New Solutions of Elastic Waves in an Elastic Rod under Finite Deformation

The nonlinear wave equation of an elastic rod under finite deformation is solved by the extended mapping method. Abundant new exact traveling wave solutions for this equation are obtained, which contain trigonometric function solutions, solitary wave solutions, Jacobian elliptic function solutions, and Weierstrass elliptic function solutions. The method can be used in further works to establish more entirely new solutions for other kinds of nonlinear evolution equations arising in physics

Nonlinear Correction to the Euler Buckling Formula for\ud Compressible Cylinders

Euler’s celebrated buckling formula gives the critical load N for the buckling of a slender cylindrical column with radius B and length L as\ud \ud N/(π 3B2)=(E/4)(B/L)2,\ud \ud where E is Young’s modulus. Its derivation relies on the assumptions that linear elasticity applies to this problem, and that the slenderness (B/L) is an infinitesimal quantity. Here we ask the following question: What is the first nonlinear correction in the right hand-side of this equation when terms up to (B/L)4 are kept? To answer this question, we specialize the exact solution of non-linear elasticity for the homogeneous compression of a thick cylinder with lubricated ends to the theory of third-order elasticity. In particular, we highlight the way second- and third-order constants —including Poisson’s ratio— all appear in the coefficient of (B/L)4

Beam-Induced Damage Mechanisms and their Calculation

The rapid interaction of highly energetic particle beams with matter induces dynamic responses in the impacted component. If the beam pulse is sufficiently intense, extreme conditions can be reached, such as very high pressures, changes of material density, phase transitions, intense stress waves, material fragmentation and explosions. Even at lower intensities and longer time-scales, significant effects may be induced, such as vibrations, large oscillations, and permanent deformation of the impacted components. These lectures provide an introduction to the mechanisms that govern the thermomechanical phenomena induced by the interaction between particle beams and solids and to the analytical and numerical methods that are available for assessing the response of impacted components. An overview of the design principles of such devices is also provided, along with descriptions of material selection guidelines and the experimental tests that are required to validate materials and components exposed to interactions with energetic particle beams.Comment: 69 pages, contribution to the 2014 Joint International Accelerator School: Beam Loss and Accelerator Protection, Newport Beach, CA, USA , 5-14 Nov 201

Swirling fluid flow in flexible, expandable elastic tubes: variational approach, reductions and integrability

Many engineering and physiological applications deal with situations when a fluid is moving in flexible tubes with elastic walls. In the real-life applications like blood flow, there is often an additional complexity of vorticity being present in the fluid. We present a theory for the dynamics of interaction of fluids and structures. The equations are derived using the variational principle, with the incompressibility constraint of the fluid giving rise to a pressure-like term. In order to connect this work with the previous literature, we consider the case of inextensible and unshearable tube with a straight centerline. In the absence of vorticity, our model reduces to previous models considered in the literature, yielding the equations of conservation of fluid momentum, wall momentum and the fluid volume. We show that even when the vorticity is present, but is kept at a constant value, the case of an inextensible, unshearable and straight tube with elastics walls carrying a fluid allows an alternative formulation, reducing to a single compact equation for the back-to-labels map instead of three conservation equations. That single equation shows interesting instability in solutions when the vorticity exceeds a certain threshold. Furthermore, the equation in stable regime can be reduced to Boussinesq-type, KdV and Monge-Amp\`ere equations equations in several appropriate limits, namely, the first two in the limit of long time and length scales and the third one in the additional limit of the small cross-sectional area. For the unstable regime, we numerical solutions demonstrate the spontaneous appearance of large oscillations in the cross-sectional area.Comment: 57 pages, 11 figures. arXiv admin note: text overlap with arXiv:1805.1102

Attenuation of stress waves in single and multi-layered structures

Analytical and experimental studies were made of the attenuation of the stress waves during passage through single and multilayer structures. The investigation included studies on elastic and plastic stress wave propagation in the composites and those on shock mitigating material characteristics such as dynamic stress-strain relations and energy absorbing properties. The results of the studies are applied to methods for reducing the stresses imposed on a spacecraft during planetary or ocean landings

Nonlinear Correction to the Euler Buckling Formula for Compressed Cylinders with Guided-Guided End Conditions

Euler's celebrated buckling formula gives the critical load $N$ for the buckling of a slender cylindrical column with radius $B$ and length $L$ as $$N / (\pi^3 B^2) = (E/4)(B/L)^2,$$ where $E$ is Young's modulus. Its derivation relies on the assumptions that linear elasticity applies to this problem, and that the slenderness $(B/L)$ is an infinitesimal quantity. Here we ask the following question: What is the first nonlinear correction in the right hand-side of this equation when terms up to $(B/L)^4$ are kept? To answer this question, we specialize the exact solution of incremental non-linear elasticity for the homogeneous compression of a thick compressible cylinder with lubricated ends to the theory of third-order elasticity. In particular, we highlight the way second- and third-order constants ---including Poisson's ratio--- all appear in the coefficient of $(B/L)^4$.Comment: 12 page

Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity

The aim of this paper is to compare a hyperelastic with a hypoelastic model describing the Eulerian dynamics of solids in the context of non-linear elastoplastic deformations. Specifically, we consider the well-known hypoelastic Wilkins model, which is compared against a hyperelastic model based on the work of Godunov and Romenski. First, we discuss some general conceptual differences between the two approaches. Second, a detailed study of both models is proposed, where differences are made evident at the aid of deriving a hypoelastic-type model corresponding to the hyperelastic model and a particular equation of state used in this paper. Third, using the same high order ADER Finite Volume and Discontinuous Galerkin methods on fixed and moving unstructured meshes for both models, a wide range of numerical benchmark test problems has been solved. The numerical solutions obtained for the two different models are directly compared with each other. For small elastic deformations, the two models produce very similar solutions that are close to each other. However, if large elastic or elastoplastic deformations occur, the solutions present larger differences.Comment: 14 figure