Viscoelastic Problem Research Articles

General Decay of a Nonlinear Viscoelastic Wave Equation with Balakrishnân-Taylor Damping and a Delay Involving Variable Exponents

This paper was aimed at investigating the stability of the following viscoelastic problem with Balakrishnân-Taylor damping and variable-exponent nonlinear time delay term u t t − M ∇ u 2 2 Δ u + α t ∫ 0 t g t − s Δ u s d s + μ 1 u t p . − 2 u t + μ 2 u t t − τ p . − 2 u t t − τ = 0 in Ω × ℝ + , where Ω is a bounded domain of ℝ n , p . : Ω ¯ ⟶ ℝ is a measurable function, g > 0 is a memory kernel that decays exponentially, α ≥ 0 is the potential, and M ∇ u 2 2 = a + b ∇ u t 2 2 + σ ∫ Ω ∇ u ∇ u t d x for some constants a > 0 , b ≥ 0 , and σ > 0 . Under some assumptions on the relaxation function, we use some suitable Lyapunov functionals to derive the general decay estimate for the energy. The problem considered is novel and meaningful because of the presence of the flutter panel equation and the spillover problem including memory and variable-exponent time delay control. Our result generalizes and improves previous conclusion in the literature.

An adaptive recursive SBFE algorithm for the statistical analysis of stochastic viscoelastic problems

Integrating the advantages of Taylor series expansion and TPAA -SBFEM (temporally adaptive recursive algorithm-scaled boundary finite element method), two efficient algorithms are proposed to solve stochastic viscoelastic problems, in which the constitutive parameters are presented by random variables or random fields. In addition to the reliable platform provided by TPAA-SBFEM to gain the solution of deterministic problem, a piecewise recursive adaptive algorithm is developed to obtain the derivatives required in the Taylor series expansion, which is able to provide a stable temporal solution accuracy for different step sizes. For the random variable model, SBFEM appears in BEM form with the unknowns only along the boundary of a super element; for the random field model which is approximated by Karhunen–Loève (K–L) expansion, SBFEM operates as FEM with its own advantages. Numerical tests are conducted to verify the accuracy and efficiency of the proposed approach, in comparison with Monte Carlo simulation, computational cost can significantly be reduced with sufficient accuracy for the stochastic viscoelastic problems with relatively small random dispersity.

Smoothed finite element approach for viscoelastic behaviors of general shell structures

This study focuses on the viscoelastic analysis of laminated composite shell structures under long-term creep mechanical loading. An advanced finite element technique named as cell/element-based smoothed discrete shear gap method (CS-DSG3) is employed to obtain the numerical solutions for both elastic and viscoelastic problems owing to its accuracy and rapid convergence. In the CS-DSG3 shell element, each triangular shell element is divided into three DSG3 subtriangles to avoid transverse shear locking. Subsequently, the total element strain is computed from the three partial strains of the subtriangles using a smoothing technique. To overcome the difficulty in the constitutive equations in integral forms for a viscoelastic material, all formulations are transformed into the Laplace domain using the convolution theorem. Finally, time-dependent deformations are obtained and converted back to the real-time domain using inverse Laplace techniques. For validation, various numerical examples of a pinched cylinder, clamped hyperbolic paraboloid, and hemispherical shell formed of isotropic elastic, isotropic viscoelastic, and anisotropic composite materials are selected to investigate creep behavior under mechanical loading. The present study extends the finite element simulation of anisotropic viscoelastic shell structures to achieve high accuracy and efficiency based on the Laplace transform and CS-DSG3.

RI-IGABEM for 3D viscoelastic problems with body force

A novel radial integration IGABEM (RI-IGABEM) is proposed to solve the 3D viscoelastic problems with body force in this paper. The derivation of boundary integral equations for the viscoelastic problems is proposed. To reduce computing costs, the time-dependent shear modulus is expressed in the form of Prony series, and the memory stress is calculated by using the genetic integral. Since the fundamental solution of elastostatics is applied to derive the boundary integral equations, which leads to the existence of domain integrals. To ensure that this algorithm only needs to discretize the boundary, we use radial integration method (RIM) to transform the domain integrals related to body force and memory stress into equivalent boundary integrals through applied points. Moreover, the strongly singular integral of the boundary integral equation is solved by the rigid-body displacement technique after a simple transformation between control points and collocation points, and the power series expansion method is used to solve the weakly singular integrals in RI-IGABEM. Furthermore, the 3D surface traction recovery method (TRM) of the viscoelastic mechanics is proposed to solve the strain and stress of the boundary points, and the regularized strain and stress integral equations are given in this work to solve the strain and stress of the internal points. Some 3D examples are applied to prove the effectiveness of the present method for the viscoelastic problems with body force.

