3-dimensional solid

7.1. 3-dimensional solid#

Starting from strong form of equilibrium equations, inner compatibility and congruence with essential boundary conditions, it’s possible to:

derive a weak formulation of the problem
derive energy theorems, with a proper choice of the test function involved in the weak formulation.

7.1.1. Strong formulation of the problem#

Indefinite equilibrium and natural boundary conditions on $S_N$.

\[\begin{split}\begin{cases} \nabla \cdot \boldsymbol \sigma + \overline{\mathbf{f}} = \mathbf{0} & \text{ in $V$} \\ \hat{\mathbf{n}} \cdot \boldsymbol\sigma = \overline{\mathbf{t}}_{\mathbf{n}} & \text{on $S_N$} \end{cases}\end{split}\]

Internal congruence and compatibility with essential constraints on $S_D$.

\[\begin{split}\begin{cases} \boldsymbol\varepsilon = \nabla^S \mathbf{s} = \dfrac{1}{2} \left( \nabla \mathbf{s} + \nabla^T \mathbf{s} \right) & \text{in $V$} \\ \mathbf{s} = \overline{\mathbf{s}} & \text{on $S_D$} \end{cases}\end{split}\]

Other boundary conditions - e.g. Robin. Beside essential boundary conditions (prescribing the displacement) and natural boundary condtions (prescribing the stress vector), other boundary conditions may exist. As an example, Robin boundary conditions are defined as a boundary condition prescribing a linear combination of displacement and stress, and may represent flexible constraints. The most general affine relation between dispacement and stress vector reads

\[\mathbf{a} \cdot \mathbf{s} = \mathbf{b} \cdot \boldsymbol\sigma \cdot \hat{\mathbf{n}} + \mathbf{c} \qquad \text{on $S_R$} \ ,\]

having exploited here the symmetry (todo) of the stress tensor $\boldsymbol\sigma$. If $\boldsymbol\alpha$ is invertible, the latter relation may be written in the form

\[\mathbf{s} = \mathbf{b} \cdot \boldsymbol\sigma \cdot \hat{\mathbf{n}} + \mathbf{c} \qquad \text{on $S_R$} \ .\]

Linear elastic constitutive equation. Constitutive equation of linear elastic media in the regime of small displacement reads

\[\boldsymbol\varepsilon = \mathbf{D} : \boldsymbol\sigma + \boldsymbol\alpha \, \Delta T \ ,\]

being the latter the contribution of thermal strains. The “inverse” relation reads

\[\boldsymbol\sigma = \mathbf{C} : \boldsymbol\varepsilon - \boldsymbol\beta \, \Delta T \ .\]

If temperature field is prescribed and known it can be treated as a forcing, otherwise its an unkown physical quantity and the internal energy (or temperature) balance equation needs to be solved along with the mechanical equilibrium equation.

7.1.2. Weak formulations of the problem#

7.1.2.1. Weak formulation of equilibrium conditions#

For every1 function $\mathbf{w}$

\[\begin{split}\begin{aligned} 0 & = \int_{V} \mathbf{w} \cdot \left\{ \nabla \cdot \boldsymbol\sigma + \overline{\boldsymbol{f}} \right\} = \\ & = \int_{V} w_j \left\{ \sigma_{ij/i} + \overline{f}_j \right\} = \\ & = \oint_{\partial V} n_i \sigma_{ij} w_j - \int_{V} w_{j/i} \sigma_{ij} + \int_{V} w_j \overline{f}_j = \\ & = - \int_{V} w_{j/i} \sigma_{ij} + \int_{V} w_j \overline{f}_j + \int_{S_N} w_j \overline{t}_n + \int_{\partial V / S_N} w_j t_j = \\ & = - \int_{V} \dfrac{1}{2} \left( w_{j/i} + w_{i/j} \right) \sigma_{ij} + \int_{V} w_j \overline{f}_j + \int_{S_N} w_j \overline{t}_n + \int_{\partial V / S_N} w_j t_j = \\ \end{aligned}\end{split}\]

having exploited symmetry of stress tensor $\boldsymbol\sigma$.

