Linear Elastic Material Behavior Research Articles

Efficient and accurate uncertainty quantification in engineering simulations using time-separated stochastic mechanics

A robust method for uncertainty quantification is undeniably leading to a greater certainty in simulation results and more sustainable designs. The inherent uncertainties of the world around us render everything stochastic, from material parameters, over geometries, up to forces. Consequently, the results of engineering simulations should reflect this randomness. Many methods have been developed for uncertainty quantification for linear elastic material behavior. However, real-life structure often exhibit inelastic material behavior such as visco-plasticity. Inelastic material behavior is described by additional internal variables with accompanying differential equations. This increases the complexity for the computation of stochastic quantities, e.g., expectation and standard deviation, drastically. The time-separated stochastic mechanics is a novel method for the uncertainty quantification of inelastic materials. It is based on a separation of all fields into a sum of products of time-dependent but deterministic and stochastic but time-independent terms. Only a low number of deterministic finite element simulations are then required to track the effect of (in)homogeneous material fluctuations on stress and internal variables. Despite the low computational effort the results are often indistinguishable from reference Monte Carlo simulations for a variety of boundary conditions and loading scenarios.

Design of railway transition zones: A novel energy-based criterion

Railway transition zones (RTZs) experience higher rates of degradation compared to open tracks, which leads to increased maintenance costs and reduced availability. Despite existing literature on railway track assessment and maintenance, effective design solutions for RTZs are still limited. Therefore, a robust design criterion is required to develop effective solutions. This paper presents a two-step approach for the formulation of a preliminary-design criterion to delay the onset of processes leading to uneven track geometry in RTZs. Firstly, a systematic analysis of each track component in a RTZ is performed by examining spatial and temporal variations in kinematic responses, stresses and energies using a finite element model of an embankment-bridge transition. Secondly, the study proposes an energy-based criterion to be assessed using a model with linear elastic material behavior and states that an amplification in the total train energy in the proximity of the transition interface is an indicator of increased (and thus non-uniform) degradation in RTZs compared to the open tracks. The correlation between the total strain energy (assessed in the model with linear material behaviour) and the permanent irreversible deformations is demonstrated using a model with non-linear elastoplastic material behavior of the ballast layer. In the end, it is claimed that minimising the magnitude of total strain energy will lead to reduced degradation and a uniform distribution of total strain energy in each trackbed layer along the longitudinal direction of the track will ensure uniformity in the track geometry.

A closed-form solution for evaluating the Findley critical plane factor

The fatigue assessment of structural components is a significant topic investigated both in the academia and industry. Despite the significant progress in comprehension over the past few decades, fatigue damage remains a significant challenge, often leading to unexpected component failures. One commonly used approach for fatigue assessment is the critical plane analysis, which aids in identifying the critical location and early crack propagation direction in a component. However, the conventional method for calculating critical plane factors is computationally demanding and is typically utilized only when the critical regions of the component are already known. In situations where the critical areas are difficult to be identified due to complex geometry, loads, or constraints, a more efficient method is required for evaluating critical plane factors. This research paper introduces an analytical algorithm to efficiently evaluates the widely used Findley critical plane factor. The algorithm operates within the framework of linear-elastic material behavior and proportional loading conditions, relying on tensor invariants and coordinate transformation laws. The algorithm has been tested on different component geometries, including a box-welded joint and a tubular specimen, subjected to proportional loading conditions such as tension, torsion, and a combination of them. The analytical method allowed a significant reduction in computation time while providing the exact solution of critical plane factor and critical plane orientations.

A new mechanism based cohesive zone model for Mode I delamination coupled with fiber bridging of composite laminates

Based on identification of the two distinguishing delamination mechanisms within the two delamination zones associated with Mode I fracture toughness testing of composite laminates using the well-known ASTM standard double cantilever specimen (DCB), a new mechanism based cohesive zone model (MB-CZM) is proposed in this work. Overcoming the limitations with the widely used superposed cohesive zone models, the proposed MB-CZM develops two traction-separation relations to individually represent the two distinctive delamination mechanisms. One for the quasi-brittle linear elastic behavior of composite material and another for the nonlinear characteristics of fiber bridging which is commonly simplified with tri-linear to multi-linear approximation in the previous cohesive zone models (CZMs). Energy decomposition is carried out based on different damage and toughening mechanisms associated with delamination initiation and propagation. The proposed new MB-CZM is implemented in the finite element analysis via two UMAT subroutines and used in the numerical simulations. The good agreement of the simulation results with the experimental results provides the verification and demonstration of the capabilities of the proposed MB-CZM.

