Displacement Boundary Conditions Research Articles

Displacement Reconstruction Based on Physics-Informed DeepONet Regularizing Geometric Differential Equations of Beam or Plate

Physics-informed DeepONet (PI_DeepONet) is utilized for the reconstruction task of structural displacement based on measured strain. For beam and plate structures, the PI_DeepONet is built by regularizing the strain–displacement relation and boundary conditions, referred to as geometric differential equations (GDEs) in this paper, and the training datasets are constructed by modeling strain functions with mean-zero Gaussian random fields. For the GDEs with more than one Neumann boundary condition, an algorithm is proposed to balance the interplay between different loss terms. The algorithm updates the weight of each loss term adaptively using the back-propagated gradient statistics during the training process. The trained network essentially serves as a solution operator of GDEs, which directly maps the strain function to the displacement function. We demonstrate the application of the proposed method in the displacement reconstruction of Euler–Bernoulli beams and Kirchhoff plates, without any paired strain–displacement observations. The PI_DeepONet exhibits remarkable precision in the displacement reconstruction, with the reconstructed results achieving a close proximity, surpassing 99%, to the finite element calculations.

Mechanics of inner core debonding of composite sandwich beam with CFRP hexagonal honeycomb

The inner core debonding of composite sandwich beams with carbon fiber reinforced polymer (CFRP) hexagonal honeycomb core is investigated theoretically and experimentally. Based on the principle of minimum potential energy, a theoretical model is established to reveal the core debonding mechanism with Timoshenko’s beam theory and Extended High-Order Sandwich Panel Theory (EHSAPT) for composite face sheets and CFRP hexagonal honeycomb core, respectively. The theoretical approach of core debonding simulation is based on the framework of the cohesive zone model (CZM). By constructing the constitutive relation of the core interface as the discontinuous stress and displacement boundary conditions, the vertical core debonding can be simulated overcoming the singularity of equilibrium equations. The crack propagation process is decomposed into three stages with the consideration of the double crack propagation mechanism. In experiments, the regular and reinforced hexagonal honeycomb sandwich beams are designed and fabricated, and the quantification of the defect degree of the core bonding interface is achieved by equivalent energy release rate obtained from the homogenized hexagonal honeycomb. The double cantilever beam (DCB) experiments are performed to determine the equivalent energy release rate and reveal the core debonding mechanism of CFRP hexagonal honeycomb sandwich beams. Comparing the load–displacement curves and debonding deformation patterns indicates that the established theoretical model can effectively predict the core debonding mechanical response. Finally, parametric analysis is carried out to discuss the impact of structural dimension parameters, defect location and defect degree on the debonding modes of CFRP hexagonal honeycomb sandwich beams.

Increasing accuracy of 3-D geomechanical-numerical models

SUMMARY The current crustal stress field is of key importance to understand geodynamic processes and to assess stability aspects during subsurface usage. To provide a 3-D continuous description of the stress state, linear elastic forward geomechanical-numerical models are used. These models solve the equilibrium of forces between gravitational volume forces and surfaces forces imposed mainly by plate tectonics. The latter are responsible for the horizontal stress anisotropy and impose the inverse problem to estimate horizontal displacement boundary conditions that provide a fit best to horizontal stress magnitude data within the model volume. However, horizontal stress magnitude data have high uncertainties and they are sparse, clustered and not necessarily representative for a larger rock volume. Even when Bayesian statistics are incorporated and additional stress information such as borehole failure observations or formation integrity test are used to further constrain the solution space, this approach may result in a low accuracy of the model results, that is the result is not correct. Here, we present an alternative approach that removes the dependence of the solution space based on stress magnitude data to avoid potential low accuracy. Initially, a solution space that contains all stress states that are physically reasonable is defined. Stress magnitude data and the additional stress information are then used in a Bayesian framework to evaluate which solutions are more likely than others. We first show and validate our approach with a generic truth model and then apply it to a case study of the Molasse foreland basin of the Alps in Southern Germany. The results show that the model's ability to predict a reliable stress state is increasing while the number of likely solutions may also increase, and that outlier of stress magnitude data can be identified. This alternative approach results in a substantial increase in computational speed as we perform most of the calculations analytically.

