Dynamic Elasticity Problems Research Articles

Конечно-элементное моделирование нелинейных задач упругости в абсолютных узловых координатах на неструктурированных шестигранных сетках

We consider a finite element approach to solving the problem of elasticity theory in terms of absolute nodal coordinate formulation (ANCF), in which large body displacements are described in a global reference frame without using any local coordinate system. The main feature of the method is the absence of gyroscopic effects and, as a result, constancy of the mass matrix and the vector of the generalized force of gravity. In contrast to the traditional ANCF approach, the sets of nodal degrees of freedom of the finite element are formed only on the basis of absolute coordinates of nodes, which allows us to solve the problem using, in particular, unstructured hexahedral meshes. To construct the stiffness matrix, we apply a second-order automatic differentiation algorithm, which ensures its symmetrical form (Hessian matrix) and is analytically accurate in calculating the derivative. This approach also makes it possible to carry out calculations for models of hyperelastic materials without the corresponding Piola-Kirchhoff tensor. It has been shown that within the framework of discretization of the equation of motion, along with the well-known Newmark numerical integration scheme, it is possible to use the HTT-α scheme, which is unconditionally stable, second-order accurate and dissipative for high frequencies. We present several examples of solving static and dynamic elasticity problems for compressible and incompressible models of hyperelastic materials, in which the functions of internal energy density of the body are specified in terms of the deformation gradient.

A novel meshfree method for investigating the impact of transverse joints quality on Xiaowan arch dam model

This paper presents a novel boundary collocation meshfree approach—space–time boundary moving least squares (BMLS) method to investigate the impact of transverse joints quality on the static and dynamic responses of a Xiaowan arch dam model. The proposed approach utilizes the space–time boundary moving least squares (BMLS) method based on one-dimensional moving least squares (MLS) technique to construct approximation formulas for solving three-dimensional elastic dynamic problems. Two models of Xiaowan arch dam, one with transverse joints and one without, were analyzed using the BMLS method to analyze the impact of transverse joint quality on the stability of the arch dam model. The numerical results suggest that the model with transverse joints already has good integrity when the ratio of the elastic modulus at the transverse joints to the elastic modulus of the dam material is 0.6. These findings demonstrate the promising development prospect of the BMLS method in practical engineering applications.

MOMENT FINITE ELEMENT FOR SOLVING 3D ELASTICITY PROBLEMS

The implementation of a new 8-node finite element for solving 3D elasticity static problems is considered. The idea of this finite element is based on the projection of a rare mesh FEM scheme into a space of lower dimension. The resulting scheme of the 8-node hexahedral finite element has a number of features compared to the scheme of the classical 8-node polyline element. This is one integration point in the element compared to eight in the classical element, as well as the presence of four parameters that allow you to adjust the convergence of the computational process. In a given finite element, the stresses as well as their moments (three bending and one torsional) are assumed to be constant within the element. Also, the continuity of displacement fields is preserved only in the nodes of finite elements. This circumstance is not a disadvantage and is characteristic of many new numerical FEM schemes (suffice it to mention the recently actively developed direction of the discontinuous Galerkin method). The issues of using this finite element in solving the well-known problem of increased shear stiffness (shear locking) of a number of well-known FEM schemes are discussed. The problem of hourglass instability, which is typical for FEM schemes with reduced integration and a number of other numerical schemes, in particular, the Wilkins scheme, which is popular in solving dynamic elasticity and plasticity problems, is also considered. The implementation of a technique for the numerical solution of three-dimensional static problems of the theory of elasticity on the basis of a given finite element within the framework of the traditional FEM technique using a vector-matrix notation is described. The results of solving test static problems of elasticity theory are presented.

Discretization Corrected Particle Strength Exchange for Steady State Linear Elasticity

Discretization corrected particle strength exchange (DC PSE) is a particle based spatial differential operator designed to solve meshless continuum mechanics problems. DC PSE is a spatial gradient operator that can discretize a computational domain with randomly distributed particles, provided that each particle has enough neighboring particles. In contrast, conventional methods such as the standard finite difference method require the computational domain to be discretized into a Cartesian grid. In linear elasticity simulations, especially steady state cases, this domain is mostly discretized using mesh-based methods such as finite element. However, while particle methods such as smoothed particle hydrodynamics (SPH) have been widely applied to solve dynamic elasticity problems, they have rarely been used in steady state simulations. In this study, a DC PSE operator was used to solve steady linear elasticity problems in a two-dimensional domain. The result of the DC PSE numerical simulation was compared to numerical results, empirical formula results, and results from conventional commercial finite element software, respectively.