The relevance of analytical formulations predicting stiffener tripping

In search for the governing failure mode of externally pressurized ring-stiffened cylinders the focus is, due to its loading and slender geometry, drawn to elastic buckling. However, the hydrostatic pressure loading and imperfections cause nonlinear behaviour, and excessive deformation results in plastic failure, denoted by collapse. This implies that a true bifurcation point is often hardly observed, and this raises the question whether it is worthwhile to spend much effort to determine the elastic buckling pressures other than to ensure they are well above the collapse pressure? This paper focuses on the elastic buckling of ring frames, generally referred to as frame tripping. Some latest attempts in improving the accuracy of the analytical formulations will be questioned. Firstly, this study will compare non-linear buckling with linear-elastic material behaviour to elastic buckling by means of finite element analysis. Secondly, the non-linear elastic-plastic effects will be included too. Doing so, the effect of the geometrical non-linearity and material elastic-plastic non-linearity will be visible separately and will give an indication of the relevance of frame tripping results predicted by analytical elastic buckling formulations. Literature offers a variety on analytical formulations, each with an underlying idea to remove undesired assumptions that impair results. Even recently comprehensive formulations are published to establish an accurate value of the elastic tripping pressures. This paper shows whether those methods are still relevant in modern pressure hull design or not.

Advanced finite element analyses to compute the J-integral for delaminated composite plates

This study concerns several possible ways of computing the J-integral in the cases of laminated composite plate structures containing delamination. On the one hand, two special types of plate finite element families are provided for advanced fracture mechanical analyses. On the other hand, a numerical algorithm is introduced also that utilizes results derived from solid element models built in commercial finite element software. In order to create plate elements, the first-order (FSDT) and the third-order shear deformation theories (TSDT) are applied with two equivalent single layers. A subparametric element family is created by using the Hermite interpolation with the FSDT that grants the continuity of the derivatives at the element borders. Besides, an isoparametric family is created also with the TSDT employing the Lagrange interpolation scheme. This paper aims to present the compact formulations of calculating the J-integral around the delamination fronts for both plate element families. The calculation of the J-integral is implemented as a matrix multiplication by taking advantage of the properties of plate transition elements. The models yield improved fracture mechanical analyses for delaminated composite plates with linear elastic material behaviour. In addition, computing the J-integral is not available in most commercial finite element software for laminated orthotropic materials, therefore, an appropriate algorithm is presented here for this purpose. To conclude, the models and algorithms are applied to two case studies and subjected to a comprehensive comparison with analytical and formerly used finite element solutions.

Study of the Impact of Hard Rock Pillar Behavior on the Raise Caving Mining System

Raise caving is a novel mining method with an active stress control approach, which is suitable for deep mining environments. The application of raise caving is investigated for LKAB’s Kiruna mine. This mining method consists of two different phases. In the de-stressing phase, inclined de-stressing slots are created with a minimum amount of infrastructure and long-raise bore holes. The de-stressing slots provide stress shadows for subsequent large scale mining activities. These slots are separated by massive pillars, with width to height ratios of up to 10 and a length of several hundreds of meters, which are required to control stresses and seismicity and prevent hangingwall caving. Due to the stress redistribution in the mine, pillars and abutment areas become highly stressed. In the so-called “production phase”, which follows the de-stressing phase, large scale mineral extraction is conducted. As pillars are highly stressed after the de-stressing phase, it is important to de-stress them in the production phase to be able to extract them in a safe manner. Overall, pillars play a decisive role for the success of the raise caving mining method. Hence, studies on pillars and their impact on the raise caving mining method are of great importance. The first part of the contribution outlines the importance of pillars in the raise caving mining method on a conceptual basis. Numerical simulations for different mine layouts and different pillar behavior are then conducted to study the impact of pillar behavior on the system. The stress redistribution in the mining system and the infrastructure stability, with the help of the RCF-value, are analyzed. For this purpose, simulations with a linear elastic material behavior are a well-suited method. Consequently, the de-stressing of the pillar due to crushing cannot be analyzed with a linear elastic model. To do so, a method for de-stressing the pillar has to be implemented into the simulation. The behavior for pillars with a different width to height ratio is implemented with the help of pre-calculated stress-strain curves from available knowledge on pillar strength and behavior. Since mining is a progressive process, the sequence of mining plays an important role for the stress development in pillars and abutment areas and is also included in the simulations. Different ways for pillar de-stressing are outlined and discussed. Furthermore, the influence of stoping in the production phase is discussed in this contribution.

