General Loading Conditions Research Articles

Emergence of tension-compression asymmetry from a complete phase-field approach to brittle fracture

The classical variational approach to brittle fracture propagation does not distinguish between strain energy accumulation in tension versus compression and consequently results in physically unrealistic cracking under compression. A variety of energy splits have been proposed as a possible remedy. However, a unique energy split that can describe this asymmetry for general loading conditions has not been found. The main objective of this paper is to show that a complete phase-field theory of brittle fracture nucleation and propagation, one that accounts for the material strength at large, can naturally capture the tension-compression asymmetry without an energy split. One such theory has been recently proposed by Kumar et al. (2018). Over the past few years, several studies have shown that this theory is capable of accurately describing fracture nucleation and propagation for materials soft and hard under arbitrary monotonic loading conditions. However, a systematic study of the tension-compression asymmetry that emerges from this theory has not yet been reported. This paper does precisely that. In particular, this paper reports a comprehensive study of crack propagation in two problems, one involving a symmetric tension-compression state and the other involving larger compressive stresses at the crack tip. The results are compared with popular energy splits used in literature. The results show that, remarkably, for the second problem, only the complete theory is able to produce experimentally consistent results.

An efficient moving-mesh strategy for predicting crack propagation in unidirectional composites: Application to materials reinforced with aligned CNTs

This paper presents an efficient numerical approach for reproducing the process of crack propagation inside unidirectional composites subjected to general loading conditions, with special reference to epoxy materials enhanced with embedded aligned CNTs. This approach involves a traditional FE framework improved by the Moving Mesh (MM) technique based on the Arbitrary Lagrangian-Eulerian (ALE) formulation and the Interaction Integral Method (M−integral). The MM serves as a powerful numerical tool to simulate the discrete crack advance with minimal remeshing, thus reducing the computational complexities. Instead, the M−Integral method, formulated for generally anisotropic materials, has been employed to extract the mixed-mode Stress Intensity Factors (SIFs), which are necessary to define the crack onset condition and propagation direction on the basis of the modified Maximum Hoop Stress Criterion. The proposed strategy includes the extended rule of mixtures to evaluate the homogenized elastic properties of nano-reinforced composites. The validity of the proposed methodology has been assessed through comparisons with experimental data and numerical results available in the literature.

A quadrilateral inverse plate element for real-time shape-sensing and structural health monitoring of thin plate structures

The inverse finite element method (iFEM) emerged as a powerful tool in shape-sensing and structural health monitoring (SHM) applications with distinct advantages over existing methodologies. In this study, a quadrilateral inverse-plate element is formulated via a sub-parametric approach using bi-linear and non-conforming cubic Hermite basis functions for engineering structures, which can be modeled as thin plates. Numerical validation involves dense and assumed sparse sensor arrangements for in-plane, out-of-plane, and mixed general loading conditions. iFEM analysis reveals efficient monotonic convergence to analytical and high-fidelity finite element reference solutions. After successful numerical validation, defect detection analysis is performed considering minute geometric discontinuities and structural stiffness reduction because of latent subsurface defects under tensile and transverse loading conditions. The inverse formulation successfully resolves the presence of simulated defects under a sparse sensor arrangement. The proposed inverse-plate element is accurate in the full-field reconstruction of shape-sensing profiles and reliable in defect identification and quantification in thin plate structures.

Criterion for unhomogeneous yielding of porous materials

A criterion is developed for the unhomogeneous yielding of materials containing arbitrarily oriented ellipsoidal voids. The criterion is built upon classical estimates for pure pressure and pure shear. A data-driven approach is then followed to incorporate the effects of void shape and orientation. A large number of micromechanical unit cell results are used to calibrate the yield criterion. A key feature of the criterion is that it predicts a significant reduction of the effective shear yield strength due to mere void inclination, with the reduction increasing with the void dimension perpendicular to the shear. The coupling between tension and shear deformation results in an apparent rotation of the yield surface, which provides a sound micromechanical basis for predicting void closure in shear among other new features. Once supplemented with evolution equations of relevant internal parameters, the resulting constitutive formulation will enable ductile failure simulations heretofore impossible to carry out on a sound physical basis for general loading conditions.

