Three-dimensional Material Model Research Articles

A computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy

We present a computational framework to obtain nonlinearly elastic constitutive relations of special Cosserat rods with surface energy. The framework allows the rod’s cross-section to have arbitrary shape and the rod material to obey arbitrary three-dimensional material model for its bulk volume and arbitrary surface energy model for its lateral surface. The kinematics of the Helical Cauchy–Born rule is used to first construct a family of six-parameter (corresponding to the six strain measures of rod theory) helical rod configurations in which the generated three-dimensional strain field is uniform in the rod’s arc-length coordinate. This uniformity allows the three-dimensional nonlinear equations of elasticity to reduce to just the rod’s cross-sectional plane wherein the cross-section’s boundary also experiences traction due to the inherent surface energy in the rod’s lateral surface. The solution of this cross-sectional problem yields the induced in-plane displacement and out-of-plane warping in the cross-section for arbitrarily prescribed strain measures of the rod. We then derive expressions for the induced internal contact force, moment and stiffnesses of the rod in terms of the solution of the deformed cross-section and the prescribed strains of the rod. A finite element formulation is presented to solve the nonlinear cross-sectional deformation problem and further obtain the induced internal contact force, moment and stiffnesses. The presented framework will be useful for modeling of nanorods where surface energy plays a dominant role. Several numerical examples are presented using the presented framework to illustrate the effect of surface energy parameters on the deformation of the nanorod’s cross-section and the nanorod’s bending, torsional, extensional and shearing stiffnesses. We also derive improved analytical formulas for the extensional and bending stiffnesses of isotropic rectangular nanorods in their reference state. These formulas reduce to the existing widely used formulas for a special choice of material parameters, i.e., when the surface Poisson’s ratio and the bulk Poisson’s ratio match thus highlighting the limitation of the existing formulas.

Nonlinear three-dimensional anisotropic material model for failure analysis of timber

In this paper, a constitutive material model is proposed to model the nonlinear mechanical response of timber, as elastoplastic orthotropic material under three-dimensional (3D) stress state. An associated flow rule model based on Hoffmann yield criterion and plastic potential is adopted for describing the plasticity of timber under compression. Isotropic strain hardening during plastic deformation is incorporated into Hoffman yield criterion by assuming that equivalent yield stress is a function of the equivalent plastic strain. A stress-based continuum damage formulation with four independent failure criteria in tension and compression is used. The proposed model is implemented as a user material subroutine UMAT in ABAQUS. The results obtained in numerical simulations are evaluated and compared with experimental results. A good agreement has been found between numerical simulations and the experimental results. Therefore, the proposed constitutive model can be used further in numerical simulations of the anisotropic behavior of timber.

A three-dimensional mechano-electrochemical material model of mechanosensing hydrogels

Mechanotransduction is the initiation of an electrochemical signal as a result of mechanical stimuli. It is found predominately in biological tissue and its mechanisms are well documented. In the gel like tissues of the body, such as articular cartilage and intervertebral discs, mechanotransduction regulates matrix building and degrading processes as well as keeping the tissues adequately hydrated, both with the aim of minimizing degradation. The electrochemical responses to mechanical loading at constant volume of an inanimate hydrogel could assist in the understanding of these processes.There is considerable evidence that the modulus of hydrated tissues and hydrogels depend explicitly on ionic concentration. By modeling the mechano-electrochemical relationship of a hydrogel, the coupling of the elastic and electrochemical energies can be quantified. In turn, the mechanisms that govern this phenomenon can be better understood. This study modifies the Flory-Rehner theory of gels, using material-specific experimental data as input. The results show up to a 11% difference in equilibrium swelling magnitude compared to the Flory-Rehner model. Furthermore, under isochoric deformation, an increase in electrical potential is shown with increasing shear strain, something which is not possible with conventional Flory-Rehner and Donnan theory. This aligns the continuum model presented here more closely with both experiment and microscopic theories. The mechanosensing capabilities as well as varying swelling responses in different solution concentrations highlight the models potential applications in both biological and technological settings.

