Small Strain Assumption Research Articles

Globally stable quasistatic evolution for a coupled elastoplastic–damage model

We show the existence of globally stable quasistatic evolutions for a rate-independent material model with elastoplasticity and incomplete damage, in small strain assumptions. The main feature of our model is that the scalar internal variable which describes the damage affects both the elastic tensor and the plastic yield surface. It is also possible to require that the history of plastic strain up to the current state influences the future evolution of damage.

Constitutive Modelling of Damage Evolution and Martensitic Transformation in 316L Stainless Steel

Abstract In this work, the constitutive model, derived with the use of thermodynamic of irreversible processes framework is presented. The model is derived under the assumption of small strains. Plastic strain induced martensitic phase transformation is considered in the austenitic matrix where the volume fraction of the martensite is reflected by a scalar parameter. The austenitic matrix is assumed as the elastic-plastic material and martensitic phase is assumed as randomly distributed and randomly oriented inclusions. Both phases are affected by damage evolution but there is no distinction in the model between damage in austenite and martensite.

Large Deformations of a Soft Porous Material

Compressing a porous material will decrease the volume of the pore space, driving fluid out. Similarly, injecting fluid into a porous material can expand the pore space, distorting the solid skeleton. This poromechanical coupling has applications ranging from tissue mechanics to hydrogeology. The classical theory of linear poroelasticity captures this coupling by combining Darcy's law with Terzaghi's effective stress and linear elasticity in a linearized kinematic framework. Linear poroelasticity is a good model for very small deformations, but it becomes increasingly inappropriate for moderate to large deformations, which are common in the context of phenomena such as swelling and damage, and for soft materials such as gels and tissues. The well-known theory of large-deformation poroelasticity combines Darcy's law with Terzaghi's effective stress and nonlinear elasticity in a rigorous kinematic framework. This theory has been used extensively in biomechanics to model large elastic deformations in soft tissues, and in geomechanics to model large elastoplastic deformations in soils. Here, we first provide an overview and discussion of this theory with an emphasis on the physics of poromechanical coupling. We present the large-deformation theory in an Eulerian framework to minimize the mathematical complexity, and we show how this nonlinear theory simplifies to linear poroelasticity under the assumption of small strain. We then compare the predictions of linear poroelasticity with those of large-deformation poroelasticity in the context of two uniaxial model problems: Fluid outflow driven by an applied mechanical load (the consolidation problem) and compression driven by a steady fluid throughflow. We use these problems to explore the steady and dynamical errors associated with the linear model in both situations, as well as the impact of introducing a deformation-dependent permeability.

Finite element modeling and analysis of piezo-integrated composite structures under large applied electric fields

In this article, we focus on static finite element (FE) simulation of piezoelectric laminated composite plates and shells, considering the nonlinear constitutive behavior of piezoelectric materials under large applied electric fields. Under the assumptions of small strains and large electric fields, the second-order nonlinear constitutive equations are used in the variational principle approach, to develop a nonlinear FE model. Numerical simulations are performed to study the effect of material nonlinearity for piezoelectric bimorph and laminated composite plates as well as cylindrical shells. In comparison to the experimental investigations existing in the literature, the results predicted by the present model agree very well. The importance of the present nonlinear model is highlighted especially in large applied electric fields, and it is shown that the difference between the results simulated by linear and nonlinear constitutive FE models cannot be omitted.

Viscous approximation of quasistatic evolutions for a coupled elastoplastic-damage model

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions for elastoplastic materials with incomplete damage affecting both the elastic tensor and the plastic yield surface, in a softening framework and in small strain assumptions.