Crease instability in Gent-Gent hyperelastic materials

We study the crease instability of an incompressible Gent-Gent elastomer under general loading conditions. The Gent-Gent model has two distinct characteristics: the logarithmic nonlinearity triggered by I2 energy term and the strain hardening due to limiting chain extensibility. By performing finite element analysis combined with an evaluation of surface tension, we investigate the effects of the two characteristics on the critical creasing onset and the creasing onset delayed by surface tension. The results show that uniaxial compression induces logarithmic nonlinearity that accelerates the critical creasing onset, whereas equibiaxial compression hastens the initiation of strain hardening that delays the critical creasing onset. In addition, logarithmic nonlinearity mitigates the delay in creasing onset caused by surface tension under the uniaxial condition, and strain hardening suppresses creasing in cooperation with surface tension under the equibiaxial condition. Conversely, under the plane strain condition, two characteristics have little effect on the creasing onset delayed by surface tension.

A Griffith description of fracture for non-monotonic loading with application to fatigue

With the fundamental objective of establishing the universality of the Griffith energy competition to describe the growth of large cracks in solids not just under monotonic but under general loading conditions, this paper puts forth a generalization of the classical Griffith energy competition in nominally elastic brittle materials to arbitrary non-monotonic quasistatic loading conditions, which include monotonic and cyclic loadings as special cases. Centered around experimental observations, the idea consists in: (i) viewing the critical energy release rate Gcnot as a material constant but rather as a material function of both space X and time t, (ii) one that decreases in value as the loading progresses, this solely within a small region Ωℓ(t) around crack fronts, with the characteristic size ℓ of such a region being material specific, and (iii) with the decrease in value of Gc being dependent on the history of the elastic fields in Ωℓ(t). By construction, the proposed Griffith formulation is able to describe any Paris-law behavior of the growth of large cracks in nominally elastic brittle materials for the limiting case when the loading is cyclic. For the opposite limiting case when the loading is monotonic, the formulation reduces to the classical Griffith formulation. Additional properties of the proposed formulation are illustrated via a parametric analysis and direct comparisons with representative fatigue fracture experiments on a ceramic, mortar, and PMMA.

Contact force and pressure analysis of the three-row roller pitch bearing in a large-scale wind turbine

Pitch bearings are the key components of wind turbines (WTs). The reliability of pitch bearings affects the electric energy output efficiency of WTs. Due to its large loading ability, constant contact angle and high stiffness, the three-row roller slewing bearing (TRSB) is becoming a more attractive choice as a pitch bearing, especially in large-scale WTs. The contact force and pressure distributions significantly affect the fatigue life of a three-row roller pitch bearing (TRPB). Therefore, it is essential to propose a precise calculation method for the contact force and pressure. The present work establishes a quasi-static five degree-of-freedom (DOF) mathematical model for TRPBs, in which the bearing under general loading conditions is considered. The influences of combined external forces, overturning moments and inner ring misalignment angle on the contact force and pressure distributions of TRPBs are comprehensively investigated. According to the analysis results, the overturning moment and axial force can significantly affect the number of load-carrying rollers in each row. Furthermore, a slight variation of the inner ring angular misalignment has a significant effect on the contact force and pressure of TRPBs. Finally, it is more prone to pressure concentration in the non-crowned roller than that in the crowned roller, especially in the case of the inner ring with a misalignment angle. Since the high calculation efficiency and accuracy, the proposed method has remarkable potential applications in the fatigue life prediction and design of TRPBs.

Safety-Constrained Stagewise Optimization of Droop Control Parameters for Isolated Microgrids

To improve the primary droop control performance under variable load conditions in the isolated microgrid, this paper proposes a safety-constrained stagewise optimization method for P/f and Q/V droop control parameters. In the first stage, these parameters are determined through look-ahead distributionally robust optimization so that they are robust against general uncertain and variable load conditions. To avoid the excessive pursuit of robustness at the expense of economic benefits, the second and third stages are responsible for addressing accidental extreme disturbances that lead to unacceptable droop control deviations, where the parameters determined in the first stage are modified adaptively according to the contemporary system state. Specifically, the second and third stages are pertinent to Q/V and P/f droop control parameter adaptations, respectively, considering different requirements for voltage and frequency regulation. This design also contributes to simplifying the parameter modification problem. A measurement-driven decentralized method is utilized to quickly accomplish the two adaptation stages, yielding a quick remedy of the frequency and voltage violations. Case studies show that our proposed method can maintain microgrid security and is robust to uncertain load variations without sacrificing economic benefits.

