Reduction of Three-Dimensional Elasticity to the Kirchhoff Plate Equation
This appendix proves Theorem 34.64 of Continuum Mechanics and Elasticity: that the transverse deflection \(w(x,y,t)\) of a thin isotropic plate obeys \(\rho h\,\pp_{t}^{2}w+D\nabla^{4}w=q\) with the flexural rigidity Equation (34.57), and that a free edge carries exactly two boundary conditions rather than the three a naive force balance demands. The reduction has three stages and each is a separate piece of work: a kinematic hypothesis about how the material moves, a constitutive statement that the plate is in plane stress and not plane strain, and a variational calculation whose boundary terms are the interesting part of the answer.
Everything used is in the book. The constitutive law is Equation (34.21), proved in Isotropic and Cubic Elastic Tensors; the conversions among the moduli are Proposition 34.33; the flexural rigidity is Definition 34.63; the variational machinery is that of Calculus of Variations, and in particular the fundamental lemma Lemma 20.18 and the natural boundary conditions Theorem 20.31. Nothing is quoted from outside the treatise. One point of bookkeeping is worth making in advance: Theorem 20.38 states the Euler–Poisson equation for a functional of one independent variable, while the plate functional depends on two, so the corresponding equation is derived here by carrying out the two integrations by parts explicitly — which is necessary anyway, because the boundary terms those integrations produce are the content of The free edge: three conditions are one too many.
Throughout, Greek indices \(\alpha,\beta,\gamma\) run over the in-plane values \(x,y\) and are summed when repeated; Latin indices run over \(x,y,z\). The mid-surface is \(z=0\), the thickness is \(h\) and the material occupies \(\abs{z}\le h/2\). Overdots are \(\pp_{t}\).
Kinematics: the Kirchhoff hypothesis
The plate is said to deform in the Kirchhoff manner when the mid-surface points move only transversely and material lines initially normal to the mid-surface remain straight, normal and unstretched. Writing \(w(x,y,t)\) for the transverse displacement of the mid-surface, this is
so that a cross-section rotates through the angle \(\pp_{\alpha}w\), as in the rod of Proposition 34.51. All three displacements carry \(\mathrm{m}\) and \(w\) is assumed small enough that the strain Definition 34.6 may be linearised. Rests on Definition 34.6 and Proposition 34.51.
Under Equation (A57.1) the in-plane strains are proportional to the curvatures of the deflected mid-surface and the transverse shear strains vanish identically,
while \(u_{zz}\) is not determined by the hypothesis and is fixed instead by the constitutive statement of Plane stress, not plane strain. Rests on Definitions 34.6 and A57.1.
Derives Lemma A57.2. From Definition 34.6, \(u_{\alpha\beta}=\tfrac{1}{2} \left(\pp_{\alpha}u_{\beta}+\pp_{\beta}u_{\alpha}\right) =-\tfrac{1}{2}z\left(\pp_{\alpha}\pp_{\beta}w +\pp_{\beta}\pp_{\alpha}w\right) =-z\,\pp_{\alpha}\pp_{\beta}w\), the two mixed partials being equal for a twice continuously differentiable \(w\). For the transverse shear,
That the shear vanishes is exactly the statement that normals stay normal, and it is the hypothesis rather than a consequence: a real plate carries a transverse shear stress, and must, since the transverse load has to reach the supports somehow. What Equation (A57.3) asserts is that the shear strain it produces is negligible against the bending strain, which is true to relative order \(\left(h/L\right)^{2}\) for a plate of lateral dimension \(L\).
The remaining component, \(u_{zz}=\pp_{z}u_{z}\), would be zero if Equation (A57.1) were read as an exact statement about every material point. It must not be so read. The ansatz is a statement about the in-plane displacement field and about the rotation of normals; the thickness of the plate does change, by an amount of the same order as the in-plane strains, and \(u_{zz}\) is left free here so that the correct constitutive condition can determine it.
∎Plane stress, not plane strain
This is the step at which a plate differs from a two-dimensional block of material, and it is where the factor \(\left(1-\nu^{2}\right)^{-1}\) of Equation (34.57) is born.
Let the faces \(z=\pm h/2\) be free of traction. Then throughout a thin plate \(\sigma_{zz}\) is negligible, and eliminating \(u_{zz}\) from Equation (34.21) by \(\sigma_{zz}=0\) gives
Both \(\lambda,\mu\) and \(E,\nu\) are the constants of Definition 34.31, related by Proposition 34.33, and every quantity in Equation (A57.4) except \(\nu\) carries \(\mathrm{Pa}\). Rests on Equation (34.21), Proposition 34.33 and Definition 34.63.
