Local form of the work theorem

The local form of the work theorem is is the well known work-energy theorem of the theory of structures, locally defined in a local domain or region, used in the development of numerical methods. In an arbitrary local region, the work theorem establishes an energy relationship between a statically-admissible stress field and an independent kinematically-admissible strain field.

This method was first presented by Oliveira and Portela (2016),^[1] for the development of two new meshfree methods: The generalized-strain mesh-free (GSMF) formulation and the rigid-body displacement mesh-free (RBDMF) formulation.

Development

Let $\Omega$ be the domain of a body and $\Gamma$ its boundary subdivided in ${\displaystyle \Gamma _{u}}$ and $\Gamma _{t}$ that is ${\displaystyle \Gamma =\Gamma _{u}\cup \Gamma _{t}}$. The mixed fundamental boundary value problem of linear elastostatics aims to determine the distribution of stresses ${\boldsymbol {\sigma }}$, strains $\boldsymbol{\varepsilon}$ and displacements $\mathbf {u}$ throughout the body, when it has constrained displacements ${\displaystyle {\overline {\mathbf {u} }}}$ defined on ${\displaystyle \Gamma _{u}}$ and is loaded by an external system of distributed surface and body forces with densities denoted by ${\displaystyle {\overline {\mathbf {t} }}}$ on $\Gamma _{t}$ and $\mathbf {b}$ in $\Omega$, respectively.

A totally admissible elastic field is the solution of the posed problem that simultaneously satisfies the kinematic admissibility and the static admissibility. If this solution exists, see Fredholm (1906)^[2] and Fichera (1965),^[3] it can be shown that it is unique, provided linearity and stability of the material are admitted.

The general work theorem establishes an energy relationship between any statically-admissible stress field and any kinematically-admissible strain field that can be defined in the body. Derived as a weighted residual statement, the work theorem serves as a unifying basis for the formulation of numerical models in continuum mechanics, as seen in Brebbia (1985).^[4]

In the domain of the body, consider a statically-admissible stress field that is

${\displaystyle \mathbf {L} ^{T}{\boldsymbol {\sigma }}+\mathbf {b} =\mathbf {0} }$

in the domain $\Omega$, with boundary conditions

${\displaystyle \mathbf {t} =\mathbf {n} \,{\boldsymbol {\sigma }}={\overline {\mathbf {t} }}}$

on the static boundary ${\displaystyle \Gamma _{t}}$, in which the vector ${\boldsymbol {\sigma }}$ represents the stress components; $\mathbf {L}$ is a matrix differential operator; the vector $\mathbf {t}$ represent the traction components; ${\displaystyle {\overline {\mathbf {t} }}}$ represent prescribed values of tractions and $\mathbf {n}$ represents the outward unit normal components to the boundary.

In the global domain $\Omega$, consider an arbitrary local subdomain ${\displaystyle \Omega _{Q}}$, centered at the point $Q$, with boundary ${\displaystyle \Gamma _{Q}=\Gamma _{Qi}\cup \Gamma _{Qt}\cup \Gamma _{Qu}}$, in which ${\displaystyle \Gamma _{Qi}}$ is the interior local boundary, while ${\displaystyle \Gamma _{Qt}}$ and ${\displaystyle \Gamma _{Qu}}$ are local boundaries that respectively share a global boundary. Due to its arbitrariness, this local domain can be overlapping with other similar subdomains. For the local domain ${\displaystyle \Omega _{Q}}$, the strong form of the weighted-residual equation is written as

${\displaystyle \int \limits _{\Omega _{Q}}\left(\mathbf {L} ^{T}{\boldsymbol {\sigma }}+\mathbf {b} \right)^{T}\mathbf {W} _{\Omega }\,d\Omega +\int \limits _{\Gamma _{Qt}}\left(\mathbf {t} -{\overline {\mathbf {t} }}\right)^{T}\mathbf {W} _{\Gamma }\,d\Gamma =\mathbf {0} ,}$

in which ${\displaystyle \mathbf {W} _{\Omega }}$ and ${\displaystyle \mathbf {W} _{\Gamma }}$ are arbitrary weighting functions defined, respectively in $\Omega$ and on $\Gamma$. When the domain term is integrated by parts, the following local weak form of the weighted residual equation is obtained

${\displaystyle \int \limits _{\Gamma _{Q}}\left(\mathbf {n} {\boldsymbol {\sigma }}\right)^{T}\mathbf {W} _{\Omega }\,d\Gamma -\int \limits _{\Omega _{Q}}\left({\boldsymbol {\sigma }}^{T}\,\mathbf {L} \mathbf {W} _{\Omega }-\mathbf {b} ^{T}\mathbf {W} _{\Omega }\right)\,d\Omega +\int \limits _{\Gamma _{Qt}}\left(\mathbf {t} -{\overline {\mathbf {t} }}\right)^{T}\mathbf {W} _{\Gamma }\,d\Gamma =\mathbf {0} }$

which now requires continuity of ${\displaystyle \mathbf {W} _{\Omega }}$, as an admissibility condition for integrability. For the sake of convenience, the arbitrary weighting function ${\displaystyle \mathbf {W} _{\Gamma }}$ is chosen as

