Displacement Derivatives Research Articles

Analysis of Concrete Tensile Failure using Dynamic Particle Difference Method under High Loading Rates

This study presents the use of the dynamic particle difference method (PDM) to analyze tensile failure in concrete subjected to high loading rates. In general, strong form-based meshfree methods suffer from limitations pertaining to the material modeling of concrete because concrete exhibits both softening and damage behaviors that initiate crack growth under an impact load. These methods are generally based on the direct discretization of the governing equations, such as Navier's equation, which involves second-order differentiation. However, conventional material models are based on the first-order derivatives of displacement. The newly developed dynamic PDM can effectively address the limitations of material modeling using a combination of first-order derivative approximations. This circumvents the requirement for high-order derivative approximations, which are essential in strong formulations, such as the finite difference method and point collocation method. The strain rate effect caused by an extremely high loading speed was successfully modeled by accurately reflecting the energy dissipations that result from the cohesive property of the concrete and brittle cracking. Although the developed method incorporates the elastic constitutive relation, it enables the effective modeling of these nonlinear effects. In addition, it can reduce the computational effort. The proportional damping algorithm simulates the effect of velocity in the equation of motion and the cohesion effect in the concrete material. The damage model and the visibility criterion adequately handle crack initiation and propagation in the concrete member. Furthermore, it is noteworthy that the final discrete forms of the dynamic PDM are similar to the integrands of the weak form in the conventional finite element formulation. We ascertained that the stiffness and mass proportional damping effects are related to the inertia and strain of the material, respectively. It was confirmed that the location and direction of crack propagation in concrete varied with the strain rate. Hence, the accuracy and robustness of the proposed method were successfully verified by simulation, and the strain-rate dependency of concrete fracture was efficiently simulated using the proposed method.

Isogeometric Collocation Method to solve the strong form equation of UI-RM Plate Theory

This work presents the formulation of the isogeometric collocation method to solve the strong form equation of a unified and integrated approach of Reissner Mindlin plate theory (UI-RM). In this plate theory model, the total displacement is expressed in terms of bending and shear displacements. Rotations, curvatures, and shear strains are represented as the first, the second, and the third derivatives of the bending displacement, respectively. The proposed formulation is free from shear locking in the Kirchhoff limit and is equally applicable to thin and thick plates. The displacement field is approximated using the B-splines functions, and the strong form equation of the fourth-order is solved using the collocation approach. The convergence properties and accuracy are demonstrated with square plate problems of thin and thick plates with different boundary conditions. Two approaches are used for convergence tests, e.g., increasing the polynomial degree (NELT = 1✕1 with p = 4, 5, 6, 7) and increasing the number of element (NELT = 1✕1, 2✕2, 3✕3, 4✕4 with p = 4) with the number of control variable (NCV) is used as a comparable equivalent variable. Compared with DKMQ element of a 64✕64 mesh as the reference for all L/h, the problem analysis with isogeometric collocation on UI-RM plate theory exhibits satisfying results.

Further results on advanced robust iterative learning control and modeling of robotic systems

Recently, calculus of general order [Formula: see text] has attracted attention in scientific literature, where fractional operators are often used for control issues and the modeling of the dynamics of complex systems. In this work, some attention will be devoted to the problem of viscous friction in robotic joints. The calculus of general order and the calculus of variations are utilized for the modeling of viscous friction which is extended to the fractional derivative of the angular displacement. In addition, to solve the output tracking problem of a robotic manipulator with three DOFs with revolute joints in the presence of model uncertainties, robust advanced iterative learning control (AILC) is introduced. First, a feedback linearization procedure of a nonlinear robotic system is applied. Then, the proposed intelligent feedforward-feedback AILC algorithm is introduced. The convergence of the proposed AILC scheme is established in the time domain in detail. Finally, simulations on the given robotic arm system confirm the effectiveness of the robust AILC method.

Generalized Finite Difference Method for Plate Bending Analysis of Functionally Graded Materials

In this paper, an easy-to-implement domain-type meshless method—the generalized finite difference method (GFDM)—is applied to simulate the bending behavior of functionally graded (FG) plates. Based on the first-order shear deformation theory (FSDT) and Hamilton’s principle, the governing equations and constrained boundary conditions of functionally graded plates are derived. Based on the multivariate Taylor series and the weighted moving least-squares technique, the partial derivative of the underdetermined displacement at a certain node can be represented by a linear combination of the displacements at its adjacent nodes in the GFDM implementation. A certain node of the local support domain is formed according to the rule of “the shortest distance”. The proposed GFDM provides the sparse resultant matrix, which overcomes the highly ill-conditioned resultant matrix issue encountered in most of the meshless collocation methods. In addition, the studies show that irregular distribution of structural nodes has hardly any impact on the numerical performance of the generalized finite difference method for FG plate bending behavior. The method is a truly meshless approach. The numerical accuracy and efficiency of the GFDM are firstly verified through some benchmark examples, with different shapes and constrained boundary conditions. Then, the effects of material parameters and thickness on FG plate bending behavior are numerically investigated.

