Branching of stretch histories in biaxially loaded nonlinear viscoelastic fiber-reinforced sheets

2018 ◽  
Vol 24 (3) ◽  
pp. 807-827 ◽  
Author(s):  
Alan Wineman
Keyword(s):  
Isotropic Material ◽  
Transversely Isotropic ◽  
Solution Branch ◽  
Fiber Reinforced ◽  
Governing Equations ◽  
Nonlinear Viscoelastic ◽  
Tensile Forces ◽  
Single Integral ◽  
Biaxial Creep ◽  
Integral Constitutive Equation

This work considers an experiment in which a nonlinear viscoelastic square sheet, subjected to uniformly distributed tensile forces on its edge surfaces, undergoes a homogeneous biaxial extensional creep history. The sheet is fiber-reinforced with the direction of reinforcement either normal to the midplane of the sheet or parallel to one of its edges. The possibility is considered that when the governing equations are solved for the biaxial creep response, there may be a time during the deformation when a second solution branch can form. Thus, if equal biaxial forces are applied to the sheet, its deformed states are squares until some time when they become rectangular. The material is modeled using the Pipkin–Rogers nonlinear single integral constitutive equation for a transversely isotropic material. A condition is derived to determine the time when the solution to the governing equations forms a new branch. Numerical examples are presented for fibers oriented normal to the sheet and in the plane of the sheet.

Large Axisymmetric Deformation of a Nonlinear Viscoelastic Membrane Due to Spinning

1972 ◽  
Vol 39 (4) ◽  
pp. 946-952 ◽  
Author(s):  
A. S. Wineman
Keyword(s):  
Relaxation Times ◽  
Numerical Procedure ◽  
Axisymmetric Deformation ◽  
Time Step ◽  
First Order ◽  
Nonlinear Viscoelastic ◽  
Single Integral ◽  
Shear Effects ◽  
Integral Constitutive Equation ◽  
Derivatives Of

The problem of a viscoelastic membrane undergoing large planar deformation due to spinning is solved. The membrane, rigidly bonded at its inner boundary and traction-free at the outer, consists of a nonlinear viscoelastic solid whose behavior is modeled by a nonlinear single integral constitutive equation. The model includes the possibility that the relaxation time is influenced by the amount of stretching. Spinning is considered to be sufficiently slow so that shear effects can be neglected. The formulation uses radial and circumferential stretch ratios as dependent variables. These satisfy a system of first-order nonlinear partial-differential integral equations. The numerical procedure at each time step obtains the current spatial derivatives of the stretch ratios in terms of the current and previously determined stretch ratios. This gives essentially a first-order nonlinear system of differential equations, of the same structure as that obtained in the elastic version of the problem [1], which is integrated numerically. Solutions are obtained for both strain-dependent and strain-independent relaxation times.

Fiber symmetry

2017 ◽  
Author(s):  
David J. Steigmann
Keyword(s):  
Composite Materials ◽  
Constitutive Equation ◽  
Transversely Isotropic ◽  
Fiber Reinforced ◽  
Fiber Reinforced Materials ◽  
Reinforced Materials

This chapter develops the general constitutive equation for transversely isotropic, fiber-reinforced materials. Applications include composite materials and bio-elasticity.

THE ANALYSIS OF DYNAMICAL RESPONSE OF TRANSVERSELY ISOTROPIC MATERIAL UNDER BLASTING LOAD

2008 ◽  
Vol 22 (09n11) ◽  
pp. 1443-1448
Author(s):  
YUE-XIU WU ◽  
QUAN-SHENG LIU
Keyword(s):  
Isotropic Material ◽  
Peak Velocity ◽  
Transversely Isotropic ◽  
Normal Direction ◽  
Dynamical Response ◽  
Transversely Isotropic Material ◽  
Peak Acceleration ◽  
Loading Direction ◽  
Blasting Load ◽  
Explosion Load

To understand the dynamic response of transversely isotropic material under explosion load, the analysis is done with the help of ABAQUS software and the constitutive equations of transversely isotropic material with different angle of isotropic section. The result is given: when the angle of isotropic section is settled, the velocity and acceleration of measure points decrease with the increasing distance from the explosion borehole. The velocity and acceleration in the loading direction are larger than those in the normal direction of the loading direction and their attenuation are much faster. When the angle of isotropic section is variable, the evolution curves of peak velocity and peak acceleration in the loading direction with the increasing angles are notching parabolic curves. They get their minimum values when the angle is equal to 45 degree. But the evolution curves of peak velocity and peak acceleration in the normal direction of the loading direction with the increasing angles are overhead parabolic curves. They get their maximum values when the angle is equal to 45 degree.

Wrinkling of a Fiber-Reinforced Membrane

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.

Mixed State Hamiltonian Element for Plane Transversely Isotropic Magnetoelectroelastic Solids

2013 ◽  
Vol 275-277 ◽  
pp. 1978-1983
Author(s):  
Xiao Chuan Li ◽  
Jin Shuang Zhang
Keyword(s):  
Mixed State ◽  
Hamiltonian Formulation ◽  
Transversely Isotropic ◽  
Laminated Plates ◽  
Governing Equations ◽  
Time Coordinate ◽  
First Order ◽  
Linear Elements ◽  
Propagation Matrix ◽  
Magnetoelectroelastic Solids

Hamiltonian dual equation of plane transversely isotropic magnetoelectroelastic solids is derived from variational principle and mixed state Hamiltonian elementary equations are established. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate. Then the x-direction is discreted with the linear elements to obtain the state-vector governing equations, which are a set of first order differential equations in z and are solved by the analytical approach. Because present approach is analytic in z direction, there is no restriction on the thickness of plate through the use of the present element. Using the propagation matrix method, the approach can be extended to analyze the problems of magnetoelectroelastic laminated plates. Present semi-analytical method of mixed Hamiltonian element has wide application area.

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