7.1.2.2. Weak formulation of congruence conditions#

For every 2$^{nd}$ order tensor function $\boldsymbol\Omega$

\[\begin{split}\begin{aligned} 0 & = \int_V \boldsymbol\Omega : \left\{ \boldsymbol\varepsilon - \dfrac{1}{2} \left( \nabla \mathbf{s} + \nabla^T \mathbf{s} \right) \right\} = \\ & = \int_V \Omega_{ij} \left\{ \varepsilon_{ij} - \frac{1}{2}\left( s_{i/j} + s_{j/i} \right) \right\} = \\ & = \int_V \dfrac{1}{2} \left\{ \Omega_{ij/j} s_i + \Omega_{ij/i} s_j \right\} + \int_{V} \Omega_{ij} \varepsilon_{ij} - \oint_{\partial V} \dfrac{1}{2} \left\{ n_j \Omega_{ij} s_i + n_i \Omega_{ij} s_j \right\} \ . \end{aligned}\end{split}\]

If $\boldsymbol\Omega$ is chosen to be symmetric,

\[\begin{split}\begin{aligned} 0 & = \int_V \Omega_{ij/j} s_i + \int_{V} \Omega_{ij} \varepsilon_{ij} - \oint_{\partial V} n_i \Omega_{ij} s_j = \\ & = \int_V \Omega_{ij/j} s_i + \int_{V} \Omega_{ij} \varepsilon_{ij} - \int_{S_D} n_i \Omega_{ij} \overline{s}_j - \int_{\partial V / S_D} n_i \Omega_{ij} s_j \ . \end{aligned}\end{split}\]

7.1.2.3. Existence and uniqueness of the solution#

Theorem 7.1 (Existence and uniqueness of the solution of the small-strain, small-displacement elastic problem)

Under the assumptions …, there exists a unique solution of the elastic problem that is at the same time congruent and equilibrated.

In structural mechanics, it’s quite common to deal with congruent displacement and strain fields, and equilibrated stress fields. Stress and strain fields are related via the constitutive law of the medium. A congruent displacement and strain field may produce non-equilibrated stress fields; an equilibrated stress field may produce non-congruent displacement field. Under the assumptions… it’s possible to prove that the lienar elastic problem has a unique solution corresponding to the congruent strain and displacement and equilibrated stress fields, i.e. the unique set of fields simultaneously satisfying equilibrium equations and constraints.

The proof of existence of a solution requires some functional analysis tools (not developed here, yet). The proof of uniqueness and the necessary conditions can be discussed with a proof by contradiction, assuming that two solutions of the same problem exists and evaluating the norm of the difference of the solution on the whole domain, quantifying the difference between these solutions, and eventually assessing that this norm is identically zero under certain assumptions.

The proof follows a similar procedure as the proof of uniqueness of the solution of elliptic problems.

Proof.

Equilibrium equation for statics can be written as a function of strain and displacement, exploiting the constitutive equation $\sigma_{ij} = C_{ijkl} \varepsilon_{kl} - \beta_{ij} \Delta T$,

\[\begin{split}\begin{aligned} 0 & = \sigma_{ij/i} + f_j = \left( C_{ijkl} \varepsilon_{kl} - \beta_{ij} \Delta T \right)_{/i} + f_j && \text{in $V$} \\ g_i & = s_i && \text{on $S_D$} \\ t_j & = n_i \sigma_{ij} = n_i ( C_{ijkl} \varepsilon_{kl} - \beta_{ij} \Delta T ) && \text{on $S_N$} \\ \end{aligned}\end{split}\]

Let’s assume two strain fields exists s.t. their solution of the elastic problem, and let’s define $\delta \boldsymbol\varepsilon = \boldsymbol\varepsilon_2(\mathbf{r}) - \boldsymbol\varepsilon_1(\mathbf{r})$. This difference satisfies the homogeneous problem

\[\begin{split}\begin{aligned} 0 & = C_{ijkl} \delta \varepsilon_{kl} && \text{in $V$} \\ 0 & = \delta s_i && \text{on $S_D$} \\ 0 & = n_i \, C_{ijkl} \delta \varepsilon_{kl} && \text{on $S_N$} \ , \end{aligned}\end{split}\]

given a prescribed temperature field $\Delta T$.

