Deformation Formulation Research Articles

A new analysis of stresses in arteries based on an Eulerian formulation of growth in tissues

The simple analysis of stresses in arteries considers the artery as an incompressible elastic circular cylindrical tube. When the intact artery is unloaded and cut transmurally, it springs open to a nearly circular cylindrical sector. The resulting nonzero opening angle indicates the existence of residual stresses in the unloaded intact artery. This paper presents a new analysis of stresses in arteries based on an Eulerian formulation of elastic deformations in soft tissues that models the inelastic process of remodeling (homeostasis) towards its homeostatic state. Within the context of this new analysis it is not necessary to model details of homeostasis. Instead, it is possible to directly determine the loaded homeostatic state, which is considered as the reference configuration of the artery, and to specify the geometry and associated values of the circumferential and axial stresses by matching measurements of the geometry of the unloaded open artery. The predicted geometry of the unloaded intact artery is shown to be accurate relative to the measurements. Also, predictions of the stress distributions for remodeling due to hypertension are presented.

The Role of Surface Infiltration in Hydromechanical Coupling Effects in an Unsaturated Porous Medium of Semi-Infinite Extent

Rainfall infiltration into an unsaturated region of the earth’s surface is a pervasive natural phenomenon. During the rainfall-induced seepage process, the soil skeleton can deform and the permeability can change with the water content in the unsaturated porous medium. A coupled water infiltration and deformation formulation is used to examine a problem related to the mechanics of a two-dimensional region of semi-infinite extent. The van Genuchten model is used to represent the soil-water characteristic curve. The model, incorporating coupled infiltration and deformation, was developed to resolve the coupled problem in a semi-infinite domain based on numerical methods. The numerical solution is verified by the analytical solution when the coupled effects in an unsaturated medium of semi-infinite extent are considered. The computational results show that a numerical procedure can be employed to examine the semi-infinite unsaturated seepage incorporating coupled water infiltration and deformation. The analysis indicates that the coupling effect is significantly influenced by the boundary conditions of the problem and varies with the duration of water infiltration.

A basic study of peridynamics for elast-plastic analysis

Peridynamics (PD) was reformulated as a new continuum mechanics technique by Silling . So far many brittle fracture problems have been performed and the effectiveness is examined, it also has a potential for ductile fracture problem. This paper presents a formulation and discretization for elastic-plastic deformation for ductile fracture. Plastic strain increments are assumed by force state increments in each bond in ordinary state based PD unlike considering stress increments in classical continuum mechanics theory. The validity of plastic deformation is examined by simple problem as a rectangular plate applied cyclic tensile/compressive loads.

Size dependent dynamic behavior of electrostatically actuated microbridges

In this paper, based on the size dependent theories, an accurate investigation for nonlinear dynamic analysis of electrostatically actuated microbridges is presented by using the nonlinear finite element formulation, analytical method and numerical approach. The microbridge used as a microresonator, is driven by simultaneously applied DC and AC voltages. In the framework of modified couple stress theory (MCST), the nonlinear equation of dynamic motion for this system is derived using extended Hamilton's principle. Thereupon, the nonlinear partial differential equation is converted to the nonlinear ordinary equation by Galerkin's method and solved by analytical and numerical techniques. The analytical solution for nonlinear vibration of the microresonators is obtained by perturbation theory. Findings show that the numerical and analytical results are in good agreement with each other. In order to make further verification of the analytical expression, a nonlinear nonclassical finite element formulation for dynamic deformation of the system is presented to calculate the time history results. The good agreement observed between the results of analytical method and nonlinear finite element simulation demonstrates that the procedures used in the analytical method such as Taylor expansion, one mode analysis and perturbation expansion in multiple scale approach are appropriate. By determining the frequency-response curves, the effects of size dependency on the amplitude of vibration and frequency position of the peak response are studied. Results show that considering the size dependent theory leads to notable decreasing of the amplitude of nonlinear vibration and approaching the frequency value of peak response to linear resonance frequency.

