Stress Wave Propagation Problems Research Articles

Development of BB model and investigation of P-wave propagation across jointed rock masses using CDEM

This paper describes an advanced coupling method of finite element and discrete element called the Continuum-Discontinuum Element Method (CDEM). In this study, the Barton-Bandis model(BB model) for joint non-linear deformation is introduced into the CDEM to investigate the propagation of stress waves in jointed rock masses. The paper provides a detailed account of the development process of the BB model. Then, the developed model is utilized to simulate several representative cases of stress wave propagation in jointed rock masses, revealing the principles of stress wave propagation in jointed rock masses. The findings of this study demonstrate that the simulation results of the CDEM exhibit favorable consistency with experimental and theoretical results. This confirms the capability of the CDEM to accurately and effectively simulate stress wave propagation in jointed rock masses, thereby expanding the applicability of the CDEM. This research provides powerful support for further understanding and analysis of stress wave propagation problems in jointed rock masses.

Formulation for wave propagation in dissipative media and its application to absorbing layers in elastoplastic analysis using mathematical programming

AbstractIn this article, we propose a new solution scheme for modeling elastoplastic problems with stress wave propagation in dissipative media. The scheme is founded on a generalized Hellinger–Reissner (HR) variational principle. The principle renders the discretized boundary‐value problem into an equivalent second‐order cone programming (SOCP) problem that can be resolved in mathematical programming using the advanced optimization algorithm—the interior point method. In such a way, the developed method not only inherits admirable features of the SOCP‐based finite element method in solving elastoplastic problems but also enables the enforcement of absorbing layers (i.e., Caughey absorbing layer), which is essential in modeling stress wave propagation problems, to absorb wave energy. The proposed scheme is validated via the comparison between analytical and numerical results for seismic wave propagation in dissipative media. Its application to elastoplastic dynamic problems with stress wave propagation is also illustrated to demonstrate its efficiency.

Time Domain Implementation of Consistent Boundaries for Continuum Two-Dimensional Wave Propagation Problems Using ABAQUS User-Defined Elements

The present paper details the finite element implementation procedure for applying consistent boundaries for two-dimensional continuum soil-structure interaction systems. Consistent boundary conditions are ensured by applying rigorously calculated interaction forces at the truncated soil-structure interface. These calculations involve the evaluation of convolution integral on acceleration unit-impulse response functions for all boundary degrees of freedom for all time steps. The interactive or run-time evaluation of the interaction forces based on the past response history, is successfully carried out using user-defined element formulation in ABAQUS. This precise implementation of this boundary element procedure is demonstrated through benchmark problems and demonstrated through typical stress-wave propagation problems.

Boundary element modeling of fractional nonlinear generalized photothermal stress wave propagation problems in FG anisotropic smart semiconductors

The main aim of this article is to develop an efficient boundary element method (BEM) modeling of the fractional nonlinear generalized photo thermal stress wave propagation problems in the context of functionally graded (FG) anisotropic smart semiconductors. Due to nonlinearity, fractional order heat conduction and strongly anisotropy of mechanical properties, the governing equations system of such problems is often very difficult to solve using classical analytical methods. Therefore, a reliable and efficient coupling scheme based on BEM was proposed to address this challenge, where, the Cartesian transformation method (CTM) has been implemented to calculate the domain integrals, and the generalized modified shift-splitting (GMSS) has been implemented for solving the linear systems arising from BEM. The calculation findings are depicted in graphical forms to display the impacts of temperature-dependent, anisotropy, piezoelectric, graded parameter and fractional parameter on the nonlinear photo thermal stress wave propagation in the considered structure. The numerical findings confirm the consistency and efficacy of the developed modeling methodology.