Cyclic elastoplasticity and XFEM based fatigue life assessment of out-of-plane gusset welded joints

Conventional design methods for high-cycle fatigue failure of engineering components often assume a linear-elastic material behaviour. The different driving forces of crack initiation and propagation fail to be considered, resulting in less accurate predictions of fatigue life. To address this issue, the present study proposes a novel method to predict the fatigue life of out-of-plane gusset welded joints based on cyclic elastoplasticity and extended finite element method (XFEM). Incorporating the welding residual stress simulated by thermal–mechanical analysis, the cyclic stress–strain response of out-of-plane gusset welded joints was assessed by Fatigue SS (FSS) constitutive model. The crack initiation life was therefore predicted by the local strain approach considering the mean stress correction. Based on the converged cyclic stress–strain response, the crack propagation behaviour and life were predicted by XFEM in combination with Virtual Crack Closure Technique (VCCT). The influence of mean stress, mixed-mode crack and residual stress on crack initiation and propagation life was also investigated. The results show that the proposed method can predict the fatigue life of out-of-plane gusset welded joints under the pure tension loading with high accuracy against experimental results.

Prediction of moisture-induced cracks in wooden cross sections using finite element simulations

Wood absorbs and desorbs moisture due to its hygroscopic behavior, leading to moisture gradients in timber elements as well as swelling and shrinkage. These processes are constrained due to the orthotropic material properties of wood, leading to moisture-induced stresses, which can cause crack initiation and propagation. A significant amount of the damage in timber constructions indoors can be related to changes of the moisture content (MC). However, more information is needed about the correlation between moisture changes or gradients and specific damage characteristics, like crack depths. Thus, based on numerical simulations, the crack depth development within two solid timber and one glued laminated timber (GLT) cross section over time for different relative humidity (RH) reductions and initial MCs is analyzed. For this purpose, a multi-Fickian transport model is used to determine moisture fields, which are then used as loads in a subsequent stress simulation, where linear elastic material behavior is considered. An extended finite element approach, supported by a multisurface failure criterion defining the failure behavior, allows for the simulation of moisture-induced discrete cracking. Based on simulation results, correlations between potential maximum crack depths and moisture gradients in indoor climate conditions are derived, which enables the prediction of crack depths in wood. Finally, it is shown that the initial MC level significantly influences the maximum crack depth that can be expected.

A length scale insensitive phase field model based on geometric function for brittle materials

The commonly used phase field model has a broad application in static and dynamic scenarios with brittle materials. Following the assumption of irreversibility of fracture, monotonically increasing phase and tensile strain energy are applied in the commonly used phase field model. By doing so, a clear stiffness reduction, however, is observed in stress–strain curve before reaching fracture strength, which conflicts with the linear elastic behavior of brittle materials. The limitation has also been mentioned in publications when length scale is the only parameter to adjust fracture strength. To solve these problems, a generalized quadratic geometric function is proposed to build a length scale insensitive phase field model. Governing equations and one-dimensional analytical solutions (homogeneous and nonhomogeneous solutions) of this length scale insensitive phase field model are presented. By adjusting the extra parameter in the generalized quadratic geometric function and zeroing negative phase, different phase profile (ranged from exponential profile to linear profile) and fracture strength can be obtained, which could be more suitable to cover different types of fractures. Meanwhile, a narrower crack band and linear elastic behavior can be guaranteed. LS-DYNA user-defined element and user-defined material subroutines are used to implement the proposed length scale insensitive phase field model. A tensile test successfully verifies the properties of a narrower crack band and linear elastic behavior. Three representative numerical examples show the feasibility of this model to simulate fracture propagation, including a Mode Ⅰ failure of wedge splitting test, a mixed-mode failure of l-shaped panel test and a mixed-mode failure of notched beam test. From the crack band and force–displacement curve, the proposed model can handle fracture propagation better for brittle materials.

