Perfectly plastic plates: a variational definition

Archive for Rational Mechanics and Analysis, 2006

We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼h β , where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼h α . Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Kármán theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)3→ℝ3, the L 2 distance of ∇v from a single rotation is bounded by a multiple of the L 2 distance from the set SO(3) of all rotations.

Journal of Applied Mathematics and Mechanics, 1985

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1997

ein dgemeines Vetfahren fir die w e W P i g e Hcrteitung w n Phttmtheorien wrgesdJasm Die Methodik basiert oqf &r Annahme, da$ Plattenthcoeen bei d r e i d i m i o & Elmtizitlit durch Einfihrung p m & Nebededingungen ubcr die Dejamtioprr-w d Spannungsftlder ausgejfihrt w d e n fitwen. Me w i h n g m l l e Theofic der Lagrangeschn Mdtiplihtmen w i d eingeftihd, urn die Variationsprir&c ~~n , die artf dem Hu-Washim Fud+tdonal tiaeiwcn und Elmtizitiitaproblense mil iVebenbedt?igungen batirnnaen. Sowolrl dte Schubdefmtions-Plattcw theotie erster Ordnung ah ~c h die Lo-Christensen-Wu-Plattentheorie h o k r Wnung werden laergeleifet. Die Bwtimmungsgledchtmgen wcrrdcn wieder abgekitet und die RiiekPrd~ngsfeider, die ah eine Folge &r e i n d w r d e n Nebenbedingungen anf.stehen, werden dargestdt. Wenn d i u e RGckwirbngsfelder in Rechnung geskut wcrden, teQen sich die Gkdchgawichtj-, Kongruewund M a~l e t d r u n g m ah ex& &Ut. In thia paper a general procedure for a rationu1 d~u a t i o n of phte theories is proposed Th methodolofi w b u c d on the conjecture that plate theoties can be camcd out from the three-dimensioned elastdcity by imposing witoble cowhints on #ie strain 4 #tress &l&. The powerfrJ Lagrange mdtipliers theovy ia adopted to deriue the vard~atio~l prineipka, baaed on #c Hu-Wwhim functiod, govcming the consbradncd ehtic* pmblems. Both the fir&o&r #hear &fmfwn plate theory, and the higher-order Lo-Chrishm WU plate thwy are derived. The g o m i n g e q w t i o~ are recop a d , and the reactive fields, o d n g G ( a comepence of the imposed constraints, are casried out. When thtsc r e d w fields are a m into account, the equalih~um, congruence, and constituiiut equaiio~le turn out to be tmctly satiarj5ed. MSC (1991): 73K10 1. Introduction A plate is a flat three-dimensional body, having one dimension smaller than the other two. Of course, tbe governing equations for this special body are the classical equatioiis of the Coutinuum Mechanics. B-use of the complexity in solving the elastostatic problem of a three-dimensional M y , ad hoc simplifying hypotheses are often introduced. Hence, different plate theories are obtained from the thr*dimensional theory depending on the different assumptions adopted. Sometimes the assumptions have a mechanical basis, since there is a good understanding of the physical behavior of a plate, sometimes tbey have a mathenlatical basis.

Journal of Theoretical and Applied Mechanics, 1982

1999

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

This paper derives a finite-strain plate theory consistent with the principle of stationary threedimensional potential energy under general loadings with a fourth-order error. Starting from the three-dimensional nonlinear elasticity (with both geometrical and material nonlinearity) and by a series expansion, we deduce a vector plate equation with three unknowns, which exhibits the local force-balance structure. The success relies on using the three-dimensional field equations and bottom traction condition to derive exact recursion relations for the coefficients. Associated weak formulations are considered, leading to a two-dimensional virtual work principle. An alternative approach based on a two-dimensional truncated energy is also provided, which is less consistent than the first plate theory but has the advantage of the existence of a twodimensional energy function. As an example, we consider the pure bending problem of a hyperelastic block. The comparison between the analytical plate solution and available exact one shows that the plate theory gives second-order correct results. Compared with existing plate theories, it appears that the present one has a number of advantages, including the consistency, order of correctness, generality of loadings, applicability to finite-strain problems and no involvement of non-physical quantities.

arXiv (Cornell University), 2022

A new mathematical formulation for the constitutive laws governing elastic perfectly plastic materials is proposed here. In particular, it is shown that the elastic strain rate and the plastic strain rate form an orthogonal decomposition with respect to the tangent cone and the normal cone of the yield domain. It is also shown that the stress rate can be seen as the projection on the tangent cone of the elastic stress tensor. This approach leads to a coherent mathematical formulation of the elasto-plastic laws and simplifies the resulting system for the associated flow evolution equations. The cases of one or two yields functions are treated in detail. The practical examples of the von Mises and Tresca yield criteria are worked out in detail to demonstrate the usefulness of the new formalism in applications. Contents 1. Introduction 2. Reformulating elasto-plasticity equations 3. Explicit constitutive laws in the case of one or two yield functions 3.1. The main practical results 3.2. A simple lemma on projections 3.3. The case of one saturated yield function: Proof of Theorem 3.1 3.4. The case of two saturated yield function: Proof of Theorem 3.2 4. Examples: Von Mises and Tresca criteria 4.1. Von Mises criterion 4.2. Tresca criterion 5. Concluding remarks 5.1. Main results 5.2. General case of spectral yield functions 5.3. Outlook Acknowledgement Appendix A. Appendix: proof of Lemma 4.4 References

1999

We consider the derivation of two-dimensional models for the bending and stretching of a thin three-dimensional linearly elastic plate us- ing variational methods. Specically we consider restriction of the trial space in two dierent forms of the Hellinger-Reissner variational principle for 3-D elasticity to functions with a specied polynomial dependence in the trans- verse direction. Using this approach many dierent

Mathematics and Mechanics of Solids, 2018

This work derives an exact two-dimensional plate theory for heterogeneous plates consistent with the principle of stationary three-dimensional potential energy under general loading. We do not take any hypotheses about the shape of the heterogeneity. We start from three-dimensional linear elasticity and by using the Fourier series expansion in the thickness direction of the displacement field with respect to a basis of scaled Legendre polynomials. We deduce an exact two-dimensional model expressed in power-series in the ratio between the thickness of the plate and a characteristic measurement of its mid-plane. Then we can derive an approximative theory by neglecting in the expression of potential energy all terms that contain a power of this ratio that is higher than a given truncation power for getting to an approximative two-dimensional problem. In the last section, we show that the solution of the approximation problem only differs from the exact solution by a difference of the same order of the neglected terms in the potential energy. A similar result when we truncate the displacement field can be also established. This model can be a starting point to formulate a two-dimensional homogenized boundary value problem for highly heterogeneous periodic plates.

International Journal of Non-Linear Mechanics, 2007

Non-linear plate theory for thin prismatic elastic bodies is obtained by estimating the total three-dimensional strain energy generated in response to a given deformation in terms of the small plate thickness. The Euler equations for the estimate of the energy are regarded as the equilibrium equations for the thin plate. Included among them are algebraic formulae connecting the gradients of the midsurface deformation to the through-thickness derivatives of the three-dimensional deformation. These are solvable provided that the three-dimensional strain energy is strongly elliptic at equilibrium. This framework yields restrictions of the Kirchhoff-Love type that are usually imposed as constraints in alternative formulations. In the present approach they emerge as consequences of the stationarity of the energy without the need for any a priori restrictions on the three-dimensional deformation apart from a certain degree of differentiability in the direction normal to the plate. ᭧