Small Strain Assumption Research Articles

On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions

Expressions for thermodynamic potentials (internal energy, Helmholtz energy, Gibbs energy and enthalpy) of a thermoelastic material are developed under the assumption of small strains and finite changes in the thermal variable (temperature or entropy). The literature provides expressions for the Helmholtz energy in terms of strain and temperature, most often as expansions to the second order in strain and to a higher order in temperature changes, which ensures an affine stress–strain relation and a certain temperature dependence of the moduli of the material. Expressions are here developed for the four potentials in terms of all four possible pairs of independent variables. First, an expression is obtained for each potential as a quadratic function of its natural mechanical variable with coefficients depending on its natural thermal variable that are identified in terms of the moduli of the material. The form of the coefficients’ dependence on the thermal variable is not specified beforehand so as to obtain the most general expressions compatible with an affine stress–strain relation. Then, from each potential expressed in terms of its natural variables, expressions are derived for the other three potentials in terms of these same variables using the Gibbs–Helmholtz equations. The paper provides a thermodynamic framework for the constitutive modeling of thermoelastic materials undergoing small strains but finite changes in the thermal variables, the properties of which are liable to depend on the thermal variables.

Non-ideal effects in indentation testing of soft tissues

Indentation has several advantages as a loading mode for determining constitutive behavior of soft, biological tissues. However, indentation induces a complex, spatially heterogeneous deformation field that creates analytical challenges for the calculation of constitutive parameters. As a result, investigators commonly assume small indentation depths and large sample thicknesses to simplify analysis and then restrict indentation depth and sample geometry to satisfy these assumptions. These restrictions limit experimental resolution in some fields, such as brain biomechanics. However, recent experimental evidence suggests that conventionally applied limits are in fact excessively conservative. We conducted a parametric study of indentation loading with various indenter geometries, surface interface conditions, sample compressibility, sample geometry and indentation depth to quantitatively describe the deviation from previous treatments that results from violation of the assumptions of small indentation depth and large sample thickness. We found that the classical solution was surprisingly robust to violation of the assumption of small strain but highly sensitive to violation of the assumption of large sample thickness, particularly if the indenter was cylindrical. The ramifications of these findings for design of indentation experiments are discussed and correction factors are presented to allow future investigators to account for these effects without recreating our finite element models.

Analysis of Large Deformation Wound Roll Models

Almost all winding models incorporate the assumption of small linear deformations and strain in their development. These models treat the addition of a layer of web to a winding roll with linear analysis using linear strain theory. Very few winding models have been developed that incorporate large deformation theory although many models treat material nonlinearity. Tissue and nonwoven webs are highly extensible in-plane and highly compressible in the thickness dimension when compared to paper, plastic film, and metal foil webs. Winding models that embody large deformation theory should apply to all web materials. Such models may be wasteful in computation time for web materials such as paper, film, and foils where models that employ small deformation theory may provide sufficient accuracy. This would appear deterministic based upon the extensibility and compressibility of a web material, but the issue becomes more complex due to limitations in tension that can be exerted on the webs. Herein, a large deformation winding model will be developed. Results from this model will be used to benchmark results from other small and large deformation models, and with laboratory test data, a review of all results will be used to determine when or if large deformation winding models are required.

Experiments on the effective compliance in the radial–tangential plane of Norway spruce

Analysis and design of wood based composites and structures need reliable experimental compliance of their constituting materials. Therefore, an experimental method is presented to determine all of the effective in-plane elastic parameters and applied to Norway spruce specimens in the radial tangential RT plane due to the use of this species in wide variety of engineering applications. For the purpose, uni-axial compression experiments are conducted on the specimens for different material orientations under the assumptions of small strains and small rotations. Boundary extremes of the specimens are allowed to rotate for minimizing the boundary artifacts and providing near-ideal specimen deformation. During these experiments, digital image correlation is used to obtain the displacement and strain fields whilst the stress is computed through the applied load and specimen geometry. In addition to the common practice of tabulated parameters for each nominal material orientation, the experiment results are further processed through the transformation rule and least squares function. Consequently, the effective elastic parameters in the RT plane are obtained by taking general anisotropic linear elasticity into account and the material classification is performed. By means of the current method, it is expected to advance the effective in-plane compliance measurements of natural cellular materials.

