Alexander Lion

On the large deformation behaviour of reinforced rubber at different temperatures

Abstract This essay investigates the temperature dependence of the mechanical properties of a filler-loaded tread compound experimentally and proposes a physically based method to represent this behaviour in the framework of non-linear continuum thermomechanics. To this end, we realise a series of monotonic and cyclic strain controlled tests on cylindrical specimens in tension at different temperature levels. The experimental data show the isothermal mechanical behaviour to be mainly influenced by non-linear elasticity in combination with non-linear rate dependence and weak equilibrium hysteresis. We observe that the rate sensitivity of the material depends strongly on the temperature : at low temperature levels, the rate sensitivity is essentially higher than at high temperatures. The elastic properties of the material depend comparatively less on the temperature. Nevertheless, higher temperature levels lead to higher equilibrium stresses. In order to represent the material behaviour, we start with a multiplicative split of the deformation gradient into a mechanical and a thermal part as proposed by Lu and Pister (1975). Physically, this idea corresponds to a stress-free thermal expansion followed by an isothermal stress-producing deformation. We suppose the thermal part of the deformation gradient to be isotropic. As a consequence of this, the velocity gradient decomposes additively into a pure thermal and a pure mechanical part. By using these elements, we exploit the Clausius Duhem inequality and assume the so-called ‘mechanical second Piola Kirchhoff stress tensor’ to be a functional of the ‘mechanical Greens strain tensor’. In a further step, we define this functional by a system of constitutive equations which are based on a rheological model. The evolution equations for the internal variables are formulated by using the concept of dual variables proposed by Haupt and Tsakmakis (1989, 1996). The rate sensitivity is modelled by a stress and temperature dependent viscosity function. The elastic part of the equilibrium stress is described by entropy elasticity in combination with a modified Mooney Rivlin strain energy function. The equilibrium hysteresis effects are represented by rate independent plasticity in arclength representation as proposed by Valanis (1971). The constitutive model is compatible with the dissipation principle of thermodynamics and describes the general trend of the experimental data fairly well.

Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models

Abstract This essay deals with a physical approach to formulate constitutive laws of finite thermoviscoplasticity. As proposed, for example, by Besdo (Besdo D., 1980. Zur Formulierung von Stoffgesetzen fur plastisch anistrope/elastisch isotrope Medien im Dehnungsraum, Zeitschr. angew. Math. Mech., 60, 101–103) or Negahban (Negahban, M., 1995. A study of thermodynamic restrictions, constraint conditions and material symmetry in fully strain-space theories of plasticity, Int. J. Plast., 11, 679–724) the whole theory is formulated in the strain space. For the sake of clarity and owing to the stringency required when choosing appropriate internal variables and evolution laws, the layout of the theory is dictated by rheological models. Combined with the concept of dual variables proposed by Haupt and Tsakmakis (Haupt, P., Tsakmakis, C., 1989. On the application of dual variables in continuum mechanics, Continuum Mech. Thermodyn. 1, 165–196), this method ensures the compatibility of the constitutive theory with the second law of thermodynamics. To illustrate the train of thought, we begin with the formulation of a uniaxial model of thermoviscoplasticity and restrict ourselves to kinematic hardening. In order to take thermal strains into account, we dissect the total strain into a thermal and a mechanical part. The mechanical deformation is the driving force for the stress and the thermal strain is a function of the temperature. In addition, we divide the mechanical strain into an elastic and an inelastic part. The stress depends only on the elastic strain, whereas the inelastic deformation is a functional of the process history. It corresponds to that part of strain which remains if the stress is reduced to zero. For the purpose of describing kinematic hardening with internal variables of strain type we introduce a further decomposition and dissect the inelastic deformation into two parts which has a motivation on the microscopic scale. The first part can be interpreted as the spatial average of local lattice deformations caused by dislocation fields (cf. Bruhns, O.T., Lehmann, T., Pape, A., 1992. On the description of transient cyclic hardening behaviour of mild steel CK15. Int. J. Plast. 8, 331–359) and the second can be attributed to inelastic slip processes on the microscale. Based on these ideas, it is straightforward to specify the free energy and to satisfy the dissipation principle in the form of the Clausius–Duhem inequality. The potential relations for the stress, the kinematic hardening variable and the entropy as well as the evolution laws for the internal variables are sufficient conditions for the non-negativity of the entropy production. Based on simplifying assumptions, we find that the Armstrong–Frederick model of hardening is incorporated as a special case. Subsequently, we transfer the structure of the theory to finite non-isothermal deformations. To this end we apply the thermomechanical decomposition of the deformation gradient as proposed by Lu and Pister (Lu, S.C.H., Pister, K.D., 1975. Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids. Int. J. Solids Struc. 11, 927–934). As prompted above, we define two further decompositions: the first one dissects the mechanical part of the deformation gradient into an elastic and an inelastic part and the second one splits the inelastic part into two further sections. Its first part can be interpreted as an averaged elastic lattice deformation which is caused by dislocations and its second part as an averaged plastic strain due to local plastic slip effects. To develop the constitutive relations for the free energy, the stress, the internal variables and the entropy we consider the rheological model in combination with the concept of dual variables and evaluate the Clausius–Duhem inequality.

