A comparative study is presented to solve the inverse problem in elasticity for the shear modulus (stiffness) distribution utilizing two constitutive equations: (1) linear elasticity assuming small strain theory, and (2) finite elasticity with a hyperelastic neo-Hookean material model. Assuming that a material undergoes large deformations and material nonlinearity is assumed negligible, the inverse solution using (2) is anticipated to yield better results than (1). Given the fact that solving a linear elastic model is significantly faster than a nonlinear model and more robust numerically, we posed the following question: How accurately could we map the shear modulus distribution with a linear elastic model using small strain theory for a specimen undergoing large deformations? To this end, experimental displacement data of a silicone composite sample containing two stiff inclusions of different sizes under uniaxial displacement controlled extension were acquired using a digital image correlation system. The silicone based composite was modeled both as a linear elastic solid under infinitesimal strains and as a neo-Hookean hyperelastic solid that takes into account geometrically nonlinear finite deformations. We observed that the mapped shear modulus contrast, determined by solving an inverse problem, between inclusion and background was higher for the linear elastic model as compared to that of the hyperelastic one. A similar trend was observed for simulated experiments, where synthetically computed displacement data were produced and the inverse problem solved using both, the linear elastic model and the neo-Hookean material model. In addition, it was observed that the inverse problem solution was inclusion size-sensitive. Consequently, an 1-D model was introduced to broaden our understanding of this issue. This 1-D analysis revealed that by using a linear elastic approach, the overestimation of the shear modulus contrast between inclusion and background increases with the increase of external loads and target shear modulus contrast. Finally, this investigation provides valuable information on the validity of the assumption for utilizing linear elasticity in solving inverse problems for the spatial distribution of shear modulus associated with soft solids undergoing large deformations. Thus, this work could be of importance to characterize mechanical property variations of polymer based materials such as rubbers or in elasticity imaging of tissues for pathology.
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A NEW GOVERNING EQUATION FOR WEB TENSION BY EMPLOYING A NEO-HOOKEAN MATERIAL MODEL
In this work, we derive a governing equation for web tension in a span by
employing a Neo-Hookean material model that is applicable for transport of web
materials under both small and large strains. This governing equation may be
employed to study the evolution of tension within a span as well as propagation of
tension variations from span to span as the web is transported in the machine.
First, we find the stretch in a web span and relate it to web tension via a
Neo-Hookean material model; the Neo-Hookean model is linear for small strain
and nonlinear otherwise. Second, we conduct a dimensional analysis by defining
several key coefficients that aid in grouping the machine and web material
parameters separately in order to obtain a compact system of governing equations;
this representation may be utilized to efficiently study the impact of web and
roller properties on transport behavior.
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- Award ID(s):
- 1635636
- NSF-PAR ID:
- 10110994
- Date Published:
- Journal Name:
- Proceedings of the Fifteenth International Conference on Web Handling
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract The most widely-used representation of the compressible, isotropic, neo-Hookean hyperelastic model is considered in this paper. The version under investigation is that which is implemented in the commercial finite element software ABAQUS, ANSYS and COMSOL. Transverse stretch solutions are obtained for the following homogeneous deformations: uniaxial loading, equibiaxial loading in plane stress, and uniaxial loading in plane strain. The ground-state Poisson’s ratio is used to parameterize the constitutive model, and stress solutions are computed numerically for the physically permitted range of its values. Despite its broad application to a number of engineering problems, the physical limitations of the model, particularly in the small to moderate stretch regimes, are not explored. In this work, we describe and analyze results and make some critical observations, underlining the model’s advantages and limitations. For example, a snap-back feature of the transverse stretch is identified in uniaxial compression, a physically undesirable behavior unless validated by experimental data. The domain of this non-unique solution is determined in terms of the ground-state Poisson’s ratio and the state of stretch and stress. The analyses we perform are essential to enable the understanding of the characteristics of the standard, compressible, isotropic, neo-Hookean model used in ABAQUS, ANSYS and COMSOL. In addition, our results provide a framework for the parameter-fitting procedure needed to characterize this standard, compressible, isotropic neo-Hookean model in terms of experimental data.
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In this paper we derive spatially dependent transfer functions for web span lateral dynamics which provide web lateral position and slope as outputs at any location in the span; the inputs are guide roller displacement, web lateral position disturbances from upstream spans, and disturbances due to misaligned rollers. This is in sharp contrast to the existing approach where only web lateral position response is available on the rollers. We describe the inherent drawbacks of the existing approach and how the new approach overcomes them. The new approach relies on taking the 1D Laplace transform with respect to the temporal variable of both the web governing equation and the boundary conditions. One can also obtain the web slope at any location within the web span with the proposed approach. A general span lateral transfer function, which is an explicit function of the spatial position along the span, is obtained first followed by its application to different intermediate guide configurations.more » « less
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Accepted Manuscript1.0
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