Finite volume based asymptotic homogenization of viscoelastic unidirectional composites

The previously developed finite volume based asymptotic homogenization theory designed for unidirectional composites with finite-sized microstructures is further extended to non-ageing linear viscoelastic region via correspondence principle. The theory is a unified framework able to accommodate general loadings, both out-of-plane and in-plane. It first transforms the problems in time domain to Laplace domain via elastic–viscoelastic correspondence principle, solves the problems in transformed region, and transforms the solutions back to time domain through numerical inversion technique. In Laplace domain, the microscale problems are solved up to the third term of expansion, and the homogenized stiffness tensors of all orders are passed to the macroscale, leading to the solutions of macroscale displacements. The method’s capability is assessed through comparison with direct numerical solution, showing increasing accuracy with increasing truncation order and decreasing scale ratio, and further enhancement by reconstructing boundary layer. The robustness of theory is verified through parametric studies. The possibility of reducing the highest order of spatial derivative of homogenized displacement is also investigated. The finite volume based methodology marks the theory’s uniqueness among the existing higher order asymptotic homogenization methods, providing an efficient alternative to solve the viscoelastic composite structural problems.

Finite element approximation and analysis of a viscoelastic scalar wave equation with internal variable formulations

We consider linear scalar wave equations with a hereditary integral term of the kind used to model viscoelastic solids. The kernel in this Volterra integral is a sum of decaying exponentials (The so-called Maxwell, or Zener model) and this allows the introduction of one of two types of families of internal variables, each of which evolve according to an ordinary differential equation (ODE). There is one such ODE for each decaying exponential, and the introduction of these ODEs means that the Volterra integral can be removed from the governing equation. The two types of internal variable are distinguished by whether the unknown appears in the Volterra integral, or whether its time derivative appears; we call the resulting problems the displacement and velocity forms. We define fully discrete formulations for each of these forms by using continuous Galerkin finite element approximations in space and an implicit ‘Crank–Nicolson’ type of finite difference method in time. We prove stability and a priori bounds, and using the FEniCS environment, https://fenicsproject.org/ (The FEniCS project version 1.5, Archive of Numerical Software, 3 (100), 9–23, 2015.) give some numerical results. These bounds do not require Grönwall’s inequality and so can be regarded to be of high quality, allowing confidence in long time integration without an a priori exponential build up of error. As far as we are aware this is the first time that these two formulations have been described together with accompanying proofs of such high quality stability and error bounds. The extension of the results to vector-valued viscoelasticity problems is straightforward and summarised at the end. The numerical results are reproducible by acquiring the python sources from https://github.com/Yongseok7717, or by running a custom built docker container (instructions are given).

Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

A system of a first order history-dependent evolutionary variational– hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem for history dependent operators, results on the well-posedness of the system are proved. Existence, uniqueness, continuous dependence of the solution on the data, and the solution regularity are established. Two applications of dynamic problems from contact mechanics illustrate the abstract results. First application is a unilateral viscoplastic frictionless contact problem which leads to a hemivariational inequality for the velocity field, and the second one deals with a viscoelastic frictional contact problem which is described by a variational inequality.

Numerical analysis of fractional viscoelastic fluid problem solved by finite difference scheme

The finite difference schemes for equations depicting the fluid flow, heat and mass transfer of viscoelastic fluid in porous media based on fractional constitutive model are analyzed, when Soret and Dufour effects are taken into account. Different from the conventional fractional partial differential equations, the governing momentum equation of this system not only contains convection terms, but also a term: Dtβ0(∂2u/∂y2), which are great challenges when we do the stability analysis. In addition, the temperature equation and concentration equation are coupled with each other, which increases the difficulty of numerical analysis. By using the energy method, the stability and convergence of the finite difference schemes employed for the governing equations are obtained. The accuracy of the scheme for momentum equation is O(τ+hx2+hy2), while the one for temperature and concentration equations is O(τ2+hx2+hy). Finally, numerical examples are presented to verify methods and demonstrate the analysis effectiveness.

Viscoelastic and Viscoplastic Glucose Theory (VGT #5): Three Developed Postprandial Plasma Glucose (PPG) Stress-Strain Diagrams for the PPG Levels of <140 mg/dL, 140- 170 mg/dL, and >170 mg/dL, to Illustrate the Correlation between Biomedical Diabetes Conditions and Biophysical Illustrations of Glucose Behaviors Using Linear Elasticity Theory (Time-Independent), Plasticity Theory (Time-Independent), and Viscoplasticity Theory (Time-Dependent) Based on GH-Method: Math-Physical Medicine (No.