Unique solvability of an equilibrium problem for a Kirchhoff-Love plate with a crack along the boundary of a flat rigid inclusion

A new model of an equilibrium problem for a Kirchhoff-Love plate with a flat cylindrical rigid inclusion and an interfacial crack is considered. As in previous works, we consider a rigid inclusion defined with the help of a cylindrical surface, but unlike the known models relating to the crack theory, we suppose that traces of derivatives of vertical displacements (deflections) satisfy certain boundary conditions. These conditions determine constant angles of normal fibers along an entire flat cylindrical inclusion. The interfacial crack is located on the boundary of the rigid inclusion. A condition of mutual non-penetration of opposite crack faces is given as an inequality on the crack curve. We prove the existence and uniqueness of a solution for this variational problem.

Jerk within the Context of Science and Engineering—A Systematic Review

Rapid changes in forces and the resulting changes in acceleration, jerk and higher order derivatives can have undesired consequences beyond the effect of the forces themselves. Jerk can cause injuries in humans and racing animals and induce fatigue cracks in metals and other materials, which may ultimately lead to structure failures. This is a reason that it is used within standards for limits states. Examples of standards which use jerk include amusement rides and lifts. Despite its use in standards and many science and engineering applications, jerk is rarely discussed in university science and engineering textbooks and it remains a relatively unfamiliar concept even in engineering. This paper presents a literature review of the jerk and higher derivatives of displacement, from terminology and historical background to standards, measurements and current applications.

Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the strain-gradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first- and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

Data-Driven Computational Simulation in Bone Mechanics.

The data-driven approach was formally introduced in the field of computational mechanics just a few years ago, but it has gained increasing interest and application as disruptive technology in many other fields of physics and engineering. Although the fundamental bases of the method have been already settled, there are still many challenges to solve, which are often inherently linked to the problem at hand. In this paper, the data-driven methodology is applied to a particular problem in tissue biomechanics, a context where this approach is particularly suitable due to the difficulty in establishing accurate and general constitutive models, due to the intrinsic intra and inter-individual variability of the microstructure and associated mechanical properties of biological tissues. The problem addressed here corresponds to the characterization and mechanical simulation of a piece of cortical bone tissue. Cortical horse bone tissue was mechanically tested using a biaxial machine. The displacement field was obtained by means of digital image correlation and then transformed into strains by approximating the displacement derivatives in the bone virtual geometric image. These results, together with the approximated stress state, assumed as uniform in the small pieces tested, were used as input in the flowchart of the data-driven methodology to solve several numerical examples, which were compared with the corresponding classical model-based fitted solution. From these results, we conclude that the data-driven methodology is a useful tool to directly simulate problems of biomechanical interest without the imposition (model-free) of complex spatial and individually-varying constitutive laws. The presented data-driven approach recovers the natural spatial variation of the solution, resulting from the complex structure of bone tissue, i.e. heterogeneity, microstructural hierarchy and multifactorial architecture, making it possible to add the intrinsic stochasticity of biological tissues into the data set and into the numerical approach.

Real-time and High Quality Ultrasound Elastography Using Convolutional Neural Network by Incorporating Analytic Signal.

Convolutional Neural Networks (CNN) have been extensively used for many computer vision applications including optical flow estimation. Although CNNs have been very successful in optical flow problem, they have been rarely used for displacement estimation in Ultrasound Elastography (USE) due to vast differences between ultrasound data and computer vision images. In USE, a main goal is to obtain the strain image which is the derivative of the axial displacement in axial direction; therefore, a very accurate displacement estimation is required. Radio Frequency (RF) data is needed to obtain accurate displacement estimation. RF data contains high frequency contents which cannot be downsampled without significant loss of information, in contrast to computer vision images. We propose a novel technique to utilize LiteFlowNet for USE. For the first time, we incorporate analytic signal to improve the quality of the displacement estimation. We show that this network with the designed inputs is more suitable for USE compared to more complex networks such as FlowNet2. The network is adopted to our application and it is compared with FlowNet2 and a state-of-the-art elastography method (GLUE). The results show that this network performs well and comparable to GLUE. Furthermore, not only this network is faster and has lower memory footprint compared to FlowNet2, but also it obtains higher quality strain images which makes it suitable for portable and real-time elastography devices.