Thermomechanical initial post-buckling of Timoshenko's beam on elastic foundations

ABSTRACTThe objective of this work is to propose an analytical solution for the thermomechanical initial post-buckling response of a thick beam resting on a linear elastic foundation and subject to a uniform temperature rise throughout its cross-section. The thermal strain is assumed to follow a linear law with the temperature rise and the material properties are considered temperature independent. The beam cross-section geometrical properties are constant along the beam length, and the formulation is consistent with the small strain assumption. The beam ends are assumed pinned and immovable, thermal expansion is not allowed and as a consequence compressive forces arise and the beam may buckle. If the temperature is increased further, the beam continues to deflect laterally, hence the problem is geometrically non-linear. In addition, the model is appropriate to describe the behavior of short beams as it takes into account transverse shear deformations. The governing equations are derived and made non-dimensional, and it is seen that three non-dimensional parameters control the thermomechanical initial post-buckling problem: The elastic foundation stiffness, the beam slenderness ratio and the beam cross-section shear coefficient. A classical perturbation method is applied to the non-linear set of differential governing equations, therefore the critical buckling temperatures (loads) and modes and the initial post-buckling behavior may be analytically determined. The change in length, reaction forces at the supports and geometric configurations are obtained as a function of temperature, the elastic foundation.

Comparison of a mixed least‐squares formulation using different approximation spaces

AbstractThe main goal of the present work is the comparison of the performance of a least‐squares mixed finite element formulation where the solution variables (displacements and stresses) are interpolated using different approximation spaces. Basis for the formulation is a weak form resulting from the minimization of a least‐squares functional, compare e.g. [1]. As suitable functions for $H^1(\cal{B})$ standard interpolation polynomials of Lagrangian type are chosen. For the conforming discretization of the Sobolev space $H({\rm div}, \cal{B})$ vector‐valued Raviart‐Thomas interpolation functions, see also [2], are used. The resulting elements are named as PmPk and RTmPk. Here m (stresses) and k (displacements) denote the approximation order of the particular interpolation function. For the comparison we consider a two‐dimensional cantilever beam under plain strain conditions and small strain assumptions. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Second-Order Computational Homogenization Approach Using Higher-Order Gradients at Microlevel

Realistic description of heterogeneous material behavior demands more accurate modeling at macroscopic and microscopic scales. To observe strain localization phenomena and material softening occurring at the microstructural level, an analysis on the microlevel is unavoidable. Multiscale techniques employing several homogenization schemes can be found in literature. Widely used second-order homogenization requires C1 continuity at the macrolevel, while standard C0 continuity has usually been hold at microlevel. However, due to the C1-C0 transition macroscale variables cannot be defined fully consistently. The present contribution is concerned with a multiscale second-order computational homogenization employing C1 continuity at both scales under assumptions of small strains and linear elastic material behavior. All algorithms derived are implemented into the FE software ABAQUS. The numerical efficiency and accuracy of the proposed computational strategy is demonstrated by modeling three point bending test of the notched specimen.

Variational formulations for the linear viscoelastic problem in the time domain

Abstract Under the assumption of small displacements and strains, we formulate new variational principles for the linear viscoelastic hereditary problem, extending the well-known Hu-Washizu, Hellinger-Reissner, Total Potential Energy, and Complementary Energy principles related to the purely elastic problem. In addition, a new global minimum formulation is derived, giving an energetic interpretation. The new formulations are based on a convolutive bilinear form of the Stieltjes type and on the division of the time domain into two equal parts, with the resulting decomposition of the variables and of the equations governing the problem. In particular, the global minimum principle is achieved by virtue of the positive definiteness of a part of the split constitutive law operator and by means of a partial Legendre transform, and is then used to provide bounds of the overall mechanical properties of viscoelastic composite materials.

Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors (SCFs). IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the small strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.

Cauchy elasticity with dislocations in the small strain assumption

In this letter it is shown how the singularities created by dislocations in an elastic body must not preclude from a linear approach. Cauchy elasticity is considered and hence no variational approach is needed and the displacement field only appears in a second step, as an interpretation of the one of the two model variables, arisen from a Beltrami decomposition of the strain tensor. The second model variable obeys nonclassical PDEs relying on the incompatibility operator, and whose well posedness is here addressed.

