Undeformed Configuration Research Articles

A principal axis formulation for nonlinear magnetoelastic deformations: Isotropic bodies

In this work a new set of principal axis invariants is proposed in order to study the problem of considering large magneto-elastic deformations, for bodies that are isotropic in the un-deformed configuration when no external magnetic induction is applied. The new set of invariants has clear physical meanings and may have an experimental advantage over the standard invariants used in many previous works in this area. The principal axis invariant formulation is also shown to be more general. Some simple boundary value problems are solved, such as the simple shear, and the biaxial extension of a slab, where with the use of these new invariants, it is possible to study in a much simpler manner the effect of different types of deformations on the response of the material. An illustrated simple specific constitutive equation is proposed which compares well with experiment. • A new set of invariants are proposed to be used in the modelling of magneto-active elastomers. • The new set of invariants has clearer physical meanings than the standard set of Rivlin and Spencer. • Some simple boundary value problems are solved considering the new formulation. • A general form for the total energy function has been proposed.

Constitutive Models for Uniaxially-Post-Buckled Square Lattices

The objective of this paper is to analyse the mechanical stress-strain response of an ini-tially uniaxially-post-buckled periodic structure. This requires selecting a representative volume ele-ment (RVE) that contains all the micro-structural features and responds as the infinite medium when subjected to uniform load and boundary conditions. Selection is done by analysing the stress-strain response for macro-cells of increasing size as well as identifying dominant length scales with a 2D-FFT. For a uniaxially–post buckled square lattice, the RVE is found to be a Y3x3 (3x3 unit cells for the undeformed configuration). A study for the post-buckled behaviour of the pre-stressed RVE, with in-ternal components modelled as long slender beams, beyond elastic buckling is presented. Incremental constitutive models are then formulated analytically and numerically for loads above the critical load.

A Method for Predicting Collagen Fiber Realignment in Non-Planar Tissue Surfaces as Applied to Glenohumeral Capsule During Clinically Relevant Deformation

Previously developed experimental methods to characterize micro-structural tissue changes under planar mechanical loading may not be applicable for clinically relevant cases. Such limitation stems from the fact that soft tissues, represented by two-dimensional surfaces, generally do not undergo planar deformations in vivo. To address the problem, a method was developed to directly predict changes in the collagen fiber distribution of nonplanar tissue surfaces following 3D deformation. Assuming that the collagen fiber distribution was known in the un-deformed configuration via experimental methods, changes in the fiber distribution were predicted using 3D deformation. As this method was solely based on kinematics and did not require solving the stress balance equations, the computational efforts were much reduced. In other words, with the assumption of affine deformation, the deformed collagen fiber distribution was calculated using only the deformation gradient tensor (obtained via an in-plane convective curvilinear coordinate system) and the associated un-deformed collagen fiber distribution. The new method was then applied to the glenohumeral capsule during simulated clinical exams. To quantify deformation, positional markers were attached to the capsule and their 3D coordinates were recorded in the reference position and three clinically relevant joint positions. Our results showed that at 60deg of external rotation, the glenoid side of the posterior axillary pouch had significant changes in fiber distribution in comparison to the other sub-regions. The larger degree of collagen fiber alignment on the glenoid side suggests that this region is more prone to injury. It also compares well with previous experimental and clinical studies indicating maximum principle strains to be greater on the glenoid compared to the humeral side. An advantage of the new method is that it can also be easily applied to map experimentally measured collagen fiber distribution (obtained via methods that require flattening of tissue) to their in vivo nonplanar configuration. Thus, the new method could be applied to many other nonplanar fibrous tissues such as the ocular shell, heart valves, and blood vessels.

Evolution of anisotropy in soft tissue

The phenomenological approach to the modelling of the mechanical response of arteries usually assumes a reduced form of the strain-energy function in order to reduce the mathematical complexity of the model. A common approach eschews the full basis of seven invariants for the strain-energy function in favour of a reduced set of only three invariants. It is shown that this reduced form is not consistent with the corresponding full linear theory based on infinitesimal strains. It is proposed that compatibility with the linear theory is an essential feature of any nonlinear model of arterial response. Two approaches towards ensuring such compatibility are proposed. The first is that the nonlinear theory reduces to the full six-constant linear theory, without any restrictions being imposed on the constants. An alternative modelling strategy whereby an anisotropic material is compatible with a simpler material in the linear limit is also proposed. In particular, necessary and sufficient conditions are obtained for a nonlinear anisotropic material to be compatible with an isotropic material for infinitesimal deformations. Materials that satisfy these conditions should be useful in the modelling of the crimped collagen fibres in the undeformed configuration.

