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# Comments on Mindlin plate theory

**Dear all:**

**The advantages, discrepancies and shortcomings of Kirchhoff plate theory (Kirchhoff, 1850) are well known. The study on Reissner-Mindlin plate theory (Reissner, 1945; Mindlin, 1951) seems not as in-depth as the study on the Kirchhoff plate theory. New theories aimed at improving the discrepancies and/or shortcomings of Mindlin plate theory are continuously being proposed. Here the advantages, discrepancies and shortcomings, and advances in research of the two well known plate theories are listed. Experts and scholars on this topic are invited and encouraged to present their comments to make in-depth understanding of the Mindlin plate theory.**

**Advantages, discrepancies and shortcomings of Kirchhoff plate theory:**

**Advantages:** (1) easiness of analytical treatment. (2) free from numerical ill-conditioning problems. (3) only one quantity *w* is used to represent the state of deformation, numerical computation is efficient. **Discrepancies:** the shear strain is ignored first but the same thing gets called up later. **Shortcomings:** (1) its application is limited to thin plates. (2) requiring C1 continuity for shape functions in the finite element method.

**Advantages, discrepancies and shortcomings of Mindlin plate theory:**

**Advantages:** (1) simpler to implement in the finite element method because C0 shape functions are used to interpolate the two rotations and the lateral displacement. (2) thick plate elements perform as well as the best of the thin plate form (L/t=1000). (3) all ‘robust’ element of the thick plate kind can be easily mapped isoparametrically and performance remains excellent and convergent. (4) thick pate elements are capable of yielding results not obtainable with thin plate theory (such as the *hard* and *soft* simply supported conditions). **Discrepancies:** does not allow the bending and shearing deflections to be determined uniquely (Endo and Kimura N., 2007; Xing and Liu, 2009a,b). **Shortcomings:** (1) presented more difficulties in analytical treatment. (2) ‘shear locking’ in the thin limit. (3) requiring a shear coefficient to satisfy the constitutive relationship between shear stress and shear strain.

**Advances in research of thin plate theory:**

(1) discrete Kirchhoff elements are used to overcome the C1 continuity requirements. (2) Xing and Liu (2009c,d) solved new exact solutions for free vibrations of thin rectangular plates. (3) new numerical methods such as the differential quadriture method (Xing and Liu, 2009e; Xing and Liu, 2010) are applied to solve thin plate problems.

**Advances in research of Mindlin plate theory:**

(1) many higher-order shear deformation theories have been investigated (Noor and Burton, 1989; Levinson, 1980), and the proposed theories have generally been successful in avoiding the need for hypotheses. (2) selective reduced integration method is used for Reissner-Mindlin plate problem (Bathe and Dvorkin, 1985). (3) Gorman and Ding (1999) proposed the superposition method to solve free vibrations of Mindlin plates. (4) Endo and Kimura (2007) and Shimpi and Patel (2006) proposed an alternative formulation. (5) Shahrokh and Arsanjani (2005) solved exact characteristic equations for some of classical boundary conditions of vibrating rectangular Mindlin plates. (6) Xing and Liu (2009a,b) solved new closed-form solution for free vibrations of rectangular Mindlin plates.

**Please present your comments to make in-depth understanding of the Mindlin plate theory.**

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**Email: liuboshanxi@126.com**

**The Solid Mechanics Research Center, Beihang University,**

**100191 Beijing, China**

**A letter from Professor Endo:**

Professor Endo said in a letter to me that: "In that paper (**Endo and Kimura N., 2007**) we insist that the bending deflection and shearing one are to be recognized as the physical entities which should be distinguished in the Timoshenko-type theories, i.e., Timoshenko beam and Mindlin plate, otherwise we can not keep the consistency with the related traditional methodology of static analysis. However, the above our opinions have hardly acquired until now the understanding of the major group of academic community (at least, in Japan)."

**A letter from Professor Gorman:**

Professor Gorman said in a letter to me that: "With regard to the Mindlin plate analysis let make two points. First, there are two distinct problems in volved in the analysis. The first is to correctly formulate the differential equation (equations ?) for use in the analysis. The second is to obtain a solution which satisfies exactly the governing differential equation(s), and the prescribed boundary conditions correctly. This is a boundary value problem, we are told. The two aspects of the problem should not be confused."