Geometrical nonlinear formulation of a Molecular Mechanics model applied to the structural analysis of single-walled carbon nanotubes

In this paper, the post-critical behavior and buckling modes of single-walled carbon nanotubes are analyzed via a Molecular Mechanics model. The main target is to develop a general formulation for the model, which has been simplified under small strains assumption, and to implement a versatile tool for the structural analysis of carbon nanotubes in the framework of geometrical nonlinearity. For this purpose, a mechanical formulation able to reproduce any load configuration and supporting conditions has been derived by using an energy approach. Then, an incremental-iterative solution procedure has been implemented in order to trace several nonlinear equilibrium paths and to obtain the corresponding critical strains of clamped–clamped nanotubes under compressive, flexural and torsional loading distributions. The model shows a good numerical performance and results in agreement with previous atomistic works. Two interatomic potentials have been adopted in order to find out the influence of different constitutive relationships on the final nonlinear response. We have concluded that the choice of the potential function has no significant effect on the final buckling strains. Our results confirm that the final buckling response is strongly determined by geometrical imperfections in the nanotube, which can be well reproduced in the proposed model, but are much more difficult to handle in continuum models.

A finite elasto‐plasticity model for thick adhesive layers based on generalized stress‐strain measure

AbstractThe collective term adhesives includes a wide field of materials with a diversity of different material properties. Regarding high‐strength adhesives, the assumption of small strains often holds according to their brittle behavior. The experience with plasticity models based on the additive decomposition into elastic and inelastic strains indicates an appropriate approach to characterize such materials. In some cases, due to a more ductile material response, the assumption of infinitesimal strains is not valid anymore. In particular this is the case for high‐strength adhesives with additives like rubber. But ductile behavior is also observed for specific stress states in one adhesive, e.g. when the behavior for tensile is quite brittle while large shear‐strains could appear. The objective of this work is to overcome the theoretical restriction of small strains and to archive the practical experiences. For the failure criterion two stress invariants are used, which involves the hydrostatic pressure as well as the deviator stress state. The flowrule is introduced for the evolution of the inelastic variables. Herein the flow rule has to be of non‐associated type to ensure the thermodynamical consistency of the model. The plasticity model also includes hardening as well as softening. The presented finite strain model makes use of the fact that the eigenvalues for Green‐Lagrange strains and generalized strains are the same. Thus the limit of applicability is extended to finite strains. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multi-site micromechanics of composite materials with temperature-dependent constituents under small strain and finite thermal perturbation assumptions

This study presents mean-field based micromechanics models to predict the effective thermoelastic properties, namely, elasticities, thermal expansions and heat capacity, of thermoelastic composite materials with temperature-dependent constituents under finite temperature changes and small strain assumptions. First, the Helmholtz potential density for small strain finite thermoelasticity has been presented. The fundamental solution based on Green’s function of elasticity problem has been used to derive the general expressions for the elastic and thermal strains concentration tensors with temperature-dependent material properties. A family of mean-field based micromechanics models (multi-site Eshelby dilute model, Mori–Tanaka model and self-consistent model) has been presented. The models are general enough to account for the morphology and topology textures of the microstructure of a thermoelastic composite. Numerical examples based on the multi-site Mori–Tanaka model are used to quantify the differences of the effective properties based on the small strain finite thermoelasticity in comparison to the linear thermoelasticity. The predictions of the multi-site Mori–Tanaka micromechanics model are also compared to those of the VAMUCH, a finite element based micromechanics model.

Experimental and numerical analysis about the cyclic behavior of the 304L and 316L stainless steels at 350 °C

In a previous study, we have demonstrated that cyclic accumulation of the inelastic strain exhibited by 304L SS at room temperature under tension–compression stress control is mostly due to creep (Taleb and Cailletaud, 2011). The same result in the same conditions is pointed out for 316L SS (Taleb, 2013a). In the present paper, the cyclic behavior of both 304L and 316L stainless steels at 350°C is investigated. Creep is not significant at this temperature. In addition to tension–compression tests, the effect of non-proportional loading paths (axial-torsion) is considered for both stress and strain controlled conditions. The study suggests that ratcheting is very small with the different mean stress and amplitude used remaining into the assumption of small strains; this observation may be linked to the large cyclic hardening exhibited by both materials. However ratcheting seems more important under non-proportional loading path compared to the equivalent tension–compression conditions. A multi-mechanism model has been used to simulate the whole experimental data base. After the identification process of the material parameters conducted by considering only strain controlled experiments, its predictive capabilities have been evaluated on the stress controlled tests. The model presents a very good quantitative agreement with the quasi absence of ratcheting. However, the model fails in describing the over-hardening (mostly isotropic) observed under monotonic loading when the maximum strain is large (about 4%).