A rotation-free beam element for beam and cable analyses

The rotation-free or RF element method represents a non-conventional finite element method in which the rotations are not used as dofs and the element interpolation domains are overlapping. Its obvious advantage is that the complication of finite rotations can be avoided. In this paper, the relatively unexplored RF plane beam element recently formulated by the authors in the course of developing a RF triangle for thin-shell analyses is revisited. Comparing with other RF plane beam elements, the present one is simple and physical yet its accuracy remains competitive. Using a corotational approach and the small strain assumption, its tangent bending stiffness matrix can be approximated by a constant matrix which does not require updating in geometric nonlinear analyses. The element is here extended to spatial cable analyses in which the torsional stiffness can often be neglected and the sectional properties are isotropic. Under the same nodal distributions, it is seen that the present element can tolerate much larger load increment and time step under static and dynamic analyses, respectively, than the two-node thin beam finite element.

Revisiting the method to obtain the mechanical properties of hydrided fuel cladding in the hoop direction

The method reported in the literature to calculate the stress–strain curve of nuclear fuel cladding from ring tensile test is revisited in this paper and a new alternative is presented. In the former method, two universal curves are introduced under the assumption of small strain. In this paper it is shown that these curves are not universal, but material-dependent if geometric nonlinearity is taken into account. The new method is valid beyond small strains, takes geometric nonlinearity into consideration and does not need universal curves. The stress–strain curves in the hoop direction are determined by combining numerical calculations with experimental results in a convergent loop. To this end, ring tensile tests were performed in unirradiated hydrogen-charged samples. The agreement among the simulations and the experimental results is excellent for the range of concentrations tested (up to 2000wppm hydrogen). The calculated stress–strain curves show that the mechanical properties do not depend strongly on the hydrogen concentration, and that no noticeable strain hardening occurs. However, ductility decreases with the hydrogen concentration, especially beyond 500wppm hydrogen. The fractographic results indicate that as-received samples fail in a ductile fashion, whereas quasicleavage is observed in the hydrogen-charged samples.

Lattice refinement strategies

This article quantitatively reconciles crystallographic and mechanics approaches to lattice refinement as part of X-ray diffraction procedures. The equivalence between the refinement based on unit-cell parameters to that based on a lattice deformation tensor is established from a fixed reference configuration. Justification for the small strain assumption, commonly employed in X-ray diffraction based stress analysis, is also derived. It is shown that relations based on infinitesimal strains are correct to within an error of quadratic order in strain. This error may be important to consider for high-precision or high-strain experiments. It is hoped that these results are of use for facilitating communication and collaboration between crystallography and experimental mechanics communities, for studies where X-ray diffraction data are the fundamental measurement.

A self-consistent approach describing the strain induced anisotropy: Case of yield surface evolution

The main goal of this work is to describe the evolution (expansion, translation, distortion and rotation) of the yield surface using a recent developed self-consistent model with small strains assumption. In fact, the ellipsoidal inclusion case in the simplified non-incremental interaction scheme has been developed by Abdul-Latif and Radi (2010) [4] showing the dependency of the elasto-inelastic behavior of polycrystals on the grain shape. The concept of yield surface is very important for evaluating the material work-hardening behavior beyond the yield point. Hence, a particular attention is made on the description of initial and subsequent yield surfaces. To show the predictive abilities of this model, these surfaces are simulated and then evaluated as a model response for different values of key model parameters (such as the grain aspect ratio α = a/ b where a and b are the half-axes of the grain and the viscous parameter γ) subjected to radial proportional loading. Then, the effect of the two parameters on these surfaces will be discussed. Subsequent yield surfaces showed different proportions of translation, expansion, distortion and rotation. Moreover, the influence of the pre-loading nature on the subsequent yield surface evolution and its rotation will be then examined under three different loading conditions of tension, torsion, and combined tension–torsion proportional loading path. Some granular responses for pre-selected grains are also illustrated and discussed.