A displacement‐based finite element formulation for general polyhedra using harmonic shape functions

SUMMARYA displacement‐based continuous‐Galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics. The polyhedra can have an arbitrary number of vertices or faces. The faces of the polyhedra can have an arbitrary number of edges and can be nonplanar. The polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex‐averaged centroid. Conforming shape functions are constructed using harmonic functions defined on the undeformed configuration, thus requiring the use of a total‐Lagrangian finite element formulation in large deformation applications. For nonlinear applications with computationally intensive constitutive models, it is important to minimize the number of element quadrature points. For this reason, an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices. As a first step toward verifying the element behavior in the general nonlinear setting, several linear verification examples are presented using both random Voronoi meshes and distorted hexahedral meshes. For the hexahedral meshes, results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation. The element behavior in the nearly incompressible regime is also examined. Copyright © 2013 John Wiley & Sons, Ltd.

On a recursive formulation for solving inverse form finding problems in isotropic elastoplasticity

BackgroundInverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. The deformed configuration of the functional component, the applied forces and boundary conditions are given and the undeformed configuration of this component is sought.MethodsIn this paper we present a new recursive formulation for solving inverse form finding problems for isotropic elastoplastic materials, based on an inverse mechanical formulation written in the logarithmic strain space. First, the inverse mechanical formulation is applied to the target deformed configuration of the workpiece with the set of internal variables set to zero. Subsequently a direct mechanical formulation is performed on the resulting undeformed configuration, which will capture the path-dependency in elastoplasticity. The so obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the inverse mechanical formulation is started again with the target deformed configuration and the current state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached.ResultsIn our three numerical examples in isotropic elastoplasticity, the convergence was reached after five, six and nine iterations, respectively, when the set of internal variables is initialised to zero at the beginning of the computation. It was also found that when the initial set of internal variables is initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. Different starting values for the set of internal variables have no influence on the obtained undeformed configuration, if convergence can be achieved.ConclusionsWith the presented recursive formulation we are able to find an appropriate undeformed configuration for isotropic elastoplastic materials, when only the deformed configuration, the applied forces and boundary conditions are given. An initial homogeneous set of internal variables equal to zero should be considered for such problems.

Size-dependent surface energy density of typically fcc metallic nanomaterials

The surface energy density of nano-sized structural elements exhibits an obvious size-dependent feature. To study this interesting phenomenon, atomistic calculations are carried out in the present paper for different face-centered-cubic (fcc) metallic nano-slabs. Lagrangian and Eulerian descriptions are adopted, respectively, in order to find the varying trends of surface energy densities in an initially un-deformed configuration and a current one. It is found that the Lagrangian surface energy density increases monotonically with an increase of the nano-slab’s thickness in the former no matter what the surface orientation is; while the variation of the Eulerian one is indefinite. The surface relaxation parameters are further simulated for differently oriented surfaces, which gives a very good explanation for the differences between the Lagrangian and Eulerian surface energy densities. The results in this paper should be a useful supplement to theoretical studies on the surface/interface effect of nanomaterials.

Total Lagrangian particle method for the large-deformation analyses of solids and curved shells

A high-accuracy smoothed-particle hydrodynamics (SPH) method for a large-deformation solid is developed from two aspects: improving the completeness of the approximation function and deducing the governing equation in the undeformed configuration. On the basis of Combescure’s researches and our previous studies, the SPH shell theory based on the Mindlin–Ressiner plate is detailed, which has overcome the unbridgeable drawbacks of solid modeling of thin structures and pushed forward the engineering applications of solving large-deformation and other nonlinear issues. Afterward, several treatments of SPH solid and shell are carried out, including the hourglass mode, boundary conditions and irregular structures; moreover, the corresponding validations are also conducted to reveal the feasibility and effectiveness of the proposed treatments. Finally, the accuracy of the present SPH program is verified further through two benchmarks of a curved shell.