Application of fractional continuum mechanics to rate independent plasticity

In the paper the generalisation of classical rate independent plasticity using fractional calculus is presented. This new formulation is non-local due to properties of applied fractional differential operator during definition of kinematics. In the description small fractional strains assumption is hold together with additive decomposition of total fractional strains into elastic and plastic parts. Classical local rate independent plasticity is recovered as a special case.

Static and dynamic FE analysis of piezoelectric integrated thin-walled composite structures with large rotations

A geometrically nonlinear finite element (FE) model based on large rotation shell theory is developed for static and dynamic analysis of piezoelectric integrated thin-walled structures with cross-ply or angle-ply laminates. The implemented large rotation theory has six kinematic parameters expressed by five nodal degrees of freedom (DOFs) based on first-order shear deformation (FOSD) hypothesis. An eight-node shell element with five mechanical DOFs and one electrical DOF is employed. Due to the assumption of small strains and weak electric potential, linear constitutive equations and constant electric field through the thickness are considered. The large rotation piezoelectric coupled FE model is validated by one static benchmark problem and afterwards applied to the static and dynamic analysis of piezoelectric integrated smart plates and shells composed of cross-ply or angle-ply laminates.

Large-strain elastic viscoplastic consolidation analysis of very soft clay layers with vertical drains under preloading

A very soft clay layer is highly compressible and exhibits significant creep under loading. The classical linearly elastic constitutive relationship and small-strain assumption are not suitable for the consolidation analysis of very soft clays. This paper presents a new large-strain consolidation model that incorporates the Yin–Graham elastic viscoplastic (EVP) constitutive equation for use in studying the consolidation of very soft clay layers with vertical drains under preloading. First, the large-strain fluid continuity equation and the EVP constitutive equation are incorporated into a quadratic differential equation of pore-water pressure and its integral terms. Second, the alternating-direction implicit (ADI) method and virtual node method are adopted to obtain the finite difference solution. A computer program named “BSSDS” is developed for large-strain EVP consolidation analysis of clay layers with vertical drains, taking into account the complicated in situ conditions, such as resistance of vertical drains, smear effects, variation of permeability with void ratio, and multilayered soils. Third, the new large-strain numerical method is applied to the consolidation modeling of very soft clay layers with vertical drains under preloading at a site that is part of the Hong Kong – Shenzhen Western Corridor Link Project. It is found that the foundation settlements of the new large-strain EVP consolidation model have good agreement with the measured data. Finally, three different consolidation models are used to calculate the average degree of consolidation and settlements of the clay layers. The analysis shows that it is essential to consider both large-strain compression and creep effects in the analysis of very soft clay layers with vertical drains under loading.

A 2D Cosserat finite element based on a damage-plastic model for brittle materials

A 2D Cosserat model with a damage-plastic isotropic constitutive law for brittle-like materials is presented. Different damaging mechanisms in tension and compression are considered. The plastic flow is described by introducing a suitable yield function and evolution laws in terms of effective stresses. Small strain and displacement assumptions hold. A 4-node finite element is formulated with three degrees-of-freedom for each node and a predictor–corrector procedure is used to solve the damage-plastic evolutionary problem. The lateral response of walls under monotonic and cyclic horizontal actions is analyzed and a satisfactory description of the global response curves and damaging mechanisms is obtained.

Multi-scale modelling of AISI H11 martensitic tool steel surface anisotropic mechanical behaviour

In this work, a numerical investigation is carried out on the anisotropic and heterogeneous behaviour of the AISI H11 martensitic tool steel surface using finite element method and a multi-scale approach. An elasto-viscoplastic model that considers nonlinear isotropic and kinematic hardenings is implemented in the finite elements code ABAQUS using small strain assumption. The parameters of the constitutive equations are identified using macroscopic quasi-static and cyclic material responses by the mean of a localization rule. Virtual realistic microstructures, consisting of laths and grains, are generated using particular Voronoi tessellations. These microstructures consider the specific crystallographic orientations α’/γ. Finite element investigation is then performed. The local heterogeneous and anisotropic behaviour of the surface as well as the subsurface is shown under quasi-static and cyclic mechanical loadings. The laths morphology and crystallographic orientation have an important impact on the local mechanical fields.