CREEP AND RELAXATION - FUNDAMENTAL PROCESSES IN BIOLOGICAL MATERIAL MODELING
Author: Nicolae Mazilu
Published on Tuesday, January 8th, 2008 in category ProtoQuant
V. Conclusions
The previous analysis points out to an attractive representation of the experimental loading curve for uniaxial tests. The principle of this representation is roughly indicated in Figure 7: the experimental loading curve is a succession of minute creep and relaxation processes, intentionally exaggerated in the Figure 7 for the sake of illustrating the point. In reality we can imagine a path of minute steps – horizontal as well as vertical – covering closely the experimental curve. In fact there may be many such paths – sample paths in the language of stochastic processes [9] – and what we actually observe is some stochastic tendency of these paths. The main point of the previous analysis is that this stochastic tendency is well represented – at least for biological tissues – by a straightforward identification of the times of the two component physical processes – creep and relaxation. The physical interpretation of the parameters l0, a, b, as given previously in this work, strongly supports this view.
One might be tempted to jump into the conclusion – usually shared among the theorists but mainly among experimentalists – that equation (22) is to be taken as a plain constitutive law. This is actually the view on the finality of loading experiments. While this fact is true to a large extent – as our own experience shows – it may not be the case in general. This conclusion comes about even to a speculative level. For, a constitutive law is an assessment about the relation of two 3×3 matrices, two eigenvalues of which are related by equation (22). This is indeed a rational relation, even though not a homography as it is apriori supposed to be. All we know is that (22) includes structural changes, which are surely reflected in the relation among the three eigenvalues of the strain in a real experiment. The incompressibility condition, usually taken into consideration at this stage of building the constitutive equations, besides proving itself possibly not true in all cases, is not nearly enough to put concordance between the physical state of a sample and its particular geometry. In spite of this, however, the equation (22) can be successfully used to build the behavior of some ideal sample shapes representing limit states of strain in ideal experiments: uniaxial state, biaxial state, planar state, etc.
The main point we want to make with this work is, nevertheless, that the conclusion of [1] that creep results cannot be extracted from relaxation data – or vice versa – is true no matter what label we attach to a biological material (linear, nonlinear, etc). In spite of this however the everlasting hope to be able to predict creep curves from relaxation data [2], or vice versa, may also be true with one simple amendment. Namely, this duality goes through the usual experimental loading curve, which plays a precise part here: it is the sample path of equal creep and relaxation times. Our opinion is that it is this last feature of the quasistatic experimental loading curve which allows one to pass from creep to relaxation and vice – versa. Thus, in order to be able to recover creep data from relaxation experiments, we must supplement the relaxation data with the experimental quasistatic loading curve. The very process of recovering may be quite elaborate at times so we reserve the procedure for a further communication.
VI. References
[1] Thornton, G. M, Oliynyk, A., Frank, C. B., Shrive, N. G.: Ligament Creep Cannot be Predicted from Stress Relaxation at Low Stress: A Biomechanical Study of the Rabbit Medial Collateral Ligament, Journal of Orthopaedic Research, Vol. 15 (1997) pp. 652 – 656
[2] Lakes, R. S., Vanderby, R.: Interrelation of Creep and Relaxation: A Modeling Approach for Ligaments, Journal of Biomechanical Engineering (Trans ASME), Vol. 121 (1999), p. 612
[3] Burnside, W. S., Panton, A. W.: The Theory of Equations, Dover Publications 1960
[4] Vossoughi, J., Conway, T. A., Mazilu, N.: Thermodynamics and Experimental Solid Mechanics of Soft Tissues, Russian Journal of Biomechanics, No. 1-2 (1997) pp. 25-37
[5] Gittus, J.: Creep, Viscoelasticity and Creep Fracture in Solids, J. Wiley, New York 1975
[6] Valenta, J.: Biomechanics, Elsevier 1993
[7] Conway, T. A., Mazilu, N.: A Rationale for Constitutive Laws of Stable Materials, in ‘Constitutive Laws, Experiments and Numerical Implementation’, A. M. Rajendran, R. C. Batra Editors, International Center for Numerical Methods in Engineering (CIMNE), Barcelona 1996
[8] Yamada, H.: Strength of Biological Materials, R. E. Krieger Publishing Co., Inc., Huntington, N. Y. 1973
[9] Øksendal, B.: Stochastic Differential Equations, Springer – Verlag 1998