Identification of the Controlling Mechanism for Predicting Critical Loads in Elastomeric Bearings

Assessing the stability of individual isolators is an important consideration for the design of seismic isolation systems composed of elastomeric bearings. A key component for the stability assessment is the prediction of the critical load capacity of the individual bearings in the laterally undeformed (service) configuration and at a given lateral displacement (seismic). The current procedure for estimating the critical load capacity of an elastomeric bearing at a given lateral displacement, with a bolted connection detail, uses a ratio of areas to reduce the critical load capacity from that in the laterally undeformed configuration, referred to as the reduced area method. Although the reduced area method provides a simple means for the estimate, it lacks a rigorous theoretical basis and is unable to capture the trends observed from experimental data. In this study, the capability of two analytical models for predicting critical loads and displacements in elastomeric bearings is evaluated by comparison with data from past experimental studies. A global variance-based sensitivity analysis is performed on the analytical model showing the best predictive capability to identify the model parameters to which the model prediction is most sensitive. The results of the sensitivity analysis demonstrate that the model prediction is most sensitive to the properties that control the nonlinear behavior of the rotational spring for lateral displacements greater than approximately 0.6 times bearing diameter/width. This finding suggests that the stability of elastomeric bearings at large lateral displacements is controlled by the transition from the yield moment to the ultimate moment in an individual rubber layers. A modified analytical model is proposed based on the results of this sensitivity analysis. The predictive capability of the more parsimonious modified model is shown to be similar, if not improved, by comparison to the original model.

Coupling between Transverse Vibrations and Instability Phenomena of Plates Subjected to In-Plane Loading

As it is known, the problems of free transverse vibrations and instability under in-plane loads of a plate are two different technological situations that have similarities in their approach to elastic solution. In fact, they are two eigenvalue problems in which we analyze the equilibrium situation of the plate in configurations which differ very slightly from the original, undeformed configuration. They are coupled in the event where in-plane forces are applied to the edges of the transversely vibrating plate. The presence of forces can have a significant effect on structural and mechanical performance and should be taken into account in the formulation of the dynamic problem. In this study, distributed forces of linear variation are considered and their influence on the natural frequencies and corresponding normal modes of transverse vibration is analyzed. It also analyzes their impact for the case of vibration control. The forces' magnitude is varied and the first natural frequencies of transverse vibration of rectangular thin plates with different combinations of edge conditions are obtained. The critical values of the forces which cause instability are also obtained. Due to the analytical complexity of the problem under study, the Ritz method is employed. Some numerical examples are presented.

Severe Bending of Two Aortic Stent-Grafts: An Experimental and Numerical Mechanical Analysis

Stent-grafts (SGs) are commonly used for treating abdominal aortic aneurysms (AAAs) and numerical models tend to be developed for predicting the biomechanical behavior of these devices. However, due to the complexity of SGs, it is important to validate the models. In this work, a validation of the numerical model developed in Demanget et al. (J. Mech. Behav. Biomed. Mater. 5:272-282, 2012) is presented. Two commercially available SGs were subjected to severe bending tests and their 3D geometries in undeformed and bent configurations were imaged from X-ray microtomography. Dedicated image processing subroutines were used in order to extract the stent centerlines from the 3D images. These skeletons in the undeformed configurations were used to set up SG numerical models that are subjected to the boundary conditions measured experimentally. Skeletons of imaged and deformed stents were then quantitatively compared to the numerical simulations. A good agreement is found between experiments and simulations. This validation offers promising perspectives to implementing the numerical models in a computer-aided tool and simulating the endovascular treatments.

Analysis and design of rolling-contact joints for evaluating bone plate performance

An apparatus for testing maxillofacial bone plates has been designed using a rolling contact joint. First, a free-body representation of the fracture fixation techniques utilizing bone plates is used to illustrate how rolling contact joints accurately simulate in vivo biomechanics. Next, a deterministic description of machine functional requirements is given, and is then used to drive the subsequent selection and design of machine elements. Hertz contact stress and fatigue analysis for two elements are used to ensure that the machine will both withstand loads required to deform different plates, and maintain a high cycle lifetime for testing large numbers of plates. Additionally, clinically relevant deformations are presented to illustrate how stiffness is affected after a deformation is applied, and to highlight improvements made by the machine over current testing standards, which do not adequately re-create in vivo loading conditions. The machine performed as expected and allowed for analysis of bone plates in both deformed and un-deformed configurations to be conducted. Data for deformation experiments is presented to show that the rolling-contact testing machine leads to improved loading configurations, and thus a more accurate description of plate performance. A machine for evaluation of maxillofacial bone plates has been designed, manufactured, and used to accurately simulate in vivo loading conditions to more effectively evaluate the performance of both new and existing bone plates.

