Oct 25, 1995 - We need a precise number of constitutive equations to supplement the ... This paper discusses the constitutive equations for the creep of ...
.a,fechantcs of (?omposzte J,Iater~als. Vol. J1. .Vo 6. 1995
A GENERALIZED CONSTITUTIVE EQUATION FOR CREEP OF POLYMERS AT MULTIAXIAL LOADING H. Altenbach,* J. Altenbach,** and A. Zolochevsky*** Dedicated to our colleague and friend Vitauts Tamu!~s on the occasion of his sixtieth birthday
INTRODUCTION It is well known that under certain conditions of classification the modeling of material behavior under the effect of external loading on a macroscopic material response can be ascertained bv the performance of macroscopic tests alone. At the macroscopic level of modeling the material behavior, the fundamental laws of phenomenologicaI continuum mechanics describe the response of material systems caused by external loads with a sufficient approximation. These fundamental laws :nay be divided into two groups [1, 2]: 9 The first group of laws is assumed to be approximately independent from the specific conditions of a c o n t i n u u m T h e y have the character of fundamental principles of mechanics. They are mainly formulated as balance laws of mass, linear and angular momenta, energy and entropy. 9 The second group of equations reflects the individuality in the behavior of materials caused by their different internal constitution. We need a precise number of constitutive equations to supplement the balance laws in such a way t h a t the mechanical problem can be determined. The extent to which a physical phenomenon has been understood is reproduced in a m a t h e m a t i c a l model. The continuum approach is based on the frame of a hypothesis t h a t material can be viewed as having a continuity of structure. There is a continous distribution of matter throughout the material at all times and all physical qualities of a material particle can be considered as a continuous function of the position and time. In this sense. phenomenological continuum mechanics is a field theory and the m a t h e m a t i c a l modeling is inside the well-known calculus. This paper considers some aspects of phenomenologicaI modeling of constitutive equations. The main wavs of modeling are 9 a deductive derivation of constitutive equations based on axioms and fundamental principles of the material theory [3]. 9 an inductive derivation of constitutive equations based on experimental performances [4], and 9 use of simple rheological models of material behavior and their combinations [5t. The deductive way is based on the principles of material theory. Material theory has to take into consideration the so-called constitutive axioms, e.g.. Causality, Detervrdnisrn,, Eq~ipresence, Material Objectivity, Local Actio~L Memory, Physical Admissibility, etc. and has to answer such questions as 9 how constitutive equations can be derived on a deductive way, 9 how materiaIs s v m m e t r i e s can be included,
*Martin-Luther-Universit~t Halle-Wittenberg, Institut fiJr Werkstoffwissenschaft, Germany. **Otto-von-GuerickeUniversit~t Magdeburg, Germany. ***Kharkov State University, School of Fundamental Medicine, Ukraine. Published in Mekhanika Kompozitnykh Materiatov, Vet. 31, No. 6, pp. 723-733, November-December, 1995. Original article submitted October 25, 1995 0191-566,5,/95/3t06-0511 $12.50 @1996 Plenum Publishing C o r p o r a t i o n
51 !
9 how internal constrains can be taken into account? The method of rheological modeling is based on the idea of fornmlating phenomenologicai material equations for very simple cases, which are consistent in the sense of thermomechanics, and to connect the basic theological models of the elastic Hooke element, the viscous Newton element, and the plastic St. Venant element to describe in a first approximation a more complex material behavior. In practical use this method is often restricted to small deformations and homogeneous and isotropic materials and demands two significant assumptions: 9 the volume dilatation is pure elastic; 9 the main differences in the material equations for different materials are based on shearing effects only, Engineers mainly prefer the inductive method of deriving constitutive equations to avoid the expanded way of proving the physical and mathematical consistence of the constitutive assumptions. The inductive way is based on simple experimental material tests, e.g., uniaxial loading, and their generalization on nmltiaxial loading. The p,'esent paper restricts the considerations to the inductive method of material modeling. Starting from the assumption of the existence of a potential, a generalized constitutive equation is developed. The three basic phenomenological models (elasticity. creep, plasticity) can be deduced from this general formulation also when the so
