Luis Dorfmann

Tufts University, Biomedical Engineering, Faculty Member

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Papers by Luis Dorfmann

Stress-softening and recovery of elastomers

arXiv (Cornell University), Feb 3, 2001

A constitutive model is developed for the mechanical response of elastomers at finite strains. A ... more A constitutive model is developed for the mechanical response of elastomers at finite strains. A polymer is treated as a network of linear chains linked by permanent (chemical crosslinks) and temporary (entanglements and van der Waals forces) junctions. Temporary junctions are assumed to be in two states: loose (passive) when they impose only topological constrains on available configurations of chains, and tight (active) when their effect is tantamount to that for crosslinks. Stretching of a specimen implies that some loose junctions become active, which decreases the average length of a chain. A long chain is treated as an ensemble of inextensible strands connected in sequel. Two neighboring strands are bridged by a bond which may be in two conformations: flexed (trans) and extended (cis). A bond in the flexed conformation is modeled as a linear elastic solid, whereas the mechanical energy of a bond in the extended conformation (two rigid rods directed along a straight line) is disregarded. For a virgin specimen, all bonds are in the flexed conformation. Under loading some bonds are transformed from flexed to extended conformation. Stress-strain relations for a rubbery polymer and kinetic equations for the transcis transition are derived using the laws of thermodynamics. Governing equations are determined by 5 adjustable parameters which are found by fitting experimental data in uniaxial tensile tests on natural rubber vulcanizates with various amounts of crosslinks. Fair agreement is demonstrated between results of numerical simulation and observations with the elongation ratio up to k = 8. We analyze the effects of cyclic loading, thermal annealing and recovery by swelling on the material constants.

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Nonlinear magnetoelastic deformations

Quarterly Journal of Mechanics and Applied Mathematics, Nov 1, 2004

In this paper we first summarize, in a simple form, the equilibrium equations for a solid materia... more In this paper we first summarize, in a simple form, the equilibrium equations for a solid material capable of large magnetoelastic deformations. Such equations are needed for the analysis of boundary-value problems for elastomers endowed with magnetic properties by the embedding of distributions of ferrous particles. The general constitutive law for an isotropic material in the presence of a magnetic field is described and expressed in a compact form, with either the magnetic field or the magnetic induction as the independent magnetic variable. The equations are applied, in the case of an incompressible material, to the solution of representative problems involving circular cylindrical geometry, specifically the helical shear of a circular cylindrical tube, its specializations to axial and azimuthal shear, and the problem of extension and torsion of a solid circular cylinder. For each problem a general formulation is afforded without specialization of the (isotropic) constitutive law, and then specific results are discussed briefly for special choices of such laws. It is noted, in particular, that certain restrictions may be placed on the class of constitutive laws for a considered combination of deformation and magnetic field to be admitted.

format_quoteNew formulations for axial and azimuthal shear problems yield clear results compared to previous analyses, enhancing understanding of magnetoelastic behaviors.format_quote

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Bifurcation of finitely deformed thick-walled electroelastic spherical shells subject to a radial electric field

International Journal of Non-linear Mechanics, May 1, 2020

This paper is concerned with the bifurcation analysis of a pressurized electroelastic spherical s... more This paper is concerned with the bifurcation analysis of a pressurized electroelastic spherical shell with compliant electrodes on its inner and outer boundaries. The theory of small incremental electroelastic deformations superimposed on a radially finitely deformed electroelastic thick-walled spherical shell is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect spherical shape of the shell. Specifically, axisymmetric bifurcations are analyzed, and results are obtained for three different electroelastic energy functions, namely electroelastic counterparts of the neo-Hookean, Gent and Ogden elastic energy functions. For the neo-Hookean energy function it was reported previously that for the purely mechanical case axisymmetric bifurcations are possible under external pressure only, no bifurcation solutions being possible for internally pressurized spherical shells. In the case of an electroelastic neo-Hookean model bifurcation under internal pressure becomes possible when the potential difference between the electrodes exceeds a certain value, which depends on the ratio of inner to outer undeformed radii. Results obtained for the three classes of model are significantly different and are illustrated for a range of fixed values of the potential difference. Although of less practical significance, results are also shown for fixed charges, and these are both different between the models and different from the case of fixed potential difference.

