Читать книгу Properties for Design of Composite Structures - Neil McCartney - Страница 26

Unlike the situation for fluids, the deformation (i.e. motion) of solid media is defined relative to an initial homogeneous reference state, thus enabling the introduction of the concept of strain, which takes account of both hydrostatic deformation considered in Section 2.9 and shear deformation. Following the approach in [1], consider now a homogeneous solid body with an initial configuration that occupies a region B¯, and which deforms into a different region B after the application of applied stresses and temperature changes. Let p¯ denote the position vector of some point in the undeformed body B¯ referred to Cartesian coordinates x¯, that moves to the point p in the deformed body B referred to Cartesian coordinates x. Let i¯K denote the orthogonal unit base vectors for the coordinate system x¯, and ik denote the orthogonal unit base vectors for the coordinate system x. The position vectors p¯ and p may then be expressed in the following form

(2.70)

where summations over values K, k = 1, 2, 3, are implied for repeated suffices. The corresponding infinitesimal vectors are written as

(2.71)

As

(2.72)

it follows that

(2.73)

Any vector v may be written as

(2.74)

Define δKl and δkL by the relation

(2.75)

It then follows that

(2.76)

It is clear that

(2.77)

The time-dependent deformation that transforms the undeformed region B¯ into the region B(t) may be expressed as

(2.78)

It then follows that

(2.79)

where the increments are taken at some time t such that dt = 0. In component form,

(2.80)

In the undeformed body, on using (2.71) and (2.73), the increment of arc length ds¯ is such that

(2.81)

where ckl is Cauchy’s symmetric deformation tensor. Similarly, for the deformed body, the line increment ds¯deforms to an increment ds such that

(2.82)

where CKL is Green’s symmetric deformation tensor. In dyadic form

(2.83)

where ∇¯ denotes the gradient with respect to the material coordinates x¯, and where the symmetric Green deformation tensor C may be written as

(2.84)

From (2.81) and (2.82)

(2.85)

The Eulerian and Lagrange strain tensors are defined by

(2.86)

so that (2.85) may also be written as

(2.87)

On using the relation u=x−x¯ for the displacement vector and (2.84), the Lagrangian strain tensor may be written in terms of the displacement vector as follows

(2.88)

where I is the symmetric fourth-order identity tensor (see (2.15)). The quantity E is the strain tensor that is used in finite deformation theory where there is no restriction on the degree of deformation provided that the deformation is continuous and the condition (2.18) is satisfied at all points in the system.

The invariants of the strain tensors in terms of principal stretches are given by the relations

(2.89)

(2.90)

(2.91)

It is clear that

(2.92)

It will be very useful to introduce here the principal values CJ, J = 1, 2, 3, of Green’s deformation tensor defined using the following relations

(2.93)

such that the symmetric tensor C may be written in the form

(2.94)

The quantities νJ,J=1,2,3, are orthogonal unit vectors defining the directions of the principal values. They have the following properties

(2.95)

The polar decomposition principle (see, for example, [2, Section 1.5]) states that the deformation gradient may be expressed in the following forms (dyadic and tensor)

(2.96)

where R is the orthogonal rigid rotation tensor having the properties R.RT=RT.R=I with det​(R)=±1, and where U and V are positive-definite symmetric right and left stretch tensors.

It follows from (2.84) and (2.96) that in tensor form

(2.97)

so that in dyadic form

(2.98)

The symmetric tensors U and V have common eigenvalues λJ but different mutually orthogonal eigenvectors νJ such that

(2.99)

The eigenvalues λJ, J = 1, 2, 3, are the principal stretches and

(2.100)

It then follows that the principal values CJ of the C may be written in terms of the principal stretches λJ as follows

(2.101)

It follows from (2.94) that

(2.102)

and on multiplying by νK using the properties (2.95) it can be shown that

(2.103)

In terms of the principal values C₁, C₂ and C₃ of the tensor C and E₁, E₂ and E₃ of the tensor E, the corresponding invariants may be written as (see [1, Section 1.10])

(2.104)

It can be shown that

(2.105)

where ρ0 is the uniform mass density before deformation has occurred at some reference temperature T₀ and reference pressure p0.