Problems Of Linear Viscoelasticity Research Articles

Existence of weak solutions to a Cahn–Hilliard–Biot system

We prove existence of weak solutions to a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot’s equations for poroelasticity, including phase-field dependent material properties, with the Cahn–Hilliard equation to model the evolution of the solid, and is further augmented by a visco-elastic regularization of Kelvin–Voigt type. To obtain this result, we approximate the problem in two steps, where first a semi-Galerkin ansatz is employed to show existence of weak solutions to regularized systems, for which later on compactness arguments allow limit passage. Notably, we also establish a maximal regularity theory for linear visco-elastic problems.

Generalization of the variable separation method for solving boundary value problems of linear viscoelasticity of kinds I and III

Generalization of the variable separation method for solving boundary value problems of linear viscoelasticity of kinds I and III

NON-STATIONARY WAVES IN A FUNCTIONALLY GRADED VISCOELASTIC PLANE-PARALLEL LAYER2

The propagation of nonstationary disturbances in a plane-parallel layer of infinite extent, which arise as a result of normal dynamic loading of one of its surfaces when the other surface is stationary, is investigated. The material of the layer is assumed to be functionally graded and, at the same time, viscoelastic. The hereditary properties of such a material are taken into account using linear integral Boltzmann–Volterra relations with regular kernels in the form of partial sums of the Prony's series. The peculiarity of this work is that here the parameters of the relaxation kernels are considered continuous functions of the transverse coordinate in the same way as other physical and mechanical characteristics. A method was used for the study, which consists in replacing a functionally graded material with an approximating layered homogeneous structure with continuity conditions at the contact of homogeneous layers. The solution of the nonstationary dynamic linear viscoelasticity problem for a package of plane homogeneous viscoelastic layers is presented in a special form, which greatly simplifies its numerical implementation, especially with a large number of layers with different hereditary properties. This made it possible to successfully apply this method and carry out a series of calculations using an efficient algorithm. Transient wave processes are investigated in the case when the parameters of a functionally graded material are non-monotonic functions of the transverse coordinate, symmetrical with respect to the middle surface of the layer. A comparison of transient wave processes for different types of these functions is carried out. The convergence of the calculation results with an increase in the number of approximating homogeneous layers with a continuous dependence of the external load on time is confirmed. The significant influence of both heterogeneity and viscosity of the material on nonstationary wave processes has been established.

An efficient displacement-based isogeometric formulation for geometrically exact viscoelastic beams

We propose a novel approach to the linear viscoelastic problem of shear-deformable geometrically exact beams. The generalized Maxwell model for one-dimensional solids is here efficiently extended to the case of arbitrarily curved beams undergoing finite displacement and rotations. High efficiency is achieved by combining a series of distinguishing features, that are: (i) the formulation is displacement-based, therefore no additional unknowns, other than incremental displacements and rotations, are needed for the internal variables associated with the rate-dependent material; (ii) the governing equations are discretized in space using the isogeometric collocation method, meaning that elements integration is totally bypassed; (iii) finite rotations are updated using the incremental rotation vector, leading to two main benefits: minimum number of rotation unknowns (the three components of the incremental rotation vector) and no singularity problems; (iv) the same SO(3)-consistent linearization of the governing equations and update procedures as for non-rate-dependent linear elastic material can be used; (v) a standard second-order accurate time integration scheme is made consistent with the underlying geometric structure of the kinematic problem. Moreover, taking full advantage of the isogeometric analysis features, the formulation permits accurately representing beams and beam structures with highly complex initial shape and topology, paving the way for a large number of potential applications in the field of architectured materials, meta-materials, morphing/programmable objects, topological optimizations, etc. Numerical applications are finally presented in order to demonstrate attributes and potentialities of the proposed formulation.