The author has studied strength of materials and theory of elasticity through his undergraduate courses at the University of Iowa. He also conducted research work to earn a master’s degree in Biomechanics under Professor James Andrews. At that time, he used the spring and dashpot models to simulate the behaviors of human joints, bones, muscles, and tendons in order to investigate the human-weapon interactions during the Vietnam war era. Later, he went to MIT to pursue his PhD study under Professor Norman Jones, who taught him theory of plasticity and dynamic plastic behaviors of various structure elements. He also took additional graduate courses in the field of fluid dynamics and thermodynamics. Since then, many advancements have been made in the biomechanics branch, especially with human body tissues that possess certain viscoelastic characteristics, such as bones, muscles, cartilages, tendons (connect bone to muscle), ligaments (connect bone to bone), fascia, and skin. For example, the author suffered plantar fasciitis for many years. He understood that the night splint dorsiflexes forefoot, at the back of the foot, increases plantar fascia tension to offer stress-relaxation for the pain. This model of muscles and tendons connecting the lower leg and foot is a form of viscoelastic study and problem solving. However, when dealing with human internal organs, it is not easy to conduct live experiments to obtain accurate measurements for the biomedical material properties. Although blood itself is a viscous (time-dependent) material, its viscosity factor may fall between water, honey, syrup, or gel. The author’s research subject is “glucose” where the carbohydrates and sugar amount is produced by the liver and is carried by red blood cells, not the blood itself. It is nearly impossible to measure material geometry or certain engineering properties of glucose, for example, to determine the viscosity of “glucose”. Therefore, the best he could do is to apply the “concept of viscoelasticity and/or viscoplasticity” to construct an analogy model of time-dependent glucose behaviors.

Viscoelastic and Viscoplastic Glucose Theory (VGT #2): Applying the Concept of Viscoelasticity and Viscoplasticity to Conduct a “Time-Dependent Glucose Analogy” Study of An Elastic Glucose Case Covering 4,056 meals <180 mg/ dL Versus Plastic Glucose Case of 23 Meals >180 mg/dL Using Collected Postprandial Plasma Glucose Data from 4,079 Meals over 3.7-Year Period from 5/8/2018 to 1/10/2022 Based on GH-Method:

The author has studied strength of materials and theory of elasticity through his undergraduate courses at the University of Iowa. He also conducted research work to earn a master’s degree in Biomechanics under Professor James Andrews. He remembers using the spring and dashpot models to simulate the behaviors of human joints, bones, muscles, and tendons to investigate the human-weapon interactions. Later, he went to MIT to pursue his PhD study under Professor Norman Jones, who taught him theory of plasticity and dynamic plastic behaviors of structure elements, and took additional graduate courses in the field of fluid dynamics and thermodynamics. Since then, many advancements have been made in biomechanics in a few application areas, especially tissues of the human body which possess viscoelastic characteristics, such as bone, muscle, cartilage, tendon (connect bone to muscle), ligament (connect bone to bone), fascia, and skin. For example, the night splint dorsiflexes forefoot at the back of the foot increases plantar fascia tension to offer stress-relaxation for plantar fascia pain. This model of muscles and tendons connecting the lower leg and foot is a type of viscoelastic problem. However, when dealing with the human internal organs, it is not easy to conduct live experiments to obtain accurate measurements of material properties. Although blood itself is a viscous material, the viscosity factor may fall between water, honey, syrup, or gel. However, the author’s research subject is “glucose”, the sugar amount in blood carried by cells, not the blood itself. It is nearly impossible to measure material geometry or certain engineering properties of glucose, for example, the viscosity of “glucose”. Therefore, the best he could do is to apply the “concept of viscoelasticity and/or viscoplasticity” to construct an analogy model of time-dependent glucose behaviors. The author’s background includes mathematics, physics, and various engineering disciplines, not including biology and chemistry. As a result, he can only investigate the observed biomedical phenomena using his ready-learned math-physical tools. He has already conducted some investigations of glucose behaviors using elasticity theory and plasticity theory and written a few articles from his findings. In the elasticity and plasticity papers, he utilized the postprandial plasma glucose (PPG) value as the strain along with carbs/sugar amount and post-meal exercise level as the stress. In a recent email from Professor Norman Jones, he said that: “I have wondered if the use of viscoelastic/viscoplastic materials might be of some value to your studies. These phenomena embrace time-dependent behaviour and I know that you have emphasized the time-dependence of various behaviours in the body. Just a thought.” His suggestion triggered the author’s interest and desire to research the subject of glucose behaviors further using viscosity theory. This particular article is a follow-up to his paper No. 578 which studies certain generic characteristics of viscoelastic glucose behaviors. In this paper, he uses a relative glucose level (individual PPG - average PPG) as the strain and the strain rate (dε/dt) multiplied with