Continuum damage mechanics of cyclic viscoplasticity using asymptotic numerical method

Conscious efforts on reduction of greenhouse gas emissions have led to an energy transition to renewable energy, however uncertainties of renewable energy production have resulted in higher thermal cycling demands from conventional power plants. Thermal load cycling at high temperature regions of steam turbine components leads to enhanced creep-fatigue damage accumulation. It is well established that such damage mechanism is numerically best predicted by unified constitutive modeling including damage as a variable as per the formalism of continuum damage mechanics at an expense of considerable computational efforts using finite element analysis. In this paper, the non-iterative Asymptotic Numerical Method (ANM), currently limited to partial cycle analysis with linear hardening plasticity model, is proposed for the first time to address cyclic viscoplasticity problems including damage capable of handling multiple cycles and most generalized loading conditions. Regularization techniques additionally necessary to implement loading-unloading-reloading criteria and advanced constitutive models etc. are presented. The constitutive model chosen for the formulation includes the non-linear multiple back stress variable modified Chaboche model to include damage combined with modified Chaboche-Rousselier isotropic hardening model to include damage, power law for viscoplasticity, Lemaitre’s damage potential and Kachanov-Rabotnov’s creep damage law. The method is verified with defined error measures and then applied to two high pressure steam turbine rotors, one with and another without thermal stress relief groove (TSRG) at the inlet under service type loading conditions to study the beneficial effect of the TSRG on creep-fatigue damage evolution. The accumulated errors of the proposed ANM and computational time are compared to a conventional Newton-Raphson solution.

Computational and physical aspects for the occurrence of crease in an elastomer under general loading conditions

We study the occurrence of crease in an elastomer under three loading conditions from both computational and physical points of view. A two-dimensional finite element analysis, including a perturbation force-based approach, is performed to detect the stable creased state, where the crease formation is completed. A mesh convergence study for the energy release and crease depth in the creased state provides a guideline for the finite element mesh requisite for predicting the occurrence of crease. We also visualize the mesh convergence for the critical creasing onset by plotting the set of compressive strain and mesh resolution at the point where the energy release is zero. In addition, the present evaluation method of the surface tension using the energy release and surface area change for an elastomer with a traction-free surface yields a good prediction for the creasing onset delayed by surface tension without additional numerical implementations or adjustment parameters. This facilitates the comprehensive study of the creasing onset delayed by the surface tension under general loading conditions. The results reveal that tensile deformation in the out-of-plane direction encourages the retardation of the creasing onset via the extension of the width and the energy emission. We further demonstrate that energetically preferential crease interval can be used to evaluate the spacing between creases delayed by moderate surface tension.

A statistical mechanics framework for constructing nonequilibrium thermodynamic models

Abstract Far-from-equilibrium phenomena are critical to all natural and engineered systems, and essential to biological processes responsible for life. For over a century and a half, since Carnot, Clausius, Maxwell, Boltzmann, and Gibbs, among many others, laid the foundation for our understanding of equilibrium processes, scientists and engineers have dreamed of an analogous treatment of nonequilibrium systems. But despite tremendous efforts, a universal theory of nonequilibrium behavior akin to equilibrium statistical mechanics and thermodynamics has evaded description. Several methodologies have proved their ability to accurately describe complex nonequilibrium systems at the macroscopic scale, but their accuracy and predictive capacity is predicated on either phenomenological kinetic equations fit to microscopic data or on running concurrent simulations at the particle level. Instead, we provide a novel framework for deriving stand-alone macroscopic thermodynamic models directly from microscopic physics without fitting in overdamped Langevin systems. The only necessary ingredient is a functional form for a parameterized, approximate density of states, in analogy to the assumption of a uniform density of states in the equilibrium microcanonical ensemble. We highlight this framework’s effectiveness by deriving analytical approximations for evolving mechanical and thermodynamic quantities in a model of coiled-coil proteins and double-stranded DNA, thus producing, to the authors’ knowledge, the first derivation of the governing equations for a phase propagating system under general loading conditions without appeal to phenomenology. The generality of our treatment allows for application to any system described by Langevin dynamics with arbitrary interaction energies and external driving, including colloidal macromolecules, hydrogels, and biopolymers.

Stress Analysis and Life Prediction of a Ceramic Matrix Composite Vane Using a Micromechanics-Based Approach

The results of stress analysis of a ceramic matrix composite (CMC) vane using a physics-based model developed for two-dimensional woven CMCs are presented. The model considers the inherent defects and micromechanical damage in woven CMCs along with time-dependent deformation of the constituents. Predictions include damage state under general load conditions and the global deformation response of the vane. Strain-gage data from burst tests are compared to strain predictions obtained using the model. Results from time-dependent analysis and life prediction of the vane under constant loads and cyclic loads at elevated temperatures are presented. Effect of fatigue frequency on the deformation and long-term life of the vane are also discussed.