A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model

Abstract The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a fading memory kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a Dirichlet–Prony series of decaying exponentials to provide the fading memory in the Volterra kernel. We introduce two types of internal variable to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and – in time – a Crank–Nicolson method to formulate the fully discrete problems: one for each type of internal variable. We present a priori stability and error analyses without using Grönwall’s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially. In this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems. We also carry out a number of numerical experiments using the FEniCS environment (https://fenicsproject.org), describe a simulation using “real” material data, and explain how the codes can be obtained and all of the results reproduced.

A Mixed Discontinuous Galerkin Method for a Linear Viscoelasticity Problem With Strongly Imposed Symmetry

A Mixed Discontinuous Galerkin Method for a Linear Viscoelasticity Problem With Strongly Imposed Symmetry

A time-incremental homogenization method for elasto-viscoplastic particulate composites based on a modified secant formulation

Starting from the exact solution of the heterogeneous linear viscoelastic Eshelby ellipsoidal inclusion problem obtained in the time domain for an ellipsoidal inclusion embedded in an isotropic matrix (Berbenni et al., 2015), a direct time-incremental homogenization Mori–Tanaka scheme for two-phase non linear elasto-viscoplastic materials is proposed. The extension to non-linear behavior is based on a “modified secant” formulation to obtain the linear comparison composite (LCC) properties. In contrast with more classic homogenization method based on the first moment of stresses (i.e. classic secant formulation), the present approach is based on the second-order moment of stresses making use of the Hill’s lemma. Hence, a modified secant viscoplastic compliance is considered for the matrix phase. The effective behavior as well as the evolution laws of the averaged strains and stresses per phase are directly solved in the time domain, without the need of Laplace-Carson transforms. The estimates provided by the present homogenization model are compared to full-field calculations from the literature for two-phase non linear particulate composites constituted of an isotropic elasto-viscoplastic matrix phase and isotropic elastic particles, and, for radial and non-radial loadings.

Modeling Issues in Problems of the Elasticity and Viscoelasticity Theory

The development of computer technology and information technology does not exclude preliminary analytical modeling in complex problems of mechanics. The approach proposed in this work allows one to obtain reasonable simplified equations that admit analytical or numerical-analytical solutions. Authors research are devoted to method elaboration for space problems of viscoelasticity. The approach for solving problems of nonlinear elasticity theory, when final deformations or physical nonlinear properties of materials are taken into consideration, is proposed. New perturbation method for solving of nonlinear equations in particular derivatives is suggested. Such approach is allowed to reduce the solution of complicated problems of linear elasticity and viscoelasticity to subsequently solved boundary problems of potential theory. New linear problems are investigated, in particular, problems on load transference from stringer to single layer or multilayer solids, on stress-strain state of fibrous composite with crack in matrix (plate and axisymmetric problems), numerous contact problems (pressing of hard stamp in an orthotropic plate with different anisotropy). Problems of nonlinear elasticity theory on stress-strain state of a plate with a circular hole under different types of loading are solved.

Application of an Innovate Energy Balance to Investigate Viscoelastic Problems

Modeling and investigating of energy distribution especially the wasted one is very important in viscoelastic problems. In this article, an applied energy model based on separation of energy components of the system is extracted and expanded to apply in linear viscoelastic problems, although this method is applicable in nonlinear problems as well. It is assumed that the whole energy of the system can be divided into two parts: Residual and non-inertial energies. The non-inertial energy is the sum of the energies that do not depend on the inertia of the system, while residual energy is the remaining of total energy. When an amount of energy is applied to the system, by determining the non-inertial energy from a novel energy conservation equation, the residual energy can be calculated. Some basic viscoelastic examples are investigated and obtained results will be compared with the expected ones.