Emergent scale behaviour in the effective equations for an elastic metamaterial

Asymptotic homogenisation of an elastic metamaterial consisting of a series of plates interspaced by a fluid is considered. It is shown that the usual method must be extended by the inclusion of a second macroscale, the “emergent scale”, in order to correctly capture the behaviour of this metamaterial at low frequency. The leading order solutions for plane wave propagation are found and the effective constitutive equations derived. At leading order, the dispersion curves for plane wave propagation depend on the direction of propagation and the branch associated with the emergent scale is dispersive. The effective constitutive equations contain some terms, associated with the shear normal to the plates, for which stress is not proportional to strain but rather depends on the third derivative of the normal displacement with respect to transverse position. These terms are not present in the usual form of the Willis equations and the effective fields are non-analytic functions of frequency as the frequency tends to zero.

Modeling method for periodic cracks in functionally graded strips with arbitrary properties

A piecewise exponential model is developed to investigate the problems of periodic cracks in functionally graded strips with arbitrary material properties. The problem of a row of embedded cracks vertical to strip surfaces under a mode I load in a functionally graded strip with arbitrary nonhomogeneous properties is considered in a plane elasticity state. The true properties of functionally graded materials (FGMs) are approached by a series of sub-layers with material properties varying exponentially and without losing continuity. The thicknesses of different sub-layers can be set to improve calculation accuracy and efficiency. The formulation for doing so is derived using Fourier integral transformation and Fourier series, and the problem is finally reduced to an integral equation with Cauchy-type singularity. Crack-surface displacement derivative is selected as an auxiliary function and is numerically solved to calculate the stress intensity factors (SIFs) at crack tips. The results of previous research are compared with those of the current study to verify the present model. Four representative functions of material properties are considered and the SIFs are presented. The effects of crack spacing, nonhomogeneous properties and crack length on SIFs are discussed. The conclusions are beneficial to the knowledge of fracture behaviors of FGMs.

Two-dimensional elastodynamic and free vibration analysis by the method of fundamental solutions

In this paper we will present a formulation for solving elastodynamic problems in 2D using the method of fundamental solutions (MFS). The governing equation for the displacement in the elastodynamic problem is expressed by the Navier equation with an additional non-homogeneous inertial term, which involves the second derivative of displacement with respect to time. The inertial term is approximated by the Houbolt finite difference formula. In each time step, the solution for the displacement is separated into a homogeneous solution and a particular solution. The homogeneous solution is constructed in terms of the fundamental solutions of the elastostatic problem. The particular solution corresponds to the non-homogeneous inertial term, which is obtained by approximating the inertial term with radial basis functions (RBFs). In addition, a formulation based on the MFS for free vibration analysis of 2D bodies is presented in this paper. By solving several numerical examples, we have demonstrated that the proposed methods are robust and accurate. The convergence of the results with respect to the number of collocation points, the number of internal points, the value of the RBF parameter, and the time step size is also studied.

Peridynamic Analysis of Cracked Beam Under Impact

ABSTRACTSpecific conditions at the tip of a crack and discontinuities in a material are the challenges in analyzing the growth of cracks using conventional methods. In recent years, a method has been developed based on the non-local mechanics, called peridynamic theory, which has improved the analysis process of such structures. In this theory, the points of matter whose displacement or displacement derivatives are discontinuous are not distinguished from other material points. In this paper, we employed the bond-based peridynamic theory to investigate the rate of crack propagation and the path of crack growth in a beam with an initial crack due to low velocity impact. Two beams made of polymethyl-methacrylate (PMMA) and steel alloy with different projectile shapes were considered. The effects of changes in the impact velocity and the fracture toughness were studied and the obtained results were validated with other conducted studies. The crack path was predicted successfully and the branching of the crack was captured. The results confirm the ability of the peridynamic theory to model the crack growth in impact problems.

Strain microseismics: Radiation patterns, synthetics, and moment tensor resolvability with distributed acoustic sensing in isotropic media

We have derived analytical formulations for the strain field produced by a moment tensor source in homogeneous isotropic media. Such formulations are important for microseismic projects that increasingly are monitored with fiber-optic distributed acoustic sensing (DAS) systems. We find that the spatial derivative of displacement produces new terms in strain proportional to [Formula: see text] with [Formula: see text]. In viscoelastic media, the derivative also produces an additional far-field term that is scaled by a frequency-dependent factor. When comparing with full wavefield synthetic data, we observe that the new terms proportional to [Formula: see text] can be considered part of a near-field in strain, similar to the practice with the displacement formulation. Analyses of moment tensor resolvability show that full moment tensors are resolvable with P-wave information from two or more noncoplanar vertical DAS cable geometries if intermediate- and far-field terms are considered and that S-wave information alone cannot constrain full moment tensors using only vertical wells. These results mirror previous observations made with displacement measurements. Furthermore, the addition of the new terms proportional to [Formula: see text] in strain improves the moment tensor resolvability but only in the case of a single vertical array. In the case of a single deviated/horizontal well, we can, in theory, resolve a full moment tensor but a case-by-case analysis is necessary to identify regions of full constraint around the well and the necessary noise conditions to guarantee reliable solutions. Real DAS measurements also are affected by the gauge length and interrogator details. In the case of the gauge length, we observe that this operator does not change the resolvability of the problem but it does affect inversion stability. The results derived here represent theoretical limits or in some cases specific examples. Practical implementations require analyses of conditioning, noise, coupling, and the effect of gauge length on a case-by-case basis.

