Integral constitutive equations based temporal finite element modeling for the static viscoelastic problem

Torsional static and vibration analysis of functionally graded nanotube with bi-Helmholtz kernel based stress-driven nonlocal integral model

A temporal finite element method based on a mixed form of the Hamiltonian weak principle is developed for dynamics and optimal control problems. The mixed form of Hamilton's weak principle contains both displacements and momenta as primary variables that are expanded in terms of nodal values and simple polynomial shape functions. Unlike other forms of Hamilton's principle, however, time derivatives of the momenta and displacements do not appear therein; instead, only the virtual momenta and virtual displacements are differentiated with respect to time. Based on the duality that is observed to exist between the mixed form of Hamilton's weak principle and variational principles governing classical optimal control problems, a temporal finite element formulation of the latter can be developed in a rather straightforward manner. Several well-known problems in dynamics and optimal control are illustrated. The example dynamics problem involves a time-marching problem. As optimal control examples, elementary trajectory optimization problems are treated.

Studies on integral and differential constitutive equations by the present authors are summarized and some new results are presented. Applicability of various constitutive equations to some typical deformation modes is investigated. In integral equations, superiority of the BKZ model is established. But it is also known that the model fails in a prediction of stress relaxation for double‐step shear strains in which the second step strain is applied in the opposite direction to the first. A stress dependent constitutive model of integral type proposed by the present authors can describe this double‐step stress relaxation very well as well as ordinary single‐step stress relaxation and shear rate dependence of viscosity. Compared to integral equations, applicability of differential equations is limited. The Leonov model is very useful to predict viscoelastic functions in steady shear flow, but it fails in describing both stress relaxations for a step strain and after cessation of steady shear flow. The Giesekus model gives slightly better predictions for steady shear flow and for these stress relaxations, but fundamentally the Giesekus and Leonov models cannot be accepted as exact models for stress relaxation. On the other hand, the Larson model can describe both stress relaxations well, but it gives too strong shear rate dependences for viscosity and coefficient of first normal stress difference. It is concluded that further studies on stress dependent model of either integral or differential type will yield fruitful development of constitutive equations.

Two technologies, 3-D reconstruction of human temporal bones and finite element analysis of ear biomechanics, were combined to construct a computational model for describing function of the ear for sound transmission. A set of histological sections of a human temporal bone was prepared, scanned, and digitized for imaging analysis. The aligned and treated images of the middle ear components, external ear canal and middle ear cavity were constructed with attached ligaments, muscles, and tendons. A 3-D CAD model was first created which provided accurate geometric data. Based on the solid model, the finite element (FE) model of the human ear was generated with whole middle ear and ear canal. The dynamic behavior of the FE model was very close to the actual mechanical performance of human temporal bone with 90 dB SPL at the eardrum across auditory frequencies. The model will be further used for acoustic analysis of the ear. (Supported by Whitaker Foundation and OCAST).

The static crack problem of a functionally graded coating-substrate structure with an internal or edge crack perpendicular to the interface is investigated under an in-plane load. The structure is made up of a functionally graded coating with an internal or edge crack and a homogeneous substrate of finite thickness. The material properties are assumed to vary continuously from the coating to the substrate. By use of Fourier transform method, the mixed boundary value problem is reduced to a singular integral equation which can be solved numerically. During the analysis, a higher-order term is obtained in the asymptotic analysis of the singular kernel to improve the convergence efficiency of numerical integrals. The influences of material constants and the geometry parameters on the stress intensity factors (SIFs) are studied. In Part II of this paper, the transient response of the structure subjected to an in-plane impact is investigated.

Spatial static boundary-value problems of mathematical physics for a layer with a non-circular cylindrical through hole have hardly been solved using the method of singular integral equations (SIE) despite the fact that numerous fundamental theoretical issues have been developed. Methods for the calculation of spectral characteristics are also absent. The paper provides an overview of the methods for solving these problems. A new mathematical model has been constructed, and a new method based on a system of three SIEs has been developed and tested numerically. As a result of a high-precision numerical study, it was found that with an increase in the thickness of the layer, an increase in the relative circumferential stress occurs. In the case of a circular hole, a shift of the maximum relative circumferential stress from the ends to the depth of the layer is observed. In the case of an elliptical hole, with a decrease in one of the radii, an increase in the relative circumferential stress is also observed. Keywords: three-dimensional boundary-value problems, singular integral equations, numerical experiment, static bending, a through hole.

A method for the formulation of elastostatical boundary value problems as integral equations is presented, the basic idea of which consists of superimposing in a suitable fashion singular solutions for the infinite medium. Since mechanical aspects play an important role in the concept of the method, all quantities in the equations can be interpreted physically. The applicability of the method is illustrated by examples of the geometrical and statical boundary value problem ofplane elastostatics for which 32 different formulations as integral equations are established. The second aim of the paper consists of revealing an analogy between the most important notions of the singularity method, viz. between state variables and singularities. The analogy is manifested by certain symmetries of influence functions, and enables the systematical representation of the basic relations and their interpretation within a larger context.

This paper describes the block element method for spatial integral equations with a difference kernel in the boundary-value problems of continuum mechanics and mathematical physics. The basis of the proposed method is the Wiener — Hopf method, whose generalization for a spatial case is called integral factorization method. The block element method is applied to solve problems in domains with piecewise smooth boundaries containing corner points. The developed method was used to solve the contact problem for a wedge-shaped stamp occupying the first quadrant. This paper describes in detail the methods of obtaining various characteristics of the solution constructed by reversing the system of one-dimensional linear integral equations typical for dynamics and static contact problems for stamps in the form of a strip.

