Equilibrium of Masonry Bodies
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The reader is assumed to have some familiarity with the elements of linear algebra. Here we recall some results of tensor algebra and tensor analysis that will be used later on. For further details see e.g. [51], [52], [19] and [53].
1.1 Notations Italic boldface minuscules a, b, u, v,...: vectors and vector fields; x, y, z, ...: points of space. Italic boldface majuscules A, B,...: (second-order) tensors and tensor fields. Symbol Name A, C B b β C E E Ee Ef Ec E I I Lin Orth R s S
Fourth-order tensors Body Body force Thermal expansion Elasticity tensor Young’s modulus Infinitesimal strain tensor Elastic part of the strain Fracture strain Crushing strain Three-dimensional Euclidean space Second-order identity tensor Fourth-order identity tensor Space of second-order tensors Set of orthogonal tensors Set of real numbers surface force n−dimensional real vector space
2
1 Elements of Tensor Algebra and Analysis
Skw Sym Sym+ Sym− T T BC T NI T u V λ μ σt σc ψ ϑ ϑ0 ⊗ ()· ()T ∇ div tr
Space of skew-symmetric tensors Space of symmetric tensors Set of positive semidefinite symmetric tensors Set of negative semidefinite symmetric tensors Cauchy stress tensor Stress function for masonry-like materials Stress function for masonry-like materials with bounded compressive strength Stress function for masonry-like materials under non-isothermal conditions Displacement vector Three-dimensional real vector space Lam´e modulus Lam´e modulus Tensile strength Compressive strength Strain energy density Absolute temperature Reference temperature Tensor product Time derivative Transpose Gradient Divergence Trace
1.2 Finite-Dimensional Vector Spaces Let S be a real vector space; the elements of S are called vectors. An inner product (or scalar product) on S is a function ( , ) defined on S × S with values in R such that 1. (a, b) = (b, a) for each a, b ∈ S (symmetry), 2. (α1 a1 + α2 a2 , b) = α1 (a1 , b) + α2 (a2 , b) for each a1 , a2 , b ∈ S, and α1 , α2 ∈ R (bilinearity), 3. (a, a) ≥ 0 for each a ∈ S and (a, a) = 0 if and only if a = 0 (positiveness). For a ∈ S, a = (a, a) is the norm or length of a. Moreover, for a, b ∈ S, the Schwarz inequality |(a, b)| ≤ a b (1.1) and the parallelogram law
1.2 Finite-Dimensional Vector Spaces 2
2
2
a + b + a − b = 2 a + 2 b
2
3
(1.2)
both hold. Vectors a, b ∈ S are orthogonal if (a, b) = 0. Vectors u1 , ..., uk are orthonormal if 1, i = j, (1.3) (ui , uj ) = δij = 0, i = j. Orthonormal vectors are linearly independent, that is α1 u1 + ... + αk uk = 0 implies
α1 = ... = αk = 0.
(1.4)
Throughout the chapter we consider vector spaces S with an inner product and finite dimension n, and indicate by {e1 , ..., en } an orthonormal basis of S, i.e., a set of orthonormal vectors generating S. In other words, vectors e1 , ..., en satisfy (1.3), and moreover, for each u ∈ S, there exist unique n real numbers β1 , ..., βn such that u=
n
βi ei .
(1.5)
i=1
A vector space S with inner product is a metric space with the distance d(u, v) = ||u − v||, for every u, v ∈ S.
(1
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