清华大学弹性力学讲义chap2_Elasticity of Solids

2．Elasticity of Solids

References

J.H.Weiner ，Statistical mechanics of elasticity, Wiley, 1981

Green & Zerna ，Theoretical elasticity, 1968

Ashby & Jones ，Engineering materials

2.1 Definition of Elasticity Elasticity

σ

F

Figure 2.1 An elastic response.

An elastic response of the material can be abstracted mathematically as

()X F ,T σ= (2.1) where σ denotes the stress tensor, T the response function that depends only on the current values of the deformation gradient X x F ∂∂=, with X denoting the material coordinates of a point while x the spatial coordinates. If the material is homogeneous within the domain under consideration, the explicit dependence on X in (2.1) can be eliminated. Several remarks can be made to the definition in (2.1):

(1) In the claim of ()()X t X,

F ,T σ=, one pins down an elastic response as the one prtrayed by the current status of deformation, and henceforth irrelevant to the

history or the process of deformation. The strain rate plays no role in the constitutive response. A hysteresis is ruled out, as shown in Fig. 2.1. Whenever the loading is removed, the original configuration (or the “non-distorted configuration”) is recovered.

(2)For the special case of infinitesimal deformation, the response (2.1) is reduced to

()X,ε

σ=, where εdenotes the strain tensor. The response function T is not T

necessarily linear.

(3)For homogeneous materials, one has ()F

σ=in finite deformation and

T

()ε

σ=for infinitesimal deformation.

T

(4)For the even special case of infinitesimal deformation, homogeneous material and

linear elasticity, the generalized Hooke’s law ε

=is recovered, with C

σ:

C

being the fourth-rank stiffness tensor. The notations of C as the stiffness tensor and S as the compliance tensor, not the otherwise according to their initials, unfortunately became the convention in the historic development of elasticity.

The difference between the material responses at a solid state and a fluid state can be quoted as follows (Mechanics of Solids, The New Encyclopedia of Britannica, 15th edition, V ol. 23, pp. 734-747, 2002, written by J. R. Rice):

“A material is called solid rather than fluid if it can also support a substantial shearing force over the time scale of some natural process or technological application of interest.”

An elastic solid can resist the volume and shape changes, whereas a fluid can only resist the volume change but not the shape changes in a relatively long time scale. Hyperelasticity

Hyperelasticity refers to an elastic response that can be defined by a potential. The basic assumptions are: (1) the response of the elastic body only depends on its current state, not the processes to achieve it; and (2) the current state of the elastic body can be described by a tensor, such as the strain tensor εfor the special case of infinitesimal deformation. The first assumption leads to the independence of