Peridynamics is a nonlocal extension of classical solid mechanics that allows for the modeling of bodies in which discontinuities occur spontaneously. Because the peridynamic expression for the balance of linear momentum does not contain spatial derivatives and is instead based on an integral equation, it is well suited for modeling phenomena involving spatial discontinuities such as crack formation and fracture. In this study, both peridynamics and classical finite element analysis are applied to simulate material response under dynamic blast loading conditions. A combined approach is utilized in which the portion of the simulation modeled with peridynamics interacts with the finite element portion of the model via a contact algorithm. The peridynamic portion of the analysis utilizes an elastic-plastic constitutive model with linear hardening. The peridynamic interface to the constitutive model is based on the calculation of an approximate deformation gradient, requiring the suppression of possible zero-energy modes. The classical finite element portion of the model utilizes a Johnson-Cook constitutive model. Simulation results are validated by direct comparison to expanding tube experiments. The coupled modeling approach successfully captures material response at the surface of the tube and the emerging fracture pattern.

Finite element analyses of multi-dimensional, partial differential equations require accurate and complete boundary condition assignments. However, continuous temporal and spatial values along physical boundaries are rarely available when solving realistic problems based on field data. To mitigate this situation a neural network system was developed and coupled with an interpolation routine. This innovative computational utility was used to provide continuous boundary condition information in time and space along physical boundaries of interest. The neural network was trained to predict the time response at discrete boundary locations based on field measurements at those locations. Once trained the neural network system was capable of providing a continuous time history for those locations. This system was then linked to an interpolation routine which handled the spatial component of the boundary condition specifications. These coupled routines facilitated rapid deployment and testing of various finite element representations. Further, the neural network system captured the transient physics of the situation more accurately than interpolation routines used previously.

A common approach for dynamic analysis in current practice is based on a discrete time-integration scheme. This approach can be largely attributed to the absence of a true variational framework for initial value problems. To resolve this problem, a new stationary variational principle was recently established for single-degree-of-freedom oscillating systems using mixed variables, fractional derivatives and convolutions of convolutions. In this mixed convolved action, all the governing differential equations and initial conditions are recovered from the stationarity of a single functional action. Thus, the entire description of linear elastic dynamical systems is encapsulated. For its practical application to structural dynamics, this variational formalism is systemically extended to linear elastic multidegree- of-freedom systems in this study, and a corresponding weak form is numerically implemented via a quadratic temporal finite element method. The developed numerical method is symplectic and unconditionally stable with respect to a time step for the underlying conservative system. For the forced-damped vibration, a three-story shear building is used as an example to investigate the performance of the developed numerical method, which provides accurate results with good convergence characteristics.

N an era of finite element methods and supercomputers, a modal method for static aeroelastic analysis1 must be regarded as an anachronism. The method is now only of historical interest25 and does not represent the current state of the art. The direct solution6 is preferred because it avoids questions of convergence (other than a sufficient number of elements in the finite element model), and avoids the unnecessary intermediate step of a vibration analysis. Certainly a modal solution using vibration modes is valid since the modes satisfy all of the requirements of the Galerkin method. However, the general view has been that convergence in a static problem can only be achieved with a large number of vibration modes, and, as Sheena and Karpel7 have shown, the series eventually does converge: somewhere between 25 to 35 modes were required to obtain a converged divergence speed in an example using a 190 degree-of-freedom model of a small aircraft. Reference 1 has not demonstrated that only four modes have achieved convergence at any of the five sweep angles analyzed. Another series solution was utilized in the planar subsonic kernel function lifting-surface aerodynamic theory. The kernel function method assumes a series of pressure loading functions and satisfies the downwash condition of tangential flow at a number of points on the surface, usually at as many points as there are pressure loading functions, unless a least squares solution8 is used. Neither the number of pressure functions/downwash control points nor the location of the points is discussed in Ref. 1. One would expect in an application to a very high aspect ratio wing that a large number of control points would be required in the spanwise direction, particularly in a vibration mode with many nodes along the span. Convergence of the generalized aerodynamic forces in, e.g., the third bending mode, has not been demonstrated. It cannot be concluded that the EAL9 and FAST10 computer programs have been validated by the correlation study of Ref. 1. Since the aspect ratios studied are so high, the results of Fig. 4 of Diederich arid Budiansky11 might have correlated with the experimental data just as well, and those results would have made an interesting addition to Fig. 3, par

An iterative approach combined with multi-dimensional fitting of limited measured stress points to reconstruct residual stress field generated by laser shock peening

In this chapter we use the Hybrid Displacement Boundary Element Method (HDBEM) to investigate several static and dynamic field problems. The method is based on variational principles, rather than on integral equations as the BEM from Part I and II, which is based on weighted residual statements. To avoid confusion, the weighted residual method will be referred to as Direct BEM for the remainder of the book.

A new form of integral constitutive equation for uniaxial stress conditions derived from the endochronic theory is shown to predict with computational ease cyclic plasticity of stable materials subjected to symmetric stress or strain loading. The detailed procedure to determine material constants is also given in this paper, based on the derived integral stress-strain relation. Close agreement between theoretical prediction and experiments for SUS 304 stainless steel and normalized mild steel shows the validity of the present constitutive modeling.