuss the future outlook. The PINN training code and associated training data files have been made publicly available online, to enable reproducibility of our results.

Stochastic failure analysis of proximal femur using an isogeometric analysis based nonlocal gradient-enhanced damage model

Background and objectiveMedical imaging-based finite element methods are more accurate tools for fracture risk prediction than the traditional aBMD based methods. However, these methods have drawbacks like geometric errors, high computational cost, mesh-dependent results, etc. In this article, the authors have proposed an isogeometric analysis-based nonlocal gradient-enhanced damage model to overcome some of these issues. Moreover, there are uncertainties in the values of input parameters for such analysis due to various measurement errors. Hence, stochastic analysis is performed to quantify the effect of these parametric uncertainties on the fracture behavior of the proximal femur. MethodsComputed Tomography images of a patient are used to create a 2D proximal femur model with a heterogeneous description of material properties. A numerical model based on gradient-enhanced nonlocal continuum damage mechanics is used for fracture analysis of proximal femur to overcome the issues related to mesh dependency in traditional continuum damage mechanics models. Further, a multipatch isogeometric solver is developed to solve the governing equations. Monte Carlo simulations are used to understand the effect of parametric uncertainties on the fracture behavior of the proximal femur. ResultsThe developed numerical framework is used to solve the fracture problem of proximal femur under single leg stance loading conditions. The obtained results are validated by comparing the load-displacement response and the crack path with that given in the literature. Stochastic analysis is performed by considering a ±5% variation in the elastic modulus, damage initiation strain, and the neck-shaft angle values. ConclusionThe proposed numerical framework can correctly predict the damage initiation and propagation in a proximal femur. The results reveal that the heterogeneous nature of material properties of bone plays a significant role in determining the fracture characteristics of the proximal femur. Further, the results of the stochastic analysis reveal that the parametric uncertainties in the neck-shaft angle have a much more significant influence on the results of the analysis than the parametric uncertainties in the elastic modulus and damage initiation strain.

AnisoMPM

Dynamic fracture surrounds us in our day-to-day lives, but animating this phenomenon is notoriously difficult and only further complicated by anisotropic materials---those with underlying structures that dictate preferred fracture directions. Thus, we present AnisoMPM: a robust and general approach for animating the dynamic fracture of isotropic, transversely isotropic, and orthotropic materials. AnisoMPM has three core components: a technique for anisotropic damage evolution, methods for anisotropic elastic response, and a coupling approach. For anisotropic damage, we adopt a non-local continuum damage mechanics (CDM) geometric approach to crack modeling and augment this with structural tensors to encode material anisotropy. Furthermore, we discretize our damage evolution with explicit and implicit integration, giving a high degree of computational efficiency and flexibility. We also utilize a QR-decomposition based anisotropic constitutive model that is inversion safe, more efficient than SVD models, easy to implement, robust to extreme deformations, and that captures all aforementioned modes of anisotropy. Our elasto-damage coupling is enforced through an additive decomposition of our hyperelasticity into a tensile and compressive component in which damage is used to degrade the tensile contribution to allow for material separation. For extremely stiff fibered materials, we further introduce a novel Galerkin weak form discretization that enables embedded directional inextensibility. We present this as a hard-constrained grid velocity solve that poses an alternative to our anisotropic elasticity that is locking-free and can model very stiff materials.

Scale effects in the static response of a one-dimensional quasi-brittle damage lattice

In this paper, we investigate the failure of a discrete elastic-damage axial system using both a discrete and an equivalent continuum approach. The microstructured damage chain consists of a one-dimensional damage lattice with direct nearest–neighbour interactions, which is composed of a series of periodic elastic-damage springs (axial lattice system treated in terms of discrete damage mechanics). We show that the damage lattice equations are equivalent to the centred finite difference formulation of a continuum damage mechanics (CDM) evolution problem. Such a discrete damage system reveals some scale effects on both the structural strength and stiffness. The nonlocal CDM in the hardening branch and cohesive damage model in the softening branch considered here are built using a continualisation procedure applied to the nonlinear difference equations of the lattice system. With this procedure, the difference equations to be solved are approximated by higher order differential equations. Using a rational asymptotic method, the continualised model appears to be equivalent to a nonlocal CDM model in the damage propagation zone. A finite length cohesive model is obtained in the softening range. A comparison of the discrete and the continuous problems for damage chains brings out the effectiveness of the new micromechanics-based nonlocal and cohesive continuum damage model, especially for capturing scale effects.