A symplectic analytical singular element for V-notched analyses in elastic and viscoelastic plane problems

The linear elastic and viscoelastic problems with V-shaped notches under in-plane loading are investigated in this paper. We consider two different sets of boundary conditions on the radial edges: Free-Free case and Clamped-Clamped case. A novel singular finite element (SASE) with arbitrary high-order accuracy for the notched problems is proposed. Firstly, via Laplace transform, the original viscoelastic problem is transformed into a corresponding elastic one. Then we construct the SASE by using elastic symplectic eigen solutions with higher order expanding terms. The SASE can depict the characteristics of displacement fields and singular stress fields in the vicinity of the notch vertex having arbitrary opening angle. By taking advantage of the symplectic eigen solutions in the SASE, Mode I and/or Mode II notch stress intensity factors can be determined directly without any post-processing. Fine finite element meshes are not required. Numerical examples are provided to illustrate the validity and accuracy of the present method.

An Enhanced Finite Element method for Two Dimensional Linear Viscoelasticity using Complex Fourier Elements

In this paper, the finite element analysis of two-dimensional linear viscoelastic problems is performed using quadrilateral complex Fourier elements and, the results are compared with those obtained by quadrilateral classic Lagrange elements. Complex Fourier shape functions contain a shape parameter which is a constant unknown parameter adopted to enhance approximation’s accuracy. Since the iso-parametric formulation utilized in the finite element code, based on the experience of authors, it is proposed that a suitable shape parameter for each problem is adopted based on an acceptable approximation of the problem’s geometry by a complex Fourier element. Several numerical examples solved, and the results showed that the finite element solutions using complex Fourier elements have excellent agreement with analytical solutions, even though noticeable fewer elements than classic Lagrange elements are employed. Furthermore, the run-times of the executions of the developed finite element code to obtain accurate results, in the same personal computer, using classic Lagrange and complex Fourier elements compared. Run-times indicate that in the finite element analysis of viscoelastic problems, complex Fourier elements reduce computational cost efficiently in comparison to their classic counterpart.

Analytical Solution of Non-stationary Waves Propagation in Viscoelastic Layer Problem

Analytical solution of one-dimensional non-stationary waves propagation in linear viscoelastic layer problem using different methods is obtained. For hereditary layer material properties description physical relation of linear viscoelasticity in integral form is used and it is supposed that relaxation kernel has exponential form. Layer displacement calculation results for concrete initial data are represented.

A 3D‐2D asymptotic analysis of viscoelastic problem with nonlinear dissipative and source terms

The purpose of this article is to study the asymptotic analysis of the solutions of a linear viscoelastic problem with a dissipative and source terms in a three‐dimensional thin domain Ωε. Firstly, we give the strong formulation of the problem and the existence and uniqueness theorem of the weak solution. Then, we establish some estimates independent of the parameter ε. These last will be useful to obtain the limit problem with a specific weak form of the Reynolds equation.

A second-order multiscale approach for viscoelastic analysis of statistically inhomogeneous materials

An effective second-order multiscale approach is developed in this work to discuss viscoelastic properties of statistically inhomogeneous materials. In these materials, the sophisticated microscale structure of inclusions, including related shape, orientation, size, spatial distribution, volume fraction and so on, results in varying of the macroscale properties. At first, the Laplace transform is applied to the linear viscoelastic problems, and expected relaxation modulus in Laplace domain for the materials is given. Also, the second-order multiscale formulas for evaluating the viscoelastic problems of statistically inhomogeneous materials are derived. Next, the stochastic multiscale algorithm is proposed, and the expected relaxation moduli in time domain are obtained by the least-square and inverse Laplace transform. The salient features of the proposed approach are the asymptotic high-order homogenizations that do not require high-order continuity of the coarse-scale (or macroscale) solutions and can handle the random materials with complicated microstructure at a fraction of computational cost. Finally, some examples for the composites are computed by the effective algorithm, and compared with the data by the theoretical models and experimental results. The comparison illustrates that the stochastic multiscale model is efficient for determining the viscoelastic properties of the materials and shows their potential application in practical engineering calculation.