Research on elasticity behavior of Ni-Ti shape memory alloys using the reproducing kernel particle method

The unique superelastic properties of the Ni-Ti shape memory alloys (SMAs) have been found extensive practical application in many engineering fields, but the Ni-Ti SMAs exhibits elastic behavior in small load, so the elastic behavior of the Ni-Ti SMAs is studied by using meshless method of reproducing kernel particle method (RKPM) in this paper. The displacement boundary conditions are applied by utilizing the penalty function method, and the impact of penalty factor and parameter of influence domain on errors are discussed, and computational stability is analyzed. At last, the correctness of the RKPM in studying the elastic behavior of the Ni-Ti SMAs is evidenced by using three numerical examples.

Modelling principal stress orientations in the Arabian Plate using plate velocities

Abstract We model maximum principal horizontal stress orientations in the Arabian Plate using a 3D finite element approach in conjunction with plate velocities. To capture the impact of geometry and tectonics, the model considers an accurate plate boundary shape and associated deformations. Three primary geological units represent plate architecture: sedimentary cover, crust and upper mantle. The mesh resolution varies to capture important geometrical features. Subsequently, we calculate the stress field using the force balance equation. Displacement boundary conditions are evaluated as accumulative deformation. NE–SW maximum principal horizontal stress (S Hmax ) azimuths dominate in northeastern Saudi Arabia and Kuwait, whilst NW–SE to NNW–SSE define the Dead Sea area. The Red Sea Basin and Saudi Arabia's interior is characterized by north–south S Hmax azimuths. Iraq's western area shows azimuths from NNW–SSE to NW–SE due to the collision at the Zagros Mountain Range, but changes to NE–SW in the east at the Zagros fold-and-thrust belt. An extensive literature review reveals publicly available S Hmax azimuth data which augment the sparse records compiled in the World Stress Map database. Our simulated S Hmax azimuths are consistent with these data. The results further corroborate ongoing tectonic processes, deepen our understanding of in situ stress variation drivers and inform current elastic deformation mechanisms in the Arabian Plate.

A direct FE[formula omitted] method for concurrent multilevel modeling of piezoelectric materials and structures

Multiscale piezoelectric materials and structures have shown their promising potential of applications in sensor, energy harvester, actuator and so on. However, the large complexity of such materials and structures poses a significant challenge to the accurate and efficient prediction of their piezoelectric behavior which is significant for guiding and optimizing their applications. In this paper, a Direct FE2 method was proposed for concurrent multilevel modeling of the coupled electromechanical behavior of multiscale piezoelectric materials and structures. The energy equilibrium and kinematic constraints between macro- and meso-scales were satisfied by prescribing periodic boundary conditions for both displacement and electric potential, which were derived from the governing equation of FEM. Another boundary condition was prescribed to the center node of each RVE, to prevent rigid motion and shifting of electric potential of meso-scale RVEs. All these boundary conditions can be easily prescribed to the Direct FE2 using multiple points constraints (MPCs), a commonly used feature in most commercial software. The accuracy, computational efficiency and ease of numerical implementation of the proposed Direct FE2 method were validated using a series of multiscale simulations, including the quasi-static response of homogeneous piezoelectric panels, an arc honeycomb structure, a regular honeycomb structure and a piezoelectric composite panel, as well as the dynamic response of the piezoelectric composite panel. It shows that the proposed Direct FE2 method will be valuable for large-scale simulations of complicated multiscale piezoelectric materials and structures.

Elastic analytical method with machine learning for predicting the stratum displacement field induced by shallow tunneling

Support vector regression (SVR) with sparrow search algorithm (SSA) is developed as the machine learning (ML) model to predict maximum surface settlement smax caused by tunneling. A novel method for calibrating boundary conditions of analytical solution is proposed, where the maximum surface settlement derived by the analytical method is equal to smax predicted by SSA-SVR method. The elastic analytical solution for stratum displacement of a shallow tunnel is presented by the complex variable method, when the calibrated nonuniform displacement function is applied as the tunnel displacement boundary condition. The proposed analytical solution-machine learning (AM) method can predict the stratum displacement field prior to the tunnel excavation. Seventy-three tunnel engineering cases are employed to verify the rationality of the proposed SSA-SVR method in predicting smax. The value of R2 in the training and test process is 0.894 and 0.877, respectively. Taking Heathrow Express Trial Tunnel as an example, the potential of the proposed AM method in predicting stratum displacement is presented where the influence of cohesion strength, internal friction angle, Young's elastic modulus of stratum, tunnel radius and depth are considered. The proposed AM method can well predict the stratum surface settlement trough curve, vertical and horizontal displacement at different positions.