Derives Proposition A57.3. Why \(\sigma_{zz}\) and not \(u_{zz}\) vanishes. The traction on the faces is \(\sigma_{iz}n_{z}\) with \(n_{z}=\pm1\), so \(\sigma_{zz}=0\) exactly on \(z=\pm h/2\). Between the faces \(\sigma_{zz}\) is of the order of the applied load \(q\), whereas the in-plane stresses are of order \(qL^{2}/h^{2}\), larger by the square of the aspect ratio; setting \(\sigma_{zz}=0\) throughout is therefore an error of relative order \(\left(h/L\right)^{2}\), the same order already discarded in Equation (A57.3). Imposing \(u_{zz}=0\) instead — plane strain — would be a different and wrong theory: it describes a slab confined between rigid platens, and it would put \(\lambda+2\mu\) where Equation (34.57) has \(E/(1-\nu^{2})\), overstating the flexural rigidity by \(\left(\lambda+2\mu\right)\left(1-\nu^{2}\right)/E =\left(1-\nu\right)^{2}/\left(1-2\nu\right)\), a factor \(1.22\) at \(\nu=0.3\).
The elimination. The \(zz\) component of Equation (34.21) reads \(\sigma_{zz}=\lambda\left(u_{\gamma\gamma}+u_{zz}\right)+2\mu u_{zz}\), where \(u_{\gamma\gamma}=u_{xx}+u_{yy}\) is the in-plane trace. Setting this to zero gives the first of Equation (A57.4). Substituting it back into the in-plane components,
since \(\lambda\left[1-\lambda/(\lambda+2\mu)\right]=2\lambda\mu/(\lambda+2\mu)\).
In engineering constants. By Equation (34.28), \(\lambda=2\mu\nu/(1-2\nu)\) and \(2\mu=E/(1+\nu)\), so
Substituting Equation (A57.6) into Equation (A57.5) and pulling out \(2\mu=E/(1+\nu)\) gives the second of Equation (A57.4).
∎Contracting the second of Equation (A57.4) for a state of pure uniaxial in-plane strain, \(u_{xx}=\epsilon\) and \(u_{yy}=0\), gives \(\sigma_{xx}=\left[E/(1+\nu)\right]\left[1+\nu/(1-\nu)\right]\epsilon =E\epsilon/\left(1-\nu^{2}\right)\): the plate is stiffer than a bar of the same material by \(\left(1-\nu^{2}\right)^{-1}\), because its own width prevents the transverse contraction that a bar is free to make. That is the sentence Definition 34.63 states in words, and it is the whole difference between Equation (34.57) and the \(EI\) of Equation (34.49). Rests on Proposition A57.3 and Definition 34.63.
Integrating the energy across the thickness
Under Definition A57.1 and Proposition A57.3, the elastic energy of a plate occupying the region \(\Omega\) of the mid-plane is
with \(D=Eh^{3}/\left[12\left(1-\nu^{2}\right)\right]\) the flexural rigidity Equation (34.57), of SI dimension \(\mathrm{N}\,\mathrm{m}\); \(U\) is then in \(\mathrm{J}\). Rests on Proposition A57.3, Lemma A57.2 and Definition 34.63.
Derives Proposition A57.5. The density. By Equation (34.22) and Equation (34.19) the energy density is \(F=\tfrac{1}{2}\sigma_{ik}u_{ik}\). Of the nine terms, \(\sigma_{\alpha z}u_{\alpha z}\) vanishes because the strain does (Equation (A57.3)) and \(\sigma_{zz}u_{zz}\) vanishes because the stress does (Proposition A57.3), so only the in-plane block survives:
Note that the plate stores no energy in the eliminated component: the work done by \(\sigma_{zz}\) through \(u_{zz}\) is zero because \(\sigma_{zz}\) is zero, which is the reason the elimination costs nothing.
Through the thickness. Insert Equation (A57.2). Every strain carries one factor of \(z\), so every term of Equation (A57.8) carries \(z^{2}\), and
Writing \(w_{,\alpha\beta}=\pp_{\alpha}\pp_{\beta}w\) and abbreviating
the integrated energy is
The rearrangement. Expanding \(Q\) and comparing with \(P\),
because \(Q=\left(\pp_{x}^{2}w\right)^{2} +2\,\pp_{x}^{2}w\,\pp_{y}^{2}w+\left(\pp_{y}^{2}w\right)^{2}\). Write \(2G\) for the right-hand side of Equation (A57.12), so that \(P=Q+2G\) with \(G=\left(\pp_{x}\pp_{y}w\right)^{2}-\pp_{x}^{2}w\,\pp_{y}^{2}w\). Then the bracket of Equation (A57.11) is
and the prefactor of Equation (A57.11) combines with the \(\left(1-\nu\right)^{-1}\) to give
which is Equation (A57.7).