Solutions of Non-stationary Dynamic Problems of Linear Viscoelasticity

The problems of non-stationary waves propagation in linear viscoelastic bodies are considered. The issues related to the construction of the solutions of such problems by method of integral Laplace transform in time are discussed. The statements about the properties of the solution in the space of Laplace transforms, which simplify the process of finding the originals, are formulated. As an example, the solution of the problem of propagation of one-dimensional shear waves in a viscoelastic cylinder is obtained.

A three-scale asymptotic analysis for ageing linear viscoelastic problems of composites with multiple configurations

• A novel three-scale asymptotic analysis is proposed. • Composite materials with multiple periodic configurations are considered. • Ageing linear viscoelastic problems of composites are investigated. • Numerical results are given in detail to validate the proposed method. A novel three-scale asymptotic expansion is proposed in this work to investigate ageing linear viscoelastic problems of composites with multiple periodic configurations. Here, the composite structures are established by periodical distribution of local cells on microscale and mesoscale. The new three-scale asymptotic expansion formulas based on classical homogenized methods in time domain are constructed at first, and the microscale and mesoscale functions are also derived in detail. Further, two distinct homogenized parameters defined on microscale and mesoscale domains are obtained by upscaling methods, and the newly developed homogenized equations are given on whole structure in time domain. Then, the three-scale strain and stress solutions are constructed by assembling the unit cell solutions and homogenized solutions. Also, the efficient finite element algorithms based on the three-scale asymptotic expansion and homogenized method are brought forward. Finally, some representative numerical examples are evaluated to verify the proposed methods. They show that the three-scale asymptotic expansions developed in this work are effective and valid for predicting the ageing linear viscoelastic properties of the composite materials with multiple configurations.

Нестационарные процессы в подкрепленном тонкостенной оболочкой вязкоупругом цилиндре при импульсном нагружении

Solid propellant rocket motor is considered as hollow viscoelastic cylinder inserted in multilayered elastic shell-like case. The material of propellant is considered to be compressible. An estimation of maximum unsteady stresses on cylinder-shell boundary and shell under growing pressure on interior or external cylindrical surface were calculated by FEM. Four corner isoparametric finite element is utilized. Numark method to integrate by time the dynamic equations is used. The problem of linear viscoelasticity have been employing of the Schapery method. `In the case of internal pressure, the possibility of tensile radial stresses on the contact surface of the propellant-shell during the transition process has been established. The dependence of the maximum contact stresses as well as circumferential stresses in the shell on the shell thickness is established. In the case of external pressure pulse, the presence of significant tensile radial stresses on the propellant-shell interface is shown. Insignificant tensile circumferential stresses in the transient wave process are possible in the shell.

Comparison of the innovative method for linear viscoelasticity problems with standard ansys tools using simple tasks

A description of the original method for solving problems of linear viscoelasticity, based on the separation of spatial and temporal variables and the reduction of this problem to an elastic problem, is presented in the paper. We compared the solutions given by this method, as well as those obtained using standard ANSYS tools, with known analytical solutions. The results show the effectiveness of the new method.

Residual‐type a posteriori estimator for a viscoelastic contact problem with velocity constraints

AbstractWe consider a linear viscoelastic contact problem in which the non‐penetration condition is enforced by constraints in the velocities. We present a residual‐type a posteriori estimator and sketch how upper and lower bounds are derived based on the works [1,4].

Solving the Problem of Linear Viscoelasticity for Piecewise-Homogeneous Anisotropic Plates

An approximate method for solving the problem of linear viscoelasticity for thin anisotropic plates subject to transverse bending is proposed. The method of small parameter is used to reduce the problem to a sequence of boundary problems of applied theory of bending of plates solved using complex potentials. The general form of complex potentials in approximations and the boundary conditions for determining them are obtained. Problems for a plate with elliptic elastic inclusions are solved as an example. The numerical results for a plate with one, two elliptical (circular), and linear inclusions are analyzed.