Buckling response of functionally graded multilayer graphene platelet-reinforced composite plates with circular/elliptical cutouts supporting on an elastic foundation under normal and shear loads

The present article deals with the buckling response of functionally graded multilayer graphene platelet-reinforced composite (FG-GPL RC) rectangular plates with circular/elliptical cutouts resting on a Winkler-type elastic foundation under uniaxial and biaxial normal and shear loads. Rule of mixtures and the Halpin–Tsai approach are applied to obtain the effective Poisson’s ratio, mass density, and elastic modulus of the reinforced composite. The governing equations are developed by applying the third-order shear deformation plate theory. Then, the finite element procedure is used to solve the problem. Four different types of graphene platelet distributions, namely, UD, FG-X, FG-V, and FG-O, are considered. A broad range of factors such as plate aspect ratio, plate slenderness ratio, applying uniaxial and biaxial normal and shear loads to the plate, several Winkler elastic foundation stiffness parameters, different displacement boundary conditions, the effect of size of the circular cutout and orientation of the elliptical cutout, and the influence of GPL weight fraction are discussed in several tabular and graphical data in detail.

Establishment of the microstructure of porous materials and its relationship with effective mechanical properties

In this study, a porous structure for a porous liquid storage medium is generated, and the homogenization theory based on displacement boundary conditions is used to predict the effective mechanical properties. The relationship between the porous material’s macroscopic mechanical properties and microstructure is next analyzed. In order to establish the relationship between the microstructure of porous materials and their macroscopic mechanical properties, assuming that the pores grow along the z direction, a method is proposed to generate 3D open-cell porous materials based on six design parameters (i.e., the number of pores, porosity, irregularity of pore distribution, the randomness of pore growth in the x and y directions, and randomness of pore size). Since the porosity of oil-bearing materials ranges from 20 to 30%, the porosity of the RVE (Representative Volume Element) was kept under control at about 25%, and the effect of the six design factors on the mechanical properties of the RVE was investigated. Utilizing SLA 3D printing technology, specimens were produced, and compression tests were used to show how useful the results of the numerical analysis were. The results demonstrated that after the number of RVE pores reaches 9, the numerical results have good repeatability. The irregularity of the initial pore distribution has little effect on the effective mechanical properties of the RVE. At the same time, the increase in the randomness of pore growth and the randomness of pore size increases the degree of weakening of the mechanical properties in the z-direction, while reducing the degree of weakening in the x and y directions, but the latter has a smaller impact. Furthermore, there is a superimposition effect of design parameters on the RVE.

A method to analyze the axisymmetric problem of FRP laminated tubes based on singularity correction

In order to solve the problem of singularity in the stress and strain calculation process of FRP laminated tubes with arbitrary layer angles under axisymmetric thermomechanical loading, a three-dimensional elasticity theory analysis method based on singularity correction by modifying the stiffness matrix coefficients is proposed. Firstly, the basic three-dimensional elasticity theory model for FRP laminated tubes under axisymmetric thermomechanical loading is established through the displacement method. Then, the singularity plies in the laminate are determined, and the stiffness matrix coefficients of the singularity plies are modified respectively. Modifications to the displacement function coefficients, stiffness coefficients, boundary condition, and interlayer continuous condition equilibrium equations are also conducted subsequently. Verifications of the present approach are conducted by comparing the stress distributions calculated by the singularity correction method with those extracted by traditional isotropic elasticity models and then with numerical results provided by finite element analysis. Good consistency was obtained; thus, the correctness of the theory is proved from both theoretical and numerical aspects.

