A geometrically nonlinear analysis of laminated composite plates using a shear deformation theory

Giacinto Porco; Giuseppe Spadea; Raffaele Zinno

Displaying similar documents to “A geometrically nonlinear analysis of laminated composite plates using a shear deformation theory”

A geometrically nonlinear analysis of laminated composite plates using a shear deformation theory

Giacinto Porco, Giuseppe Spadea, Raffaele Zinno (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A shear deformation theory is developed to analyse the geometrically nonlinear behaviour of layered composite plates under transverse loads. The theory accounts for the transverse shear (as in the Reissner Mindlin plate theory) and large rotations (in the sense of the von Karman theory) suitable for simulating the behaviour of moderately thick plates. Square and rectangular plates are considered: the numerical results are obtained by a finite element computational procedure and are given...

Numerical approximation of the non-linear fourth-order boundary-value problem

Svobodová, Ivona

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We consider functionals of a potential energy $ℙ_{ψ} (u)$ corresponding to $𝑎𝑛 𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 - 𝑣𝑎𝑙𝑢𝑒 𝑝𝑟𝑜𝑏𝑙𝑒𝑚$ . We are dealing with $𝑎 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑡ℎ𝑖𝑛 𝑎𝑛𝑛𝑢𝑙𝑎𝑟 𝑝𝑙𝑎𝑡𝑒$ with $𝑁𝑒𝑢𝑚𝑎𝑛𝑛 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠$ . Various types of the subsoil of the plate are described by various types of the $𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒$ nonlinear term $ψ (u)$ . The aim of the paper is to find a suitable computational algorithm.

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki, P. Kacprzyk (2008)

Applicationes Mathematicae

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{j k}$ . Moreover, we assume that the stored energy function is $C^{\infty}$ with respect to x and $e_{j k}$ . In our description we assume that Piola-Kirchhoff’s stress tensor $p_{j k}$ depends on the tensor...

Gradient theory for plasticity via homogenization of discrete dislocations

Adriana Garroni, Giovanni Leoni, Marcello Ponsiglione (2010)

Journal of the European Mathematical Society

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We deduce a macroscopic strain gradient theory for plasticity from a model of discrete dislocations. We restrict our analysis to the case of a cylindrical symmetry for the crystal under study, so that the mathematical formulation will involve a two-dimensional variational problem. The dislocations are introduced as point topological defects of the strain fields, for which we compute the elastic energy stored outside the so-called core region. We show that the $Γ$ -limit of this energy (suitably...

Generalised functions of bounded deformation

Gianni Dal Maso (2013)

Journal of the European Mathematical Society

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We introduce the space $G B D$ of generalized functions of bounded deformation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for $G B D$ , which leads to a compactness result for the space $G S B D$ of generalized special functions of bounded deformation. The latter is connected to the existence of solutions to a weak formulation...

On a bifurcation problem arising in cholesteric liquid crystal theory

Carlo Greco (2017)

Commentationes Mathematicae Universitatis Carolinae

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In a cholesteric liquid crystal the director field $n (x, y, z)$ tends to form a right-angle helicoid around a twist axis in order to minimize the internal energy; however, a fixed alignment of the director field at the boundary (strong anchoring) can give rise to distorted configurations of the director field, as oblique helicoid, in order to save energy. The transition to this distorted configurations depend on the boundary conditions and on the geometry of the liquid crystal, and it is known...

On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long (2024)

Mathematica Bohemica

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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u (η_{1}, t), \dots, u (η_{q}, t)$ with $0 \leq η_{1} < η_{2} < \dots < η_{q} < 1 .$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $(P_{q})$ of (P) in which the nonlinear term contains the sum $S_{q} [u^{2}] (t) = q^{- 1} \sum_{i = 1}^{q} u^{2} (\frac{(i - 1)}{q}, t)$ . Under suitable conditions, we prove that the solution of $(P_{q})$ converges to the solution...