format_quoteFor the neo-Hookean model, bifurcation first occurs at specific dimensionless pressures, demonstrating distinct dependencies on geometric parameters.format_quote

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Modeling the response of filled elastomers at finite strains by rigid-rod networks

Archive of Applied Mechanics, 2002

A constitutive model is developed for the isothermal response of particle-reinforced elastomers a... more A constitutive model is developed for the isothermal response of particle-reinforced elastomers at ®nite strains. An amorphous rubbery polymer is treated as a network of long chains bridged to permanent junctions. A strand between two neighboring junctions is thought of as a sequence of rigid segments connected by bonds. In the stress-free state, a bond may be in one of two stable conformations:¯exed and extended. The mechanical energy of a bond in the¯exed conformation is treated as a quadratic function of the local strain, whereas that of a bond in the extended conformation is neglected. An explicit expression is developed for the free energy of a network. Stress±strain relations and kinetic equations for the concentrations of bonds in various conformations are derived using the laws of thermodynamics. In the case of small strains, these relations are reduced to the constitutive equation for the standard viscoelastic solid. At ®nite strains, the governing equations are determined by four adjustable parameters which are found by ®tting experimental data in uniaxial tensile, compressive and cyclic tests. Fair agreement is demonstrated between the observations for several ®lled and un®lled rubbery polymers and the results of numerical simulation. We discuss the effects of the straining state, ®ller content, crosslink density and temperature on the adjustable constants.

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Nonlinear Electroelastic Interactions

Springer eBooks, Dec 25, 2013

In this chapter we bring together the theory of electrostatics and the nonlinear theory of elasti... more In this chapter we bring together the theory of electrostatics and the nonlinear theory of elasticity in order to build a theory of nonlinear electroelastic interactions. For this purpose it is necessary to distinguish between different measures of stress, and this is reflected in different choices of potential energy in considering the equation of energy balance. We then introduce Lagrangian counterparts of the usual Eulerian electric and electric displacement field vectors, and this facilitates the construction of electroelastic constitutive equations that have compact general forms. Material symmetry considerations lead to the definition of an isotropic electroelastic material and explicit representations for the total Cauchy stress and the electric displacement field (if the electric field is the independent electric variable), or the electric field (if the electric displacement is the independent electric variable), in terms of invariants of the independent kinematic and electric variables. In the final section of the chapter we consider the linear specialization of the general nonlinear theory and discuss briefly the linear theory of piezoelectricity.

Anisotrope Gummilager für die Erdbebenisolation von Brücken: Versuche am Rütteltisch

AAAに適用した残留応力を有する材料のモデル化【Powered by NICT】

Journal of The Mechanical Behavior of Biomedical Materials, 2016

fib Bulletin 22. Monitoring and safety evaluation of existing concrete structures

fib Bulletins, 2003

The effect of annealing temperature on the viscoelastic response of glassy polymers

Mechanics Research Communications, 2001

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Non-axisymmetric dilatation of a thick-walled aortic aneurysmal tissue

International Journal of Non-Linear Mechanics, 2018

The objective of this paper is to use the constrained mixture theory of growth and remodeling to ... more The objective of this paper is to use the constrained mixture theory of growth and remodeling to simulate the non-axisymmetric dilatation of a thick-walled aortic aneurysmal tissue. The primary load carrying constituents of the vascular tissue are elastin and collagen and the contribution of smooth muscle cells is secondary and therefore not included. In the homeostatic state a blood vessel is in a mechanobiologically stable regime. Hence, a loss of wall material is compensated by production of new material without a significant dilatation of the artery. Using the theory we find that a local degradation of the matrix material produces a mechanobiologically unstable regime that causes aneurysm formation. It induces an increase of mass locally achieved via production of new material that exceeds the removal of old material. The combined effects of loss of elastin, degradation of existing and deposition of new collagen as well as remodeling results in a continuous enlargement of the aneurysm bulge. Numerical results are included to verify and validate the theory.

format_quoteCollagen content in aneurysmal aorta generally increases due to elastin degradation, but may not significantly change in familial cases, implying genetic factors.format_quote

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Histology and Biaxial Mechanical Behavior of Abdominal Aortic Aneurysm Tissue Samples