Thixotropic behaviour of rubber under dynamic loading histories: Experiments and theory

Abstract In the first part of this essay we investigate the thermomechanical behaviour of carbon black-filled rubber under dynamic loading conditions. The loadings consist of static predeformations which are superimposed by small sinusoidal oscillations. The frequency was varied between 0.1 Hz and 100 Hz, the deformation amplitude between 0.006 and 0.06, the temperature between 253 K and 373 K and the storage and dissipation moduli measured. The data show that the frequency dependence of the moduli is of the powerlaw type. They also depend on the deformation amplitude and the temperature. If the temperature is constant, increasing amplitudes lead to decreasing moduli. As discussed by Payne (1965), this behaviour can be interpreted in terms of a thixotropic change. If, on the other hand, the deformation amplitude is kept constant, increasing temperature levels lead to decreasing moduli. In the second part of this work we develop a constitutive theory to represent the material behaviour observed. We consider the dissipation principle of thermodynamics and formulate the model for three-dimensional finite deformations. To describe thermal expansion effects, we decompose the deformation gradient into a thermal and a mechanical part as proposed by Lu and Pister (1975). We prescribe the thermal part by a constitutive function and assume the mechanical part to be the driving force for the stress tensor. As motivated in earlier works we split the total stress into an equilibrium stress and a rate-dependent overstress. We represent the equilibrium stress using a modified Mooney-Rivlin strain energy function and the overstress by a series of Maxwell elements whose springs are of Neo-Hookean type. Both the kinematic tensors and the associated stress measures are defined by using the concept of dual variables proposed by Haupt and Tsakmakis (1989). In order to represent the thixotropic effects, the viscosities depend on the temperature and the deformation history. The history dependence is implied by an internal variable which is a measure for the deformation amplitude and has a relaxation property as discussed by Payne (1965). To determine the material parameters, we linearise the constitutive equations with respect to the static predeformation and take the analytical solution for harmonic strain-controlled loadings into account. Numerical simulations demonstrate that the constitutive theory describes all phenomena experimentally observed with a fairly good approximation. The model is compatible with the second law of thermodynamics in the form of the Clausius Duhem inequality.

Constitutive Modelling of the Glass Transition and Related Phenomena: Relaxation of Shear Stress Under Pressure

In industrial fabrication processes as well as in many applications of polymer parts, the glass transition plays a significant role. This is due to high mechanical processing speeds, high temperatures or large cooling rates. The mechanical, the thermomechanical and the caloric properties of polymers differ below and above the glass transition which is a thermoviscoelastic phenomenon. It depends on the ratio between the intrinsic time scale of the polymer and that of the thermomechanical loading process. If both scales are comparable, the material is in the glass transition region. Otherwise it is in the equilibrium or in the glassy region. In the industry, there are increasing demands to simulate fabrication processes in order to estimate the resulting behaviour of the polymer parts before they are manufactured. To this end, constitutive models of finite thermoviscoelasticity are needed which can represent the volumetric as well as the isochoric mechanical behaviour of the polymer in combination with the caloric and the thermomechanical properties. In a recent paper of the authors, the concept of a hybrid free energy has been developed. This approach will be applied in the current essay where the pressure-dependent relaxation behaviour under shear deformations is of interest.