Nonlinear transient FE analysis of piezolaminated smart structures

AbstractThe presented paper focuses on the modeling and finite element simulation of thin composite structures with integrated piezoelectric layers. Geometrically nonlinear piezolaminated finite elements are based on the Reissner‐Mindlin or third‐order transverse shear deformation hypotheses and the assumptions of small strains but moderate rotations. The numerical results are compared to results in literature. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Weighted Least‐Squares Mixed Finite Element Formulation for Quasi‐Incompressible Elastodynamics

AbstractThe purpose of this work is the application of the least‐squares finite element method to an elastodynamic, quasi‐incompressible problem under small strain assumptions. Therefore a mixed finite element based on a weighted least‐squares formulation is developed. The L2‐norm minimization of the time‐discretized residuals of the given first‐order system of partial differential equations leads to a functional depending on displacements and stresses. In the numerical example the proposed mixed element is compared to an alternative approach, which is based on a least‐squares mixed finite element with improved momentum balance, see [1]. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Failure Predictions for DP Steel Cross-die Test using Anisotropic Damage

The Lemaitre's continuum damage model is well known in the field of damage mechanics. The anisotropic damage model given by Lemaitre is relatively simple, applicable to nonproportional loads and uses only four damage parameters. The hypothesis of strain equivalence is used to map the effective stress to the nominal stress. Both the isotropic and anisotropic damage models from Lemaitre are implemented in an in-house implicit finite element code. The damage model is coupled with an elasto-plastic material model using anisotropic plasticity (Hill-48 yield criterion) and strain-rate dependent isotropic hardening. The Lemaitre continuum damage model is based on the small strain assumption; therefore, the model is implemented in an incremental co-rotational framework to make it applicable for large strains. The damage dissipation potential was slightly adapted to incorporate a different damage evolution behavior under compression and tension. A tensile test and a low-cycle fatigue test were used to determine the damage parameters. The damage evolution was modified to incorporate strain rate sensitivity by making two of the damage parameters a function of strain rate. The model is applied to predict failure in a cross-die deep drawing process, which is well known for having a wide variety of strains and strain path changes. The failure predictions obtained from the anisotropic damage models are in good agreement with the experimental results, whereas the predictions obtained from the isotropic damage model are slightly conservative. The anisotropic damage model predicts the crack direction more accurately compared to the predictions based on principal stress directions using the isotropic damage model. The set of damage parameters, determined in a uniaxial condition, gives a good failure prediction under other triaxiality conditions.

Fracture toughness of bonds using interfacial stresses in four-point bending test

Bond strength is an important factor in electronic and photonic packaging. Measuring bond characteristics, strength, and fracture toughness is particularly difficult in microelectronic devices where miniaturized bonds are used (e.g., wafer bonding and 3 dimensional integrated circuits). Since applying load directly to the bonds is proven difficult, four-point bending test with notched specimen has been typically used to facilitate fracture toughness measurements at small scale. This method is based on experiencing large peeling and shear stresses at the interface of two layers of different materials at the notch. However, there is a lack of analytical solution that can be used to determine the peeling and shear stresses at the interface of four-point bend specimen. This manuscript presents analytical modeling of peeling and shear stresses in tri-layer four-point bend specimens. Strength of materials approach with the assumption of small strains, within elastic region, is adopted in this study for evaluation of stresses and displacements. Furthermore, beam theory is utilized to develop equations that describe the moments and shear forces in the assembly due to the four-point bending loads. Second and fourth order differential equations are derived for shear and peeling stresses respectively at the interfaces of the assembly. Boundary conditions are determined based on load, supports and shear force and moment diagrams of the assembly. The problem solution is obtained by solving the governing differential equations instantaneously using standard methods. Finite-Element-based simulation is used to compare with and verify the analytical solution. This work also presents an alternative approach where the stresses at the notch are used to determine the fracture toughness. An experiment is conducted on a specimen with intermetallic material as bond material. The analytical approach is then verified with the results of four-point bending experiment and is validated using the conventional technique (which is based on energy equilibrium and presented by Eq. (60)).