Numerical modelling of flax short fibre reinforced and flax fibre fabric reinforced polymer composites

The ever-increasing demand of flax short fibre-reinforced and flax fibre fabric-reinforced polymer composites in various engineering applications calls for accurate predictions of their mechanical behaviors. In this study, numerical methods to generate and simulate mechanical properties of flax short fibre-reinforced and flax fibre fabric-reinforced polymer composites are proposed. The microstructures of short flax fibres with different fibre length-to-diameter ratios are generated by algorithm taking fiber defects (e.g. kink band) and fiber bundles into account. Bidirectional flax fabric is generated and discretized by tetrahedron 4-node finite elements. A brittle material law for fibre defects and interfacial zones of fibre bundles is proposed. Flax short fibre/polypropylene and flax fabric/epoxy composites are modeled by a non-linear plasticity model considering an isotropic hardening law and non-local continuum damage mechanics. The numerical modelling results are compared with the experimental results of these composites. This study shows that the simulation can capture the main damage mechanisms of the composites such as fibre breakage initiated at the fiber defects, damage of polymer matrix and the fibre debonding at fibre/matrix interface accurately. In addition, the simulation results exhibit good agreements with the experimental results in the aspects of elastic properties and nonlinear tensile stress-strain behavior of the short fibre and fibre fabric reinforced polymer composites.

On the failure of a discrete axial chain using a continualized nonlocal Continuum Damage Mechanics approach

SummaryThe failure of a discrete elastic‐damage axial system is investigated using both a discrete and an equivalent continuum approach. The Discrete Damage Mechanics approach is based on a microstructured model composed of a series of periodic elastic‐damage springs (axial Discrete Damage Mechanics lattice system). Such a discrete damage system can be associated with the finite difference formulation of a Continuum Damage Mechanics evolution problem. Several analytical and numerical results are presented for the tensile failure of this axial damage chain under its own weight.The nonlocal Continuum Damage Mechanics models examined in this paper are mainly built from a continualization procedure applied to centered or uncentered finite difference schemes. The asymptotic expansion of the first‐order upward difference equations leads to a first‐order nonlocal model, whereas the asymptotic expansion of the centered finite difference equations leads to a second‐order nonlocal Eringen's approach. To complete this study, a phenomenological nonlocal gradient approach is also examined and compared with the first continualization methods.A comparison of the discrete and the continuous problems for the chains shows the effectiveness of the new micromechanics‐based nonlocal Continuum Damage modeling, especially for capturing scale effects. For both continualized approaches, the length scale of the nonlocal models depends only on the cell size, while for the so‐called phenomenological approach, the length scale may depend on the loading parameter. This apparent load‐dependent length scale, already discussed in the literature with numerical arguments, is found to be sensitive to the postulated structure of the nonlocal model calibrated according to a lattice approach. Copyright © 2015 John Wiley & Sons, Ltd.

Adaptive modeling of damage growth using a coupled FEM/BEM approach

SummaryWe propose a coupled boundary element method (BEM) and a finite element method (FEM) for modelling localized damage growth in structures. BEM offers the flexibility of modelling large domains efficiently, while the non‐linear damage growth is accurately accounted by a local FEM mesh. An integral‐type nonlocal continuum damage mechanics with adapting FEM mesh is used to model multiple damage zones and follow their propagation in the structure. Strong form coupling, BEM hosted, is achieved using Lagrange multipliers. Because the non‐linearity is isolated in the FEM part of the system of equations, the system size is reduced using Schur complement approach, then the solution is obtained by a monolithic Newton method that is used to solve both domains simultaneously. The coupled BEM/FEM approach is verified by a set of convergence studies, where the reference solution is obtained by a fine FEM. In addition, the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature. Copyright © 2015 John Wiley & Sons, Ltd.

Nonlocal Continuum Damage Mechanics Approach of a Discrete Axial Chain under Non-Uniform Axial Load

The failure of a discrete elastic-damage axial system is investigated using both a discrete and anequivalent continuum approach. The discrete damage mechanics (DDM) approach is based on amicrostructured model composed of a series of periodic elastic-damage springs (axial DDM latticesystem). Such a damage discrete system can be associated with the finite difference formulation of aContinuum Damage Mechanics (CDM) evolution problem.The nonlocal CDM models considered in this paper are mainly built from a continualizationprocedure applied to centered finite difference schemes. A comparison of the discrete and thecontinuous problems for the chains shows the effectiveness of the new micromechanics-basednonlocal Continuum Damage modeling, especially for capturing scale effects.

From discrete to nonlocal continuum damage mechanics: Analysis of a lattice system in bending using a continualized approach

It is shown herein that the bending problem of a discrete damage system, also called microstructured damage system or lattice damage system, can be rigorously handled by a nonlocal continuum damage mechanics approach. It has been already shown that Eringen’s nonlocal elasticity was able to capture the scale effects induced by the discreteness of a microstructured system. This paper generalizes such results for inelastic materials and first presents some results for engineering problems modelled within continuum damage mechanics. The microstructured model is composed of rigid periodic elements connected by rotational elastic damage springs (discrete damage mechanics). Such a discrete damage system can be associated with the finite difference formulation of a continuum damage mechanics problem, i.e. the Euler–Bernoulli damage beam problem. Starting from the discrete equations of this structural problem, a continualization method leads to the formulation of an Eringen’s type nonlocal model with full coupling between nonlocal elasticity and nonlocal continuum damage mechanics. Indeed, the nonlocality appears in this continualized approach both in the constitutive law and in the damage loading function. A comparison of the discrete and the continuous problems for the cantilever shows the efficiency of the new micromechanics-based nonlocal continuum damage modelling for capturing scale effects. The length scale of the nonlocal continuum damage mechanics model is rigorously calibrated from the size of the cell of the discrete repetitive damage system. The new micromechanics-based nonlocal damage mechanics model is also analysed with respect to available nonlocal damage mechanics models.