∎The integrand \(G=\left(\pp_{x}\pp_{y}w\right)^{2}-\pp_{x}^{2}w\,\pp_{y}^{2}w\) of Equation (A57.7) is an exact divergence,
so that \(\int_{\Omega}G\,\dd A\) depends only on \(w\) and its first derivatives on \(\pp\Omega\). It therefore contributes nothing to the field equation, and nothing at all when the edge is clamped. Rests on Proposition A57.5.
Derives Lemma A57.6. Differentiate the two products: \(\pp_{x}\left(\pp_{y}w\,\pp_{x}\pp_{y}w\right) =\left(\pp_{x}\pp_{y}w\right)^{2} +\pp_{y}w\,\pp_{x}^{2}\pp_{y}w\) and \(\pp_{y}\left(\pp_{y}w\,\pp_{x}^{2}w\right) =\pp_{y}^{2}w\,\pp_{x}^{2}w +\pp_{y}w\,\pp_{y}\pp_{x}^{2}w\). The two third-derivative terms are equal and cancel in the difference, leaving \(G\). By Theorem 11.133 in the plane, \(\int_{\Omega}G\,\dd A=\oint_{\pp\Omega} \left(\pp_{y}w\,\pp_{x}\pp_{y}w\,n_{x} -\pp_{y}w\,\pp_{x}^{2}w\,n_{y}\right)\dd s\), an expression in the boundary values of \(w\) and \(\nabla w\) alone. A variation \(\delta w\) vanishing together with \(\nabla\delta w\) on \(\pp\Omega\) therefore does not change it, so it cannot enter the Euler equation; and for a clamped edge, where \(w\) and \(\pp_{n}w\) are prescribed, the whole term is a constant that may be discarded from the functional.
∎Lemma A57.6 is what makes the plate equation as simple as it is, and a reader who does not see it will suspect an error: the energy Equation (A57.7) is manifestly not proportional to \(\int(\nabla^{2}w)^{2}\), it depends on Poisson's ratio, and yet the field equation contains only the biharmonic operator and no \(\nu\) at all. The resolution is that the \(\nu\)-dependent part is a boundary functional. Geometrically, \(-G\) is the Gaussian curvature of the deflected mid-surface to leading order in \(\nabla w\) — the determinant of the Hessian — so Equation (A57.15) is the small-deflection shadow of the Gauss–Bonnet theorem: the total Gaussian curvature of a surface is determined by its boundary. Poisson's ratio does not disappear from the problem; it survives exactly where the boundary functional lives, in the edge conditions of The free edge: three conditions are one too many, which is why two plates of the same \(D\) and different \(\nu\) have the same equation and different free-edge behaviour. Rests on Lemma A57.6 and Definition 34.63.
The field equation
To leading order in \(h/L\) the kinetic energy of the plate is
with \(\rho\) the density in \(\mathrm{kg}/\mathrm{m}^{3}\), so that \(\rho h\) is a mass per unit area in \(\mathrm{kg}/\mathrm{m}^{2}\). The discarded term is the rotatory inertia of the cross-sections, \(\left(\rho h^{3}/24\right)\int_{\Omega} \abs{\nabla\dot{w}}^{2}\dd A\). Rests on Definition A57.1.
Derives Lemma A57.8. Differentiating Equation (A57.1) in time gives \(\dot{u}_{z}=\dot{w}\) and \(\dot{u}_{\alpha}=-z\,\pp_{\alpha}\dot{w}\), so
using Equation (A57.9). For a disturbance of lateral scale \(L\) one has \(\abs{\nabla\dot{w}}^{2}\sim\dot{w}^{2}/L^{2}\), so the second term is smaller than the first by \(\left(h/L\right)^{2}/12\), the order already discarded twice above, and is dropped. Retaining it, together with the transverse shear strain discarded in Equation (A57.3), gives the Mindlin–Reissner theory, whose flexural waves have a finite limiting speed instead of the unbounded \(\omega\propto k^{2}\) that Equation (34.58) predicts; the two agree for \(kh\ll1\), which is the regime in which the plate theory is being used.