Hygro-thermo-mechanical behavior of classical composites using a new trigonometrical shear strain shape function and a compact layerwise approach

The present paper discusses the hygro-thermo-mechanical behavior of multilayered composite and sandwich plates based on a Layerwise Displacement (LD) approach using the Carrera’s Unified Formulation (CUF). Legendre polynomials and a new proposed trigonometrical shear strain shape function are evaluated within a compact shear deformation formulation. Hygro-thermo-mechanical loads are applied considering linear and calculated profiles solved by the Fick moisture diffusion law equation and the Fourier heat conduction equation. The Navier solution method was used to solve the governing equations obtained by the principle of the virtual displacement (PVD). Results are in good agreement with published results in the literature. Some bending problem benchmarks considering hygro-thermo-mechanical loads applied in a coupled manner are presented.

A formulation of tooth deformation and its error compensation for worm gear transmission mechanisms

Milling head and rotary table are the key components in five-axis CNC machine tools. However, tracking and positioning precision of these rotary axes based on worm gear mechanisms are affected by the backlash, friction, elastic deformation, and other nonlinearities. Therefore, finding out a simple and universal principle to predict the tooth deformation in an initial design is quite significant. A novel deformation calculation and compensation method with finite element analysis and circular plate theory is proposed for the worm gear transmissions. When the relationship (form factor) between the solutions of circular plate theory and finite element analysis is derived, deformation calculation with finite element analysis can be replaced by multiplying the solution of circular plate theory and the form factor. Analysis and calculation results show that the presented method can provide a reference for the compensation control of the positioning errors.

Offline Predictive Control of Out-of-Plane Shape Deformation for Additive Manufacturing

Abstract Additive manufacturing (AM) is a promising direct manufacturing technology, and the geometric accuracy of AM built products is crucial to fulfill the promise of AM. Prediction and control of three-dimensional (3D) shape deformation, particularly out-of-plane geometric errors of AM built products, have been a challenging task. Although finite-element modeling has been extensively applied to predict 3D deformation and distortion, improving part accuracy based purely on such simulation still needs significant methodology development. We have been establishing an alternative strategy that can be predictive and transparent to specific AM processes based on a limited number of test cases. Successful results have been accomplished in our previous work to control in-plane (x–y plane) shape deformation through offline compensation. In this study, we aim to establish an offline out-of-plane shape deformation control approach based on limited trials of test shapes. We adopt a novel spatial deformation formulation in which both in-plane and out-of-plane geometric errors are placed under a consistent mathematical framework to enable 3D accuracy control. Under this new formulation of 3D shape deformation, we develop a prediction and offline compensation method to reduce out-of-plane geometric errors. Experimental validation is successfully conducted to validate the developed 3D shape accuracy control approach.

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

In this work, we present a new rate type formulation of large deformation generalized plasticity which is based on the consistent use of the logarithmic rate concept. For this purpose, the basic constitutive equations are initially established in a local rotationally neutralized configuration which is defined by the logarithmic spin. These are then rephrased in their spatial form, by employing some standard concepts from the tensor analysis on manifolds. Such an approach, besides being compatible with the notion of (hyper)elasticity, offers three basic advantages, namely: (i) The principle of material frame-indifference is trivially satisfied. (ii) The structure of the infinitesimal theory remains essentially unaltered. (iii) The formulation does not preclude anisotropic response. A general integration scheme for the computational implementation of generalized plasticity models which are based on the logarithmic rate is also discussed. The performance of the scheme is tested by two representative numerical examples.

Generalized Component Mode Synthesis for the Spatial Motion of Flexible Bodies With Large Rotations About One Axis1

In industrial practice, the floating frame of reference formulation (FFRF)—often combined with the component mode synthesis (CMS) in order to reduce the number of flexible degrees-of-freedom—is the common approach to describe arbitrarily shaped bodies in flexible multibody systems. Owed to the relative formulation of the flexible deformation with respect to the reference frame, the equations of motion show state-dependent nonconstant inertia terms. Such relative description, however, comes along with considerable numerical costs, since both the mass matrix and gyroscopic forces, i.e., the quadratic velocity vector, need to be evaluated in every integration step. The state dependency of the inertia terms can be avoided by employing an alternative formulation based on the mode shapes as in the classical CMS approach. In this approach, which is referred to as generalized component mode synthesis (GCMS), the total (absolute) displacements are approximated directly. Consequently, the mass matrix is constant, no quadratic velocity vector appears, and the stiffness matrix is a corotated but otherwise constant matrix. In order to represent the same flexible deformation as in the classical FFRF-based CMS, however, a comparatively large number of degrees-of-freedom is required. The approach described in the present paper makes use of the fact that a majority of components in technical systems are constrained to motions showing large rotations only about a single spatially fixed axis. For this reason, the GCMS is adapted for multibody systems that are subjected to small flexible deformations and undergo a rigid body motion showing large translations, large rotations about one axis, but small rotations otherwise. Thereby, the number of shape functions representing the flexible deformation is reduced, which further increases numerical efficiency compared to the original GCMS formulation for arbitrary rotations.