…

\[\begin{split}\begin{aligned} \int_{V} \delta \varepsilon_{ij} C_{ijkl} \delta \varepsilon_{kl} & = \int_{V} \dfrac{1}{2} \left( \delta s_{i/j} + \delta s_{j/i} \right) C_{ijkl} \delta \varepsilon_{kl} = \\ & = \int_{V} \delta s_{j/i} C_{ijkl} \delta \varepsilon_{kl} = \\ & = \int_{V} \left\{ \left( \delta s_{j} C_{ijkl} \delta \varepsilon_{kl} \right)_{/i} - \delta s_{j} \left( C_{ijkl} \delta \varepsilon_{kl} \right)_{/i} \right\} = \\ & = \oint_{\partial V} n_i \, \delta s_{j} C_{ijkl} \delta \varepsilon_{kl} - \int_V \delta s_{j} \left( C_{ijkl} \delta \varepsilon_{kl} \right)_{/i} = \\ & = \int_{S_D} n_i \, \underbrace{ \delta s_{j} }_{\delta s_j |_{S_D} = 0} C_{ijkl} \delta \varepsilon_{kl} + \int_{S_N} n_i \, \delta s_j \underbrace{ C_{ijkl} \delta \varepsilon_{kl} }_{ C_{ijkl} \delta \varepsilon |_{S_N} = 0} - \int_V \delta s_{j} \underbrace{ \left( C_{ijkl} \delta \varepsilon_{kl} \right)_{/i}}_{ = 0} = \\ & = 0 \ . \end{aligned}\end{split}\]

As $\delta \varepsilon : \mathbf{C} : \delta \varepsilon \ge 0$ for every symmetric $2^{nd}$-order tensor (todo this is related to constraints on deformation energy…), the integral condition implies $\delta \boldsymbol\varepsilon : \mathbf{C} : \delta \boldsymbol\varepsilon = 0$ (todo check it). As $\mathbf{C}$ … (todo which property required?), this further implies $\delta \boldsymbol\varepsilon = \mathbf{0}$, and thus the two solution may differ at most by a rigid motion,

\[s_2(\mathbf{r}) = s_1(\mathbf{r}) + \mathbf{a} + \boldsymbol\theta_\times \mathbf{r} \ .\]

If the problem has boundary conditions preventing rigid motion, i.e. no rigid degree of freedom, there’s no arbitrariety in the solution as the boundary conditions set $\mathbf{a} = \mathbf{0}$, and $\boldsymbol\theta = \mathbf{0}$, and thus

\[\mathbf{s}_1(\mathbf{r}) = \mathbf{s}_2(\mathbf{r}) \ .\]

Comments and todo.

todo Comment condition $\boldsymbol\varepsilon : \mathbf{C} : \boldsymbol\varepsilon = \varepsilon_{ij} C_{ijkl} \varepsilon \ge 0$, with internal energy balance equation
Show explicitly the case of a linear isotropic medium, governed by the constitutive law $\sigma_{ij} = 2 \mu \varepsilon_{ij} + \lambda \varepsilon_{ll} \delta_{ij}$, i.e.

\[C_{ijkl} = \mu \delta_{ik} \delta_{jl} + \mu \delta_{il} \delta_{jk} + \lambda \delta_{ij} \delta_{kl} \ ,\]

i.e. the most general expression of an isotropic rank-$4$ tensor relating two isotropic and symmetric rank-$2$ tensors, like the stress and small strain tensors2. In this case the double dot product of the stress and strain tensors reads

\[\boldsymbol\varepsilon : \boldsymbol\sigma = \varepsilon_{ij} \sigma_{ij} = \varepsilon_{ij} \left( 2 \mu \varepsilon_{ij} + \lambda \varepsilon_{ll} \delta_{ij} \right) = 2 \mu \varepsilon_{ij} \varepsilon_{ij} + \lambda \varepsilon_{ll} \varepsilon_{kk} \delta_{ij} = 2 \mu |\boldsymbol\varepsilon|^2 + \lambda \text{tr}\left(\boldsymbol\varepsilon\right)^2 \ge 0 \ .\]

todo Any thermodynamic condition prescribing the sign of the coefficients $\mu$, $\lambda$? Anything like minimum energy or maximum entropy principle at thermodynamic equilibrium?