Journal of biomechanical engineering, Jan 18, 2016

Abdominal aortic aneurysms (AAA) represent permanent, localized dilations of the abdominal aorta ... more Abdominal aortic aneurysms (AAA) represent permanent, localized dilations of the abdominal aorta that can be life-threatening if progressing to rupture. Evaluation of risk of rupture depends on understanding the mechanical behavior of patient AAA walls. In this project, a series of patient AAA wall tissue samples have been evaluated through a combined anamnestic, mechanical and histopathologic approach. Mechanical properties of the samples have been characterized using a novel, strain-controlled, planar biaxial testing protocol emulating the in vivo deformation of the aorta. Histologically, the tissue ultrastructure was highly disrupted. All samples showed pronounced mechanical stiffening with stretch and were notably anisotropic, with greater stiffness in the circumferential than the axial direction. However, there were significant intra-patient variations in wall stiffness and stress. In biaxial tests in which the longitudinal stretch was held constant at 1.1 as the circumferentia...

The pseudo‐elastic response of rubberlike solids

PAMM, 2004

In this paper we focus on the mechanical behaviour of filled natural rubber. Filled elastomers un... more In this paper we focus on the mechanical behaviour of filled natural rubber. Filled elastomers under cyclic loading show noticeable differences between the mechanical response under loading and unloading during the first cycles in oscillation tests. We examine the change in material response associated with the Mullins effect and with cavitation, the latter arising from hydrostatic tensile stresses of sufficient magnitude. Then, we focus on constitutive equations using the theory of pseudoelasticity. Specifically, the strain‐energy function depends on a scalar parameter, which provides a means for modifying the form, thereby reflecting stress softening observed during unloading. The dissipation of energy is also accounted for by the use of a dissipation function, which evolves with the deformation history. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Nonlinear response of an electroelastic spherical shell

International Journal of Engineering Science, 2014

In this paper the theory of nonlinear electroelasticity is used to examine radial deformations of... more In this paper the theory of nonlinear electroelasticity is used to examine radial deformations of a thick-walled spherical shell of soft dielectric material with compliant electrodes on its inner and outer surfaces combined with an internal pressure. A general expression for the dependence of the pressure on the deformation and a potential difference between the electrodes, or uniform surface charge distributions on the electrodes, is obtained in respect of a general incompressible isotropic electroelastic energy function. To illustrate the behavior of the shell specific forms of energy functions are used for numerical purposes, for either fixed potential difference or fixed charge distribution. Explicit general results are given for the limiting case of a thin-walled (or membrane) shell. Depending on the elastic part of the energy function (as distinct from that part which involves the electric field) different behaviors are obtained, including non-monotonicity of the pressure-radius relationship reminiscent of that known in the purely elastic situation, but moderated by the presence of an electric field.

format_quoteApplies theory to a thick-walled spherical shell, defining pressure-dependent deformations, crucial for practical electroelastic applications.format_quote

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Magnetoelastic Wave Propagation

Nonlinear Theory of Electroelastic and Magnetoelastic Interactions, 2013

In this chapter we analyze incremental small amplitude motions and magnetic fields superimposed o... more In this chapter we analyze incremental small amplitude motions and magnetic fields superimposed on an underlying finite deformation and magnetic field with a particular interest in the effect of the underlying configuration on the propagation of small amplitude waves. First we examine the propagation of homogeneous plane waves in an infinite medium of an incompressible magnetoelastic material where the underlying configuration corresponds to a homogenous deformation with a uniform magnetic field; no restriction is placed on the material symmetry. This involves an extension of the notion of strong ellipticity to the magnetoelastic context. The theory is then applied to a prototype model of a neo-Hookean magnetoelastic material. Following this we specialize to increments in two dimensions in a principal plane of an isotropic magnetoelastic material. This specialization is then applied to the study of surface wave propagation, first for Rayleigh-type waves on a half-space with the magnetic field either parallel to or perpendicular to the surface and then to Love-type waves with a layer of different material bonded to the half-space. Finally, we investigate the propagation of Bleustein–Gulyaev-type waves on a half-space without a layer. For each type of surface wave numerical results are obtained for the speed of wave propagation in terms of parameters associated with the underlying configuration and illustrated graphically, while for the Bleustein–Gulyaev-type waves some closed-form expressions are obtained for the wave speed in particular cases.

