MATERIALS ENGINEERING CONCEPTS - Materials for Civil and Construction Engineers - page 8

Initial
tangent
modulus

Tangent
modulus

Chord
modulus

Secant
modulus

FIGURE 1.5 Methods for
approximating modulus .

Strain

For materials that do not display any linear behavior, such as concrete
and soils, determining a Young's modulus or elastic modulus can be prob-
lematical. There are several options for arbitrarily defining the modulus for
these materials. Figure 1.5 shows four options: the initial tangent, tangent,
secant, and chord moduli. The initial tangent modulus is the slope of the
tangent of the stress-strain curve at the origin. The tangent modulus is the
slope of the tangent at a point on the stress-strain curve. The secant modu-
lus is the slope of a chord drawn between the origin and an arbitrary point
on the stress-strain curve. The chord modulus is the slope of a chord drawn
between two points on the stress-strain curve. The selection of which mod-
ulus to use for a nonlinear material depends on the stress or strain level at
which the material typically is used. Also, when determining the tangent,
secant, or chord modulus, the stress or strain levels must be defined.
Table 1.1 shows typical modulus and Poisson's ratio values for some ma-
terials at room temperature. Note that some materials have a range of mod-
ulus values rather than a distinct value. Several factors affect the modulus,
such as curing level and proportions of components of concrete or the di-
rection of loading relative to the grain of wood.

1.2.4 Elastoplastic Behavior
For some materials, as the stress applied on the specimen is increased, the
strain will proportionally increase up to a point; after this point the strain
will increase with little additional stress. In this case, the material exhibits
linear elastic behavior followed by plastic response. The stress level at
which the behavior changes from elastic to plastic is the elastic limit . When
the load is removed from the specimen, some of the deformation will be re-
covered and some of the deformation will remain as seen in Figure 1.6(a). As
discussed in Chapter 2, plastic behavior indicates permanent deformation of
the specimen so that it does not return to its original shape when the load is
removed. This indicates that when the load is applied, the atomic bonds