3D Derivations of Static Plate Theories
Synonyms
Definitions

Thin plate model: A model where the only kinematic d.o.f. is the transverse deflection. It neglects the shear energy.

Thick plate model: A model including also two inplane rotation d.o.f. and including shear deflection.
Introduction
Plates are threedimensional structures with a small dimension compared to the other two dimensions. Numerous approaches were suggested in order to replace the threedimensional problem by a twodimensional problem while guaranteeing the accuracy of the reconstructed threedimensional fields. Turning the 3D problem into a 2D plate model is known as dimensional reduction.
The approaches for deriving a plate model from 3D elasticity may be separated in two main categories: axiomatic and asymptotic approaches. Axiomatic approaches start with ad hoc assumptions on the 3D field representation of the plate, separating the outofplane coordinate from the inplane coordinates. The limitation of these approaches comes from the educated guess for the 3D field distribution. Asymptotic approaches come often after axiomatic approaches. They are based on the explicit introduction of the plate thickness, which is assumed to go to 0, in the equations of the 3D problem. Following a rather wellestablished procedure, they enable the derivation of plate models, often justifying a posteriori axiomatic approaches, and are the basis of a convergence result.
The very first and simplest model is the KirchhoffLove plate model or thinplate model (Kirchhoff, 1850; Love, 1888), where the outofplane deflection is the only kinematic degree of freedom. In this model, it is assumed that the fiber normal to the plate midsurface remains normal during the motion. In order to take into account the influence of shear energy on the deflection, several thick plate models were suggested almost simultaneously (Reissner, 1944; Hencky, 1947; Bollé, 1947). In these models, gathered here under the common denomination ReissnerHencky models, two inplane rotations are added to the kinematics. Note that the denomination ReissnerMindlin is also very common in the literature. It comes from Mindlin’s contribution based on dynamic considerations (Mindlin, 1951). Whereas all these models were historically derived axiomatically, they also have close relations with asymptotic considerations.
This chapter is dedicated to the case of a homogeneous and linear elastic plate with static loading which was the foundation of many extensions to heterogeneous plates. It recalls in detail the derivation of the thick plate model from Hencky (1947) as well as the one from Reissner (1944). Both approaches are related but yield different plate models. This choice is motivated by the following considerations. First, ReissnerHencky models are the most widely used plate models in engineering applications. Indeed, their boundary conditions seem more natural than those of the KirchhoffLove plate model. They also relax the higher regularity of the KirchhoffLove displacement required for finite elements implementations. Second, the KirchhoffLove model may be directly retrieved from these models by means of the Kirchhoff kinematic restriction “Direct Derivation of Plate Theories”.
Two modifications are made with respect to the historical contributions. First, the membrane model is also included in the present derivation at very little price. Second, the applied load is a body force uniformly distributed through the thickness instead of a force per unit surface applied only on the upper face of the plate. This choice leads to a more compact derivation and removes a higherorder coupling between the membrane and bending problems widely ignored in the historical literature. Finally, all mathematical developments are purely formal and the reader is referred to (“Mathematical Justification of Plate Models” and Ciarlet 1997) for rigorous justifications.
The 3D Problem
for regular enough u, and the constitutive law (5) are satisfied.
Kinematic Derivation of Hencky’s Plate Model
In this section, the kinematic derivation of a thick plate model from Hencky (1947) and Bollé (1947) is presented. It delivers the correct plate generalized variables. However, the constitutive equations are incorrect. It starts with the assumption of a 3D kinematically compatible displacement field. The plate model is derived from the application of the minimum potential energy principle.
Plate Kinematics
Here, U_{ α } is the membrane inplane displacement, U_{3} is the outofplane displacement, and ϕ_{ α } is the material inclination of the fiber normal to the midplane of the plate. The corresponding inplane rotation vector is θ, where θ_{1} = −ϕ_{2} and θ_{2} = ϕ_{1}.
