Viscoelastic finite-element analysis

Botha, Jones, and Brinson,Henriksen,(22) Becker et a/.,(23) and Yadagiri and Papi Reddy(24,25) reported results of viscoelastic finite-element analysis of adhesive joints. Henriksen used Schapery s(26) nonlinear viscoelastic model to verify the experimental results of Peretz and Weitsman(27) for an adhesive layer. The work of Becker et is largely... 

The present section deals with the review and extension of Schapery s single integral constitutive law to two dimensions. First, a stress operator that defines uniaxial strain as a function of current and past stress is developed. Extension to multiaxial stress state is accomplished by incorporating Poisson s effects, resulting in a constitutive matrix that consists of instantaneous compliance, Poisson s ratio, and a vector of hereditary strains. The constitutive equations thus obtained are suitable for nonlinear viscoelastic finite-element analysis. 

Next, a model joint (or thick adherend specimen) problem presented in Reference 49 is analyzed using the present program, NOVA. In this case, a linear viscoelastic finite-element analysis was carried out on the model joint under a constant applied load of 4448 N giving an average adhesive shear stress of 13.79 MPa. The specimen geometry, descretization, and boundary conditions are shown in Figure 5. The thickness of the adhesive layer is taken to be 0.254 mm. A nine-parameter solid model was used to represent the tensile creep compliance of FM-73 at 72 °C and is given by... 

S. Shrivastava and J. Tang, Large Deformation Finite Element Analysis of Non-linear Viscoelastic Membranes With Reference to Thermoforming, J. Strain Anal. 28, 31 (1993). 

Load and support conditions for individual components depend on the complete structure (or system) analysis, and are unknown to be determined in that analysis. For example, if a plastic panel is mounted into a much more rigid structure, then its support conditions can be specified with acceptable accuracy. However, if the surrounding structure has comparable flexibility to the panel, then the interface conditions will depend on the flexural analysis of the complete structure. In a more localized context, structural stiffness may be achieved by ribbing and relevant analyses may be carried out using available design formulae (usually for elastic behavior) or finite element analysis, but necessary anisotropy or viscoelasticity complicate the analysis, often beyond the ability of the design analyst. 

Liu K, Ovaert TC (2011) Poro-viscoelastic constitutive modeling of unconfined creep of hydrogels using finite element analysis with integrated optimization method. J Mech Behav Biomed Mater 4 440-450... 

Chen CJ, Kwak BM, Rim K, Falsetti HL (1980) A model for an active left ventricle deformation -formulation of a nonlinear quasi-steady finite-element analysis for orthotropic, three-dimensional myocardium. Int Conf Finite Elements in Biomechanics 2 639-655 Feit TS (1979) Diastolic pressure-volume relations and distribution of pressure and fiber extension across the wall of a model left ventricle. Biophys J 28 143-166 Ghista ND, Sandler HD (1968) An elastic viscoelastic model for the shape and the forces in the left ventricle. J Biomechanics 2 35-47... 

Pao YC, Robb RA, Ritman EL (1976) Plane-strain finite-element analysis of reconstructed diastolic left ventricular cross section. Ann Biomed Eng 4 232-249 Pao YC, Ritman EL (1977) Viscoelastic, fibrous, finite-element, dynamic analysis of beating heart. Proc Symp Appl Comp Meth, 477-486... 

Only recently, work involving the time-dependent fracture characteristics of adhesively bonded joints has been under way. Francis et discussed the effects of a viscoelastic adhesive layer, geometry, mixed-mode fracture response, mechanical load history, environmental history, and processing variations on the fracture processes of adhesively bonded joints. However, their finite-element analysis includes only linear elastic fracture mechanics. 

In a more localized context, structural stifBiess may be achieved by ribbing, and relevant analyses may be carried out using available design formulae (usually for elastic behavior) or finite element analysis. But necessary anisotropy or viscoelasticity complicate the analysis, often beyond the ability of the design analyst. 

Surface tension-driven breakup into droplets is rarely important in melt spinning, where the large viscous and elastic forces overwhelm the surface tension forces, ft is an important mechanism in the formation of the dispersed phase in polymer blends, and it is important in solution processing. The surface tension-driven breakup of a viscoelastic filament has been analyzed using both thin filament equations and a transient finite element analysis, but we will not pursue the topic here because it is not relevant to our present discussion. 

Shrivastava S, Tang J (1993) Large deformation finite element analysis of non-linear viscoelastic membranes with reference to thermoforming. J Strain Anal 28(1) 31-51... 

Erchiqui, F., Gakwaya, A, and Rachik, M. (2005) Dynamic finite element analysis of nonlinear isotropic hyperelastic and viscoelastic materials for ihermoforming appUcations. Polym. Er. Sci., 45,124—134. 

A finite-element analysis is used to perform a nonlinear dynamic transient analysis of the tunnel. Tuimel segments are modeled using beam elements that take into account shear rigidity. The joints are modeled with nonlinear hyperelastic elements. The bored tunnels at the end of both segments are incorporated in the analysis as beams on viscoelastic foundation. Influence of segment length and joint properties was then investigated parametrically. 

Finite element analysis using a simple material model (e.g., linear elasticity, hyperelasticity, linear viscoelasticity, isotropic plasticity)... 

The accuracy of Finite Element Analysis is wholly dependent on the precision of the material model employed. In the realm of elastomers, accurate material models are difficult to create due to the nonlinear behavior of the material as well as other viscoelastic effects, such as creep, stress relaxation, compression set, and cyclic softening. 

An area of continued research is in development of elastomeric material models which can incorporate cyclic behavior into one model casting aside the need to switch from non-cyclic to cyclic models. Further, other areas of research involve more accurately predicting viscoelastic behavior such as stress relaxation, creep, and most importantly for cyclic applications, compression set. As these models are developed, the accuracy and abilities of Finite Element Analysis of elastomers will improve dramatically and provide a much better method to predict the complex loading conditions over time that elastomeric parts commonly see. 

M. Kawahara and N. Takeuchi. Mixed finite element method for analysis of viscoelastic fluid flow. Comput. Fluids., 5 33, 1977. 

Baaijens FPT (1998) Mixed finite element methods for viscoelastic flow analysis a review. J Non-Newtonian Fluid Mech 79 361-85. 

In the context of viscoelastic fluid flows, numerical analysis has been performed for differential models only, and for the following types of approximations finite element methods for steady flows, finite differences in time and finite element methods in space for unsteady flows. Finite element methods are the most popular ones in numerical simulations, but some other methods like finite differences, finite volume approximations, or spectral methods are also used. 

J. Baranger and S. Wardi, Numerical analysis of a finite element method for a transient viscoelastic flow, Comput. Meth. Appl. Mech. Engrg., 125 (1995) 171-185. 

If a discrete inf-sup inequality is satisfied by the velocity and pressure elements, then the discrete problem has a unique solution converging to the solution of the continuous one (see equation (27) of the chapter Mathematical Analysis of Differential models for Viscoelastic Fluids). This condition is referred to as the Brezzi-Babuska condition and can be checked element by element. Finite element methods for viscous flows are now well established and pairs of elements satisfying the Brezzi-Babuska condition are referenced (see [6]). Two strategies are used to compute the numerical solution of these equations involving velocity and pressure. 

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