A nonlinear thermomechanical formulation for anisotropic volume and surface continua

A thermomechanical, polar continuum formulation under finite strains is proposed for anisotropic materials using a multiplicative decomposition of the deformation gradient. First, the kinematics and conservation laws for three-dimensional, polar, and nonpolar continua are obtained. Next, these kinematics and conservation laws are connected to their corresponding counterparts for surface continua, based on Kirchhoff–Love assumptions. Then the shell material models are extracted from three-dimensional material models for finite-temperature problems using established connections. The weak forms are obtained for both three-dimensional nonpolar continua and Kirchhoff–Love shells. These formulations are expressed in tensorial form so that they can be used in both curvilinear and Cartesian coordinates. They can be used to model anisotropic crystals and soft biological materials.

Reduced-Integrated 8-Node Hexahedral Solid-Shell Element for the Macroscopic Forming Simulation of Continuous Fibre-Reinforced Polymers

Finite element (FE) forming simulation offers the possibility of a detailed analysis of the deformation behaviour of continuously fibre-reinforced polymers (CFRPs) during forming, in order to predict possible manufacturing effects such as wrinkling or local changes in fibre volume content. The majority of macroscopic simulations are based on conventional two-dimensional shell elements with large aspect ratios to model the membrane and bending behaviour of thin fibrous reinforcements efficiently. However, without a three-dimensional element approach, stresses and strains in thickness direction cannot be modelled accurately. Commercially available linear 3D solid elements for this purpose are rarely suitable for forming simulations since they are subjected to several locking phenomena under bending deformation, especially with large aspect ratios. To alleviate this problem, so-called solid-shell elements based on the assumed natural strain (ANS) and enhanced assumed strain (EAS) method can be used. Therefore, a locking-free explicit reduced-integrated 8-node-hexahedron solid-shell element, based on the initial work of Schwarze and Reese (2011), is implemented in the commercially available FE solver Abaqus. Its suitability for macroscopic modelling of the forming behaviour of fibrous reinforcements is outlined in this work. The presented element combines the advantages of a locking-free out-of-plane deformation behaviour of conventional thin shell elements with the advantage of maintaining a fully three-dimensional material model and geometry description.

Chirality in the plane

It is well-known that many three-dimensional chiral material models become non-chiral when reduced to two dimensions. Chiral properties of the two-dimensional model can then be restored by adding appropriate two-dimensional chiral terms. In this paper we show how to construct a three-dimensional chiral energy function which can achieve two-dimensional chirality induced already by a chiral three-dimensional model. The key ingredient to this approach is the consideration of a nonlinear chiral energy containing only rotational parts. After formulating an appropriate energy functional, we study the equations of motion and find explicit soliton solutions displaying two-dimensional chiral properties.

Neúspěšné setkání matematiky a chemie: van ‘t Hoffovy a Sachsovy skládací modely

Three-dimensional material models of molecules were used throughout the 19th century, either functioning as a mere representation or opening new epistemic horizons. In this paper, two case studies are examined: the 1875 models of van ‘t Hoff and the 1890 models of Sachse. What is unique in these two case studies is that both models were not only folded, but were also conceptualized mathematically. When viewed in light of the chemical research of that period not only were both of these aspects, considered in their singularity, exceptional, but also taken together may be thought of as a subversion of the way molecules were chemically investigated in the 19th century. Concentrating on this unique shared characteristic in the models of van ‘t Hoff and the models of Sachse, this paper deals with the shifts and displacements between their operational methods and existence: between their technical and epistemological aspects and the fact that they were folded, which was forgotten or simply ignored in the subsequent development of chemistry.