Instability Analysis of Sand under Undrained Biaxial Loading with Rigid and Flexible Boundary

AbstractThe emergence of possible instability modes under undrained biaxial shear of sand was investigated to develop better understanding about the onset of localized and liquefaction-type solid-fluid instability modes using a generalized three-dimensional (3D) material model. These two modes are the primary instability modes that can exist in most cases while conducting undrained biaxial testing of sand at different densities, confining pressures, and boundary conditions. Using a 3D rate-independent and nonassociative constitutive model, the instability analysis was performed as a plane-strain bifurcation problem from a uniform stress-strain state. Large deformation formulation was used to simulate the biaxial test configuration with both rigid and flexible boundaries in the lateral direction. The existing theoretical framework of solid-fluid instability analysis under rigid boundaries was extended to the flexible lateral boundary condition. Interestingly, the onset of solid-fluid instability modes is s...

Two-dimensional displacement discontinuity method for transversely isotropic materials

This paper presents the fundamental solution for a two-dimensional displacement discontinuity method (DDM) for transversely isotropic elastic materials. We follow the procedures shown in the literature, in which there are some typographic errors and a lack of proper explanations for some expressions. Based on the fundamental solution of deformation due to a single point force in transversely isotropic materials, the formulation for deformation from force dipoles has been revisited. Generalised Hooke's Law is used to establish the relationship between dipole strengths and displacement discontinuities, which leads to the fundamental formulation of DDM for transversely isotropic materials. We present the full details of derivation and corrections to some expressions which have previously been presented with errors. In addition, we present the fundamental solution expressions for DDM for one situation which was not included in the literature. The fundamental solutions are implemented in an existing DDM code, FRACOD, and the method is verified by some examples with an analytic solution and finite element method. Furthermore an engineering application is simulated with the scheme.

Optimization, in vitro cytotoxicity and penetration capability of deformable nanovesicles of paclitaxel for dermal chemotherapy in Kaposi sarcoma

Although much research has been published on ways to overcome the low oral bioavailability of paclitaxel, exploration of novel drug delivery systems that can target paclitaxel deep in to the dermal areas in AIDS-related Kaposi sarcoma (KS) have not yet been reported. Our aim was to develop deformable nanovesicles of paclitaxel capable of being used in dermal chemotherapy, especially deep into the dermal areas of AIDS related KS. Deformable nanovesicular formulations (TS1–TS15) composed of soya lecithin and span80 were prepared by the rotary evaporation sonication method within the constraints of our Box–Behnken design. The formulations were subjected to vesicle characterization and ex vivo permeation. The optimized vesicular suspension was formulated as a gel and assessed for in vitro cytotoxicity and penetration characteristics by confocal laser scanning microscopy (CLSM). TS9 with vesicle size characteristics of 185.76 ± 2.15 nm, zeta potential of −23.2 mV, deformability index = 138.02 and cumulative drug permeation of 89.80 ± 1.84% was identified as the optimized formulation. TEM revealed spherical vesicles with firm boundaries that were stable at 4 °C. TS9 was developed as carbopol 934P gel (TG) and compared with the control gel (CG) made with the pure drug (paclitaxel). TG showed significantly higher (p < 0.05) in vitro drug permeation and flux compared to the CG. In vitro cytotoxicity study on KSY-1 cell lines revealed higher IC50 (≤17) for TS against IC50 ≤19 for TG. CLSM confirmed the penetrating potential of transfersomes via TG to the dermal layers of skin, the proposed target site. Conclusively, deformable nonovesicles of paclitaxel appear as a feasible alternative to the conventional formulations of paclitaxel in the management of AIDS-related KS.