7.1.2.4. Principle of virtual work#

Starting from the weak form of equilibrium conditions, and choosing $\mathbf{w}$ to be the variation of a congruent displacement field $\widetilde{s}$ with internal congruence in $V$ and compatibility with given essential constraints on $S_D$, i.e. that satisfies the conditions

\[\begin{split}\begin{aligned} \widetilde{\boldsymbol{\varepsilon}} & := \dfrac{1}{2} \left( \nabla \widetilde{\mathbf{s}} + \nabla^T \widetilde{\mathbf{s}} \right) && \text{in $V$} \\ \widetilde{\mathbf{s}} & = \overline{\mathbf{s}} && \text{on $S_D$} \ , \end{aligned}\end{split}\]

with no other conditions on $\widetilde{\boldsymbol\sigma}$ in $V$ and $\widetilde{\mathbf{t}}_{\mathbf{n}}$ on $S_N$. From the definition $\mathbf{w} = \delta \widetilde{\mathbf{s}}$, it follows

\[\begin{split}\begin{aligned} \delta \widetilde{\boldsymbol{\varepsilon}} & = \dfrac{1}{2} \left( \nabla \delta \widetilde{\mathbf{s}} + \nabla^T \delta \widetilde{\mathbf{s}} \right) && \text{in $V$} \\ \delta \widetilde{\mathbf{s}} & = \mathbf{0} && \text{on $S_D$} \ , \end{aligned}\end{split}\]

and

\[\begin{aligned} 0 & = - \int_{V} \delta \widetilde{\varepsilon}_{ij} \sigma_{ij} + \int_{V} \delta \widetilde{s}_j \overline{f}_j + \int_{S_N} \delta \widetilde{s}_j \overline{t}_j + \int_{S_R} \delta \widetilde{s}_j t_j \ . \end{aligned}\]

7.1.2.5. Principle of complementary virtual work#

Starting from the weak form of congruence conditions, and choosing $\boldsymbol\Omega$ to be the variation of an equilibrated stress field $\widetilde{\boldsymbol{\sigma}}$ due to given external loads $\widetilde{\mathbf{f}}$ in $V$ and $\widetilde{\mathbf{t}}_{\mathbf{n}}$ on $S_N$, i.e. satisfying the conditions

\[\begin{split}\begin{cases} \widetilde{\sigma}_{ij/i} + \widetilde{f}_j = 0 & \text{in $V$} \\ n_i \widetilde{\sigma}_{ij} = \widetilde{t}_j & \text{in $S_N$} \end{cases}\end{split}\]

with no other conditions on $\boldsymbol\varepsilon$ and $\mathbf{s}$ in $V$ and $S_D$. From the definition $\Omega_{ij} = \delta \widetilde{\sigma}_{ij}$, it follows

\[\begin{split}\begin{cases} \delta \widetilde{\sigma}_{ij/i} = 0 & \text{in $V$} \\ n_i \, \delta \widetilde{\sigma}_{ij} = 0 & \text{in $S_N$} \end{cases}\end{split}\]

and

\[\begin{split}\begin{aligned} 0 & = \int_V \underbrace{\delta \widetilde{\sigma}_{ij/i}}_{= 0} s_j + \int_{V} \delta \widetilde{\sigma}_{ij} \varepsilon_{ij} - \int_{S_D} n_i \delta \widetilde{\sigma}_{ij} \overline{s}_j - \int_{S_N} \underbrace{n_i \delta \widetilde{\sigma}_{ij}}_{=0} s_j - \int_{S_R} n_i \delta \widetilde{\sigma}_{ij} s_j = \\ & = \int_{V} \delta \widetilde{\sigma}_{ij} \varepsilon_{ij} - \int_{S_D} n_i \delta \widetilde{\sigma}_{ij} \overline{s}_j - \int_{S_R} n_i \delta \widetilde{\sigma}_{ij} s_j \ . \end{aligned}\end{split}\]

7.1.2.6. Principle of stationariety of total potential energy#

Choosing the (unique) solution of the elastic problem as the compatible field used in the principle of virtual work, $\widetilde{\mathbf{s}} = \mathbf{s}$, $\widetilde{\boldsymbol{\varepsilon}} = \boldsymbol{\varepsilon}$, it follows that

\[\begin{aligned} 0 & = - \int_V \delta \boldsymbol\varepsilon : \boldsymbol\sigma + \int_{V} \delta \mathbf{s} \cdot \overline{\mathbf{f}} + \int_{S_N} \delta \mathbf{s} \cdot \overline{\mathbf{t}} + \int_{S_R} \delta \mathbf{s} \cdot \mathbf{t} = \end{aligned}\]

