2.1 Modelling Tools
25
∂B of the one-dimensional domain B) may be transformed to a domain integral by the
one-dimensional format of the Gauss theorem (which follows in a one-dimensional
context simply from the fundamental lemma of calculus)
∂B
σ d dx =
B
σ
dx =⇒ 0 =
B
[b + σ
] dx.
(2.2)
Requiring finally that the above global statement holds as well for arbitrary subsets
V ⊂ B of the configuration, a process called localization, renders the corresponding
local statement for the equilibrium of forces
1
− σ
= b in B and σ = ¯
σ on ∂B
σ
.
(2.3)
The generic continuum mechanics problem is summarized in Table 2.1. The local
statement is also denoted as the strong form of the equilibrium of forces due to the
differentiability requirements posed by the presence of the σ
term. A corresponding
weak form, with reduced differentiability requirements for σ is most useful. The
standard
2 solution space is then taken as
U = {u ∈ H
1
(B) in B and u = ¯
u on ∂B
u
},
(2.4)
whereas the space of test functions, which may be interpreted as virtual displacements, is defined as
U
0
= {u ∈ H
1
(B) in B and u = 0 on ∂B
u
}.
(2.5)
Recall that H
1
(B) denotes the space of functions δu with square-integrable derivatives δu
:= δδ on B. Then multiplying the strong form by δu, integrating over B
and ∂B
σ , applying integration by parts and the one-dimensional format of the Gauss
theorem renders eventually the weak form as follows: Find u ∈ U that solves
B
δδ σ dx =
B
δu b dx +
∂B σ
δu ¯
σ d dx ∀ δu ∈ U
0
.
(2.6)
Note the differentiation has here been shifted from the stress σ to the test function
δu
. The weak form is also denoted as the Principle of Virtual Work: If equilibrium
of forces holds, the internal virtual work performed by the stresses along the virtual
strains equals the virtual work performed by the external forces along the virtual
displacements. An approximate solution to the weak form is typically determined
via the Finite Element Method.
1 In one dimension the equilibrium statement is statically determinate. Thus the stress may here be
computed directly from σ (x) = σ (x 0 ) −
x
x0 b( ¯
x) d ¯
x. Note however, that in two and three dimensions the equilibrium statement is obviously statically indeterminate.
2 The regularity that is implicit in this choice of solution and test spaces is considered as the appropriate one for a very large class of problems. However, it would exclude, say, slip line solutions in
rigid plasticity that involve displacement discontinuities.
25
∂B of the one-dimensional domain B) may be transformed to a domain integral by the
one-dimensional format of the Gauss theorem (which follows in a one-dimensional
context simply from the fundamental lemma of calculus)
∂B
σ d dx =
B
σ
dx =⇒ 0 =
B
[b + σ
] dx.
(2.2)
Requiring finally that the above global statement holds as well for arbitrary subsets
V ⊂ B of the configuration, a process called localization, renders the corresponding
local statement for the equilibrium of forces
1
− σ
= b in B and σ = ¯
σ on ∂B
σ
.
(2.3)
The generic continuum mechanics problem is summarized in Table 2.1. The local
statement is also denoted as the strong form of the equilibrium of forces due to the
differentiability requirements posed by the presence of the σ
term. A corresponding
weak form, with reduced differentiability requirements for σ is most useful. The
standard
2 solution space is then taken as
U = {u ∈ H
1
(B) in B and u = ¯
u on ∂B
u
},
(2.4)
whereas the space of test functions, which may be interpreted as virtual displacements, is defined as
U
0
= {u ∈ H
1
(B) in B and u = 0 on ∂B
u
}.
(2.5)
Recall that H
1
(B) denotes the space of functions δu with square-integrable derivatives δu
:= δδ on B. Then multiplying the strong form by δu, integrating over B
and ∂B
σ , applying integration by parts and the one-dimensional format of the Gauss
theorem renders eventually the weak form as follows: Find u ∈ U that solves
B
δδ σ dx =
B
δu b dx +
∂B σ
δu ¯
σ d dx ∀ δu ∈ U
0
.
(2.6)
Note the differentiation has here been shifted from the stress σ to the test function
δu
. The weak form is also denoted as the Principle of Virtual Work: If equilibrium
of forces holds, the internal virtual work performed by the stresses along the virtual
strains equals the virtual work performed by the external forces along the virtual
displacements. An approximate solution to the weak form is typically determined
via the Finite Element Method.
1 In one dimension the equilibrium statement is statically determinate. Thus the stress may here be
computed directly from σ (x) = σ (x 0 ) −
x
x0 b( ¯
x) d ¯
x. Note however, that in two and three dimensions the equilibrium statement is obviously statically indeterminate.
2 The regularity that is implicit in this choice of solution and test spaces is considered as the appropriate one for a very large class of problems. However, it would exclude, say, slip line solutions in
rigid plasticity that involve displacement discontinuities.