Competing fracture modes in Al-steel resistance spot welded structures: Experimental evaluation and numerical prediction

Multi-material design has been increasingly used in the automotive industry for structural lightweighting. A novel resistance spot welding technique has been developed to join aluminium to steel with acceptable joint strength. However, aluminium to steel resistance spot welds can fail in either interfacial (IF) or pull-out (PO) fracture modes. In the present study, coupon-level tensile shear and coach peel samples are tested under quasi-static condition, and component-level T-joint structures are tested under impact load. Experimental results show that in the IF fracture mode, weld failure occurs due to the facture of the intermetallic layer, while in the PO fracture mode, weld failure occurs due to a combination of different mechanisms including through-thickness tearing of the weld nugget and possibly partial fracture of the intermetallic layer. In the finite element analysis, two force-based fracture criteria are implemented via mesh-independent connector element considering a general loading condition, and whichever fracture potential reaches unity first determines the fracture mode. The proposed method is validated by experiments and has the following features: firstly, it can capture the behaviour of fracture mode transitioning from IF to PO as weld diameter increases; secondly, it can simulate the maximum load and deformation at fracture; thirdly, it can indicate the global response and absorbed energy in impact analysis with reasonable accuracy. Furthermore, a distinct advantage of the model is that it is free of non-physical calibration. The parameters in the model are obtained from geometric and material properties of the weld directly. The developed method shows desirable performance and is potentially suitable for applications in large-scale automotive structural analysis and design.

A probabilistic-phase field model for the fracture of brittle materials

We develop a computational method to determine the failure probability of brittle materials under general mechanical loading conditions. The method is a combination of two parts: (1) numerical simulations of materials with multiple cracks using phase field theory, where the complete fracture process is viewed as ‘damage percolation’ along critical paths or clusters of cracks, rather than the traditional weak-link failure mechanism of Weibull, and (2) an extension of the Batdorf statistical theory of fracture to finite domains, where it is implemented within the finite element framework. The results of phase-field simulations at the ‘percolation threshold’ are used as failure data in the Batdorf theory to determine the overall probability of failure. The input to this approach is the size distribution of cracks in a pristine material. An example is shown, where alumina samples that were previously tested by Abe and coworkers (Abe et al 2003 J. Am. Ceram. Soc. 86 1019–21) in four-point loading are compared to the results of our numerical simulations. The approach developed here has the advantage of being extendable to more complex thermomechanical loading.

A semi-analytical approach for two-dimensional frictional contact of anisotropic magneto-electro-elastic solids

In this work, we present a semi-analytical modeling approach for solving the frictional contact of two-dimensional anisotropic magneto-electro-elastic (MEE) solids. The method is established in a broad framework in which the contact body is made of generally anisotropic MEE materials, the punch head can be arbitrary, and the punch can be subjected to generalized loading conditions including mechanical forces and electric and magnetic charges. An analytical solution of generalized distributed load along the half-plane surface is derived and utilized to formulate the semi-analytical equations governing the relation between the generalized displacements and tractions. Based upon the semi-analytical equations, the semi-analytical model in which the influence matrices can be calculated explicitly and its effectiveness is established. An iterative procedure is then used for determining the contact region, contact status, generalized surface deformation and the distribution of contact tractions, as well as the punch’s indentation depth and electric and magnetic potentials. The present approach is verified by comparing its numerical results with those obtained by the existing analytical solutions and boundary element method. The influences of different factors, such as punch profile, mechanical load, electric and magnetic charges, friction coefficient and material properties, are also analyzed. The capability of this method to handle complicated geometry, such as rough surfaces, is finally discussed, showing the potential performance of the developed method for practical engineering applications.

A semi-analytical solution for the in-plane stress fields in isotropic convex finite solids with circular holes reinforced with cylindrically orthotropic rings

In this paper a semi-analytical solution for the stress distribution in a finite convex plate with a reinforced hole and subjected to a general loading condition is presented. The method described accounts for the geometrical parameters of the plate, as well as material properties and loading conditions, which can be applied internally to the hole boundary and externally to the outside boundaries of the plate. An extensive validation of the proposed solution is carried out comparing the theoretical predictions with results from various finite element analyses, showing an excellent agreement.