${\displaystyle \mathbf {W} _{\Gamma }=-\mathbf {W} _{\Omega },}$

on the boundary ${\displaystyle \Gamma _{Qt}}$. Thus, leads to

${\displaystyle \int \limits _{\Gamma _{Q}-\Gamma _{Qt}}\mathbf {t} ^{T}\mathbf {W} _{\Omega }\,d\Gamma +\int \limits _{\Gamma _{Qt}}{\overline {\mathbf {t} }}^{T}\mathbf {W} _{\Omega }\,d\Gamma -\int \limits _{\Omega _{Q}}\left({\boldsymbol {\sigma }}^{T}\,\mathbf {L} \mathbf {W} _{\Omega }-\mathbf {b} ^{T}\mathbf {W} _{\Omega }\right)\,d\Omega =\mathbf {0} .}$

Consider further an arbitrary kinematically-admissible strain field ${\displaystyle {\boldsymbol {\varepsilon }}^{*}}$, with continuous displacements ${\displaystyle \mathbf {u} ^{*}}$ and small derivatives, in order to assume geometrical linearity, defined in the global domain that is

${\displaystyle {\boldsymbol {\varepsilon }}^{*}=\mathbf {L} \,\mathbf {u} ^{*},}$

in the domain $\Omega$, with boundary conditions

${\displaystyle \mathbf {u} ^{*}={\overline {\mathbf {u} }},}$

on the kinematic boundary ${\displaystyle \Gamma _{u}}$.

When the continuous arbitrary weighting function ${\displaystyle \mathbf {W} _{\Omega }}$, is defined as

${\displaystyle \mathbf {W} _{\Omega }=\mathbf {u} ^{*},}$

the weak form, of the weighted residual equation, becomes

${\displaystyle \int \limits _{\Gamma _{Q}-\Gamma _{Qt}-\Gamma _{Qu}}\mathbf {t} ^{T}\mathbf {u} ^{*}\,d\Gamma +\int \limits _{\Gamma _{Qu}}\mathbf {t} ^{T}{\overline {\mathbf {u} }}^{*}\,d\Gamma +\int \limits _{\Gamma _{Qt}}{\overline {\mathbf {t} }}^{T}\mathbf {u} ^{*}\,d\Gamma -\int \limits _{\Omega _{Q}}\left({\boldsymbol {\sigma }}^{T}\,\mathbf {L} \mathbf {u} ^{*}-\mathbf {b} ^{T}\mathbf {u} ^{*}\right)\,d\Omega =\mathbf {0} }$

which can be written in a compact form as

${\displaystyle \int \limits _{\Gamma _{Q}}\mathbf {t} ^{T}\mathbf {u} ^{*}\,d\Gamma +\int \limits _{\Omega _{Q}}\mathbf {b} ^{T}\mathbf {u} ^{*}\,d\Omega =\int \limits _{\Omega _{Q}}{\boldsymbol {\sigma }}^{}{\boldsymbol {\varepsilon }}^{*}\,d\Omega .}$

This equation which expresses the static-kinematic duality, is the local form of the well-known work theorem, the fundamental identity of solid mechanics, see Sokolnikoff (1956).^[5] This equation is the starting point of the kinematically admissible formulations of the local meshfree methods, such as the generalized-atrain mesh-free (GSMF) formulation and the rigid-body displacement mesh-free (RBDMF) formulation.

It can be notice that the stress field ${\boldsymbol {\sigma }}$, is any one that satisfies equilibrium with the applied external forces $\mathbf {b}$ and $\mathbf {t}$, which is not necessarily the stress field that actually settles in the body. Also, the strain field ${\displaystyle {\boldsymbol {\varepsilon }}^{*}}$, is any one that is compatible with the constraints ${\displaystyle \mathbf {u} ^{*}={\overline {\mathbf {u} }}}$, which is not necessarily the strain field that actually settles in the body. This two fields are not linked by any constitutive relationship; indeed, they are completely independent as a consequence of the arbitrariness of the weighting function ${\displaystyle \mathbf {W} _{\Omega }}$. For that reason this formulation can be used under the only assumption of geometrical linearity.

See also

• Structural engineering theory
• Finite element method
• Boundary element method
• Meshfree methods
• Numerical analysis
• Computational solid mechanics

References

1. Oliveira, T. and A. Portela (2016). “Weak-Form Collocation – a Local Meshless Method in Linear Elasticity”. Engineering Analysis with Boundary Elements.
2. Fredholm, I. (1906), "Solution d'un probleme fondamental de la theorie del'elasticitee", Ark. Mat., Astr. Fysik, 2.
3. Fichera, G. (1965), "Linear Elliptic Diferential Systems and Eigenvalue Problems", Lecture Notes in Mathematics No. 8, Springer Verlag, Berlin, Heidelberg and New York.
4. Brebbia, C.A. (1985), "Variational Basis of Approximate Models in Continuum Mechanics", Proc. of the II International Conference on Variational Methods in Engineering, C.A. Brebbia and H. Tottenham (Editors), Southampton, 1985,Computational Mechanics Publications, Southampton and Springer Verlag, Berlin.
5. Sokolnikoff, I.S. (1956), "Mathematical Theory of Elasticity", McGraw-Hill, New York