A simple and efficient BEM implementation of quasistatic linear visco-elasticity

Abstract

a b s t r a c tA simple yet efficient procedure to solve quasistatic problems of special linear visco-elastic solids at small strains with equal rheological response in all tensorial components, utilizing boundary element method (BEM), is introduced. This procedure is based on the implicit discretisation in time (the so-called Rothe method) combined with a simple ''algebraic'' transformation of variables, leading to a numerically stable procedure (proved explicitly by discrete energy estimates), which can be easily implemented in a BEM code to solve initial-boundary value visco-elastic problems by using the Kelvin elastostatic fundamental solution only. It is worth mentioning that no inverse Laplace transform is required here. The formulation is straightforward for both 2D and 3D problems involving unilateral frictionless contact. Although the focus is to the simplest Kelvin-Voigt rheology, a generalization to Maxwell, Boltzmann, Jeffreys, and Burgers rheologies is proposed, discussed, and implemented in the BEM code too. A few 2D and 3D initial-boundary value problems, one of them with unilateral frictionless contact, are solved numerically.

Quasistatic normal-compliance contact problem of visco-elastic bodies with Coulomb friction implemented by QP and SGBEM

“…In most materials (like metals) the relaxation time χ>0 is small comparing to outer-loading time scale but we assume that it is rather large to affect the tangential contact with friction. We assume χ to be the same for both subdomains; note that the potential (3.11) cannot be formulated if χ would be different in each subdomain; anyhow, some considerations (and computations) can be performed for non-potential problems, too, as suggested in [40,Sect.3.3]. This allows for using the simple trick from [45, Remark 6.2], which can be extended even for more complex rheologies as shown in [40], which facilitates usage of standard static BIEs even for quasi-static evolution problems.…”

“…We assume χ to be the same for both subdomains; note that the potential (3.11) cannot be formulated if χ would be different in each subdomain; anyhow, some considerations (and computations) can be performed for non-potential problems, too, as suggested in [40,Sect.3.3]. This allows for using the simple trick from [45, Remark 6.2], which can be extended even for more complex rheologies as shown in [40], which facilitates usage of standard static BIEs even for quasi-static evolution problems. The simple viscosity ansatz (3.6) is chosen in order to exploit the reformulation of the viscoelastic problem in the bulk in terms of an recursive static elastic problem in the bulk, similarly as in [40], which is solved by the conventional elastostatic SGBEM here.…”

“…This allows for using the simple trick from [45, Remark 6.2], which can be extended even for more complex rheologies as shown in [40], which facilitates usage of standard static BIEs even for quasi-static evolution problems. The simple viscosity ansatz (3.6) is chosen in order to exploit the reformulation of the viscoelastic problem in the bulk in terms of an recursive static elastic problem in the bulk, similarly as in [40], which is solved by the conventional elastostatic SGBEM here. For this aim, let us introduce a new variable v (a 'fictitious' displacement) at the time level k, as…”

“…Sometimes the former type of rheology is referred to as solid-type and the latter as fluid-type, cf. [40].…”

Formulação posicional para descrição do comportamento mecânico de fluência em vigas e estruturas de pórtico

“…Alguns estudos recentes em viscoelasticidade apresentam ainda desenvolvimentos referentes à utilização de derivadas fracionárias na obtenção de modelos reológicos mais simples e com maior precisão na representação do comportamento complexo de materiais reais, como materiais compostos, materiais poliméricos, tecidos biológicos, entre outros (BAHRAINI et al, 2013;SHEN et al, 2013;PÉREZ ZERPA et al, 2015).…”