∎Let \(w\) extremise the action
built from Equation (A57.7) and Equation (A57.16), with \(q\) the transverse load per unit area in \(\mathrm{Pa}\). Then in the interior of \(\Omega\)
which is Equation (34.58), and the variation leaves on \(\pp\Omega\) the boundary integral Equation (A57.25) below. Rests on Proposition A57.5, Lemma A57.8 and Lemma 20.18.
Derives Theorem A57.9. Moments. It shortens every formula to introduce the bending moments per unit length of edge,
of SI dimension \(\mathrm{N}\,\mathrm{m}/\mathrm{m}=\mathrm{N}\). With the notation Equation (A57.10), the energy Equation (A57.7) is
the first equality being Equation (A57.13) read backwards, \(Q+2(1-\nu)G=(1-\nu)P+\nu Q\).
The variation. \(U\) is a quadratic form in the second derivatives of \(w\) and \(M_{\alpha\beta}\) is linear in them, so \(\delta U=-\int_{\Omega}M_{\alpha\beta}\, \pp_{\alpha}\pp_{\beta}\delta w\,\dd A\). Integrating by parts twice with Theorem 11.133,
with \(\vect{n}\) the outward unit normal of the edge. Taking the double divergence of Equation (A57.20),
the \(\nu\) cancelling exactly as Remark A57.7 promised. The kinetic term contributes \(\delta\int T\dd t=-\int\dd t\int_{\Omega} \rho h\,\pp_{t}^{2}w\,\delta w\,\dd A\) after one integration by parts in \(t\), the endpoint terms vanishing because \(\delta w\) is held zero at \(t_{1}\) and \(t_{2}\). Collecting,
where \(\mathcal{B}\) is the edge integral
For variations supported strictly inside \(\Omega\), \(\mathcal{B}\) vanishes and Lemma 20.18 applied to the area integral gives Equation (A57.19). Since \(D\) carries \(\mathrm{N}\,\mathrm{m}\) and \(\nabla^{4}w\) carries \(/\mathrm{m}^{3}\), each term of Equation (A57.19) is a pressure, as \(q\) is.
∎The free edge: three conditions are one too many
Let \(\vect{n}\) and \(\vect{s}\) be the outward unit normal and the unit tangent of \(\pp\Omega\), forming a right-handed pair, and let \(\pp_{n}=n_{\alpha}\pp_{\alpha}\) and \(\pp_{s}=s_{\alpha}\pp_{\alpha}\). Define
Explicitly, from Equation (A57.20), \(M_{nn}=-D\left[\pp_{n}^{2}w+\nu\,\pp_{s}^{2}w\right]\) for a straight edge, and \(M_{ns}=-D\left(1-\nu\right)\pp_{n}\pp_{s}w\), the trace term dropping out of the second because \(\delta_{\alpha\beta}n_{\alpha} s_{\beta}=\vect{n}\cdot\vect{s}=0\): the twisting moment measures the twist \(\pp_{n}\pp_{s}w\) of the mid-surface at the edge and nothing else. Rests on Equation (A57.20) and Proposition A57.3.
Read as a force balance, a free edge — one at which nothing is attached — carries no bending moment, no twisting moment and no transverse shear, which is three conditions: \(M_{nn}=0\), \(M_{ns}=0\), \(Q_{n}=0\). This is Poisson's count, and Remark 34.67 records that it was believed for two decades. It cannot be right. Equation (A57.19) is of fourth order in the two space variables, so along each edge exactly two conditions may be imposed; three over-determine the problem and in general leave it with no solution at all. The resolution is not to discard one of the three arbitrarily but to observe that they are not independent, which is what the variational form Equation (A57.25) makes visible and a force balance cannot. Rests on Definition A57.10 and Theorem A57.9.
Let \(\pp\Omega\) be a closed piecewise-smooth curve with corners \(c_{1},\ldots,c_{N}\). Then the edge integral Equation (A57.25) may be written
with the effective (Kirchhoff) shear and the corner force
where \(\left[\!\left[M_{ns}\right]\!\right]_{c_{j}}\) is the jump in \(M_{ns}\) across the corner. Consequently, at an edge where nothing is prescribed the natural boundary conditions (Theorem 20.31) are the two statements
together with the vanishing of \(F_{j}\) at every free corner. These are the conditions asserted by Theorem 34.64. Rests on Theorem A57.9, Definition A57.10 and Theorem 20.31.
Derives Theorem A57.12. Resolving the normal derivative. Along the edge, \(\set{\vect{n},\vect{s}}\) is an orthonormal frame of the plane, so \(\pp_{\beta}\delta w =n_{\beta}\,\pp_{n}\delta w+s_{\beta}\,\pp_{s}\delta w\) and
Substituting Equation (A57.32) and Equation (A57.28) into Equation (A57.25) gives the three-term form
which is where Poisson's three conditions appear to come from.