A novel complex variable solution for the stress and displacement fields around a shallow non-circular tunnel

A novel complex variable solution is derived in this paper for the stress and displacement fields around a shallow non-circular tunnel. The proposed method allows explicit computations in matrix form and has advantages of considerable efficiency and high accuracy. Firstly, the stress and displacement boundary conditions for the shallow non-circular tunnel are obtained using the generalized complex variable theory and conformal mapping technique. Then, the fast Fourier transform algorithm is employed to effectively perform the Fourier series fitting for these boundary conditions, and matrix equations for the potential functions in the frequency domain are established. Later, a matrix solution for the potential functions is formulated to determine the stress and displacement fields around the shallow non-circular tunnel. Finally, select numerical analyses are conducted to verify the proposed method, discuss the key parameters in the algorithm, and reveal the effect of the acceleration strategy using FFT as well as the effect of non-circular geometries on the stress and displacement fields.

On the influence of initial stress on final stress in data-calibrated numerical geomechanical models

Abstract. The prognosis of the state of stress in the subsurface can be improved by using numerical models in addition to data from stress measurements, as such models allow for consideration of variability stemming from structural complexities and an inhomogeneous distribution of rock properties. These models are generally set up in two consecutive steps. In the first step one an initial stress is established that accounts for gravity and a reasonable ratio of horizontal to vertical stress. This represents a reference stress state in the absence of tectonic stress and ensures equilibrium with gravity: i.e. no strain is produced once gravity acts. In the second step tectonic stress is included via displacement boundary conditions which induce horizontal differential stress to come up with the final stress state. Both initial stress and tectonic stress are chosen in such a way that calibration data are reproduced by the model at the locations and depths where the data were measured. We present generic models to investigate whether the choice of initial and tectonic stress affects the final state of stress in areas of the model domain where no stress data are available. We find that there is in general an ambiguity as different combinations of initial stress and tectonic stress yield the same final state of stress at the points where data are available. However, in those areas of the model domain where calibration data do not exist, these different choices of initial stress and tectonic stress produce differing results. These deviations are largest in the vicinity of lithological interfaces. We find that this ambiguity is reduced if more stress data exist, particularly in not just one lithology and at one depth. On the other hand, if more data are available, it becomes increasingly more difficult to find a combination of initial and tectonic stress to match them all. In view of the uncertainties of the data, such deviations between modelled stress and data may be expectable to some degree. However, such deviations may indicate that inelastic rock properties do play a role in some lithologies.

Stress state estimation – new data and variability assessment of model results

Abstract. The upper crust of the Earth is used more and more to transport and extract raw materials and energy. It is also used for the final disposal of radioactive waste in deep geological repositories with a particularly crucial requirement for long-term safety over 1×106 years. An accurate knowledge of the hosting rock and surrounding formation coupled with comprehensive modelling is fundamental to demonstrate the geological site has the required properties for safe and long-term underground storage. Besides other criteria, geomechanics play an important role. Even if the determination of rock properties is already a challenge, the estimation of the contemporary stress state in the upper crust is even more challenging. To predict the stress state 3-D geomechanical–numerical models are used, but high-quality underground structure, rock properties and in situ stress data for model calibration are needed. In particular, data of maximum and minimum horizontal stress magnitudes (SHmax and Shmin) are required to find suitable lateral displacement boundary conditions to produce a best-fit model. During the recent exploration phase for a deep geological repository for radioactive waste in Switzerland, a unique dataset of stress magnitude data has been acquired from eight cored boreholes. Rock mechanical properties were constrained from geophysical logging and laboratory testing. The empirically correlated rock properties were not simply averaged, but a probability distribution was provided. The stress field was explored by conducting more than 120 micro-hydraulic fracturing (MHF) and dry sleeve re-opening (SR) tests in different stratigraphic units, to estimate the magnitudes of Shmin and SHmax. We present the results of a 3-D geomechanical–numerical model that shows the best fit with respect to the measured stress magnitudes. Considering the uncertainties of the MHF/SR tests and the ones resulting from rock property variability, the model can reproduce most of the measurements. However, we do show not only the best-fit result but also a bandwidth of individual stress components within a P05–P95 probability range. We also show that there are still outliers for this bandwidth, but we will show that these can be explained by scaling effects.

