There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

http://www.christov.metacontinuum.com/My_Publications/2009/Christov_MRC09.pdf

On frame indifferent formulation of the Maxwell-Cattaneo model of finite-speed heat conduction

Thermomechanics of shape memory polymers: Uniaxial experiments and constitutive modeling

The present state-of-the-art article is devoted to the analysis of new trends and recent results carried out during the last 10 years in the field of fractional calculus application to dynamic problems of solid mechanics. This review involves the papers dealing with study of dynamic behavior of linear and nonlinear 1DOF systems, systems with two and more DOFs, as well as linear and nonlinear systems with an infinite number of degrees of freedom: vibrations of rods, beams, plates, shells, suspension combined systems, and multilayered systems. Impact response of viscoelastic rods and plates is considered as well. The results obtained in the field are critically estimated in the light of the present view of the place and role of the fractional calculus in engineering problems and practice. This articles reviews 337 papers and involves 27 figures. DOI: 10.1115/1.4000563

Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results

Ferroelectric and ferroelastic piezoceramics – modeling of electromechanical hysteresis phenomena

1 Kinematics.- 2 Balance Relations of Mechanics.- 3 Balance Relations of Thermodynamics.- 4 Objectivity.- 5 Classical Theories of Continuum Mechanics.- 6 Experimental Observation and Mathematical Modelling.- 7 General Theory of Mechanical Material Behaviour.- 8 Dual Variables.- 9 Elasticity.- 10 Viscoelasticity.- 11 Plasticity.- 12 Viscoplasticity.- 13 Constitutive Models in Thermomechanics.- References.

Continuum Mechanics and Theory of Materials

Shape memory behaviour: modelling within continuum thermomechanics

A characteristic of filled elastomers is their ability to undergo very large deformations without damaging their internal structure. The material behaviour is mainly elastic, however, elastomers show hysteresis effects leading to damping properties, which are quite important as regards their applications in various fields of mechanical engineering. A series of experiments (tension, torsion and combinations of both) was carried out on cylindrical bars made of a carbon-black filled rubber mixture. In addition to a pronounced nonlinear rate-dependence, relaxation and viscosity properties are observed as being influenced by the process histories. The behaviour of elastomeric materials is modelled on the basis of a free energy function and evolution equations for additional internal variables. Incorporating or disregarding the very small rate-independent hysteresis, the constitutive modelling may be classified under viscoplasticity or viscoelasticity. The constitutive equations are formulated for isothermal processes in a thermodynamically consistent manner. Particular attention is focused on nonlinear rate-dependence as well as on process-dependent relaxation properties. Numerical simulations on the basis of identified material parameters show that the proposed constitutive model is able to represent the main elastic and inelastic phenomena.

Viscoplasticity of elastomeric materials: experimental facts and constitutive modelling

Stress and strain tensors that arise in the expression of the stress power are called “conjugate variables”. More special is the term “dual variables” which has been introduced in connection with incremental constitutive relations of hypoelasticity and plasticity, where the rates of both tensors arise. We propose a rational rule from which the most natural form of tensor-valued kinematic and dynamic variables (strain and stress tensors) including their corresponding time rates can be deduced. Dual variables and their associated dual derivatives are characterized by the property that apart from the stress power also the incremental stress power is invariant under a group of transformations that corresponds to a set of physically reasonable intermediate configurations. We outline the precursory history of these concepts and then discuss in detail how the invariance properties can be realized in the various stress and strain measures. We finally demonstrate the concept in three different applications: The rate form of the principle of virtual work, the formulation of constitutive relations in viscoelasticity and the formulation of incremental constitutive assumptions of rate-independent plasticity.

On the application of dual variables in continuum mechanics

This paper deals with two main topics. The first one concerns the equivalence of stress algorithms, based on a Backward-Euler-step applied on viscoplastic models of Chaboche-type, and their elastoplastic counterpart. Generally, the stress algorithm yields a system of non-linear algebraic equations and the corresponding consistent tangent operator, occurring in the principle of virtual displacements, leads to a system of linear equations. This procedure can be obtained utilizing only numerical methods. The second topic concerns a special constitutive relation based on a kinematic hardening model using a sum of Armstrong/Frederick terms, which is equivalent to a multi-surface plasticity model. Applying this model a so-called problem-adapted stress algorithm is derived, where only one non-linear equation must be solved. This result is independent of the number of terms in the hardening model. Furthermore, only the viscoplastic algorithm must be implemented, since it includes the elastoplastic constitutive model as a special case. © 1997 by John Wiley & Sons, Ltd.

Peter Haupt

Papers

Continuum Mechanics and Theory of Materials

Shape memory behaviour: modelling within continuum thermomechanics

Viscoplasticity of elastomeric materials: experimental facts and constitutive modelling

On the application of dual variables in continuum mechanics

An efficient stress algorithm with applications in viscoplasticity and plasticity