Antiplane Wave Scattering from a Cylindrical Void in a Pre-Stressed Incompressible Neo-Hookean Material

AbstractAn isolated cylindrical void is located inside an incompressible nonlinear-elastic medium whose constitutive behaviour is governed by a neo-Hookean strain energy function. In-plane hydrostatic pressure is applied in the far-field so that the void changes its radius and an inhomogeneous region of deformation arises in the vicinity of the void. We consider scattering from the void in the deformed configuration due to an incident field (of small amplitude) generated by a horizontally polarized shear (SH) line source, a distance r0 (R0) away from the centre of the void in the deformed (undeformed) configuration. We show that the scattering coefficients of this scattered field are unaffected by the pre-stress (initial deformation). In particular, they depend not on the deformed void radius a or distance r0, but instead on the original void size A and original distance R0.

Biaxially stretched nematic liquid crystalline elastomers

The biaxial stretching of sheets of liquid crystalline neo-Hookean elastomer has been studied in the isotropic case. The results suggest two types of laminate structures in the process of quasiconvexification of the free energy, a fact that implies the appearance of several shear terms in the deformation gradient matrix. More that one decomposition of the deformation gradient is possible, which is consistent with a bifurcation in the undeformed configuration (λ = 1) . This situation is similar to the well-known Rivlin's problem of the triaxial symmetric traction of a neo-Hookean cube. The problem can easily be generalized for an anisotropic material by introducing a semisoft term in the free-energy expression. In this case, the horizontal plateau corresponding to the minimal energy, characteristic of the soft elasticity, disappears, and only an equilibrium condition is obtained.

A virtual experiment technique on the elementary behaviour of granular materials with discrete element method

SUMMARYMulti‐scale investigations aided by the discrete element method (DEM) play a vital role for current state‐of‐the‐art research on the elementary behaviour of granular materials. Similar to laboratory tests, there are three important aspects to be considered carefully, which are the proper stress/strain definition and measurement, the application of target loading paths and the designed experiment setup, to be addressed in the present paper. Considering the volume sensitive characteristics of granular materials, in the proposed technique, the deformation of the tested specimen is controlled and measured by deformation gradient tensor involving both the undeformed configuration and the current configuration. Definitions of Biot strain and Cauchy stress are adopted. The expressions of them in terms of contact forces and particle displacements, respectively, are derived. The boundary of the tested specimen consists of rigid massless planar units. It is suggested that the representative element uses a convex polyhedral (polygonal) shape to minimize possible boundary arching effects. General loading paths are described by directly specifying the changes in the stress/strain invariants or directions. Loading can be applied in the strain‐controlled mode by specifying the translations and rotations of the boundary units, or in the stress‐controlled mode by using a servo‐control mechanism, or in the combination of the two methods to realize mixed boundary conditions. Taking the simulation results as the natural consequences originated from a complex system, virtual experiments provide particle‐scale information database to conduct multi‐scale investigations for better understanding in granular material behaviours and possible development of the constitutive theories provided the qualitative similarity between the simulation results from virtual experiments and observations on real material behaviour. Copyright © 2011 John Wiley & Sons, Ltd.

Elasticity and permeability of porous fibre-reinforced materials under large deformations

Soft biological tissues with collagen reinforcement can be represented by a porous matrix saturated by a fluid and reinforced by a network of statistically oriented, impermeable fibres. This paper aims at determining the effect of the fibres on both the elastic properties and the permeability of the system, under large deformations, and represents the unification and generalisation of previous works in which the elasticity was studied for a pure solid, in the absence of the pore fluid, and the permeability was studied in the neighbourhood of the undeformed configuration. Throughout this work, the saturation constraint is assumed to hold, and the solid and fluid phases are assumed to be intrinsically incompressible. These hypotheses imply that the volumetric fraction of one of the two phases (e.g., the solid) is sufficient to determine the distribution of solid and fluid mass at every point of the homogenised porous medium. Overall incompressibility is achieved at compaction, i.e., when pores are closed and all the fluid has escaped. A new form of the elastic strain energy potential is proposed, based on the sum of a given “base” potential and a “correction” potential, function of the volumetric deformation, which serves solely to impose the incompressibility constraint at compaction. The large-strain overall permeability is obtained by employing a pull-back of the structure tensor to the reference configuration. This gives rise to an integral form of the permeability that needs to be calculated at each increment of deformation. The presentation is entirely covariant, so that general curvilinear coordinates can immediately be employed, if needed.