Applicability of Using GIN Method, by Considering Theoretical Approach of Grouting Design

In the practice of grouting of fractured rock, currently, empirical methods are used. Amongst them, the GIN method is popular mostly in Europe and has been tried in many projects. The concept of this method is to limit the combination of pressure and injected volume to a specific grout intensity number in order to control the energy induced in the rock fractures and to avoid uplift. However, difficulties in employing this method have been reported, which are mainly due to uncertainties in recognizing the distance of grout penetration and the state of the fractures during grouting and at the completion grouting. In this paper, the purpose has been examining the applicability of the GIN method by defining the characteristic curve of the P·V diagram (referred to here as the hyperbola) and suggesting appropriate completion criteria based on the radius of grout spread around the borehole. This will provide the chance to assign a permitted level of fracture deformation (or jacking) to the GIN by considering the formulation of fracture deformation based on grout propagation in a previously developed theoretical approach by Stille et al. (Geotech Geol Eng 30:603–624, 2012) as a part of the Real Time Grouting Control Method. Thus, in attaining the hyperbola, the identified radius of grout spread is achieved and the resulting fracture deformation at this completion point can be beneficial in improving penetrability. However, if the full extent of this deformation extends beyond the grouted zone, part of the fracture may remain un-grouted, and this will affect the sealing efficiency of the grouting program. This may be continued by selecting a smaller GIN and reducing the grouting pressure as the real time pressure–volume plot moves along the hyperbola, which will bring the fracture back to its initial state as grouting approaches the completion point, i.e. when the grout has spread to the desired distance. This hypothesis has been examined against the grouting works performed in three different real projects, for which the grouting parameters can be determined from the available grouting records. It is concluded that the GIN used in practice was much higher than the theoretically estimated values obtained through the proposed analytical solution. Furthermore, in the grouting of fractures close to the surface, the radius of grout spread impacts the GIN significantly, and only a limited grouting pressure is applicable, thus in using split spacing technique in such circumstances, different GINs should be selected for different sets of boreholes to obtain enough propagation at the maximum applicable pressure. The introduced analytical solution introduced in this paper can be a useful procedure for designing the GIN based on the grout spread. Nevertheless, it becomes complicated in dealing with fracture deformation. In a difficult grouting case where the demand for sealing is high, the recommendation is to use the proposed theoretical approach, which provides detailed information during the actual grouting procedure, by estimation of the radius of grout spread and the state of the fracture in real time.

Predicting electromyographic signals under realistic conditions using a multiscale chemo-electro-mechanical finite element model.

This paper presents a novel multiscale finite element-based framework for modelling electromyographic (EMG) signals. The framework combines (i) a biophysical description of the excitation-contraction coupling at the half-sarcomere level, (ii) a model of the action potential (AP) propagation along muscle fibres, (iii) a continuum-mechanical formulation of force generation and deformation of the muscle, and (iv) a model for predicting the intramuscular and surface EMG. Owing to the biophysical description of the half-sarcomere, the model inherently accounts for physiological properties of skeletal muscle. To demonstrate this, the influence of membrane fatigue on the EMG signal during sustained contractions is investigated. During a stimulation period of 500 ms at 100 Hz, the predicted EMG amplitude decreases by 40% and the AP propagation velocity decreases by 15%. Further, the model can take into account contraction-induced deformations of the muscle. This is demonstrated by simulating fixed-length contractions of an idealized geometry and a model of the human tibialis anterior muscle (TA). The model of the TA furthermore demonstrates that the proposed finite element model is capable of simulating realistic geometries, complex fibre architectures, and can include different types of heterogeneities. In addition, the TA model accounts for a distributed innervation zone, different fibre types and appeals to motor unit discharge times that are based on a biophysical description of the α motor neurons.