Nonlinear Electroelastic Interactions

Nonlinear Theory of Electroelastic and Magnetoelastic Interactions, 2013

Mechanical aspects of high damping rubber, 2nd Int

Variational treatment of electrostatic interaction force in atomic force microscopy

Zeitschrift für angewandte Mathematik und Physik, 2007

In this paper we introduce the mathematical model for the electrostatic interaction force between... more In this paper we introduce the mathematical model for the electrostatic interaction force between an atomic force microscope (AFM) tip and a sample surface. We formulate the electrostatic potential problem in Sobolev spaces and find the corresponding weak solution in terms of the integral potential, which can be approximated numerically by generalized Fourier series and used to find the interaction force between an AFM tip and a sample surface. The formulation of the problem in a weak (Sobolev) space setting allows us to determine the force for AFM tips of arbitrary shape. Efficiency of the method is illustrated using numerical examples for the spherical and tetrahedral AFM tips.

format_quoteVariational formulation modifies boundary integral equations to obtain weak solutions, showing effectiveness for diverse applications.format_quote

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Finite Viscoelasticity of Particle-Reinforced Elastomers: the Effect of Filler Content

Macromolecular Theory and Simulations, 2002

ABSTRACT Constitutive equations are derived for the time-dependent behavior of particle-reinforce... more ABSTRACT Constitutive equations are derived for the time-dependent behavior of particle-reinforced rubbers at isothermal loading with finite strains. An elastomer is thought of as a network of macromolecules bridged by junctions that slip with respect to the bulk material under straining. A filled rubber is treated as an ensemble of meso-regions where sliding of junctions occurs at different rates. Two types of domains are distinguished: passive, where slippage of junctions is prevented by intermolecular links, and active, where the sliding process is not affected by interchain forces. Activation of passive meso-regions under loading is modelled as mechanically-induced breakage of van der Waals links between strands. Stress–strain relations are developed using the laws of thermodynamics. For tensile relaxation tests, these equations are determined by four adjustable parameters that are found by fitting experimental data on gum rubber, unfilled rubber, and carbon black-filled natural rubber at longitudinal strains in the range from 100 to 250%. It is revealed that the rate of sliding and the concentration of active meso-regions are noticeably affected by stretching, whereas the distribution of potential energies for sliding of junctions is independent of strains, but is severely influenced by the filler content.

Micromechanical modeling of sand production and arching effects around a cavity

International Journal of Rock Mechanics and Mining Sciences, 1997

A two-dimensional discrete element method has been developed and applied to evaluate the influenc... more A two-dimensional discrete element method has been developed and applied to evaluate the influence of fluid saturation and flow rate on the stability of sand cavities. The model simulates interparticle forces around an opening, individual grain movement, and the development of arching effects. Discrete element modeling of sand movement into cavities seems to capture much of the behavior observed in the laboratory and the field. Cavity stability is clearly influenced by saturation and flow effects as cavity growth progresses in an episodic pattern of destabilization and stabilization. Parametric simulations completed with this model indicate that fluid pressure gradients have a slightly greater influence on cavity growth than capillary force, especially at relatively low gradient and capillary force magnitudes.

format_quoteArching becomes unstable at pressure gradients above 0.045 MPa/m (-2 psi/ft) when capillary force is low, indicating limits for stability.format_quote

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Electroelastic waves in a finitely deformed electroactive material

IMA Journal of Applied Mathematics, 2010

ABSTRACT The coupling between a finite deformation and an electric field is examined with particu... more ABSTRACT The coupling between a finite deformation and an electric field is examined with particular reference to the propagation of small amplitude waves in a non-linear electroelastic material based on the quasi-electrostatic approximation. The general equations governing the linearized response of electroelastic solids superimposed on a state of finite deformation in the presence of an electric field are derived along with incremental forms of the electroelastic constitutive laws and boundary conditions. Both unconstrained and incompressible materials are considered. Without restriction on the electroelastic constitutive law, the theory is first applied to the analysis of plane waves propagating in a homogeneously deformed material with an underlying uniform electric field and illustrated in the case of an isotropic material. The general equations governing 2D incremental motions are then derived and applied to the study of surface waves in a homogeneously deformed half-space of incompressible isotropic material with the electric field normal to the surface of the half-space. The dependence of the wave speed on the deformation, the electric field and the electromechanical coupling parameters is illustrated for a prototype electroelastic constitutive law. Editorial remark: Originally published in [ibid. 75, No. 3, 356–365. According to the publisher, the article was published incorrectly in this previous issue.

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