Free Vibration Analyses of Stiffened Functionally Graded Graphene-Reinforced Composite Multilayer Cylindrical Panel

In this paper, the free vibration response of a stiffened functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer cylindrical shell panel is studied for the first time. The shell is stiffened by both stringers and rings. Additionally, the effect of reinforcing the shell panel, ring and stinger with GPLs is independently studied. Halpin–Tsai relations are employed to evaluate the mechanical properties of the shell panel, rings and stringers. The first-order shear deformation shell theory, accompanied by the Lekhnitsky smeared stiffener model, using the numerical finite element method and Hamilton principle, is employed to develop the governing motion equations of the shell panel. Four different types of GPL patterns, including FG-A, FG-X, FG-O and UD, are assumed across the thickness of the shell panel, rings and stringers. The effects of different factors, including various weight fractions and patterns of GPLs nanofillers, the geometry of the shell panel and stiffeners and two displacement boundary conditions, on the natural frequencies of the shell panel, have been studied.

Computational study of the mechanical behavior of the astrocyte network and axonal compartments in the mouse optic nerve head.

Glaucoma is a blinding disease characterized by the degeneration of the retinal ganglion cell (RGC) axons at the optic nerve head (ONH). A major risk factor for glaucoma is the intraocular pressure (IOP). However, it is currently impossible to measure the IOP-induced mechanical response of the axons of the ONH. The objective of this study was to develop a computational modeling method to estimate the IOP-induced strains and stresses in the axonal compartments in the mouse astrocytic lamina (AL) of the ONH, and to investigate the effect of the structural features on the mechanical behavior. We developed experimentally informed finite element (FE) models of six mouse ALs to investigate the effect of structure on the strain responses of the astrocyte network and axonal compartments to pressure elevation. The specimen-specific geometries of the FE models were reconstructed from confocal fluorescent images of cryosections of the mouse AL acquired in a previous study that measured the structural features of the astrocytic processes and axonal compartments. The displacement fields obtained from digital volume correlation in prior inflation tests of the mouse AL were used to determine the displacement boundary conditions of the FE models. We then applied Gaussian process regression to analyze the effects of the structural features on the strain outcomes simulated for the axonal compartments. The axonal compartments experienced, on average, 6 times higher maximum principal strain but 1800 times lower maximum principal stress compared to those experienced by the astrocyte processes. The strains experienced by the axonal compartments were most sensitive to variations in the area of the axonal compartments. Larger axonal compartments that were more vertically aligned, closer to the AL center, and with lower local actin area fraction had higher strains. Understanding the factors affecting the deformation in the axonal compartments will provide insights into mechanisms of glaucomatous axonal damage.

Microstructure-Based Two-Scale Simulation of Deformation and Fracture of SiCp/2009Al Composite

Accurately predicting the formability of composites is quite important for designing the materials and optimizing the processing parameters of such components. This paper presents a microstructure-based two-scale model to analyze the deformation behavior of the SiCp/2009Al composite during upsetting. The microstructure-based micro-scale model with a displacement boundary condition that was obtained from a macro-scale model can predict particle cracking, interface failure and matrix fracture by introducing the normal stress criterion, the maximum stress ratio criterion and the shear fracture model. The simulation results show that, during unidirectional upsetting, the crack initiates in the matrix, and then propagates in the matrix, the interface and the particle. Comparing with the unidirectional upsetting, the bidirectional upsetting results in a more uniform stain distribution and a smaller maximum strain, which prevent the initiation of the crack in the matrix. The simulation results compare well with the deformation and fracture patterns observed in experiment, indicating an effective way to optimise the plastic working and designing of composites.

Displacements and stresses around a circular tunnel in an elastic medium with pseudo-static horizontal earthquake body forces

This manuscript presents an analytical solution based on the complex variable approach for computing displacements and stresses around a single circular tunnel in a linear elastic medium in the presence of pseudo-static horizontal earthquake body forces. The solution has been obtained with the imposition of uniform as well as non-uniform radial displacement boundary conditions along the tunnel’s periphery taking the ground volume loss (GVL) as a governing input parameter. The non-uniform radial displacements’ boundary condition along the tunnel periphery has been modelled by using the Gaussian distribution function. The effects of the distribution of the tunnel’s peripheral displacements, Poisson’s ratio of the continuum and the depth to radius of the tunnel on the results have been examined. Necessary validation exercise, in the presence of horizontal earthquake acceleration, has also been carried out by comparing the obtained results with that computed on the basis of the finite element method. The results were also compared with that reported in literature. It was observed that with an increase in the pseudo static acceleration, there is (i) a significant increase in the horizontal displacements, (ii) a generation of non-symmetric displacement and stress pattern, (iii) an increase in the shear stresses, and (iv) the development of high deviatoric stress zones around the tunnel periphery.