Isogeometric analysis of 3D straight beam-type structures by Carrera Unified Formulation

This paper applies the isogeometric analysis (IGA) based on unified one-dimensional (1D) models to study static, free vibration and dynamic responses of metallic and laminated composite straight beam structures. By employing the Carrera Unified Formulation (CUF), 3D displacement fields are expanded as 1D generalized displacement unknowns over the cross-section domain. 2D hierarchical Legendre expansions (HLE) are adopted in the local area for the refinement of cross-section kinematics. In contrast, B-spline functions are used to approximate 1D generalized displacement unknowns, satisfying the requirement of interelement high-order continuity. Consequently, IGA-based weak-form governing equations can be derived using the principle of virtual work and written in terms of fundamental nuclei, which are independent of the class and order of beam theory. Several geometrically linear analyses are conducted to address the enhanced capability of the proposed approach, which is prominent in the detection of shear stresses, higher-order modes and stress wave propagation problems. Besides, 3D-like behaviors can be captured by the present IGA-based CUF-HLE method with reduced computational costs compared with 3D finite element method (FEM) and FEM-based CUF-HLE method.

A scaled boundary finite element formulation for dynamic elastoplastic analysis

SummaryThis study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.

A non-oscillatory time integration method for numerical simulation of stress wave propagations

When the compressive loads are dominant in a composite structure, a tensile stress may be induced owing to the propagation of a stress wave and the interaction between an incident wave and a reflection wave, thus leading to the occurrence of cracks. Therefore, stress wave have a significant effect on the life of composite structures. In this study, a four sub-step explicit time integration scheme is proposed for solving stress wave propagation problems. This method builds on the fourth-order central difference method and a high-order derivative term to minimize the numerical oscillation. The proposed scheme possesses a first-order accuracy in the case of undamped and damped systems. Stability, accuracy, and dispersion of the proposed explicit direct time integration scheme are analyzed. Furthermore, the performance of this scheme is illustrated by the solution of a stress wave propagation and wave reflection in a one-dimensional impact problem and two-dimensional scalar wave propagation.

Directly Simulation of Second Order Hyperbolic Systems in Second Order Form via the Regularization Concept

AbstractWe present an efficient and robust method for stress wave propagation problems (second order hyperbolic systems) having discontinuities directly in their second order form. Due to the numerical dispersion around discontinuities and lack of the inherent dissipation in hyperbolic systems, proper simulation of such problems are challenging. The proposed idea is to denoise spurious oscillations by a post-processing stage from solutions obtained from higher-order grid-based methods (e.g., high-order collocation or finite-difference schemes). The denoising is done so that the solutions remain higher-order (here, second order) around discontinuities and are still free from spurious oscillations. For this purpose, improved Tikhonov regularization approach is advised. This means to let data themselves select proper denoised solutions (since there is no pre-assumptions about regularized results). The improved approach can directly be done on uniform or non-uniform sampled data in a way that the regularized results maintenance continuous derivatives up to some desired order. It is shown how to improve the smoothing method so that it remains conservative and has local estimating feature. To confirm effectiveness of the proposed approach, finally, some one and two dimensional examples will be provided. It will be shown how both the numerical (artificial) dispersion and dissipation can be controlled around discontinuous solutions and stochastic-like results.

Application of Discrete Element Method for Continuum Dynamic Problems

A new method based on the principle of minimum potential energy is presented, aiming to overcome some weakness of the present discrete element method (DEM). Our primary research is to put forward the DEM with a tight theory base and a fit technique for treating continuum dynamic problems. By using this method, we can not only extend the existing seven-disc model, but also establish a new nine-disc model in a general way. Moreover, the equivalences of two kinds of models have been verified. In addition, three numerical examples of stress wave propagation problems are given in order to validate accuracy and efficiency of the present DEM models and their algorithms. Finally, the dynamic stress concentration problem of a square plate with a circular hole is analyzed.

Unstructured grid method for stress wave propagation in elastic media

A new numerical method, named unstructured grid method, is presented to calculate two-dimensional stress wave propagation problems in elastic media. This method is based on the dynamic equilibrium equations of the investigated lumps that are formed among the auxiliary unstructured grids. The approach to the solution is obtained by calculating the node displacements and the central point stresses of the spatial grids alternately. The stability criterion of the method is given and tested numerically. Analysis of dispersion for numerical phase velocity is studied. The rule is provided to determine the spatial step size of discretization mesh so as to ensure the numerical accuracy of the unstructured grid method. Numerical results are compared with the available exact solutions and with the solutions of central difference method for demonstrating the good accuracy of the new method. The unstructured grid method possesses higher calculating speed than the central difference method. Finally, the dynamic behavior of a square plate with a hole under an impact loading is researched.

