Propagation of harmonic waves in an initially deformed cylinder of an incompressible material

This paper presents a method for modeling of pneumatic bias tire axisymmetric deformation. A previously developed model of all-steel radial tire was expanded to include the non-linear stress–strain relationship for textile cord and its thermal shrinkage. Variable cord density and cord angle in the cord-rubber bias tire composite are the major challenges in pneumatic tire modeling. The variabilities result from the tire formation process, and they were taken into account in the model. Mechanical properties of the composite were described using a technique of orthotropic reinforcement overlaying onto isotropic rubber elements, treated as a hyperelastic incompressible material. Due to large displacements, the non-linear problem was solved using total Lagrangian formulation. The model uses MSC.Marc code with implemented user subroutines, allowing for the description of the tire specific properties. The efficiency of the model was verified in the simulation of mounting and inflation of an actual bias truck tire. The shrinkage negligence effect on cord forces and on displacements was examined. A method of investigating the influence of variation of cord angle in green body plies on tire apparent lateral stiffness was proposed. The created model is stabile, ensuring convergent solutions even with large deformations. Inflated tire sizes predicted by the model are consistent with the actual tire sizes. The distinguishing feature of the developed model from other ones is the exact determination of the cord angles in a vulcanized tire and the possibility of simulation with the tire mounting on the rim and with cord thermal shrinkage taken into account. The model may be an effective tool in bias tire design.

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Second harmonic generation: Effects of the multiple reflections of the fundamental and the second harmonic waves on the Maker fringes

Optics Communications ◽

10.1016/j.optcom.2007.06.048 ◽

2007 ◽

Vol 279 (1) ◽

pp. 183-195 ◽

Cited By ~ 2

Author(s):

Gildas Tellier ◽

Christian Boisrobert

Keyword(s):

Second Harmonic Generation ◽

Harmonic Generation ◽

Second Harmonic ◽

Harmonic Waves ◽

Multiple Reflections ◽

Maker Fringes

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Perturbation method to solve the couple problem among harmonic waves

Applied Mathematics and Mechanics ◽

10.1007/bf02454507 ◽

1988 ◽

Vol 9 (11) ◽

pp. 1039-1044

Author(s):

Gao Shi-qiao ◽

Loo Wen-da

Keyword(s):

Perturbation Method ◽

Couple Problem ◽

Harmonic Waves

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A novel semi-implicit scheme for elastodynamics and wave propagation in nearly and truly incompressible solids

Acta Mechanica ◽

10.1007/s00707-020-02883-5 ◽

2021 ◽

Author(s):

Chennakesava Kadapa

Keyword(s):

Wave Propagation ◽

Wave Speed ◽

Critical Time ◽

Incompressible Material ◽

Implicit Scheme ◽

Bulk Wave ◽

Lumped Mass ◽

Time Step ◽

Material Models ◽

Shear Wave Speed

AbstractThis paper presents a novel semi-implicit scheme for elastodynamics and wave propagation problems in nearly and truly incompressible material models. The proposed methodology is based on the efficient computation of the Schur complement for the mixed displacement-pressure formulation using a lumped mass matrix for the displacement field. By treating the deviatoric stress explicitly and the pressure field implicitly, the critical time step is made to be limited by shear wave speed rather than the bulk wave speed. The convergence of the proposed scheme is demonstrated by computing error norms for the recently proposed LBB-stable BT2/BT1 element. Using the numerical examples modelled with nearly and truly incompressible Neo-Hookean and Ogden material models, it is demonstrated that the proposed semi-implicit scheme yields significant computational benefits over the fully explicit and the fully implicit schemes for finite strain elastodynamics simulations involving incompressible materials. Finally, the applicability of the proposed scheme for wave propagation problems in nearly and truly incompressible material models is illustrated.

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Wrinkling of a Fiber-Reinforced Membrane

Journal of Applied Mechanics ◽

10.1115/1.2967888 ◽

2008 ◽

Vol 76 (1) ◽

Author(s):

E. Shmoylova ◽

A. Dorfmann

Keyword(s):

Boundary Conditions ◽

Energy Function ◽

Mechanical Response ◽

Transverse Isotropy ◽

Base Material ◽

Strain Energy Function ◽

Transversely Isotropic ◽

Incompressible Material ◽

Fiber Reinforced ◽

Relaxed Energy

In this paper we investigate the response of fiber-reinforced cylindrical membranes subject to axisymmetric deformations. The membrane is considered as an incompressible material, and the phenomenon of wrinkling is taken into account by means of the relaxed energy function. Two cases are considered: transversely isotropic membranes, characterized by one family of fibers oriented in one direction, and orthotropic membranes, characterized by two family of fibers oriented in orthogonal directions. The strain-energy function is considered as the sum of two terms: The first term is associated with the isotropic properties of the base material, and the second term is used to introduce transverse isotropy or orthotropy in the mechanical response. We determine the mechanical response of the membrane as a function of fiber orientations for given boundary conditions. The objective is to find possible fiber orientations that make the membrane as stiff as possible for the given boundary conditions. Specifically, it is shown that for transversely isotropic membranes a unique fiber orientation exists, which does not affect the mechanical response, i.e., the overall behavior is identical to a nonreinforced membrane.

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A Stabilized B-Splines FEM Formulation for the Solution of an Inverse Elasticity Problem Arising in Medical Imaging

Volume 12: Mechanics of Solids, Structures and Fluids ◽

10.1115/imece2008-66700 ◽

2008 ◽

Author(s):

Carlos E. Rivas ◽

Paul E. Barbone ◽

Assad A. Oberai

Keyword(s):

Mechanical Properties ◽

Diagnostic Value ◽

Incompressible Material ◽

High Order ◽

Elasticity Problem ◽

Parameter Distribution ◽

Order Continuity ◽

B Splines ◽

Inverse Elasticity ◽

Continuous Problem

Soft tissue pathologies are often associated with changes in mechanical properties. For example, breast and other tumors usually present as stiff lumps. Imaging the spatial distribution of the mechanical properties of tissues thus reveals information of diagnostic value. Doing so, however, typically requires the solution of an inverse elasticity problem. In this work we consider the inverse elasticity problem for an incompressible material in plane stress, formulated and solved as a constrained optimization problem. We formulate this inverse problem enforcing high order continuity for our variables. Driven by the requirements for the strong and weak solutions to this problem, we assume that our data field (i.e. the measured displacement) is in H2 and our parameter distribution (i.e. the sought shear modulus distribution) is in H1. This high order regularity requirement for the data is incompatible with standard FEM. We solve this problem using a FEM formulation that is novel in two respects. First, we employ quadratic b-splines that enforce C1 continuity in our displacement field, consistent with the variational requirements of the continuous problem. Second, we include Galerkin-least-squares (GLS) stabilization in the iterative optimization formulation. GLS adds consistent stability to the discrete formulation that otherwise violates an ellipticity condition that is satisfied by the continuous problem. Computational examples validate this formulation and demonstrate numerical convergence with mesh refinement.

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