72 Yield behavior of metallic foams

We can modify the above theory in a straightforward manner to account for the effect of porosity on the yield criterion and strain-hardening law for a metallic foam. We shall assume the elastic response of the foam is given by that of an isotropic solid, with Young's modulus E and Poission's ratio v. Since foams can yield under hydrostatic loading in addition to deviatoric loading, we modify the yield criterion (7.8) to

where we define the equivalent stress o by

This definition produces a yield surface of elliptical shape in (om — oe) space, with a uniaxial yield strength (in tension and in compression) of Y, and a hydrostatic strength of v/(l + («/3)2 loj = 1-Y

The parameter a defines the aspect ratio of the ellipse: in the limit a = 0, o reduces to oe and a J2 flow theory solid is recovered. Two material properties are now involved instead of one: the uniaxial yield strength, Y, and the pressure-sensitivity coefficient, a. The property Y is measured by a simple compression test, which can also be used to measure a in the way described below.

The yield surfaces for Alporas and Duocel for compressive stress states are shown in Figure 7.1. The data have been normalized by the uniaxial compressive yield strength, so that ae = 1 and rr„, = ^ for the case of uniaxial compression. We note that the aspect ratio a of the ellipse lies in the range 1.35 to 2.08. The effect of yield surface shape is reflected in the measured

Alulight " 18%

0.4 0.6 0.8 1 Normalized mean stress

Figure 7.1 Yield surfaces for Alporas and Duocel foams. The surfaces are approximately elliptical, described by equations (7.11) and (7.12)

Alulight " 18%

0.4 0.6 0.8 1 Normalized mean stress

Figure 7.1 Yield surfaces for Alporas and Duocel foams. The surfaces are approximately elliptical, described by equations (7.11) and (7.12)

o CL

o CL

Mean stress coefficient, a

Figure 7.2 The relationship between the plastic Poisson's ratio vP and the constant a

Mean stress coefficient, a

Figure 7.2 The relationship between the plastic Poisson's ratio vP and the constant a plastic Poisson's ratio in a uniaxial compression test: the ratio of transverse strain to axial strain depends upon a, as shown in Figure 7.2. Experimental data, available for Alporas and Duocel foams, support this, (see Figure 7.2). The yield surface shape (equations (7.11) and (7.12)) is sufficiently simple for an analytical expression to be derivable for in terms of a, giving

with the inversion a = 3

It would appear that the measurement of vP in a uniaxial compression test offers a quick and simple method for estimation of the value for a, and thereby the shape of the yield surface. Preliminary experience suggests that the measurement of vP is best done by compressing a sample, with suitably lubricated loading platens, to a uniaxial strain of 20-30%.

Having defined the yield surface shape, it remains to stipulate how the yield surface evolves with strain. For simplicity, we shall assume that isotropic hardening occurs: the yield surface grows in a geometrically self-similar manner with strain; the limited measurements of the yield surface for metallic foams approximate this behavior (see, for example, Figure 7.3 for the case of Alporas with an initial relative density of 0.16). Yield surfaces are displayed for the initial state, and for 10% and 30% uniaxial pre-strain. We note that the yield surfaces are smooth and geometrically self-similar.

We assume that the strain-hardening rate scales with the uniaxial compression response as follows. The plastic strain rate is again taken to be normal to the yield surface (7.11), and specified by the analogue of (7.9), given by

1 1

1 1 10%

- •

• . Volumetric -

strain

/

_ ^

Q -

10%%.

Uniaxial

r - -o^Initial

A. strain \ • -

Y V

-

\ v -

\

i l i \

1 rn. 1

Normalized mean stress

Normalized mean stress

Figure 7.3 The evolution of the yield surface with strain for an Alporas foam with an initial relative density of 0.16

where the equivalent strain rate £ is the work conjugate of a, such that as = O-ij£0.

and can be written explicitly as

A unique hardening relation is given by the evolution of a with £. Unless otherwise stated, the uniaxial compressive stress-strain response is used to define the a — £ relation, as follows. The true stress a versus logarithmic plastic strain £P curve in uniaxial compression is written in the incremental form

where the slope h evolves with increasing stress level a. Recall that, for the case of uniaxial compression (or tension), the above definitions of a and of £ have been so normalized that a is the uniaxial stress and £ is the uniaxial plastic strain rate. The hardening law (7.17) for uniaxial loading can then be rewritten as s = a/h(a)

It is assumed that this relation holds also for general multi-axial loading. Some checks on the accuracy of this approach are given in Figure 7.4: the measured tensile, compressive and shear stress-strain curves for Alporas and Al 6101-T6 Duocel foams are shown in terms of a versus £. It is noted that

- Hydrostatic compression ,

- Uniaxial compression '

Alporas 13%

Alporas 8%

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1. True (Volumetric or Axial) strain

Figure 7.4 Demonstration of the ability of the equivalent stress a and the equivalent strain e to define uniquely the stress-strain response of Alporas CP = 0.16) and Duocel (p = 0.071) foams. The tension, compression and shear response are plotted in terms of a and e the curves almost collapse unto a unique curve for a given material, up to its peak strength.

The constitutive law for plastic flow is completely specified by the yield surface as defined by equations (7.11) and (7.12), and by the flow rule (7.15), with the definitions (7.16b) and (7.18). Explicit expressions can be given for (90/9oj) in (7.15), and for d in (7.18). They are:

An example of the use of these equations is given in Chapter 9, Section 9.5.

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