Stress wave propagation in 1-D and 2-D media using Smooth Particle Hydrodynamics method

The paper involves the study on the elastic and elasto-plastic stress wave propagation in the 1-D and 2-D solid media. The Smooth Particle Hydrodynamics equations governing the elastic and elasto-plastic large deformation dynamic response of solid structures are presented. The proposed additional stress points are introduced in the formulation to mitigate the tensile instability inherent in the SPH approach. Both incremental rate approach and leap-frog algorithm for time integration are introduced and the new solution algorithm is developed and implemented. Two examples on stress wave propagation in aluminium bar and 2-D elasto-plastic steel plate are included. Results from the proposed SPH approach are compared with available analytical values and finite element solutions. The comparison illustrates that the stress wave propagation problems can be effectively solved by the proposed SPH method. The study shows that the SPH simulation is a reliable and robust tool and can be used with confidence to treat transient dynamics such as linear and non-linear transient stress wave propagation problems.

Computational Approaches With Applications to Non-Classical and Classical Thermomechanical Problems

Although the issues are not very controversial, the fusion of both the fields of heat conduction in solids and elasticity which results in the so-called field of dynamic thermoelasticity (both non-classical and classical), dates as far back as Maxwell (1867), and has long been a subject matter of widespread research activity and interest. Although routine thermomechanical problems are primarily influenced by classical effects, issues where non-classical influences may become important are also addressed. To date, numerous applications ranging from defense, aerospace to manufacturing related problems including routine mechanical, civil, nuclear and allied engineering applications influenced by thermal-structural interactions continue to pose significant challenges both from the underlying mechanics and from a computational viewpoint. Herein is first presented an overview of non-classical and classical dynamic thermoelasticity models and equations governing these situations. Subsequently, attention is focused on computational approaches for the modeling and analysis of various classes of problems encompassing thermal-structural interactions which can be broadly classified as: i) thermally-induced stress wave propagation problems, ii) thermally-induced dynamic (inertial type) problems, and iii) the general field of thermal stresses. A variety of illustrative numerical examples encompassing non-classical and classical influences are finally presented to provide an improved understanding of the behavior of thermal-structural problems via effective unified computational developments. This review article contains 142 references.

Analytical, Computational and Experimental Investigations on Stress Wave Propagation

Wave phenomena in intensely loaded bodies are often paradoxical and unexpected. They are evoked by stress wave interactions and depend substantially on actual material properties, body shapes and dimensions, friction, etc. Their proper understanding is indispensable for the reliable evaluation of the quality of various technologies as nuclear reactor safety, armour penetration, nondestructive testing, etc. The paper presents a survey of analytical, computational and experimental techniques and methods being employed for the solutions of stress wave propagation problems in solids.

A transient finite element analysis of linear viscoelastic material model

Since the attenulation of propagating waves through soil/rock is related to the localized material properties as well as the strain developed, the commonly used Rayleigh-type damping model and its variations are not suitable for dynamic finite element analysis of such materials. A linear viscoelastic material model based on the concept of the relaxation spectrum is manipualted in place of the damping model in this paper. The method proposed by Day and Minster11 to transform the convolutional form of the stress–strain relationship to a set of differential operators using the Pade approximant method is generalized to non-scalar waves and implemented for transient finite element analyses. A time-marching scheme is also proposed to incorporate the resultant differential operators into the governing equation of motion. The accuracy related to the Pade approximant method and the time-marching scheme is investigated by critically analysing some scalar wave propagation problems. The proposed technique is further verified using two one-dimensional stress wave propagation problems and a two-dimensional transient propagating wave through an unbounded linear viscoelastic medium. Some encouraging results have been obtained using the proposed technique and guidelines for using this technique are also presented. Comparisons of analytical solutions obtained by Fourier synthesis and numerical results have been provided.