With proper scaling, it may be demonstrated that this kinematics is related to the asymptotic expansion of the 3D displacement solution of \(\mathcal {P}^{\mathrm {3D}}\) with respect to the thickness of the plate h. The membrane displacement and the outofplane displacement are the leadingorder terms of the expansion. The material rotation is related to the nextorder term of the expansion (Ciarlet and Destuynder, 1979).
Formulation of Hencky’s Plate Model
Minimum Potential Energy
Plate Generalized Stresses
Plate Equilibrium
These equilibrium equations are almost identical to those obtained from the direct derivation “Direct Derivation of Plate Theories”. Here the drilling moment vanishes, and the membrane stress and bending moment tensors are symmetric because the plate is originally assumed as a 3D Cauchy medium.
Natural Scaling of Stresses in Plates
Because the upper and lower face of the plate are actually free of stress, there is a natural scaling of stresses when, for a fixed inplane dimension L, the thickness h goes to 0.
Indeed, from the outofplane part of the 3D equilibrium equation (7), it appears that the normal stress scales like σ_{33} ∼ h^{2}f_{3} and that the transverse shear stress scales like σ_{ α3} ∼ Lhf_{3}. Furthermore, the bending equilibrium equation (24) ensures the following relation between the inplane stress and the transverse shear stress: \(\sigma _{\alpha 3} \sim \frac {h}{L}\sigma _{\alpha \beta } \). Hence Open image in new window is of order h^{0}. This is also in agreement with the inplane equilibrium which yields σ_{ αβ } ∼ Lf_{ α } also of order h^{0} and motivates the initial scaling of the load.
Finally, the inplane stresses are of order h^{0}, the transverse shear stresses are of order h^{1}, and the normal stress is of order h^{2}. A direct consequence of this observation is that at leading order in h, the plate is in a state of planestress.
Constitutive Equations
Once plate kinematically as well as statically compatible fields are derived, there remains to establish plate constitutive equations. This is usually performed, integrating through the thickness the strain energy related to the approximation of strains (15). However, whereas Hencky’s kinematics is correct asymptotically when h goes to 0, the corresponding strain field is not the leading order of the expansion of the 3D solution. Indeed, \(\varepsilon _{33}^{\mathrm {H}} =0\) corresponds to a planestrain state. It is in contradiction with the natural scaling of stresses in the plate and does not satisfy the free boundary conditions on the upper and lower face of the plate. A small outofplane displacement is required to release outofplane Poisson’s effect (see Braess et al. 2010 among others).
In most textbooks, it is arbitrarily assumed at this stage that the correct constitutive equations are those derived in a previous work from Reissner following static considerations and detailed below.
Static Derivation of Reissner Plate Model
In this derivation, Hencky’s kinematics relating plate generalized displacements and 3D displacement is dropped and another interpretation of the plate kinematics will be derived. Reissner’s model is obtained from the derivation of a statically compatible 3D stress distribution and the application of the minimum complementary energy principle.
Derivation of a Statically Compatible Stress Field
From this inplane stress distribution, a complete statically compatible stress distribution is now derived by successively integrating through the thickness of the 3D equilibrium equation (7).
Formulation of the Reissner Plate Model
Minimum of the Complementary Energy
Plate Kinematics
Remarkably, Hencky’s kinematics (12) is in agreement with the projections (38).
Constitutive Equations
Note that the membrane problem for U_{ α } generalized displacements is fully uncoupled from the bending problem for U_{3} and ϕ_{ α } generalized displacements. This is because of the monoclinic symmetry assumed in (4) and the mirror symmetry with respect to the midplane of the plate. For heterogeneous plates, this uncoupling is not always true “Anisotropic and Refined Plate Theories”.
Finally, the socalled “shear correction factor” 5/6 taking into account the nonuniform distribution of the outofplane shear stress was obtained. Note that, when dealing with heterogeneous plates, this definition is meaningless since several shear stiffness moduli may be involved in the shear force constitutive equation.
Static Boundary Conditions
Conclusion
The approaches from Hencky and Reissner for deriving a thick plate theory are often confused in the literature. Whereas they are closely related, they actually yield different plate models which suffer from different limitations.
