Logarithmic strain

Principal true or logarithmic strains ej, n and iii are defined in terms of the principal 

These strains are exactly additive, in contrast with engineering strains that are only approximately so. To illustrate this, consider two pure deformations V and V that are applied successively to give a total deformation V, so that 

V is said to be multiplicatively decomposed into V and V . When operating with finite strains, deformation gradients must be used that, beiug tensors, are combined multiplicatively. Suppose that both V and V have the same principal directions and are expressed in diagonal form, with principal extension ratios respectively A. and A. (i = I, II, HI), then 

The logarithmic strain has also been termed the natural strain by Nadai [4], and is discussed by other authors, for example Rees [5]. In terms of the engineering strain e, 

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When creep curves are plotted on logarithmic strain and time scales they are approximately straight lines so that the creep strain, edO may be expressed as... 

Such total or finite strains are known as natural or logarithmic strains the use of which was first suggested by Ludwig (1909). 

Such total finite strains are known as natural or logarithmic strains, the use of which was first suggested by Ludwig (1909). The engineering strain is frequently used as a measure of finite linear strain (e.g., the percentage elongation in a simple tensile test is usually quoted). The relationship between these two measures can be derived as follows ... 

In other applications where large-strain representation is necessary, logarithmic strains are often used. These are defined as... 

The change in logarithmic strain as the element moves from z to z + dz is... 

In the small strain limit, the logarithmic strain approaches the engineering strain (see Section 3.1.6) and comparison of this expression with Hooke s law then reveals that (i is the shear modulus. In order to validate the model over a range of material data, a plot was made of... 

This logarithmic strain measure is sometimes called the Hencky strain (1924). In the limit of small strain, it is the same as the usual ( engineering ) strain measure L/Lo — 1. 

Fig. 10. Yield stress normalized to temperature as a function of logarithmic strain rate for polycarbonate. After Bauwens-Crowet and co-workers (44) with permission.

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