Fundamentals of Linear Elasticity
In this chapter a number of concepts are introduced to describe a linear elastic body. In particular, the displacement vector, strain tensor, and stress tensor fields are introduced to define a linear elastic body which satisfies the strain-displacement r
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Fundamentals of Linear Elasticity
In this chapter a number of concepts are introduced to describe a linear elastic body. In particular, the displacement vector, strain tensor, and stress tensor fields are introduced to define a linear elastic body which satisfies the strain-displacement relations, the equations of motion, and the constitutive relations. Also, the compatibility relations, the general solutions of elastostatics, and an alternative definition of the displacement field of elastodynamics are discussed. The stored energy of an elastic body, the positive definiteness and strong ellipticity of the elasticity fourth-order tensor, and the stress-strain-temperature relations for a thermoelastic body are also discussed.
2.1 Deformation of an Elastic Body A material body B is defined as a set of elements x, called particles, for which there is a one-to-one correspondence with the points of a region κ(B) of a physical space; while a deformation of B is a map κ of B onto a region κ(B) in E 3 with det (∇κ) > 0. The point κ(x) is the place occupied by the particle x in the deformation κ, and u(x) = κ(x) − x
(2.1)
is the displacement of x. If the mapping κ depends also on time t ∈ [0, ∞), such a mapping defines a motion of B, and the displacement of x at time t is u(x, t) = κ(x, t) − x
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_2, © Springer Science+Business Media Dordrecht 2013
(2.2)
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2 Fundamentals of Linear Elasticity
By the deformation gradient and the displacement gradient we mean the tensor fields F = ∇κ and ∇u, respectively. A finite strain tensor D is defined by D=
1 T (F F − 1) 2
(2.3)
or, equivalently, by 1 D = E + (∇u)(∇uT ) 2 where E=
1 (∇u + ∇uT ) = ∇u 2
(2.4)
(2.5)
The tensor field E is called an infinitesimal strain tensor. An infinitesimal rigid displacement of B is defined by u(x) = u0 + W(x − x0 )
(2.6)
where u0 , x0 are constant vectors and W is a skew constant tensor. An infinitesimal volume change of B is defined by δv(B) =
div u dv
(2.7)
B
while div u = tr E
(2.8)
represents a dilatation field. If a deformation is not accompanied by a change of volume, that is, if δv(P) = 0 for every P ⊂ B, the displacement u is called isochoric. Kirchhoff Theorem. If two displacement fields u1 and u2 correspond to the same strain field E then (2.9) u1 − u2 = w where w is a rigid displacement field. A homogeneous displacement field is defined by u(x) = u0 + A(x − x0 )
(2.10)
where A is an arbitrary constant tensor and u0 , x0 are constant vectors. Clearly, if A is skew, (2.10) represents a rigid displacement, while for an arbitrary A u(x) = u1 (x) + u2 (x)
(2.11)
2.1 Deformation of an Elastic Body
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where u1 (x) is a rigid displacement field and u2 (x) is a displacement field corresponding to the strain E = sym A. The displacement u2 (x) of the form u2 (x) = E (x − x0 )
(2.12)
corresponds to a pure strain from x0 . Let e > 0 and let n be a unit vector. Then by substituting
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