Influence of Vertical Load on Lateral-Loaded Monopiles by Numerical Simulation

Monopiles are the most common foundation form of offshore wind turbines, which bear the vertical load, lateral load and bending moment. It remains uncertain whether the applied vertical load increases the lateral deflection of the pile. This paper investigated the influence of vertical load on the behaviour of monopiles installed in the sand under combined load using three-dimensional numerical methods. The commercial software PLAXIS was used for simulations in this paper. Monopiles were modelled as a structure incorporating linear elastic material behaviour and soil was modelled using the Hardening-Soil (HS) constitutive model. The monopiles under vertical load, lateral load and combined vertical and lateral loads were respectively studied taking into account the sequence of load application and pile slenderness ratio (L/D; L and D are the length and diameter of the pile). Results suggest that the sequence of load application plays a major role in how vertical load affects the deflection behaviour of the pile. Specifically, when L/D ratios obtained by lengthening the pile while keeping its diameter constant are 3, 5 and 8, the relationships between lateral load and the deflection behaviour of the pile under the effect of vertical load demonstrate a similar trend. Furthermore, the cause of increased lateral capacity of the pile under the action of applied vertical load in the common practical application case and in the VPL case was analyzed by studying the variation law of soil stress along the pile embedment. Results confirm that the confining effect of vertical load increases means effective stress of the soil around the pile, thus increasing soil stiffness and pile capacity.

Dynamic amplification in masonry arch railway bridges

Dynamic amplification of loads in masonry arch railway bridges is not well understood. There is a scarcity of experimental data and previous numerical studies have only addressed a few specific bridge geometries. Despite this, guidance documents provide empirical and unvalidated formulae to calculate dynamic amplification for masonry arch railway bridges. To improve our fundamental understanding of the problem, determine appropriate modelling strategies and evaluate the reliability of guidance documents, simple 2D and 3D models are explored in this paper. The 2D approach idealises key bridge components (pier, arch, fill and backing elements) with straight Timoshenko beams, springs and lumped masses. It uses an analytical dynamic stiffness formulation, which is computationally efficient and well-suited to explore a range of bridge models. The higher fidelity 3D modelling approach uses shell and solid finite elements and is used to evaluate the limitations of 2D models. In both approaches, linear-elastic material behaviour is assumed and train loads are idealised as moving vertical loads distributed over an effective area. The modelling results indicate a complex relationship between train speed and dynamic amplification that depends critically on bridge geometry and axle spacing. In general, the multi-span bridge configurations experienced higher dynamic amplification over operational train speeds. The results also higlight deficiencies in existing code provisions and demonstrates how efficient numerical models may replace these provisions.

Effect of architecture disorder on the elastic response of two-dimensional lattice materials.

We examine how disordering joint position influences the linear elastic behavior of lattice materials via numerical simulations in two-dimensional beam networks. Three distinct initial crystalline geometries are selected as representative of mechanically isotropic materials with low connectivity, mechanically isotropic materials with high connectivity, and mechanically anisotropic materials with intermediate connectivity. Introducing disorder generates spatial fluctuations in the elasticity tensor at the local (joint) scale. Proper coarse-graining reveals a well-defined continuum-level scale elasticity tensor. Increasing disorder aids in making initially anisotropic materials more isotropic. The disorder impact on the material stiffness depends on the lattice connectivity: Increasing the disorder softens lattices with high connectivity and stiffens those with low connectivity, without modifying the scaling between elastic modulus and density (linear scaling for high connectivity and cubic scaling for low connectivity). Introducing disorder in lattices with intermediate fixed connectivity reveals both scaling: the linear scaling occurs for low density, the cubic one at high density, and the crossover density increases with disorder. Contrary to classical formulations, this work demonstrates that connectivity is not the sole parameter governing elastic modulus scaling. It offers a promising route to access novel mechanical properties in lattice materials via disordering the architectures.