If the stress vector on Robin boundary reads $\mathbf{t} = - \mathbf{K} \cdot \mathbf{s} + \overline{\mathbf{h}}$, it follows

\[\begin{split}\begin{aligned} 0 & = - \int_V \delta \boldsymbol\varepsilon : \boldsymbol\sigma + \int_{V} \delta \mathbf{s} \cdot \overline{\mathbf{f}} + \int_{S_N} \delta \mathbf{s} \cdot \overline{\mathbf{t}} + \int_{S_R} \delta \mathbf{s} \cdot \left( - \mathbf{K} \cdot \mathbf{s} + \overline{\mathbf{h}} \right) = \\ & = \delta \underbrace{\left\{ - \int_V \dfrac{1}{2} \boldsymbol\varepsilon : \mathbf{C} : \boldsymbol\varepsilon + \int_{V} \boldsymbol\varepsilon : \boldsymbol\beta \Delta T + \int_{V} \mathbf{s} \cdot \overline{\mathbf{f}} + \int_{S_N} \mathbf{s} \cdot \overline{\mathbf{t}} - \int_{S_R} \dfrac{1}{2}\mathbf{s} \cdot \mathbf{K} \cdot \mathbf{s} + \int_{S_R} \mathbf{s} \cdot \overline{\mathbf{h}} \right\}}_{ =: \Pi(\boldsymbol\varepsilon, \mathbf{s})} + \int_{V} \boldsymbol\varepsilon : \boldsymbol\beta \ \delta \Delta T \ , \end{aligned}\end{split}\]

and if $\Delta T$ is prescribed, it follows $\delta \Delta T = 0$, and

\[0 = \delta \Pi(\boldsymbol\varepsilon, \mathbf{s}) \ . \]

Theorem 7.2 (Principle of stationariety of total potential energy)

Among all the equilibrated solutions, the congruent solution (and thus the unique solution of the elastic problem) is the one that makes the total potential energy stationary.

7.1.2.7. Principle of stationariety of total complementary potential energy#

Choosing the (unique) solution of the elastic problem as the equilibrated stress field in the principle of complementary virtual work, $\widetilde{\boldsymbol{\sigma}} = \boldsymbol\sigma$ with given loads $\widetilde{\mathbf{f}}$, if the displacement of the Robin boundary reads $\mathbf{s} = - \mathbf{S} \cdot \mathbf{t}_{\mathbf{n}} + \overline{\mathbf{r}}$,

\[\begin{split}\begin{aligned} 0 & = \int_{V} \delta \boldsymbol\sigma : \boldsymbol\varepsilon - \int_{S_D} \delta \mathbf{t}_{\mathbf{n}} \cdot \overline{\mathbf{s}} - \int_{S_R} \delta \mathbf{t}_{\mathbf{n}} \cdot \mathbf{s} = \\ & = \delta \underbrace{ \left\{ \int_V \dfrac{1}{2} \boldsymbol\sigma : \mathbf{D} : \boldsymbol\sigma + \int_{V} \boldsymbol\sigma : \boldsymbol\alpha \Delta T - \int_{S_D} \mathbf{t}_{\mathbf{n}} \cdot \overline{\mathbf{s}} + \int_{S_R} \dfrac{1}{2} \mathbf{t}_n \cdot \mathbf{S} \cdot \mathbf{t}_{\mathbf{n}} - \int_{S_R} \mathbf{t}_{\mathbf{n}} \cdot \overline{\mathbf{r}} \right\} }_{\Pi^*(\boldsymbol\sigma, \mathbf{t}_{\mathbf{n}})} - \int_V \boldsymbol\sigma : \boldsymbol\alpha \, \delta \Delta T \\ \end{aligned}\end{split}\]

If $\Delta T$ is prescribed, it follows $\delta \, \Delta T = 0$, and

\[\delta \Pi^*(\boldsymbol\sigma, \mathbf{t}_{\mathbf{n}}) = 0 \ .\]

Theorem 7.3 (Principle of stationariety of total complementary potential energy)

Among all the congruent solutions, the equilibrated solution (and thus the unique solution of the elastic problem) is the one that makes the total complementary potential energy stationary.

7.1.3. Classical theorems#

Uncomment (and move to beam structures?)

7.1.3.1. Maxwell-Betti#

7.1.3.2. Menabrea-Castigliano#

1: For every test function that is regular enough, meaning that everything that is written in the equatiions exist.
2: Otherwise the first and second scalar factors are different, $C_{ijkl} = \mu \delta_{ik} \delta_{jl} + \eta \delta_{il} \delta_{jk} + \lambda \delta_{ij} \delta_{kl}$.