Updated Lagrangian Taylor-SPH method for elastic dynamic problems

This paper presents a discussion on the properties of the collocation meshfree method, the Updated Lagrangian Taylor-SPH (UL-TSPH), for dynamic problems in solid mechanics. The PDEs are written in mixed form in terms of stress and velocity for the elastodynamics problems. Two sets of particles are used to discretize the partial differential equations, resulting on avoiding the tensile instability inherent to classical SPH formulations. Numerical examples ranging from propagation of a shock wave in an elastic bar to a stationary Mode-I semi-Infinite cracked plate subjected to uniaxial tension are used to assess the performance of the proposed method.

Updated Lagrangian Taylor-SPH method for elastic dynamic problems

This paper presents a discussion on the properties of the collocation meshfree method, the Updated Lagrangian Taylor-SPH (UL-TSPH), for dynamic problems in solid mechanics. The PDEs are written in mixed form in terms of stress and velocity for the elastodynamics problems. Two sets of particles are used to discretize the partial differential equations, resulting on avoiding the tensile instability inherent to classical SPH formulations. Numerical examples ranging from propagation of a shock wave in an elastic bar to a stationary Mode-I semi-Infinite cracked plate subjected to uniaxial tension are used to assess the performance of the proposed method.

Modeling of thermal stresses during hardening the product surface by thermal pulse

A particular solution of a linear variant of the dynamic thermal elasticity problem is considered in application to modeling the conditions of surface hardening of metal products by an energy pulse. The authors determined the equation of medium motion with the model of temperature pulse tested earlier for compatibility with special cases of the equations of parabolic and hyperbolic thermal conductivity. The problem of loading a flat plane of a short circular cylinder (disk) with a temperature pulse is presented. Pulse is a consequence of adopted structure of the volumetric power density of the heat flux, the time multiplier of which has the form of a single wave of the Heaviside function. Classical thermoelastic displacement potential and the method of its division into the product of independent variables functions were used to construct the thermal stress tensor. Differential equations for multiplier functions and their general solutions were found. Natural boundary conditions were set for the components of thermal stress tensor, and their tasks were solved. The obtained solutions are in the form of segments of functional series (the Bessel function in radial coordinate and the exponential function in axial coordinate). The article considers a numerical example of loading a disk made of 40KhN steel which has the mechanical properties sensitive to temperature treatment. Maple computer mathematics package was used in the calculations. Approximate solutions take into account the first 24 terms of the functional series. Estimation of the example makes it possible to explain the presence of stress peaks and stress intensity as a consequence of mutually inverse processes of temperature stress growth and reduction of elasticity coefficients with temperature rise. The numerical example warns against relying only on estimates of solutions to thermoelasticity problems without taking into account the plastic and viscous properties of the material.

Stress state of an elastic rectangular domain under steady load

The dynamic elasticity problem for rectangular domain is presented at this paper. The conditions of ideal contact are given at the lateral sides and bottom edge of the domain; the upper edge is loaded by normal stress. Problem is formulated as the boundary value problem in steady-state case. It was necessary to find the wave field of the rectangular domain. The Fourier transform was applied to formulated problem and reduced it to one dimensional vector boundary problem. The last one was solved with the method based on matrix differential calculations which was successfully applied earlier to solve the analogous static problem Pozhylenkov O. V. (2019). According to this procedure of the solving the exact solution of the vector boundary problem was constructed in transform domain in the explicit form. Application of the inverse Fourier transform finalized the deriving of the stated problem solution. The analyses of wave field of rectangular domain depending on different load types, frequency values and domain size was done.

On a Method in Dynamic Elasticity Problems for Heterogeneous Wedge-Shaped Medium

The method of analysis of steady oscillations arising in the piecewise homogeneous wedge-shaped medium composed by two homogeneous elastic wedges with different mechanical and geometric characteristics is presented. Method is based on the distributions’ integral transform technique and allows reconstructing the wave field in the whole medium by displacement oscillations given in the domain on the boundary of the medium. The problem in question is reduced to a boundary integral equation (BIA). Solvability problems of the BIA are examined and the structure of its solution is established.

