A Stroh Formalism for Small-on-Large Problems in Spherical Polar Coordinates

Martin, P. A.

doi:10.1007/s10659-019-09730-2

Cited by 2 publications

(5 citation statements)

References 16 publications

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“…We assume the following separation of variables for the incremental fields [37] { ur (r, θ ), Ṡrr (r, θ ), φ(r, θ ), Ḋr (r, θ)} = {U r (r), Σ rr (r), Φ(r), r (r)}P m (cos θ),…”

Section: (B) Stroh Formulationmentioning

confidence: 99%

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Tunable morphing of electroactive dielectric-elastomer balloons

Su,

Riccobelli,

Chen

et al. 2023

Proc. R. Soc. A.

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Designing smart devices with tunable shapes has important applications in industrial manufacture. In this paper, we investigate the nonlinear deformation and the morphological transitions between buckling, necking and snap-through instabilities of layered dielectric elastomer (DE) balloons in response to an applied radial voltage and an inner pressure. We propose a general mathematical theory of nonlinear electro-elasticity able to account for finite inhomogeneous strains provoked by the electro-mechanical coupling. We investigate the onsets of morphological transitions of the spherically symmetric balloons using the surface impedance matrix method. Moreover, we study the nonlinear evolution of the bifurcated branches through finite-element numerical simulations. Our analysis demonstrates the possibility to design tunable DE spheres, where the onset of buckling and necking can be controlled by geometrical and mechanical properties of the passive elastic layers. Relevant applications include soft robotics and mechanical actuators.

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“…We assume the following separation of variables for the incremental fields [37] { ur (r, θ ), Ṡrr (r, θ ), φ(r, θ ), Ḋr (r, θ)} = {U r (r), Σ rr (r), Φ(r), r (r)}P m (cos θ),…”

Section: (B) Stroh Formulationmentioning

confidence: 99%

“…We assume the following separation of variables for the incremental fields [37] falsefalse{u˙r(r,θ),S˙rr(r,θ),ϕ˙(r,θ),D˙r(r,θ)falsefalse} 1em=falsefalse{Ur(r),Σrr(r),Φ(r),Δr(r)falsefalse}Pmfalse(cos⁡θfalse),and2em falsefalse{u˙θ(r,θ),S˙rθ(r,θ)falsefalse}={Uθfalse(rfalse)...…”

Section: Linear Stability Analysismentioning

confidence: 99%

Tunable morphing of electroactive dielectric-elastomer balloons

Su,

Riccobelli,

Chen

et al. 2023

Proc. R. Soc. A.

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“…We could build a system of coupled first‐order differential equations, and this could be done in a variety of ways. For example, there may be merit in developing a Stroh‐like formulation, as has been done in anisotropic elastodynamics 10 and for small‐on‐large problems arising in nonlinear elasticity 11 …”

Section: Discussion and Prospectsmentioning

confidence: 99%

“…But, from Equation () and the partial differential equation satisfied by Y , 11, eq. 4.13

scriptD Y \equiv \frac{1}{\sin θ} \frac{\partial}{\partial θ} ((), \sin θ \frac{\partial Y}{\partial θ}) + \frac{1}{\sin^{2} θ} \frac{\partial^{2} Y}{\partial Φ^{2}} = - λ^{2} Y,

we obtain

\frac{\partial}{\partial θ} false(C \sin θ false) - \frac{\partial B}{\partial Φ} = 0, \frac{\partial}{\partial θ} false(B \sin θ false) + \frac{\partial C}{\partial Φ} = - λ Y \sin θ .

Hence,

div bold-italicv = ((), {\overset{V}{overset}}_{P} - false(λ false/ r false) V_{B}) Y,

where we have introduced the shorthand notation

\overset{f}{overset} = \frac{1}{r^{2}} \frac{normald}{normald r} false(r^{2} f false) = f^{'} false(r false) + \frac{2}{r} f false(r false) .

…”

Section: Use Of Spherical Polar Coordinatesmentioning

confidence: 99%

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Acoustic scattering in a rarefied gas: Solving the R13 equations in spherical polar coordinates

Martin

2020

Math Methods in App Sciences

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In some circumstances, sound waves in a rarefied gas can be studied using a linearised form of the regularised 13‐moment equations of Struchtrup and Torrilhon. We build solutions of those equations in spherical polar coordinates using vector spherical harmonics. We first solve a reduced system of equations (with 11 unknowns) after introduction of a force vector (divergence of the stress). We then show that the stresses themselves can be recovered by solving five additional equations. The results obtained are expected to be useful for problems such as acoustic scattering of a plane wave by a sphere in a rarefied gas.

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A Stroh Formalism for Small-on-Large Problems in Spherical Polar Coordinates

Cited by 2 publications

References 16 publications

Tunable morphing of electroactive dielectric-elastomer balloons

Tunable morphing of electroactive dielectric-elastomer balloons

Acoustic scattering in a rarefied gas: Solving the R13 equations in spherical polar coordinates

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