A geometric nonlinear rotation‐free triangle and its application to drape simulation

SUMMARYIn this paper, a rotation‐free triangle is formulated. Unlike the thin and degenerated shell finite element models, rotation‐free triangles employ translational displacements as the only nodal DOFs. Compared with the existing rotation‐free triangles, the present triangle is simple and physical yet its accuracy remains competitive. Using a corotational approach and the small strain assumption, the tangential bending stiffness matrix of the present triangle can be approximated by a constant matrix that does not have to be updated regardless of the displacement magnitude. This unique feature suggests that the triangle is a good candidate for fabric drape simulation in which fabric sheets are often flat initially and the displacement is much larger than those in conventional shell problems. Nonlinear shell and fabric drape examples are examined to demonstrate the efficacy of the formulation. Copyright © 2011 John Wiley & Sons, Ltd.

Use of the mark-tracking method for optical fiber characterization

The mark-tracking method was used in the uniaxial tensile test to determine the elastic properties of optical fibers. The mark-tracking method is based on the follow-up of two markers on the specimen with the help of an image processing technique. It allows us to determine the true strain with respect to the small strains assumption (≤1%) or the finite strains (>1%) without any impact of the rigid solid movement or pulley fiber sliding on the measured strain. Both coated optical fiber and stripped fiber were subjected to the uniaxial tensile test and the cantilever beam bending test. The Young’ modulus results of the stripped fiber were found to be very similar for both tests. Thus, the mark-tracking method is adaptable to the tensile test of optical fibers and the elastic behaviors of both coated optical fiber and stripped fiber are found to be non-linear. Their Young's moduli are 22 and 79 GPa, respectively. These results revealed that those coatings play a mechanical role in fiber elongation.

A Phantom-Node Method for Predicting Residual Strength in Shell Structures with a Single Crack Based on a Crack Tip Opening Angle Criterion

This paper develops a phantom-node method for MITC 3-node shell element. According to the phantomnode method, overlapping paired elements are used to represent an element cut through by a crack. In which, each element is created by extending a part of the cracked element to a phantom domain until it becomes a completely standard finite element. As a result, separate assumption for transverse shear strain of the MITC 3-node element to remove “shear locking” is employed straightforwardly. The development describes the crack in both thin and thick shell structures and is no need to remesh as the crack propagates. The method is applied to numerically simulate a fracture test of an internally pressurized cylindrical shell conducted by (Keesecker et al. 2003). Material constitutive law follows the J2-plasticity with linear isotropic hardening and the small strain assumption. The crack tip opening angle (CTOA) criterion, one of the fracture criteria, characterizes the crack growth condition. A predicted crack growth curve of pressure versus half-crack length is comparable to experimental measurements and other numerical results.

3D mode discrete element method: Elastic model

For block systems that satisfy small strain, finite displacement, and finite rotation conditions, a three-dimensional mode-deformable discrete element method (3MDEM) is presented. The 3MDEM is an efficient numerical method that simulates the mechanical behavior of nonlinear, large deformation, and dynamic problems. Under the assumption of small strain, the motion of points in a deformable block can be decomposed into the sum of the block's rigid body motion and the deformation of the block itself. The kinematic equations for deformable blocks are derived. The deformation of blocks can be expressed by the combination of a series of deformation modes, which can be decoupled under given orthogonal conditions. The results of the simulation verify that good accuracy can be obtained under the small strain condition using the presented method, compared with the finite element method. Under the condition of finite deformation, reasonably similar results are obtained by the proposed method and a commercial three-dimensional deformable distinct element code, 3DEC. In the case of verification examples, the 3MDEM is found to be more efficient than 3DEC.