An alternative approach for quasi‐static large deformation analysis of saturated porous media using meshfree method

SummaryAn alternative coupled large deformation formulation combined with a meshfree approach is proposed for flow–deformation analysis of saturated porous media. The formulation proposed is based on the Updated Lagrangian (UL) approach, except that the spatial derivatives are defined with respect to the configuration of the medium at the last time step rather than the configuration at the last iteration. In this way, the Cauchy stresses are calculated directly, rendering the second Piola–Kirchhoff stress tensor not necessary for the numerical solution of the equilibrium equations. Moreover, in contrast with the UL approach, the nodal shape function derivatives are calculated once in each time step and stored for use in subsequent iterations, which reduces the computational cost of the algorithm. Stress objectivity is satisfied using the Jaumann stress rate, and the spatial discretisation of the governing equations is achieved using the standard Galerkin method. The equations of equilibrium are satisfied directly, and the nonlinear parts of the system matrix are derived independent of the stresses of the medium resulting in a stable numerical algorithm. Temporal discretisation is effected based on a three‐point approximation technique that avoids spurious ripple effects and has second‐order accuracy. The radial point interpolation method is used to construct the shape functions. The application of the formulation and the significance of large deformation effects on the numerical results are demonstrated through several numerical examples. Copyright © 2015 John Wiley &amp; Sons, Ltd.

High Resolution, Large Deformation 3D Traction Force Microscopy

Traction force microscopy (TFM) is a powerful approach of quantifying cell-material interactions, which over the last two decades has contributed significantly to our understanding of cellular mechanosensing and mechanotransduction. In addition, recent advances in three-dimensional (3D) imaging and traction force analysis (3D TFM) have highlighted the significance of the third dimension in influencing various cellular processes. Yet irrespective of dimensionality almost all TFM approaches have relied on a linear elastic theory framework to calculate cell surface tractions.This talk presents a new high-resolution 3D TFM algorithm, which utilizes a large deformation formulation to quantify cellular displacement fields with unprecedented resolution. The results feature some of the first experimental evidence that cells are indeed capable of exerting large material deformations, which require the formulation of a new theoretical TFM framework to accurately calculate traction forces. Based on our previous 3D TFM technique we reformulate our approach to accurately account for large material deformation and quantitatively contrast and compare both linear and large deformation frameworks as a function of the applied cell deformation. Particular attention is paid in estimating the accuracy penalty associated with utilizing a traditional linear elastic approach in the presence of large deformation gradients.

On the feasibility of polyurethane based 3D dosimeters with optical CT for dosimetric verification of low energy photon brachytherapy seeds.

To investigate the feasibility of and challenges yet to be addressed to measure dose from low energy (effective energy <50 keV) brachytherapy sources (Pd-103, Cs-131, and I-125) using polyurethane based 3D dosimeters with optical CT. The authors' evaluation used the following sources: models 200 (Pd-103), CS-1 Rev2 (Cs-131), and 6711 (I-125). The authors used the Monte Carlo radiation transport code MCNP5, simulations with the ScanSim optical tomography simulation software, and experimental measurements with PRESAGE(®) dosimeters/optical CT to investigate the following: (1) the water equivalency of conventional (density = 1.065 g/cm(3)) and deformable (density = 1.02 g/cm(3)) formulations of polyurethane dosimeters, (2) the scatter conditions necessary to achieve accurate dosimetry for low energy photon seeds, (3) the change in photon energy spectrum within the dosimeter as a function of distance from the source in order to determine potential energy sensitivity effects, (4) the optimal delivered dose to balance optical transmission (per projection) with signal to noise ratio in the reconstructed dose distribution, and (5) the magnitude and characteristics of artifacts due to the presence of a channel in the dosimeter. Monte Carlo simulations were performed using both conventional and deformable dosimeter formulations. For verification, 2.8 Gy at 1 cm was delivered in 92 h using an I-125 source to a PRESAGE(®) dosimeter with conventional formulation and a central channel with 0.0425 cm radius for source placement. The dose distribution was reconstructed with 0.02 and 0.04 cm(3) voxel size using the Duke midsized optical CT scanner (DMOS). While the conventional formulation overattenuates dose from all three sources compared to water, the current deformable formulation has nearly water equivalent attenuation properties for Cs-131 and I-125, while underattenuating for Pd-103. The energy spectrum of each source is relatively stable within the first 5 cm especially for I-125. The inherent assumption of radial symmetry in the TG43 geometry leads to a linear increase in sample points within the 3D dosimeter as a function of distance away from the source, which partially offsets the decreasing signal. Simulations of dose reconstruction using optical CT showed the feasibility of reconstructing dose out to a radius of 10 cm without saturating projection images using an optimal dose and high dynamic range scanning; the simulations also predicted that reconstruction artifacts at the channel surface due to a small discrepancy in refractive index should be negligible. Agreement of the measured with calculated radial dose function for I-125 was within 5% between 0.3 and 2.5 cm from the source, and the median difference of measured from calculated anisotropy function was within 5% between 0.3 and 2.0 cm from the source. 3D dosimetry using polyurethane dosimeters with optical CT looks to be a promising application to verify dosimetric distributions surrounding low energy brachytherapy sources.