Displacement Calculus

In this work, we establish a theory of Calculus based on the new concept of displacement. We develop all the concepts and results necessary to go from the definition to differential equations, starting with topology and measure and moving on to differentiation and integration. We find interesting notions on the way, such as the integral with respect to a path of measures or the displacement derivative. We relate both of these two concepts by a Fundamental Theorem of Calculus. Finally, we develop the necessary framework in order to study displacement equations by relating them to Stieltjes differential equations.

Elastoplastic Model of Rocks with Internal Self-Balancing Stresses. Continuum Approximation

The authors construct a continuum approximation for the model of a medium with internal structure and internal self-balancing stresses. The presupposition of diffeomorphism, i.e. existence of partial space derivatives of displacements is largely slackened. The information brought to the closed model of a geomedium by this presupposition is comparable with the information brought by the constitutive equations. The new model includes the local bends of grains of the skeleton, plasticity and the elastic strains of the binding medium in the pore space. The model is of a gradient type.

A four-node C$$^{0}$$ tetrahedral element based on the node-based smoothing technique for the modified couple stress theory

In this paper, a four-node $$C^{0}$$ tetrahedral element for the modified couple stress theory is proposed. Since the governing equations are the fourth-order differential equations, the first-order derivative of displacement or rotation should be approximated by a continuous function. In the proposed element, nodal rotations are defined using the node-based smoothing technique. Continuous rotation fields are defined with the shape functions and nodal rotations. Both the displacement field and the rotation field are expressed solely in terms of the displacement degrees of freedom. The element stiffness matrix is calculated using the newly defined rotation field. To prevent the increase of calculation cost due to increase of the bandwidth of the stiffness matrix, the preconditioned conjugate gradient method is introduced. The performance of the proposed element is evaluated through various numerical examples.

Construction of the bending model of micropolar elastic thin beams with a circular axis and its implementation using finite element method

This paper considers the problem of transition from the system of two-dimensional equations of the micropolar (moment) theory of elasticity in a thin curved area to the one-dimensional system of equations describing deformation of the micropolar elastic thin beam with a circular axis. During the transition process, Timoshenko's hypotheses generalized to the micropolar case are applied. As a result, the applied model (with independent fields of displacements and rotation) of a micropolar elastic thin beam with a circular axis has been constructed. It is shown that the model includes the law of conservation of energy, energy theorems and variation principles. All main functionals for the model of the micropolar elastic thin beam with a circular axis are obtained from the functional of the two-dimensional micropolar theory of elasticity, containing only the first derivatives of displacements and rotations. The finite element method (FEM) is taken to study the boundary problems of statics and dynamics of applied model of the micropolar elastic thin beam with a circular axis. The basic concepts and stages of the FEM are formulated: the discretization, the selection of basic nodal unknowns, the approximation of the solution, and the construction of the basic FEM equations. The finite-element solutions of some problems of statics and problems on natural vibrations of beams with a circular axis are considered according to the micropolar theory of elasticity. A comparative analysis with similar problems of beams with a circular axis according to the classical theory of elasticity is carried out. Based on the results, some effective properties of beams with a circular axis have been established when considering their deformations in the context of the micropolar theory of elasticity.

基于彩色相机的双波长剪切散斑干涉法同步测量面内外位移导数

基于彩色相机的双波长剪切散斑干涉法同步测量面内外位移导数

Adaptive Variable Structure Control With Neuron for Path Tracking of Beaver AUV

In order to improve path tracking performance of Beaver AUV, a new method for path tracking controller design is proposed basing on variable structure control theory and nonlinear dynamics model of Beaver. Switching surface of variable structure controllers is founded with corresponding error and error derivative of surge displacement and yaw angle. Exponential approach law is adopted to derive control algorithm for surge and yaw in the process of path tracking. Chattering phenomenon is restrained due to adaptive neurons which are used for approaching speed adjusting online. Simulation results of sinusoidal path tracking, circular path tracking and planning path tracking show that tracking precision of Beaver AUV is improved more evidently through variable structure control with adaptive approaching speed compared with variable structure control using fixed approaching speed. The method proposed in this article is effective and feasible.