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Strain measures used in general motions are most simply understood by first considering the concept of strain in one dimension and then generalizing this to arbitrary motions by using the polar decomposition theorem just derived.
We already have a measure of deformation—the stretch ratio . In fact, is itself an adequate measure of “strain” for a number of problems. To see where it is useful and where not, first notice that the unstrained value of is 1.0. A typical soft rubber component (such as a rubber band) can change length by a large factor when it is loaded, so the stretch ratio would often have values of 2 or more. In contrast, a typical structural steel component will be designed to respond elastically to its working loads. Such a material has an elastic modulus of about 200 × 103 MPa (30 × 106 lb/in2) at room temperature and a yield stress of about 200 MPa (30 × 103 lb/in2), so the stretch at yield will be about 1.001 in tension, 0.999 in compression. The stretch ratio is an unsatisfactory way of measuring deformation for this case because the numbers of interest begin in the fourth significant digit. To avoid this inconvenience, the concept of strain is introduced, the basic idea being that the strain is zero at , when the material is “unstrained.” In one dimension, along some “gauge length” , we define strain as a function of the stretch ratio, , of that gauge length:
The objective of introducing the concept of strain is that the function f is chosen for convenience. To see what this implies, suppose is expanded in a Taylor series about the unstrained state:
We must have , so at (this was the main reason for introducing this idea of “strain” instead of just using the stretch ratio). In addition, we choose at so that for small strains we have the usual definition of strain as the “change in length per unit length.” This ensures that, in one dimension, all strain measures defined in this way will give the same numerical value to the order of the approximation when strains are small (because then the higher-order terms in the Taylor series are all negligible)—regardless of the magnitude of any rigid body rotation. Finally, we require that for all physically reasonable values of (that, is for all ) so that strain increases monotonically with stretch; hence, to each value of stretch there corresponds a unique value of strain. (The choice of is arbitrary: we could equally well choose , implying that the strain is positive in compression when . This alternative choice is often made in geomechanics textbooks because geotechnical problems usually involve compressive stress and strain. The choice is a matter of convenience. In ABAQUS we always use the convention that positive direct strains represent tension when . This choice is retained consistently in ABAQUS, including in the geotechnical options.)
With these reasonable restrictions ( and at , and for all ), many strain measures are possible, and several are commonly used. Some examples are
All of these strains satisfy the basic restrictions. Obviously many strain functions are possible: the choice is strictly a matter of convenience. Since strain is usually the link between the kinematic and the constitutive theories, the convenience of this choice in the context of finite elements is based on two considerations: the ease with which the strain can be computed from the displacements, since the latter are usually the basic variables in the finite element model, and the appropriateness of the strain measure with respect to the particular constitutive model. For example, as mentioned above, it appears that log strain is particularly appropriate to plasticity, while large-strain elasticity analysis (for rubbers and similar materials) can be done quite satisfactorily without ever using any “strain” measure except the stretch ratio .
Having defined the basic concept of “strain” in one dimension, we now generalize the idea to three dimensions. In Deformation, Section 1.4.1, we established that the deforming part of the motion in the immediate neighborhood of a material point is completely characterized by six variables: the three principal stretch ratios , , and and the orientation of the three principal stretch directions in the current (or in the reference) configuration. This immediately gives the generalization of the one-dimensional strain function introduced above. We first choose the function f that will be used as the strain measure. will be the strain along the first principal direction, ; will be the strain along ; and will be the strain along .
The matrix
completely characterizes the state of strain at the material point. Notice the resemblance to the definition of the stretch matrix, Equation 1.4.1–10: we might consider to be defined by the matrix functionIn Equation 1.4.2–2 we have written the matrix by using the principal strain directions in the current configuration. We could equally have begun with the polar decomposition into a stretch followed by rotation of the principal directions of stretch: would be defined in a similar way and would then be associated with its principal directions in the reference configuration. ABAQUS generally reports strains referred to directions in the current configuration. There is no obvious reason for this choice: either approach would suffice so long as the user knows which is being used. The strain measures reported by ABAQUS are enumerated in Conventions, Section 1.2.2 of the ABAQUS Analysis User's Manual.
In a finite element code the deformation gradient is usually computed at each material calculation point from the displacement solution at the nodes of each element and the interpolation function chosen for the element. We now need an algorithm to obtain , given a choice of strain measure. This algorithm is available immediately from Equation 1.4.1–12: the eigenvalues and eigenvectors of the matrix are ; and ; and , and . We can then calculate , etc. for the function f chosen as the strain measure and, thus, construct
This algorithm also gives principal strain and stretch values—often a useful output because they give a concise description of the state of deformation at a point. However, the algorithm requires computation of the eigenvalues and eigenvectors of a matrix at each of many points in the model at each of many iterations, which involves some computational cost. Thus, it would be useful if could be computed less expensively from , which is possible only for certain choices of the strain measure, . We now consider one such possibility.
The unit matrix can be written as
Green's strain was defined in one dimension as
Comparing this one-dimensional definition with Equation 1.4.2–2 and Equation 1.4.2–3, we see that
Green's strain matrix is, thus, available directly from the deformation gradient without first having to solve for the principal directions. This advantage makes Green's strain computationally attractive. Recall that strain is the link between the kinematics and the constitutive theory, so the strain choice should be optimal based on the two considerations of convenience and appropriateness. We have already suggested that logarithmic strain is the most appropriate for elastic-plastic or elastic-viscoplastic materials in which the elastic strains are always small (because the yield stress is small compared to the elastic modulus), so it appears that the computational convenience of Green's strain cannot be used to advantage. However, the choice of a strain function, , was restricted so that, for small strains but arbitrary rotations, all strain measures are the same to the order of the approximation. Thus, for such cases Green's strain is a very convenient choice for computing the strain. The small-strain, large-rotation approximation is often useful—especially in structural problems (shells and beams) because there the thinness or slenderness of the members often allows large rotations to occur with quite small-strains—and Green's strain is commonly used in large-rotation, small-strain formulations for such problems as shell buckling.
Finally, it is worth remarking that the familiar “small-strain” measure used in most elementary elasticity textbooks,