Viscoelastic Glucose Theory (VGT #3): Similarity and Difference of Two StressStrain Diagrams for Viscoelastic Glucose Behaviors from Continuous Glucose Monitor Sensor Measured and Perturbation Theory Produced Postprandial Plasma Glucoses with “Time-Dependent” Characteristics of a Synthesized Postprandial Plasma Glucose Waveform Generated from 4,115 Meals over a 3.7-Year Period from 5/8/2018 to 1/18/2022 based on GH-Method: MathPhysical Medicine (No. 580)

The author has studied strength of materials and theory of elasticity through his undergraduate courses at the University of Iowa. He also conducted research work to earn a master’s degree in Biomechanics under Professor James Andrews. He remembers using the spring and dashpot models to simulate the behaviors of human joints, bones, muscles, and tendons in order to investigate the human-weapon interactions. Later, he went to MIT to pursue his PhD study under Professor Norman Jones, who taught him theory of plasticity and dynamic plastic behaviors of various structure elements. He also took additional graduate courses in the field of fluid dynamics and thermodynamics. Since then, many advancements have been made in a few application areas of biomechanics, especially tissues in the human body which possess viscoelastic characteristics, such as bones, muscles, cartilages, tendons (connect bone to muscle), ligaments (connect bone to bone), fascia, and skin. For example, the author suffered plantar fasciitis for many years. He understood that the night splint dorsiflexes forefoot, at the back of the foot, increases plantar fascia tension to offer stress-relaxation for the pain. This model of muscles and tendons connecting the lower leg and foot is a form of viscoelastic problem. However, when dealing with the human internal organs, it is not easy to conduct live experiments to obtain accurate measurements for the biomedical material properties. Although blood itself is a viscous (time-dependent) material, the viscosity factor may fall between water, honey, syrup, or gel. However, the author’s research subject is “glucose”, the carbohydrates and sugar amount in blood produced by the liver and carried by red blood cells, not the blood itself. It is nearly impossible to measure material geometry or certain engineering properties of glucose, for example, to determine the viscosity of “glucose”. Therefore, the best he could do is to apply the “concept of viscoelasticity and/or viscoplasticity” to construct an analogy model of time-dependent glucose behaviors.

Viscoelastic Glucose Theory (VGT #1): Applying the Concept of Viscoelasticity Theory to Conduct a “Glucose Analogy” Study and Illustrate Certain Viscoelastic Characteristics of Time-Dependent Glucose Using Continuous Glucose Monitoring (CGM) Sensor Device Collected Postprandial Plasma Glucose (PPG) Data of 4,056 Elastic Glucoses (<180 mg/dL) within 3.7 Years from 5/8/2018 to 1/10/2022 Based on GH-Method: Math-Physical Medicine (No. 578)

The author has studied the strength of materials and theory of elasticity from undergraduate courses at the University of Iowa. He also conducted research work to earn a master’s degree in Biomechanics under Professor James Andrews. He remembers that he used both spring and dashpot models to simulate the behaviors of human bone, muscle, and tendon to investigate the human-weapon interactions. Later on, he went to MIT to pursue his Ph.D. study under Professor Norman Jones who taught him the theory of plasticity and dynamic plastic behaviors of various structural elements. Furthermore, he took some graduate courses at MIT in the field of fluid dynamics and thermodynamics. Since then, biomechanics has made advancements in a few application areas, especially tissues of the human body which possess viscoelastic characteristics, such as bone, muscle, cartilage, tendon (connect bone to muscle), ligament (connect bone to bone), fascia and skin. For example, the night splint dorsiflexes forefoot on rear foot increasing plantar fascia tension to offer stress-relaxation of plantar fascia pain. This model of muscles and tendons connecting lower-leg and foot is a kind of viscoelastic problem. However, when we deal with human internal organs, it is not easy to conduct live experiments to obtain some accurate measurements of material properties. Although blood itself is a viscous material which viscosity factor may sit between water and honey, syrup, or gel. But, the author’s research subject is “glucose”, the sugar amount inside of blood or carried by blood cells, not the blood itself. It is near impossible to measure the material geometry or engineering properties of glucose, for example, the viscosity of “glucose”. Therefore, the best he could do is to apply the concept of viscoelasticity and viscoplasticity to construct an analogy motor to study the glucose behaviors which are time-dependent. The author’s background covers mathematics, physics, and various engineering disciplines, not including biology and chemistry. As a result, he can only investigate the observed biomedical phenomena using his ready-learned math-physical tools.