Analytical Solution for Radial Deformations of Functionally Graded Isotropic and Incompressible Second-Order Elastic Hollow Spheres

We analytically analyze radial expansion/contraction of a hollow sphere composed of a second-order elastic, isotropic, incompressible and inhomogeneous material to delineate differences and similarities between solutions of the first- and the second-order problems. The two elastic moduli are assumed to be either affine or power-law functions of the radial coordinate R in the undeformed reference configuration. For the affine variation of the shear modulus μ, the hoop stress for the linear elastic (or the first-order) problem at the point R=(R ou R in (R ou +R in )/2)1/3 is independent of the slope of the μ vs. R line. Here R in and R ou equal, respectively, the inner and the outer radius of the sphere in the reference configuration. For μ(R)∝R n , for the linear problem, the hoop stress is constant in the sphere for n=1. However, no such results are found for the second-order (i.e., materially nonlinear) problem. Whereas for the first-order problem the shear modulus influences only the radial displacement and not the stresses, for the second-order problem the two elastic constants affect both the radial displacement and the stresses. In a very thick homogeneous hollow sphere subjected only to pressure on the outer surface, the hoop stress at a point on the inner surface depends upon values of the two elastic moduli. Thus conclusions drawn from the analysis of the first-order problem do not hold for the second-order problem. Closed form solutions for the displacement and stresses for the first-order and the second-order problems provided herein can be used to verify solutions of the problem obtained by using numerical methods.

Propagation of waves in an incompressible transversely isotropic elastic solid with initial stress: Biot revisited

In this paper, the general constitutive equation for a transversely isotropic hyperelastic solid in the presence of initial stress is derived, based on the theory of invariants. In the general finite deformation case for a compressible material this requires 18 invariants (17 for an incompressible material). The equations governing infinitesimal motions superimposed on a finite deformation are then used in conjunction with the constitutive law to examine the propagation of both homogeneous plane waves and, with the restriction to two dimensions, Rayleigh surface waves. For this purpose we consider incompressible materials and a restricted set of invariants that is sufficient to capture both the effects of initial stress and transverse isotropy. Moreover, the equations are specialized to the undeformed configuration in order to compare with the classical formulation of Biot. One feature of the general theory is that the speeds of homogeneous plane waves and surface waves depend nonlinearly on the initial stress, in contrast to the situation of the more specialized isotropic and orthotropic theories of Biot. The speeds of (homogeneous plane) shear waves and Rayleigh waves in an incompressible material are obtained and the significant differences from Biot's results for both isotropic and transversely isotropic materials are highlighted with calculations based on a specific form of strain-energy function.

A deformational and configurational framework for geometrically non-linear continuum thermomechanics coupled to diffusion

A general framework encompassing both the (conventional) deformational and configurational settings of continuum mechanics is presented. A systematic application of balance principles over a migrating control volume in the undeformed configuration of the continuum body yields the system of governing equations in the bulk, on the surface and on a coherent interface within the continuum. The equations governing the response of the bulk agree with those of the conventional deformational approach. The localised balance equations are expressed in the configurational setting using a pull-back operator and reformulated in terms of the Eshelby stress. The configurational expression of the dissipation elucidates the energy loss associated with configurational changes. The general framework is introduced by considering the problem of coupled deformation, heat conduction and species diffusion within a geometrically non-linear continuum body intersected by a coherent interface. The nature of the coupling is emphasised throughout the presentation and via an example.

Initial stresses in elastic solids: Constitutive laws and acoustoelasticity

On the basis of the nonlinear theory of elasticity, the general constitutive equation for an isotropic hyperelastic solid in the presence of initial stress is derived. This derivation involves invariants that couple the deformation with the initial stress and in general, for a compressible material, it requires 10 invariants, reducing to 9 for an incompressible material. Expressions for the Cauchy and nominal stress tensors in a finitely deformed configuration are given along with the elasticity tensor and its specialization to the initially stressed undeformed configuration. The equations governing infinitesimal motions superimposed on a finite deformation are then used to study the combined effects of initial stress and finite deformation on the propagation of homogeneous plane waves in a homogeneously deformed and initially stressed solid of infinite extent. This general framework allows for various different specializations, which make contact with earlier works. In particular, connections with results derived within Biot's classical theory are highlighted. The general results are also specialized to the case of a small initial stress and a small pre-deformation, i.e. to the evaluation of the acoustoelastic effect. Here the formulas derived for the wave speeds cover the case of a second-order elastic solid without initial stress and subject to a uniaxial tension [Hughes and Kelly, Phys. Rev. 92 (1953) 1145] and are consistent with results for an undeformed solid subject to a residual stress [Man and Lu, J. Elasticity 17 (1987) 159]. These formulas provide a basis for acoustic evaluation of the second- and third-order elasticity constants and of the residual stresses. The results are further illustrated in respect of a prototype model of nonlinear elasticity with initial stress, allowing for both finite deformation and nonlinear dependence on the initial stress.