Metal Foams: A Design Guide

Chapter 6
Design formulae for simple structures
The formulae assembled here are useful for approximate analyses of the
response of structures to load (more details can be found in Young, (1989).
Each involves one or more material property. Results for solid metals appear
under heading (a) in each section. The properties of metal foams differ greatly
from those of solid metals. Comments on the consequences of these differences
appear under heading (b) in each section.
6.1 Constitutive equations for mechanical response
(a) Isotropic solids
The behavior of a component when it is loaded depends on the mechanism by
which it deforms. A beam loaded in bending may deflect elastically; it may
yield plastically; it may deform by creep; and it may fracture in a brittle or in
a ductile way. The equation which describes the material response is known
as a constitutive equation, which differ for each mechanism. The constitutive
equation contains one or more material properties: Young’s modulus, E, and
Poisson’s ratio, , are the material properties which enter the constitutive
equation for linear-elastic deformation; the elastic limit, y, isthematerial
property which enters the constitutive equation for plastic flow; the hardness,
H, enters contact problems; the toughness JIC enters that for brittle fracture.
Information about these properties can be found in Chapter 2.
The common constitutive equations for mechanical deformation are listed
in Table 6.1. In each case the equation for uniaxial loading by a tensile stress,
, is given first; below it is the equation for multi-axial loading by principal
stresses 1, 2 and 3, always chosen so that 1 is the most tensile and 3 the
most compressive (or least tensile) stress. They are the basic equations which
determine mechanical response.
(b) Metal foams
Metal foams are approximately linear-elastic at very small strains. In the linearelastic
region Hooke’s law (top box, Table 6.1) applies. Because they change