The meshless analysis of elastic dynamic problem based on radial basis reproducing kernel particle method

In this article, introducing the method of radial basis function (RBF) into reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is presented. The RRKPM has higher computational accuracy and better convergence than the RKPM. Then the RRKPM is used in elastic dynamic problem, and the weak form of integral is used to find the discretized governing equations. The penalty method is used to determine the essential boundary condition, and the two-point difference method is applied to discretizing the time. The RRKPM for elastic dynamic problem is constructed, and the accuracy and effectiveness of the RRKPM for elastic dynamic problem are verified by the numerical examples.

A 2D frequency-domain wave based method for dynamic analysis of orthotropic solids

In this paper, a new version of the wave based method (WBM) is proposed for the dynamic analysis of orthotropic solids in the frequency domain. Different from the conventional WBM in general elastic dynamic problems, a set of new wave functions in vector forms are derived in the present WBM to effectively avoid decomposing the governing equations into Helmholtz equations. Particularly, scaling factors are introduced to improve the properties of the wave functions. With the new wave functions satisfying the governing equations, the present method not only inherits the advantages of the conventional WBM, but also eliminates its restrictions of the application to orthotropic dynamic problems. In comparison with the results of the finite element method (FEM), the efficiency, accuracy and convergence rate of the present method are verified by examples with different geometries, materials and boundary conditions.

New approach of analytical inversion of Laplace transform for some cases

Laplace transform is a useful tool for solving of dynamic elasticity problems. However, the problem of analytical inversion of Laplace transform has not yet completely solved. Therefore, it is relevant to consider the new methods that allow to derive the analytical form of the original function by the known transform.In this paper, the new method of analytical inversion of Laplace transform for the transforms of the certain form containing exponents in the denominator that linearly depend on Laplace transform parameter is proposed. The cases of correlation between the exponential indices are considered. The theorem is proved according to which the transform is expanded into the Taylor series, and the original function is derived by term-by-term application of the inverse Laplace transform. The correctness of the term-by-term application of the inverse Laplace transform is proved. The results derived by the use of the new method are verified by comparing them with the previously known formulas. The originals of Laplace transforms that were not found in the literature are derived.

P.105 Adjunctive treatment of aripiprazole to olanzapine for weight reduction in patients with schizophrenia

In this article, introducing the method of radial basis function (RBF) into reproducing kernel particle method (RKPM), and the radial basis reproducing kernel particle method (RRKPM) is presented. The RRKPM has higher computational accuracy and better convergence than the RKPM. Then the RRKPM is used in elastic dynamic problem, and the weak form of integral is used to find the discretized governing equations. The penalty method is used to determine the essential boundary condition, and the two-point difference method is applied to discretizing the time. The RRKPM for elastic dynamic problem is constructed, and the accuracy and effectiveness of the RRKPM for elastic dynamic problem are verified by the numerical examples.

Dynamic loading of crack-weakened complex-shaped bodies

Dynamic elasticity problems in the 2D formulation are solved by the method of lines using the Crank-Nicolson implicit finite-difference scheme for time-based integration and the finite element method for solving boundary problems. The applied variant of the finite element method includes two types of elements: isoparametric elements for modeling curvilinear boundaries and cohesion elements ensuring the compliance with Barenblatt’s postulates. It is shown how to apply the method to the strength calculations by the Petrov-Morozov method.

Helmholtz Decomposition and Boundary Element Method Applied to Dynamic Linear Elastic Problems

The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the longitudinal and transverse fields satisfy scalar Helmholtz equations that can be solved using a desingularized boundary element method (BEM) framework. The curl free longitudinal and divergence free transversal conditions can also be cast as additional scalar Helmholtz equations. When compared to other BEM implementations, the current framework leads to smaller matrix dimensions and a simpler conceptual approach. The numerical implementation of this approach is benchmarked against the 3D elastic wave field generated by a rigid vibrating sphere embedded in an infinite linear elastic medium for which the analytical solution has been derived. Examples of focussed 3D elastic waves generated by a vibrating bowl-shaped rigid object with convex and concave surfaces are also considered. In the static zero frequency limit, the Helmholtz decomposition becomes non-unique, and both the longitudinal and transverse components contain divergent terms that are proportional to the inverse square of the frequency. However, these divergences are equal and opposite so that their sum, that is the displacement field that reflects the physics of the problem, remains finite in the zero frequency limit.