Quasistatic fracture using nonlinear‐nonlocal elastostatics with explicit tangent stiffness matrix

AbstractWe apply a nonlinear‐nonlocal field theory for numerical calculation of quasistatic fracture. The model is given by a regularized nonlinear pairwise potential in a peridynamic formulation. The potential function is given by an explicit formula with an explicit first and second derivatives. This fact allows us to write the entries of the tangent stiffness matrix explicitly thereby saving computational costs during the assembly of the tangent stiffness matrix. We validate our approach against classical continuum mechanics for the linear elastic material behavior. In addition, we compare our approach to a state‐based peridynamic model that uses standard numerical derivations to assemble the tangent stiffness matrix. The numerical experiments show that for elastic material behavior our approach agrees with both classical continuum mechanics and the state‐based model. The fracture model is applied to produce a fracture simulation for a ASTM E8 like tension test. We conclude with an example of crack growth in a pre‐cracked square plate. For the pre‐cracked plate, we investigated load in force (soft loading) and load in displacement (hard loading). Our approach is novel in that only bond softening is used as opposed to bond breaking. For the fracture simulation we have shown that our approach works with and without initial damage for two common test problems.

Investigation of conventional and asphaltic underlayment track behaviors subjected to freight train load: Mechanistic approach

The load characteristics in terms of the carrying capacity and the train operation speed limit Indonesia's conventional track capability. In this study, the behaviors of Indonesia's conventional tracks and the proposed asphaltic underlayment tracks under cyclic loading conditions of the low-speed Babaranjang (long-chain coal) freight trains have been evaluated using the finite element package Abaqus/CAE considering linear elastic behavior of materials. The numerical modeling was applied on three different layer thicknesses of sub-ballast in Indonesia's conventional tracks and three different layer thicknesses of asphalt in asphaltic underlayment tracks to investigate the vertical compressive stress, horizontal-vertical-maximum strain, and deformation behaviors. According to the numerical simulation, the measurements indicate that the asphaltic underlayment tracks are superior to Indonesia's conventional tracks. Furthermore, a 15 cm of asphalt layer in the asphaltic underlayment tracks demonstrates promising performance in the future of Indonesia's railway systems. The asphaltic underlayment tracks are expected to serve the heavier and higher speed Babaranjang freight trains with longer service life and lower life cycle cost than Indonesia's conventional tracks.

Digital image correlation–based analysis of hygroscopic expansion in Herrnholz granite

Groundwater-induced deformation of natural rock slopes is typically interpreted to be related to changes in pore pressure. Rock, however, is a porous medium that has been shown to expand on contact with water. Here, we present insights into surface deformation of free-standing 115 × 65 × 35 mm granite prisms with water continuously infiltrating from the upper surface over 16 – 24 hours under ambient laboratory environmental conditions. Digital image correlation analysis indicates a mean extensional strain due to wetting of approximately (4 – 4.7) × 10 −4 , equivalent to an internal pressure of 40 – 47 MPa, assuming linear-elastic material behaviour. Nanoscale adsorption theory indicates the contribution from adsorption of water onto free surfaces may be 18.4 MPa, while capillary condensation generates pressure up to 30.9 MPa. It is likely a majority of strains is the result of water entering micro pores with radius less than 5 nm, which account for 1.16% of the porosity. Our relatively simple experimental setup may serve as a useful tool for analyzing moisture-induced deformation in a range of geological materials.

A modified bond model for describing isotropic linear elastic material behaviour with the discrete element method

Recently, the discrete element method is increasingly being used for describing the behaviour of isotropic linear elastic materials. However, the common bond models employed to describe the interaction between particles restrict the range of Poisson’s ratio that can be represented. In this paper, to overcome the restriction, a modified bond model that includes the coupling of shear strain energy of neighbouring bonds is proposed. The coupling is described by a multi-bond term that enables the model to distinguish between shear deformations and rigid-body rotations. The positive definiteness of the strain energy function of the modified bond model is verified. To validate the model, uniaxial tension, pure shear and pure bending tests are performed. Comparison of the particle displacements with continuum mechanics solution demonstrates the ability of the model to describe the behaviour of isotropic linear elastic material for values of Poisson’s ratio in the range 0 le nu < 0.5.