The kinematic derivation from Hencky is probably the most straightforward but leads to incorrect estimates of the local stresses as well as the plate’s constitutive equations. The constitutive equations derived by Reissner are commonly used to correct Hencky’s model.
The extension of this model to the case of laminated plates was early performed (Yang et al., 1966). This approach is referred to as firstorder shear deformation theory and suffers even more critically from the inconsistencies encountered for homogeneous plates. The advantage of this approach is that its kinematics may be extended to large displacements and rotations.
A natural strategy for solving these inconsistencies is to enrich the plate kinematics so that it can accommodate free boundaries at the upper and lower face of the plate. This is the main concept behind hierarchical models (Babuška and Li, 1992; Paumier and Raoult, 1997; Alessandrini et al., 1999) where the 3D displacement is assumed as a polynomial of the outofplane coordinate and each monomial is multiplied by an inplane function being a generalized plate displacement. However, this requires more plate kinematic degrees of freedom than those of ReissnerHencky models.
The static derivation from Reissner leads to a very accurate model in the framework of static linear elasticity and it was observed empirically that it converges faster than KirchhoffLove model in some specific configurations (Lebée and Sab, 2017b). However, its rigorous extension to laminated plates requires the introduction of numerous additional plate degrees of freedom and is impractical for engineering applications (Lebée and Sab, 2017a).
References
 Alessandrini SM, Arnold DN, Falk RS, Madureira AL (1999) Derivation and justification of plate models by variational methods. In: Fortin M (ed) Plates and shells, vol 21. American Mathematical Society, Providence, pp 1–21CrossRefGoogle Scholar
 Babuška I, Li L (1992) Thepversion of the finiteelement method in the plate modelling problem. Commun Appl Numer Methods 8(1):17–26MathSciNetCrossRefzbMATHGoogle Scholar
 Bollé L (1947) Contribution au problème linéaire de flexion d’une plaque élastique. Bulletin technique de la Suisse romande 73(21):281–285Google Scholar
 Braess D, Sauter S, Schwab C (2010) On the justification of plate models. J Elast 103(1):53–71MathSciNetCrossRefzbMATHGoogle Scholar
 Ciarlet PG (1997) Mathematical elasticity – volume II: theory of plates. Elsevier Science Bv, AmsterdamzbMATHGoogle Scholar
 Ciarlet PG, Destuynder P (1979) Justification Of the 2dimensional linear plate model. Journal de Mecanique 18(2):315–344MathSciNetzbMATHGoogle Scholar
 Hencky H (1947) Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ingenieur Archiv 16(1):72–76CrossRefzbMATHGoogle Scholar
 Kirchhoff G (1850) Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für die reine und angewandte Mathematik (Crelles Journal) 1850(40):51–88CrossRefGoogle Scholar
 Lebée A, Sab K (2017a) On the generalization of Reissner plate theory to laminated plates, part I: theory. J Elast 126(1):39–66MathSciNetCrossRefzbMATHGoogle Scholar
 Lebée A, Sab K (2017b) On the generalization of Reissner plate theory to laminated plates, part II: comparison with the bendinggradient theory. J Elast 126(1):67–94MathSciNetCrossRefzbMATHGoogle Scholar
 Love AEH (1888) The small free vibrations and deformation of a thin elastic shell. Philos Trans R Soc Lond A 179:491–546CrossRefzbMATHGoogle Scholar
 Mindlin R (1951) Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J Appl Mech 18:31–38zbMATHGoogle Scholar
 Paumier JC, Raoult A (1997) Asymptotic consistency of the polynomial approximation in the linearized plate theory. Application to the ReissnerMindlin model. Elast Viscoelast Optim Control 2:203–214MathSciNetzbMATHGoogle Scholar
 Reissner E (1944) On the theory of bending of elastic plates. J Math Phys 23:184–191MathSciNetCrossRefzbMATHGoogle Scholar
 Yang P, Norris CH, Stavsky Y (1966) Elastic wave propagation in heterogeneous plates. Int J Solids Struct 2(4):665–684CrossRefGoogle Scholar