Influence of mutual position of a noncircular tunnel and a dynamic impact source on stress distribution in lining

Based on the analytical solution of the corresponding dynamic plane elasticity problem, general regularities of the formation of maximum stresses in lining of a noncircular cross-section tunnel are found and studied at different locations of a dynamic impact source. The paper shows that the closely spaced position of the source has a significant effect on the stress state of the underground structure, which in turn depends on the frequencies of the waves radiated by this source. It is revealed that when the dynamic impact source is removed deep in rock mass for a distance of the order of seven average radii of the tunnel cross section, the estimated stresses in the tunnel lining coincide with the stresses found from the solution of the problem on the wave propagation in the corresponding direction from infinity.

Thermomechanical Extended Layerwise Method for laminated composite plates with multiple delaminations and transverse cracks

A Thermomechanical Extended Layerwise Method (TELM) is developed for the laminated composite plates with multiple delaminations and transverse cracks. The discontinuity of displacements and temperature induced by multiple delaminations is simulated by strong discontinuous function while the discontinuity of strain and temperature gradient between dissimilar layers is modeled by a weak discontinuous function. Transverse cracks are modeled using classical Extended Finite Element Method (XFEM). The coupled thermomechanical variational principle is employed to derive the Euler equations and the discrete forms. Since the displacement and temperature fields are solved simultaneously, a fully coupled time integration method is developed based on the Newmark integration algorithm and Crank-Nicolson scheme. The strain energy release rate for multiple delamination fronts and stress intensity factors for transverse cracks are calculated by the Virtual Crack Closure Technique and the Interaction Integral Method, respectively. The proposed method is applied to the steady-state thermomechanical problems, elastic and thermomechanical dynamic problems for isotropic and composite plates with transverse cracks and multiple delaminations.

Evaluation of the stiffness matrix in static and dynamic elasticity problems

A general evaluation technique (GET) for the stiffness matrix in the finite element methods (FEM) using a modified integration rule with alternate integration points $$r\in [0, 1]$$ rather than the standard Gauss points is proposed. The GET is examined using quadrilateral elements for elasticity problems. For the first time, we have found that the desired softening and stiffening effect can be achieved with adjustments of the integration point r. This allows the FEM model to achieve better accuracy and handle special problems, such as hourglass instability and volumetric locking. Ideal regions for the integration point r are found to overcome the hourglass and volumetric locking issues for the overestimation problems. In addition, the exact solution of the FEM model with optimal r value in terms of strain energy can be always obtained for general overestimation problems of elasticity. More importantly, the optimal integration point r obtained from the static case can be directly applied to dynamic problems to improve the transient displacement significantly. The intensive numerical examples including the static, dynamic, compressible and nearly incompressible material problems are analyzed to verify the accuracy and properties of the GET. Furthermore, the implementation of the GET is extremely simple without increasing computational cost.

Dynamic mass density of resonant metamaterials with homogeneous inclusions.

The occurrence of a negative dynamic mass density is a striking property of metamaterials. It appears when an inner local resonance is present. Results coming from an asymptotic theory are recalled briefly, showing the scaling of physical properties leading to inner resonance in elastic composites containing homogeneous soft inclusions, with negligible scattering of waves traveling through the matrix. This appears for a large contrast of elastic properties between matrix and inclusion. The frequency-dependent dynamic mass density depends on the resonance frequencies of the inner inclusions and on their related participation factors. Having solved the dynamic elasticity problem, these physical quantities are provided in the case of homogeneous cylindrical and spherical inclusions. It is shown that numerous resonance frequencies do not contribute to the dynamic mass density or have small participation factors, which simplifies significantly the physics involved in the concerned inner resonance phenomena. Finally, non-dimensional resonance frequencies and participation factors are given for both cases of inclusions as functions of the Poisson's ratio, defining completely the dynamic mass density.

A Three-Dimensional BEM for Dynamic Analysis of Anisotropic Elastic Multi-Connected Bodies

In this paper, the direct boundary element method in the Laplace domain is applied for the solution of three-dimensional transient dynamic problems of anisotropic elasticity in multi-connected domains. The formulation is based upon the integral representations of anisotropic dynamic fundamental solutions. As numerical example the problem of an anisotropic elastic prismatic solid with cubic cavity is investigated.