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Constitutive modelling of PMMA-based bone cement: a functional model of viscoelasticity and its approximation for time domain investigations

Lion A., Yagimli B., Baroud G., Goerke U.

Archives of Mechanics

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2008

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Vol. 60, nr 3

221-242

EN

To represent the mechanical behaviour of polyethylmethacrylate-based bone cement, a constitutive approach of finite linear viscoelasticity is formulated and identified. Motivated by the experimental data of storage and loss modulus, the model is based on a three-dimensional functional in integral representation. In the investigated frequency range, the master curve of the loss modulus is constant and that of the storage modulus increases linearly with the logarithm of the frequency. This behaviour corresponds to a viscoelastic fluid, and can be described by a continuous relaxation spectrum. For numerical simulations which are planned in future, the constitutive functional is approximated by a discrete spectrum. To this end, an earlier-developed method to approximate continuous relaxation spectra in limited time or frequency ranges by discrete ones is applied.

2

On the phenomenological representation of curing phenomena in continuum mechanics

Lion A., Hofer P.

Archives of Mechanics

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2007

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Vol. 59, nr 1

59-89

EN

To simulate curing phenomena, for example for the purpose of optimising the manufacturing processes or to calculate the stress distribution in adhesive seams, constitutive models representing the thermomechanically-coupled behaviour of adhesives are required. During the curing reaction, the adhesive changes its thermomechanical material behaviour from a viscous fluid to a viscoelastic solid. This phase transition is an exothermal chemical reaction which is accompanied by thermal expansion, chemical shrinkage and changes in temperature. In this essay we develop a physically-theory of finite strain thermoviscoelasticity to represent these phenomena. To this end, we introduce a multiplicative split of the deformation gradient into a thermal, a chemical and a mechanical part. We define the coordinate of chemical reaction determined by an evolution equation to describe the temporal behaviour of the curing reaction. The energy of the model contains an additional term, the chemically-stored free energy, which depends on this internal variable. The mechanical behaviour of the adhesive is modelled using a constitutive approach of finite thermoviscoelasticity and the viscosities are functions of the coordinate of chemical reaction. We show that the model is compatible with the Clausius-Duhem inequality, derive the equation of heat conduction and illustrate the physical properties of the theory by a numerical example.

3

Thermomechanically consistent formulations of the standard linear solid using fractional derivatives

Lion A.

Archives of Mechanics

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2001

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Vol. 53, nr 3

253-273

EN

We study the thermomechanical properties of a frequently used fractional generalisation of the standard linear solid. Its mathematical structure arises from an ordinary linear differential equation between stress and strain when replacing the first order time rates by fractional derivatives of the order .... If the parameters ... and ... are not further restricted, the model leads to an unphysical behaviour. In the case of harmonic deformations the dissipation modulus can become negative. This corresponds to a negative entropy production and violates the second law of thermodynamics. Then we propose two generalisations of the standard linear solid which are based on a so-called thermodynamically consistent fractional rheological element. It possesses a non-negative free energy and rate of dissipation for arbitrary deformation processes and is compatible with the second law of thermodynamics. The differential equations between stress and strain of the proposed generalisations contain also fractional derivatives of different orders but both the dynamic moduli and the relaxation spectra are non-negative functions of their arguments. No restrictions on the material parameters are required.