A numerical investigation of surface crevasse propagation in glaciers using nonlocal continuum damage mechanics

We use a nonlocal viscoelastic damage model to investigate the conditions that enable water‐free surface crevasse propagation in grounded marine‐terminating glaciers. Our simulations, on idealized rectangular ice slabs in contact with the ocean, show that crevasses propagate faster in thicker ice slabs. We find that: (1) the fraction of ice slab thickness penetrated by surface crevasses decreases with increasing seawater depth near the terminus; (2) a no‐slip (fixed) basal boundary condition retards crevasse growth; and (3) crevasses form closer to the terminus when the seawater depth is larger or when the glacier base is fixed to the bedrock, which could lead to calving of smaller icebergs. However, water‐free surface crevasses can penetrate (nearly) the entire ice thickness only in thicker ice slabs terminating in shallow seawater depths. This leads us to the conclusion that surface crevasses alone are not responsible for calving events in marine‐terminating and thin glaciers.

The role of curing stresses in subsequent response, damage and failure of textile polymer composites

An integrated computational framework for textile polymer composites is introduced. A novel polymer curing model is used in connection with modeling the polymer curing process during manufacturing of textile composites. The model is based on the notion of polymer networks that are continuously formed in a body of changing shape due to changes in temperature, chemistry and external loads. Nonlinear material behavior is incorporated through nonlocal continuum damage mechanics that preserves mesh objectivity in calculations that go beyond maximum loads. The integrated model is applied to the curing of plain weave textile composites made from carbon fiber tows and Epon™862 resin. The mechanical and chemical properties are measured during curing using concurrent Brillouin and Raman light scattering. It is shown that significant stresses can develop during cure. The effect of these stresses on the manufactured part performance, when subsequent service loads are applied, is evaluated and a reduction in ultimate load, in agreement with experimental observations, is observed.

I am 80 going on 18: exercise and the fountain of youth

aging is associated with physiological decline. To slow or reverse this decline, humans have searched for various “fountains of youth” since the dawn of history ([4][1]). Most recently, aging has become medicalized and interventions, such as caloric restriction, stem cells, hormonal anti-aging

A nonlocal continuum damage mechanics approach to simulation of creep fracture in ice sheets

We present a Lagrangian finite element formulation aimed at modeling creep fracture in ice-sheets using nonlocal continuum damage mechanics. The proposed formulation is based on a thermo-viscoelastic constitutive model and a creep damage model for polycrystalline ice with different behavior in tension and compression. In this paper, mainly, we detail the nonlocal numerical implementation of the constitutive damage model into commercial finite element codes (e.g. Abaqus), wherein a procedure to handle the abrupt failure (rupture) of ice under tension is proposed. Then, we present numerical examples of creep fracture under four-point bending, uniaxial tension, and biaxial tension in order to illustrate the viability of the current approach. Finally, we present simulations of creep crack propagation in idealized rectangular ice slabs so as to estimate calving rates at low deformation rates. The examples presented demonstrate the mesh size and mesh directionality independence of the proposed nonlocal implementation.

Multi-scale failure analysis of plain-woven composites

A numerical model capable of dealing with progressive degradation of plain woven composites in a computationally efficient manner is presented in this article. A semi-analytical homogenization method is used to derive effective properties of the composite from the material properties of the constituents. The progressive failure is described using nonlocal continuum damage mechanics where the driving internal variable for the damage is the nonlocal strain. The model was implemented into Abaqus/Explicit, where the failure of a longitudinal tension and an open hole tension specimens were simulated in a multi-scale manner and verified experimentally.

Size effect predictions by fracture models for a refractory ceramic

Refractory ceramics used as kiln furniture are designed to operate at elevated temperatures with a high thermal shock resistance. In practice, however, the material fails due to thermal fatigue after a limited number of cycles. To predict this failure behaviour, it is generally not possible to use linear elastic fracture mechanics due to the fact that the coarse grained, porous material shows a dissipative mechanical behaviour. Differently sized specimens are tested to determine the size effect associated with this material. Four different finite element models widely used for concrete modelling are investigated on their ability to describe this nonlinear failure behaviour, in particular the associated size effect phenomenon. The results of the initially promising Nonlocal Continuum Damage Mechanics (CDM) model indicate that this approach cannot properly describe the observed size effect. The other three models (Adapted Local CDM, Fictitious Crack and Smeared Crack), however, give comparable results with a good description of the observed size effect phenomenon.

Probabilistic Nonlocal Damage Model for Continua with Random Field Properties

Nonlocal continuum damage mechanics and random field theory are used to model the stochastic damage behavior of a strainsoftening material with random field properties. The randomness in the damage...