Eliminating plasticity effects in the measurement of residual stress by using the hole-drilling method

Abstract When the hole-drilling method is used to measure residual stress, plasticity effects occur if the residual stress exceeds 60 % of the yield stress. Plasticity effects result in an overestimation of the residual stress as the hole-drilling method is based on the linear elastic behavior of materials. To overcome this problem, a simplified iterative method based on an existing iterative method and using the optimizer tools of the ANSYS program is proposed in this paper. The proposed method is verified through numerical and experimental studies. The numerical case study reveals that, for the case studied, the proposed method can achieve the same accuracy as the original iterative method and has a higher convergence ability. The experimental study demonstrates that the values of the maximum principal stress calculated using the American Society of Testing Materials standard is generally greater than the true values and that the error magnitude increases with an increase in stress levels. Using the proposed method, errors are reduced for the maximum principal stress while increasing slightly for the minimum principal stress. The proposed method has a high convergence ability; for the case studied, only two or three iterations are needed to achieve acceptable accuracy.

Topology optimization of structural frames considering material nonlinearity and time-varying excitation

An approach for the topology optimization of structures composed of nonlinear beam elements under time-varying excitation is presented. Central to this approach is a hysteretic beam finite element model that accounts for distributed plasticity and axial-moment interaction through appropriate hysteretic interpolation functions and yield/capacity function, respectively. Nonlinearity is represented via the hysteretic variables for curvature and axial deformations that evolve according to first order nonlinear ordinary differential equations (ODEs), referred to as evolution equations, and the yield function. Hence, the governing dynamic equilibrium equations and hysteretic evolution equations can thus be concisely presented as a system of first-order nonlinear ODEs that can be solved using general ODE solvers without the need for linearization. The approach is applied for the design of frame structures with an objective to minimize the total volume in the domain, such that the maximum displacement at specified node(s) satisfies a specified constraint (i.e., drift limit) for the given excitation. The maximum displacement is approximated using the p-norm and thus permits the completion of the analytical sensitivities required for gradient-based updating. Several numerical examples are presented to demonstrate the approach for the design of structural frames subjected to pulse, harmonic, and seismic base excitation. Topologies obtained using the suggested, nonlinear approach are compared to solutions obtained from topology optimization problems assuming linear-elastic material behavior. These comparisons show that although similarities between the designs exist, in general the nonlinear designs differ in composition and, importantly, outperform the linear designs when assessed by nonlinear dynamic analysis.

Microstructure simulation using self‐consistent clustering analysis

AbstractTo capture all the individual microstructural effects of complex and heterogeneous materials in structural finite element simulations, a two‐scale simulation approach is necessary. Since the computational effort of such two‐scale simulations is extremely high, different methods exist to overcome this problem. In terms of a FFT‐based microscale simulation, one possibility is to use a reduced set of frequencies leading to a reduced numerical solution of the Lippmann‐Schwinger equation [?]. In a post‐processing step, highly resolved microstructural fields may then be reconstructed by using the compressed sensing technique [?]. Since the stress evaluation of this method is in real space and therefore not reduced, it is most beneficial in terms of linear elastic material behavior. Another very recent method to reduce the computational effort of a microscale simulation is the self‐consistent clustering analysis [?,?]. Such a self‐consistent clustering analysis is split into an offline and an online stage. Within the offline stage, the material points of the high‐fidelity representation of the unit cell are grouped into clusters with similar material behavior. Thereafter, in the online stage, a self‐consistent clustering analysis is used to solve the boundary value problem by a clustered Lippmann‐Schwinger equation. Since the generation of clusters may be based on linear elastic simulations, we propose to use a reduced set of frequencies for these simulations to improve the efficiency of the total algorithm. Elastic and elasto‐plastic composites are investigated in a small strain setting as representative simulation examples.