Shape optimization of non-rigid origami leading to emerging bistability

Origami structures are a type of thin-walled flexible structures that undergo large deformation. Their inherent non-rigid characteristic empowers them with great potential for applications in space structures, metamaterials, and robotics. To understand the large deformation of origami structures, consideration of nonlinear mechanics is necessary. However, most origami structures are still designed by pure geometric approaches without considering their non-rigid behavior. In this work, we propose a computational design framework that incorporates nonlinear mechanics into the design procedure of origami. Guided by minimization of stored energy, under prescribed displacement boundary conditions, we optimize the configuration of origami structures. Difficulties arise due to the complex energy landscape of origami structures that inevitably induces bifurcation. We develop strategies to keep track of a stable deformation branch during the optimization process. A surprising outcome is that our approach naturally leads to self-emerging bistable structures, which is demonstrated by a series of numerical examples. We believe that our new approach would make substantial contribution to computational design of non-rigid origami structures, benefiting applications in origami-inspired solutions for science and engineering.

A multiple-data-based direct method for inverse problem in three-dimensional linear elasticity

Estimating the unknown Young’s modulus and boundary conditions of an elastic object from locally observed boundary data is an inverse problem which is usually solved by iterative methods. In this paper, a multiple-data-based direct (MD) method is proposed to solve this inverse problem without iterations. The proposed method applies a global normalized objective function and a global regularization coefficient τg for energy-like regularization to ensure the convergence of the unknown displacement boundary conditions. A coarse-mesh based preprocessing algorithm is proposed to speed up the determination of the regularization coefficient. A multi-force-position method is proposed to obtain observation data, which improves accuracy of the MD method when the number of experiments is limited. Both numerical and physical experiments were conducted. Compared with the single-data-based direct (SD) methods proposed in our previous work, the MD method yields more accurate estimation of Young’s modulus and boundary conditions in both numerical and physical experiments.

Exact Dirichlet boundary Physics-informed Neural Network EPINN for solid mechanics

Physics-informed neural networks (PINNs) have been rapidly developed for solving partial differential equations. The Exact Dirichlet boundary condition Physics-informed Neural Network (EPINN) is proposed to achieve efficient simulation of solid mechanics problems based on the principle of least work with notably reduced training time. There are five major building features in the EPINN framework. First, for the 1D solid mechanics problem, the neural networks are formulated to exactly replicate the shape function of linear or quadratic truss elements. Second, for 2D and 3D problems, the tensor decomposition was adopted to build the solution field without the need of generating the finite element mesh of complicated structures to reduce the number of trainable weights in the PINN framework. Third, the principle of least work was adopted to formulate the loss function. Fourth, the exact Dirichlet boundary condition (i.e., displacement boundary condition) was implemented. Finally, the meshless finite difference (MFD) was adopted to calculate gradient information efficiently. By minimizing the total energy of the system, the loss function is selected to be the same as the total work of the system, which is the total strain energy minus the external work done on the Neumann boundary conditions (i.e., force boundary conditions). The exact Dirichlet boundary condition was implemented as a hard constraint compared to the soft constraint (i.e., added as additional terms in the loss function), which exactly meets the requirement of the principle of least work. The EPINN framework is implemented in the Nvidia Modulus platform and GPU-based supercomputer and has achieved notably reduced training time compared to the conventional PINN framework for solid mechanics problems. Typical numerical examples are presented. The convergence of EPINN is reported and the training time of EPINN is compared to conventional PINN architecture and finite element solvers. Compared to conventional PINN architecture, EPINN achieved a speedup of more than 13 times for 1D problems and more than 126 times for 3D problems. The simulation results show that EPINN can even reach the convergence speed of finite element software. In addition, the prospective implementations of the proposed EPINN framework in solid mechanics are proposed, including nonlinear time-dependent simulation and super-resolution network.