Viscoelastic and Viscoplastic Glucose Theory (VGT #4): Three Developed Postprandial Plasma Glucose Stress-Strain Diagrams for Normal, Pre-Diabetes, and Severe Diabetes Levels to Illustrate the Correlation Between Biomedical Conditions of the Disease and Biophysics Behaviors of Glucose Based on Linear Elasticity Theory (TimeIndependent), Plasticity Theory (Time-Independent), and Viscoplasticity Theory (Time-Dependent) Utilizing GH-Method: Math-Physical Medicine (No. 581)

The author has studied strength of materials and theory of elasticity through his undergraduate courses at the University of Iowa. He also conducted research work to earn a master’s degree in Biomechanics under Professor James Andrews. At that time, he used the spring and dashpot models to simulate the behaviors of human joints, bones, muscles, and tendons in order to investigate the human-weapon interactions during the Vietnam war era. Later, he went to MIT to pursue his PhD study under Professor Norman Jones, who taught him theory of plasticity and dynamic plastic behaviors of various structure elements. He also took additional graduate courses in the field of fluid dynamics and thermodynamics. Since then, many advancements have been made in biomechanics, especially with human body tissues that possess viscoelastic characteristics, such as bones, muscles, cartilages, tendons (connect bone to muscle), ligaments (connect bone to bone), fascia, and skin. For example, the author suffered plantar fasciitis for many years. He understood that the night splint dorsiflexes forefoot, at the back of the foot, increases plantar fascia tension to offer stress-relaxation for the pain. This model of muscles and tendons connecting the lower leg and foot is a form of viscoelastic problem. However, when dealing with human internal organs, it is not easy to conduct live experiments to obtain accurate measurements for the biomedical material properties. Although blood itself is a viscous (time-dependent) material, its viscosity factor may fall between water, honey, syrup, or gel. The author’s research subject is “glucose” where the carbohydrates and sugar amount is produced by the liver and is carried by red blood cells, not the blood itself. It is nearly impossible to measure material geometry or certain engineering properties of glucose, for example, to determine the viscosity of “glucose”. Therefore, the best he could do is to apply the “concept of viscoelasticity and/or viscoplasticity” to construct an analogy model of time-dependent glucose behaviors.

An Extended Reynolds Equation for Non-Newtonian Lubrication With the Upper Convected Maxwell Model

Abstract The upper convected Maxwell (UCM) model is one of the ways to describe lubricant’s viscoelasticity. In this paper, a new modified Reynolds equation for the UCM model is derived. This equation may be called extended as compared to its analogs because it incorporates the complete structure of the upper convected derivative without the explicit omitting of any term or using the perturbation technique. Then, a numerical scheme for solving the viscoelastic lubrication problems with the employment of the derived equation is described. A mixed Euler–Lagrange approach is used here: the constitutive rheological equations are resolved by a semi-Lagrangian technique and the extended Reynolds equation is discretized by the finite volume method. A constant surface slope problem is considered as a test case. Excellent agreement is achieved between the numerical solution at low Deborah number and one of the approximate solutions. The results of simulations with different types of time derivative used in Maxwell model both for two- and three-dimensional cases are also discussed.

Dynamic viscoelastic unilateral constrained contact problems with thermal effects

A new model that describes a dynamic frictional contact between a viscoelastic body and an obstacle is investigated in this paper. We consider a nonlinear viscoelastic constitutive law which involves a convex subdifferential inclusion term and thermal effects. The contact condition is modeled with unilateral constraint condition for a version of normal velocity. The boundary conditions that describe the contact, friction and heat flux are govern by the generalized Clarke multivalued subdifferential. We derive a coupled system of two nonlinear first order evolution inclusions problems, which consists of a parabolic variational-hemivariational inequality for the displacement and a hemivariational inequality of parabolic type for the temperature. Then, the unique weak solvability of the contact problem is obtained by virtue of a fixed point theorem and the surjectivity result of multivalued maps. Finally, we deliver a continuous dependence result on a coupled system when the data are subjected to perturbations.