Influence of nonlinear material behavior on the effect of inter-fiber cracks in composite laminates

This paper presents a numerical study evaluating the influence of nonlinear material behavior on the stiffness reduction due to inter-fiber cracks. It is shown that the relation between crack density and damage variables changes as the load level increases. A three-dimensional nonlinear material model is used in a representative volume element study to accurately calculate the local stress distribution around cracks. The studies show that as the load increases the stress recovers to the far-field level within a shorter distance, and consequently the plastic deformation causes a decrease in the crack-induced stiffness reduction. To provide damage variables for a coupled elastoplastic-damage material approach at the homogenized ply level, the results of a systematic representative volume element study are used for interpolation. For verification, the response curves of several laminates influenced by inter-fiber cracks are simulated and compared to experimental results. The presented procedure enables the constitutive laminate response under large deformations to be accurately predicted.

Acoustoelastic Coefficients in Thick Steel Plates under Normal and Shear Stresses

Ultrasonic monitoring of the integrity of structural materials utilizes the acoustoelasticity of materials (stress-dependence of ultrasonic wave velocity) obtained from the higher order strain terms in the constitutive equations. This method has been applied to measure residual and applied stresses in thin metals under principal stresses without shear. Here, acoustoelastic coefficients are determined in thick steel plates in normal, orthogonal, and angled directions by means of an array of ultrasonic sensors. A three-dimensional material model is developed which includes Murnaghan hyper-elasticity and can determine the effects of plate thickness and excitation frequency on the acoustoelastic coefficients. This model is experimentally validated by tensile loading of a thick steel plate by measuring the ultrasonic signals in three directions. Numerical and experimental results agree within the measurement uncertainties of each method. The 1.0 MHz ultrasonic frequency has the highest resolution for measuring normal and shear stresses in structural plates typically used in highway bridges.

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...

Three-dimensional nonlinear constitutive model for composites

A three-dimensional nonlinear material model for fiber reinforced polymers is developed. The constitutive behavior is formulated based on mechanisms to allow for a minimum of material parameters. As a result, the model is able to capture complex nonlinear deformations under arbitrary loading conditions. Mechanisms like fiber reorientation induced by deformation and the hydrostatic sensitivity of the resin are considered. The constitutive model includes two independent non-associative flow rules to describe yielding due to transverse and longitudinal shear loading, respectively. In order to verify the capabilities of the model, transverse and off-axis compression tests superimposed with various hydrostatic pressures are simulated. The parameters controlling the hydrostatic sensitivity on the material response are discussed. Analyzes of angle-ply tension tests under atmospheric conditions show, that considering hydrostatic sensitivity is not relevant to simulate their response. Neglecting hydrostatic sensitivity results in a different characterization of material parameters, which compensate its influence. Uniaxial tests with plane stress states are not suitable to derive the hydrostatic sensitivity parameters. Therefore, tests under high hydrostatic pressure are mandatory.

A projection method to extract biological membrane models from 3D material models

This paper presents a projection method for deriving membrane models from the corresponding three-dimensional material models. As a particular example the anisotropic Holzapfel–Gasser–Ogden model is considered. The projection procedure is based on the kinematical and constitutive assumptions of a general membrane theory, considering the membrane to be a general two-dimensional manifold. By assuming zero transverse stress, the Lagrange multiplier associated with the incompressibility constraint can be eliminated from the formulation. The resulting nonlinear model is discretized and linearized within the finite element method. Several numerical examples are shown, considering quadratic Lagrange and NURBS finite elements. These show that the proposed model is in very good agreement with analytical solutions and with full 3D finite element computations.

Meso-mechanically motivated modeling of layered fiber reinforced composites accounting for delamination

Carbon fiber reinforced plastics are mostly used in laminates, which consist of several very thin layers stacked over each other. In this work, these layers are composed of either unidirectional fibers or textile reinforcements embedded in an epoxy resin. To model the orthotropic material behavior of such layers, a meso-mechanically motivated material model is used, which is based on structural tensors which represent the fiber orientations. This fully three-dimensional material model is incorporated into a solid-shell finite element, which is particularly suited for application to thin shell-like structures, because all occurring locking phenomena are cured by implementing both the EAS and the ANS concept in combination with a new concept of reduced integration with hourglass stabilization. The latter leads to high computational efficiency, still representing satisfactorily the through-the-thickness stress distribution, since the number of integration points in thickness direction can be chosen arbitrarily. Furthermore, in the presented cases, delamination is one of the most important failure mechanisms and thus needs to be considered. This is done by combining a cohesive zone-like continuum damage model with the solid-shell element formulation, which covers the interaction of different failure modes.