A NUMERICAL TECHNIQUE FOR DYNAMIC COUPLED THERMOELASTICITY PROBLEMS WITH RELAXATION TIMES

A numerical method is proposed for Green and Lindsay' s dynamic thermoelasticity problems. The semidiscrete equations resulting from standard finite element formulations are mathematically manipulated so that the-resulting coupled heat equation and elastic equations both become symmetric. Such process allows for taking advantage of the traditional uncoupled thermal-structural solution procedures. Unconditionally stable time integration schemes are recommended to solve these matrix equations. The proposed technique is very effective and versatile in solving both finite speed thermal wave and stress wave propagation problems

On heat displacement based hybrid transfinite element formulations for uncoupled/coupled thermally induced stress wave propagation

The present paper describes the evaluation of heat displacement based hybrid transfinite element formulations for their applicability to coupled/uncoupled thermally induced stress wave propagation problems. The hybrid formulations developed herein utilize the concept of heat displacement which is related to temperature changes in the manner mechanical displacement is related to strain. Therein, transform methods in conjunction with classical Galerkin schemes and contemporary finite element formulations are employed for predicting the dynamic response of coupled/uncoupled thermally induced stress wave propagations. In particular, the test cases analyzed pertain to the well-known Danilovskaya's models of coupled/uncoupled thermoelasticity. Applicability to these models is demonstrated via a unified hybrid thermal-structural approach, thereby retaining the advantages of heat displacement and transfinite element formulations.

A finite element formulation for transient analysis of viscoplastic solids with application to stress wave propagation problems

A brief note on a finite element formulation for the transient analysis of viscoplastic solids is presented. Attention is confined to small strains in the present discussion and the emphasis is on stress wave propagation problems. The algorithm is based on straightforward explicit integration of both the equations of motion and the plastic rate equations. The explicit central difference method, which appears suitable for wave propagation problems, is used for integration of the equation of motion. The stress update is accomplished by means of a forward gradient scheme based on an estimate of the plastic flow over a time increment (Peirce, Shih and Needleman, Comput. Struct. 18, 875–887, 1984). A number of simple numerical examples are presented to illustrate the method.

Strain softening of rock, soil and concrete — a review article

This paper explores the physical bases of strain softening in rock, soil and concrete, in order to determine if such softening is a true property of these materials, and examines the effect that strain softening has on stress wave propagation in softening materials. It is concluded that the strain softening inferred in the usual manner from conventional laboratory tests is not a true material property of rock, soil and concrete and that strain softening can cause stress wave propagation problems for some softening materials to become ill-posed.

Stress wave propagation in rods of elastic-viscoplastic material

A number of uniaxial stress wave propagation problems are solved based on the unified, multi-dimensional, elastic-viscoplastic constitutive equations of Bodner-Partom and a finite difference numerical procedure. Solutions are obtained for cases of a velocity imposed for a time period or indefinitely at the end of a semi-infinite bar, and for the condition of a finite bar subjected to a high velocity superimposed on an applied low velocity after a time interval. Work-hardening is taken to be isotropic for stress of constant sign, while an isochoric, anisotropic work-hardening formulation is employed for problems involving stresses of reversed sign due to unloading or reflections. The numerical exercises are based on constants for a strongly strain rate sensitive material, titanium, and the results indicate good qualitative agreement with a wide range of experimental observations.

Numerical methods for nonlinear stress wave propagation (abstract)

The partial differential equations characterizing stress wave propagation problems include conservation of volume, mass, momentum, and energy plus the constitutive relations for the materials involved. Generally these constitutive relations are nonlinear and numerical methods are required for solutions in various coordinate systems and spatial dimensions. The philosophy of stress wave propagation code development and application at Sandia Laboratories is given in this paper. The way different methods are developed, utilized, and tested is illustrated by many examples in which code solutions are compared to analytic solutions, experiments, and solutions from other codes.