Comparing Different Stabilization Strategies for Reduced Order Modeling of Viscoelastic Fluid Flow Problems

Comparing Different Stabilization Strategies for Reduced Order Modeling of Viscoelastic Fluid Flow Problems

Transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids

Consider the transient thermal stress analysis for a material that is anisotropic (direction-dependent), viscoelastic (time-dependent), and thermorheologically simple (temperature-dependent). To solve this problem, three fundamental theories are employed: (1) complex variable Stroh formalism for anisotropic elasticity, (2) elastic-viscoelastic correspondence principle (EVCP) or time-stepping method (TSM) for viscoelasticity, and (3) time-temperature superposition principle (TTSP) for thermorheologically simple materials. By using the Stroh formalism, all different kinds of anisotropic elastic materials can be analyzed by using the same formulae. With EVCP and TSM, the problem of viscoelasticity can be solved via its corresponding problem of elasticity. By TTSP, the temperature-dependent constitutive relation can be expressed by using the reference temperature and the reduced time defined through shift factor. Thus, through the combined use of these theories, the complicated problems of transient anisotropic thermo-viscoelasticity can be solved by using the available solution tool for anisotropic elasticity such as the boundary element method (BEM). Based upon this concept, two novel BEMs are developed for the transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids. One is BEM-EVCP, and the other is BEM-TSM. The final results show that both of them can provide accurate solutions, and BEM-TSM is much more efficient than BEM-EVCP.

Modeling fracture in viscoelastic materials using a modified incremental meshfree RPIM and DIC technique

In the present work, fracture mechanics of linear viscoelastic materials is investigated numerically and experimentally. Herein, an effective and accurate meshfree method is developed to analyze fracture mechanics in viscoelastic problems. To this end, an incremental approach with higher-order finite difference (FD) schemes for analysis of viscoelastic materials in the time domain, along with an improved meshfree method based on the global weak formulation, i.e. the radial point interpolation method (RPIM), is employed. An accurate and efficient integration approach with minimum computational cost, i.e. the background decomposition method (BDM), is also used to compute the domain integrals. To evaluate the accuracy and efficiency of the proposed numerical method, first, two example problems are provided and the robustness of the presented method is assessed. Then, a non-contact full-field optical method, i.e. the digital image correlation technique, is utilized to investigate the viscoelastic fields of a polymer, polymethyl-methacrylate (PMMA), during crack propagation in mode I. Also, in order to properly model the fracture behavior of this material by the meshfree RPIM, the mechanical properties of the PMMA, including the Prony viscoelastic parameters, are obtained by an ASTM test method. Moreover, the obtained results from the DIC test and also the load-displacement curve of the cracked specimen are compared with the results of the proposed meshfree RPIM and a very close agreement is observed.

Uniqueness and continuous dependence for a viscoelastic problem with memory in domains with time dependent cracks

We study some hyperbolic partial integro-differential systems in domains with time dependent cracks. In particular, we give conditions on the cracks which imply the uniqueness of the solution with prescribed initial-boundary conditions, and its continuous dependence on the cracks.

Cattaneo–Friedrich and Crank–Nicolson analysis of upper-convected Maxwell fluid along a vertical plate

The concept of fractional derivative is used to solve a variety of viscoelastic fluid problems. However, researchers mostly overlooked the consequences of nonlinear convection in the fractional viscoelastic fluid models and were concerned only with situations where the governing equations are linear. Most importantly, the nonlinear fluid models, whether classical or fractional, are solved for steady-state conditions. To overcome these limitations, this research presents unsteady flow and heat transfer of nonlinear fractional upper-convected Maxwell (UCM) viscoelastic fluid along a vertical plate. The governing equations of the fractional Maxwell fluid are developed by introducing Friedrich shear stress and Cattaneo heat flux models to the classical UCM fluid model. An additional feature to the invention of the constructed fractional model is the consequence of an external magnetic field. Moreover, the considered model comprises nonlinear, coupled, fractional partial differential equations. Therefore, a numerical scheme is developed with the aid of the L1-approximation of Caputo derivative and the Crank–Nicolson method. The effects of different regulating parameters on fluid features have been thoroughly investigated. The obtained results are exhibited graphically and discussed in detail. It is observed that the skin friction increases for the velocity relaxation time parameter, but an opposite behavior is observed against the velocity fractional derivative parameter. Moreover, a significant enhancement is noticed in the Nusselt number for increasing estimates of the Prandtl number.