XFEM\\GFEM based approach for topology optimization of extruded beams with enhanced buckling resistance

In this paper, an efficient formulation for the topology optimization of extruded beams is developed considering linearized buckling, displacement, and stress constraints. The proposed method relies on a computationally efficient extended\\generalized finite element (XFEM\\GFEM) formulation for the forward problem, recently developed for the analysis of beams. In this method, a coarse 3D finite element (FE) mesh is enhanced with global enrichment functions, enabling a substantial reduction in the number of degrees of freedom with comparable accuracy to a traditional 3D FE fine mesh. The material distribution at the cross-section is optimized utilizing the material density approach with the Solid Isotropic Material with Penalization (SIMP) method.Our objective is to minimize the total weight of the beam, subject to performance constraints including buckling, displacement, and stress. The forward problem enables the application of general loading and boundary conditions. Thus, both buckling and lateral–torsional buckling are automatically considered. The optimization problem is solved using a gradient-based optimization approach, with analytical gradients derived by using the adjoint variable method. Several numerical examples are presented to investigate the accuracy and efficiency of the proposed formulation. A parametric study showed that the optimized cross-section is strongly dependent on the length and permissible buckling load. In addition, it is shown that a simultaneous formulation leads to significantly different designs, with different structural behavior, compared to results obtained by considering only a subset of the performance constraints. Finally, we verify the model on a beam warping problem under pure torsion, in which the cross-section optimization converges to a known analytical solution. Compared with traditional FEM, two orders of magnitude in terms of execution time are reported in favor of the proposed XFEM\\GFEM approach.

Explicit analytical solutions for arbitrarily laminated composite beams with coupled stretching-bending and transverse shear deformation

Consider the coupling of stretching-bending deformation induced by the arbitrarily laminated composites, and the transverse shear deformation caused by the thick beams. A mathematical model based upon Timoshenko beam theory is established for the general laminated composite beams, and a proper replacement of stiffness is suggested for narrow or wide beam. By using the state-space approach, a general solution is expressed in terms of matrix exponential, which is applicable for any possible loading and boundary conditions. Expand the matrix exponential through Taylor series, an explicit analytical solution is obtained for the arbitrarily laminated composite beams under general loading conditions. With the obtained general solutions, several analytical solutions are derived explicitly such as the Green's functions for an infinite laminated composite thick beam, and the solutions for some typical beam problems. For example, a cantilever beam, a simply supported beam, or a fixed-end beam, subjected to uniformly distributed load, concentrated force/moment, or sinusoidal load, etc. To cover more complicated beam structures, we develop the boundary element method by using the obtained Green's functions. Three numerical examples are implemented to investigate the influences of stacking sequences and beam geometry, and to validate the applicability of boundary element method.

A Micro-Macro mixed GTN-MMC model to study plasticity and fracture of AISI 4340 steel

Experimental and numerical studies were conducted to study the plastic and fracture behaviors of AISI 4340 steel (with three different heat treatments) under multiaxial stress loading conditions, including axial symmetric and plane strain loadings. The classical Gurson-Tvergaard-Needleman (GTN) plasticity model was extended and calibrated to consider Lode angle dependence on the material's matrix plastic strength especially needed for plane strain loadings. The 3D fracture loci of AISI 4340 steels were previously calibrated by the Modified Mohr-Coulomb (MMC) model. Considering the microvoid growth and nucleation mechanisms, GTN model is combined with MMC model to derive a new analytical solution of microvoid volume fraction at fracture (ff) under general loading conditions. The ff should NOT be a constant, and it depends on different stress states. The modified GTN-MMC model was implemented using VMAT in Abaqus, and it gives a very good correlation between experimental data and finite element simulations.

The use of FFT algorithm for circumferential coordinate in axisymmetric analysis of solids by generalized finite difference method

A formulation is presented for GFDM analysis of axisymmetric viscoelastic bodies under general loading conditions. The proposed formulation reduces the dimension of analysis by one by using complex Fourier series for circumferential coordinate ((θ) dependency) of field variables. For θ (Fourier) transform computations, the discrete Fourier transform formulas with FFT algorithm are employed. Radiational effects are accounted for through the use of perfectly matched layer (PML) with the equation being modified in accordance with the proposed formulation. For the assessment of the formulation, several sample problems are presented. The results indicate its capability in estimating the response even for that requiring sensitive analysis, such as, those in wave propagation analysis.