On boundary value problems for systems of nonlinear generalized ordinary differential equations

Malkhaz Ashordia (2017)

Czechoslovak Mathematical Journal

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A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $d x = d A (t) \cdot f (t, x), h (x) = 0$ is established, where $f : [a, b] \times ℝ^{n} \to ℝ^{n}$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A : [a, b] \to ℝ^{n \times n}$ with bounded total variation components, and $h : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x (t_{1} (x)) = ℬ (x) \cdot x (t_{2} (x)) + c_{0},$ where $t_{i} : {BV}_{s} ([a, b], ℝ^{n}) \to [a, b]$ $(i = 1, 2)$ and $ℬ : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ are continuous...

A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

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We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( $P_{λ}$ ) ⎩ $u_{∣ \partial Ω} = 0$ where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( $P_{λ}$ ) admits a non-zero, non-negative strong solution $u_{λ} \in ⋂_{p \geq 2} W^{2, p} (Ω)$ such that $l i m_{λ \to 0 ⁺} | | u_{λ} {| |}_{W^{2, p} (Ω)} = 0$ for all p ≥ 2. Moreover, the function $λ \mapsto I_{λ} (u_{λ})$ is negative and decreasing in ]0,λ*[, where $I_{λ}$ is the energy functional related to ( $P_{λ}$ ). ...

Local-in-time existence for the non-resistive incompressible magneto-micropolar fluids

Peixin Zhang, Mingxuan Zhu (2022)

Applications of Mathematics

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We establish the local-in-time existence of a solution to the non-resistive magneto-micropolar fluids with the initial data $u_{0} \in H^{s - 1 + ε}$ , $w_{0} \in H^{s - 1}$ and $b_{0} \in H^{s}$ for $s > \frac{3}{2}$ and any $0 < ε < 1$ . The initial regularity of the micro-rotational velocity $w$ is weaker than velocity of the fluid $u$ .

Homogenization of a three-phase composites of double-porosity type

Ahmed Boughammoura, Yousra Braham (2021)

Czechoslovak Mathematical Journal

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In this work we consider a diffusion problem in a periodic composite having three phases: matrix, fibers and interphase. The heat conductivities of the medium vary periodically with a period of size $ε^{β}$ ( $ε > 0$ and $β > 0$ ) in the transverse directions of the fibers. In addition, we assume that the conductivity of the interphase material and the anisotropy contrast of the material in the fibers are of the same order $ε^{2}$ (the so-called double-porosity type scaling) while the matrix material has a conductivity...

Perturbed nonlinear degenerate problems in $ℝ^{N}$

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $d i v (x, \nabla u) + {a (x) | u |}^{p - 2} u = {g (x) | u |}^{p - 2} u + h (x) {| u |}^{s - 1} u$ in $ℝ^{N}$ ⎨ ⎩ u > 0, $l i m_{| x | \to \infty} u (x) = 0$ , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

Spreading and vanishing in nonlinear diffusion problems with free boundaries

Yihong Du, Bendong Lou (2015)

Journal of the European Mathematical Society

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We study nonlinear diffusion problems of the form $u_{t} = u_{x x} + f (u)$ with free boundaries. Such problems may be used to describe the spreading of a biological or chemical species, with the free boundary representing the expanding front. For special $f (u)$ of the Fisher-KPP type, the problem was investigated by Du and Lin [DL]. Here we consider much more general nonlinear terms. For any $f (u)$ which is $C^{1}$ and satisfies $f (0) = 0$ , we show that the omega limit set $ω (u)$ of every bounded positive solution is determined by a stationary...

Results on linear and nonlinear hyperbolic boundary value problems at resonance

Michael W. Smiley (1980)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si considera l’equazione non lineare nell'incognita $u(t,x)$ ((1,1) del testo) soddisfatta in un cilindro $G=(o,T)\times\Omega$ ( $\Omega$ dominio limitato di $\bf{R}^{n}$ ) con condizioni al contorno tipo Dirichlet o Neumann sulla superficie laterale di $G$ e con relazioni omogenee fra $u$ e $u_{t}$ sulle basi. Si stabiliscono per la (1) e nel caso di risonanza alcuni teoremi di perturbazione.