A Least‐Squares Mixed Finite Element for Quasi‐Incompressible Elastodynamics

AbstractIn the present work, a mixed finite element based on a modified least‐squares formulation is proposed. Here, we consider the time‐dependent equations for quasi‐incompressible elastodynamics under small strain assumptions. The main goal is to obtain an accurate approximation of both displacements and stresses in particular for the lowest‐order element. Basis for the element formulation is a weak form resulting from a least‐squares method. The L2‐norm minimization of the time‐discretized residuals of the given first‐order system leads to a functional depending on approximations for displacements and stresses. By introducing a time‐independent displacement test function, a weak form is derived. A numerical example concerning quasi‐incompressible elasticity shows the performance of the approach for the lowest‐order element RT0P1as. (© 2010 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Earthquake induced ground strains in the presence of strong lateral soil heterogeneities

Transient ground strains are recognized to govern the response of buried elongated structures, such as pipelines and tunnels, under seismic wave propagation. Since a direct measure of ground strains is not generally available, simplified formulas relating peak ground strain to peak ground velocity, and based on 1D wave propagation theory in homogeneous media, are typically used for seismic design. Although they are adopted by most of the available technical guidelines, the use of these formulas may be questionable in complex realistic situations as either in the presence of strong lateral discontinuities, or in the epicentral area of large earthquakes, or in sites where relevant site amplification effects and spatial incoherency of ground motion are expected. To provide a contribution to overcome the previous limitations, a simplified formula relating peak ground longitudinal strain to peak ground velocity is proposed in this paper, as a function of the geometrical and dynamic parameters which have the major influence on strain evaluation. The formula has been obtained under small-strain assumptions, so that it can reasonably be applied under linear or moderately non-linear soil behaviour. The adequacy of this formula in the most common case of vertically propagating S-waves has been checked against 2D numerical solutions by Spectral Elements (SE) for representative geological cross-sections in Parkway Valley (New Zealand) and in the cities of Catania (Italy) and Thessaloniki (Greece). The shear strain and the longitudinal strain variability with depth is also investigated, through some qualitative examples and comparisons with analytical formulas.

Study of the mechanical behavior of the optical fiber by a mark-tracking method

The mark-tracking method was used in the uniaxial tensile test to determine the elastic properties of optical fibers. The mark-tracking method is based on the followup of two markers on the specimen with the help of an image processing technique. It allows us to determine the true strain with respect to the small strains assumption (e 1%) or the finite strains (e>1%) without any impact of the rigid solid movement neither pulley fiber sliding on the measured strain. Optical fibers used in this study are commercial Verrillon single mode silica fibers, 125 µm in diameter with a two layers 62.5 µm thick epoxy-acrylate polymer coating. Both as-received optical fiber and stripped fiber were subjected to the uniaxial tensile test and the cantilever beam bending test. The stripped fiber Young’s modulus results under both tests were found to be very similar. Thus, the mark-tracking method is adaptable to the tensile test of optical fibers and the elastic behavior of both as-received optical fiber and stripped fiber is found to be linear. Their Young’s modulus are 22GPa and 83GPa, respectively. These results revealed that those coatings are playing a mechanical role in the fiber elongation.

Determinist-Probabilistic Concept in Modeling Fatigue Damage Through a Micromechanical Approach

Motivated by a micromechanical determinist-probabilistic model coupled with damage recently developed by the authors, a new generalization is proposed to describe the nonlinear elasto-inelastic cyclic strain-stress behavior of polycrystals notably under biaxial cyclic loading paths. In this context, this generalization considers a compressible and linear anisotropic granular elastic strain behavior coupled with damage. The model is expressed in the framework of the time dependent plasticity for a small strain assumption. It is assumed that a damage variable initiates at the mesoscopic (granular) level where the plastic strain localization phenomenon takes place. The associated thermodynamic force of the damage variable is determined using the concept of total granular energy (elastic and inelastic). The transition of the elastic strain from the single to the polycrystal is modified due to its explicit coupling with damage. Comparisons between predicted and experimental results are conducted describing the low-cycle fatigue behavior of the aluminum alloy 2024 under different complex cyclic loading paths. It is demonstrated that the model has a reasonable ability in describing the cyclic behavior of this alloy. Qualitatively, the model is tested under different cyclic loading paths with stress-controlled condition describing especially the ratcheting behavior of the alloy. In fact, the effects of the applied mean stress on the predicted overall elasto-inelastic behavior and on the fatigue life are carefully studied. It shows the dependence of the fatigue life on the mean stress value.