Kirchhoff G., 1850. Über das Gleichqewicht und die Bewegung einer elastichen Scheibe. *J. Reine und Angewandte Mathematik*, 40:51-88.

Reissner,E., 1945. The effect of transverse shear deformation on the bending of elastic plates. *Trans. ASME Journal of Applied Mechanics*, 12:A69-A77.

Mindlin R.D., 1951. Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. *Journal of Applied Mechanics*, 18(1):31–38.

Endo M., Kimura N., 2007. An alternative formulation of the boundary value problem for the Timoshenko beam and Mindlin plate. *Journal of Sound and Vibration*, 301:355–373

Xing Yufeng, Liu Bo, 2009a. Characteristic equations and Closed-form solutions for free vibrations of rectangular Mindlin plates. *Acta Mechanica Solida Sinica*, 22(2):125-136.

Xing Yufeng, Liu Bo, 2009b. Closed form solutions for free vibrations of rectangular Mindlin plates. *Acta Mechanica Solida*, 25:689–698.

Gorman D.J., Ding W., 1999. Accurate free vibration analysis of point supported Mindlin plates by the superposition method. *Journal of Sound and Vibration*, 219:265–277.

Xing Yufeng, Liu Bo, 2009c. New exact solutions for free vibrations of rectangular thin plates by symplectic dual method. Acta Mech Sin, 25:265–270.

Xing Yufeng, Liu Bo, 2009d. New exact solutions for free vibrations of thin orthotropic rectangular plates. Composite Structures, 89:567–574.

Xing Yufeng, Liu Bo, 2009e. High-accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain. *Int. J. Numer. Meth. Engng.* 80:1718–1742.

Xing Yufeng, Liu Bo, 2010. A differential quadrature finite element method. *International Journal of Applied Mechanics*, 2:1-20.

Noor A.K., Burton W.S., 1989. Assessment of shear deformation theories for multilayered composite plates. *Applied Mechanics Review*, 42(1):1–12.

Levinson M., 1980. An accurate, simple theory of the statistics and dynamics of elastic plates. *Mechanics Research Communications*, 7(6):343–350.

Bathe KJ, Dvorkin EN., 1985. A four-node plate bending element based on Mindlin-Reissner plate theory and mixed interpolation. *Int J Numer Meth Engng*, 21:367-83.

Shimpi R.P., Patel H.G., 2006. Free vibrations of plate using two variable refined plate theory. *Journal of Sound and Vibration*, 296:979–999.

Shahrokh Hosseini Hashemi, M. Arsanjani, 2005. Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates. *International Journal of Solids and Structures*, 42:819–853.

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## Comments

## Discrepancies: the shear strain is ignored but the shear forces

Just curious... Correct me if I am wrong...

Why do you call it a discrepancy that "the shear strain is ignored but the shear forces appear in equilibrium equation"? Would you argue the same on the whole rigid-body mechanics?

Equilibrium equations are established involving forces only. When relating forces to deformations, assumptions are made, which we call constitutive laws. There are constitutive laws more suitable than others when applied to specific cases...

I would agree with you if the shear strain is ignored first but the same thing gets called up later.

All the bests,

Bo

## Thank you for your correcting my mistakes

Thank you very much for your comments and your correcting my mistakes. I am not sure of some of the comments I presented indeed. I have corrected the sentence according to your suggestion. Thank you.

Sincerely yours,

Bo Liu

## Actually, it should be more

Actually, it should be more correctly stated as transverse shear strains are ignored in kinematics to formulate 3D strains in terms of plate strains, later transverse shear forces are used to derive the equilibrium equations using a Newtonian approach. If you use other approaches such as variational approach to derive these equations, you do not need shear strain and shear stress. Please refer to my lecture notes on three ways to derive the classical plate model at https://cdmhub.org/resources/390. And also assumptions are not necessary to derive Kirchoff theory or R-M theory, and shear correction factor is not absolutely needed in R-M theory. See my paper at https://cdmhub.org/resources/924 entitled

Mathematical construction of a Reissner–Mindlin plate theory for composite laminates