Pressure solution creep of random packs of spheres

AbstractWe performed numerical calculations of compaction in aggregates of spherical grains, using Lehner and Leroy's (2004, hereinafter LL) constitutive model of pressure solution at grain contacts. That model is founded on a local definition of the thermodynamic driving force and leads to a fully coupled formulation of elastic deformation, dissolution, and diffusive transport along the grain boundaries. The initial geometry of the aggregate was generated by random packing of spheres with a small standard deviation of the diameters. During the simulations, isostatic loading was applied. The elastic displacements at the contacts were calculated according to Digby's (1981) nonlinear contact force model, and deformation by dissolution was evaluated using the LL formulation. The aggregate strain and porosity were tracked as a function of time for fixed temperature, applied effective pressure, and grain size. We also monitored values of the average and standard deviation of total load at each contact, the coordination number for packing, and the statistics of the contact dimensions. Because the simulations explicitly exclude processes such as fracturing, plastic flow, and transport owing to surface curvature, they can be used to test the influence of relative changes in the kinetics of dissolution and diffusion processes caused by contact growth and packing rearrangements. We found that the simulated strain data could be empirically fitted by two successive power laws of the form, εx ∝ tξ, where ξ was equal to 1 at very early times, but dropped to as low as 0.3 at longer times. The apparent sensitivity of strain rate to stress found in the simulations was much lower than predicted from constitutive laws that assume a single dominant process driven by average macroscopic loads. Likewise, the apparent activation enthalpy obtained from the simulated data was intermediate between that assumed for dissolution and diffusion, and, further, tended to decrease with time. These results are similar to the experimental observations of Visser et al.'s (2012), who used an aggregate geometry and physical conditions closely resembling the present numerical simulations.

Numerical model of elastic laminated glass beams under finite strain

Laminated glass structures are formed by stiff layers of glass connected with a compliant plastic interlayer. Due to their slenderness and heterogeneity, they exhibit a complex mechanical response that is difficult to capture by single-layer models even in the elastic range. The purpose of this paper is to introduce an efficient and reliable finite element approach to the simulation of the immediate response of laminated glass beams. It proceeds from a refined plate theory due to Mau (1973), as we treat each layer independently and enforce the compatibility by the Lagrange multipliers. At the layer level, we adopt the finite-strain shear deformable formulation of Reissner (1972) and the numerical framework by Ibrahimbegović and Frey (1993). The resulting system is solved by the Newton method with consistent linearization. By comparing the model predictions against available experimental data, analytical methods and two-dimensional finite element simulations, we demonstrate that the proposed formulation is reliable and provides accuracy comparable to the detailed two-dimensional finite element analyzes. As such, it offers a convenient basis to incorporate more refined constitutive description of the interlayer.

High Resolution, Large Deformation 3D Traction Force Microscopy

Traction Force Microscopy (TFM) is a powerful approach for quantifying cell-material interactions that over the last two decades has contributed significantly to our understanding of cellular mechanosensing and mechanotransduction. In addition, recent advances in three-dimensional (3D) imaging and traction force analysis (3D TFM) have highlighted the significance of the third dimension in influencing various cellular processes. Yet irrespective of dimensionality, almost all TFM approaches have relied on a linear elastic theory framework to calculate cell surface tractions. Here we present a new high resolution 3D TFM algorithm which utilizes a large deformation formulation to quantify cellular displacement fields with unprecedented resolution. The results feature some of the first experimental evidence that cells are indeed capable of exerting large material deformations, which require the formulation of a new theoretical TFM framework to accurately calculate the traction forces. Based on our previous 3D TFM technique, we reformulate our approach to accurately account for large material deformation and quantitatively contrast and compare both linear and large deformation frameworks as a function of the applied cell deformation. Particular attention is paid in estimating the accuracy penalty associated with utilizing a traditional linear elastic approach in the presence of large deformation gradients.