## 61 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, v, are the material properties which enter the constitutive equation for linear-elastic deformation; the elastic limit, ay, is the material 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, a, is given first; below it is the equation for multi-axial loading by principal stresses a\, a2 and a3, always chosen so that a\ is the most tensile and a3 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 linear-elastic region Hooke's law (top box, Table 6.1) applies. Because they change

Table 6.1 Constitutive equations for mechanical response

Isotropic solids: elastic deformation

Table 6.1 Constitutive equations for mechanical response

Isotropic solids: elastic deformation

Uniaxial |
_ ai £l ~ E |

General |
a1 v £i = — - -(0-2+0-3) EE |

Isotropic solids: plastic deformation | |

Uniaxial |
a1 ^ ay |

General |
ae ^ ay (Von Mises) with o2e = \[{a 1 - 0-2)2 + (02 - 0-3)2 + (03 - ai)2] |

Metal foams: elastic deformation |

Uniaxial As isotropic solids - though some foams are

General anisotripic

Metal foams: plastic deformation | ||

Uniaxial |
a1 ^ ay | |

General |
a ^ ay | |

with a2 D |
1 9 [o-2 + cc2o-2] (1 + (a/3) ) | |

and am d |
5(0-1 +0-2 +0-3) |

Material properties

### Material properties

E = Young's modulus ay = Yield strength v = Poisson's ratio a = Yield constant volume when deformed plastically (unlike fully dense metals), a hydrostatic pressure influences yielding. A constitutive equation which describes their plastic response is listed in Table 6.1. It differs fundamentally from those for fully dense solids. Details are given in Chapter 7.

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