The three are two. The quantities \(\delta w\) and \(\pp_{n}\delta w\) may be prescribed independently on the edge, but \(\pp_{s}\delta w\) may not: it is the derivative of \(\delta w\) along the edge, and is determined by \(\delta w\) there. Integrate that term by parts along the curve. On each smooth arc,
Summing Equation (A57.34) over the arcs of the closed curve, the endpoint terms of consecutive arcs meet at each corner \(c_{j}\) and combine into \(-M_{ns}^{-}\delta w(c_{j})+M_{ns}^{+}\delta w(c_{j})\), where \(\pm\) denote the limits taken along the outgoing and incoming arc; that is \(F_{j}\,\delta w(c_{j})\) with \(F_{j}\) the jump of Equation (A57.30). On a smooth closed curve, \(M_{ns}\) is continuous and every such term cancels. Collecting with the \(Q_{n}\delta w\) term of Equation (A57.33) gives Equation (A57.29).
The conditions. In Equation (A57.29) the three factors \(\pp_{n}\delta w\) on the edge, \(\delta w\) on the edge and \(\delta w\) at each corner are now independently assignable. By Lemma 20.18 applied on the edge and then at the finitely many corners, \(\delta S=0\) for all admissible variations requires each coefficient to vanish wherever its variation is free. Where the plate is clamped, \(\delta w=\pp_{n}\delta w=0\) and nothing is required of \(M_{nn}\) or \(V_{n}\); where it is simply supported, \(\delta w=0\) but \(\pp_{n}\delta w\) is free, giving the single natural condition \(M_{nn}=0\); where it is free, both are free and Equation (A57.31) follows, with \(F_{j}=0\) at a free corner. In every case the count is two conditions per edge, as the order of Equation (A57.19) requires.
∎\(F_{j}\) is not an artefact of the variational bookkeeping. Take a square plate, simply supported on all four edges — resting on knife edges, free to rotate — and load it uniformly. Along each edge \(\delta w=0\), so the edge integral of Equation (A57.29) imposes only \(M_{nn}=0\); but at a corner the tangent turns through a right angle, the roles of \(n\) and \(s\) are exchanged, and \(M_{ns}\) changes sign, so \(F_{j}=\left[\!\left[M_{ns}\right]\!\right]=2M_{ns}\) does not vanish. A downward concentrated force of that magnitude must be supplied at each corner. If the supports cannot pull down — if the plate merely rests on them — the corners lift off, which is exactly what a uniformly loaded square glass pane does, and why a concrete floor slab supported on four walls is reinforced diagonally at its corners. The corner force also explains the sign of the effect: the plate is attempting to become anticlastic, curving up along one diagonal while it curves down along the other, and it is \(M_{ns}\) that measures that twist. Rests on Theorem A57.12 and Equation (A57.30).
The fourth-order equation is Germain's [Germain:1821] and the two correct edge conditions are Kirchhoff's [Kirchhoff:1850], obtained exactly as above, from the variation of the energy rather than from a force balance; Poisson's three-condition count is in his memoir on elastic bodies [Poisson:1829]. Three approximations were made and each is of relative order \(\left(h/L\right)^{2}\): the transverse shear strain was set to zero (Equation (A57.3)), the transverse normal stress was set to zero (Proposition A57.3), and the rotatory inertia was dropped (Lemma A57.8). Two of the three are removed by the Mindlin–Reissner theory, and none of them is a small-deflection assumption: the linearisation of the strain in Definition A57.1 is a separate hypothesis, and it is the one that fails first for a thin plate, since a deflection of order \(h\) already stretches the mid-surface and brings in the membrane terms of Remark 34.68.
Reduction of Three-Dimensional Elasticity to the Kirchhoff Plate Equation discharges the derivation owed at Theorem 34.64 of Continuum Mechanics and Elasticity. The three stages announced in that chapter are Plane stress, not plane strain (plane stress, and the birth of the factor \(1-\nu^{2}\) in Equation (34.57)), Integrating the energy across the thickness together with The field equation (the thickness integration and the variation, giving Equation (34.58)), and The free edge: three conditions are one too many (Kirchhoff's resolution of the edge count, with the concentrated corner force). Everything the chapter builds on Equation (34.58) — the Chladni figures of Phenomenon 34.65, the scaling Equation (34.59), and the historical reading of Remark 34.67 — rests on the equation proved here, and the isotropic constitutive law it starts from is Equation (34.21), proved in Isotropic and Cubic Elastic Tensors.