Finite Element Analysis of Layered Fiber Composite Structures Accounting for the Material’s Microstructure and Delamination

The present paper focuses on composite structures which consist of several layers of carbon fiber reinforced plastics (CFRP). For such layered composite structures, delamination constitutes one of the major failure modes. Predicting its initiation is essential for the design of these composites. Evaluating stress-strength relation based onset criteria requires an accurate representation of the through-the-thickness stress distribution, which can be particularly delicate in the case of shell-like structures. Thus, in this paper, a solid-shell finite element formulation is utilized which allows to incorporate a fully three-dimensional material model while still being suitable for applications involving thin structures. Moreover, locking phenomena are cured by using both the EAS and the ANS concept, and numerical efficiency is ensured through reduced integration. The proposed anisotropic material model accounts for the material’s micro-structure by using the concept of structural tensors. It is validated by comparison to experimental data as well as by application to numerical examples.

Development of constitutive model for laminated veneer lumber using digital image correlation technique

This paper describes the development of a three-dimensional constitutive model for laminated veneer lumber (LVL) needed for new developments using this material. The LVL was manufactured in New Zealand from Radiata Pine. Experimental testing has been performed according to European timber testing standards. Block compression testing has resulted in modulus of elasticity values in the three material directions. Digital image correlation (DIC) technique has been used to determine the six Poisson’s ratios. Shear testing, whereby timber specimens were glued between two steel plates, has given stiffness values using DIC measurements. Experimental testing results have been compared with values found in literature. Results from this experimental testing programme have made it possible to create a three-dimensional elastic material model of LVL for the use in finite element analysis programmes. Although the material properties do not result in a symmetrical constitutive matrix, only minor adjustments are needed to gain the benefits of a symmetrical matrix.

Modeling Fatigue Damage of Concrete

Numerical model for fatigue crack propagation within the framework of finite element smeared crack analysis is presented. It concentrates on modeling of fatigue behavior of material under tensile load which causes initiation and growth of cracks in concrete. The fatigue material model is an extension of existing static three-dimensional fracture-plastic material model, and as such it has been implemented into the ATENA Finite Element software package. The developed model has been used to model experiments with high-cycle loading of three point bending concrete specimens tested by collaborating institutions. Analysis results are compared to the measurements.

Damage mechanism and mode of square reinforced concrete slab subjected to blast loading

Reinforcement concrete is the principle material for military engineering and nuclear power plant. However, impact and explosions could completely destroy such structures, causing tremendous casualties and property loss. Therefore, analyzing the damage mechanism and mode of the structures under blast loading is significant. The objective of this paper is to investigate the dynamics response and damage mechanism of three 1000mm×1000mm×400mm reinforcement concrete slabs subjected to 400mm standoff distance close-in explosions using LS-DYNA software and compare with experimental tests. A three-dimensional material model including explosive, air and reinforcement concrete slab with separated modeling method taking into account Arbitrary Lagrange–Euler, multiple materials algorithms and fluid–structure coupling interaction, is created to simulate the experiments. The sophisticated concrete and reinforcement bar material models, considering the strain rate effects, dynamic increasing factor and equation of state for concrete, are applied in simulating the damage mechanism and dynamic response. From the numerical results and comparison of the experimental data, it can be seen that the numerical results and experimental data shows a favorable agreement and the present model can still give a reliable prediction of the damage characteristic for the reinforcement concrete slabs. It also observed that the increase of the amount of the explosive can change the failure mode of the slab.

A three-dimensional wood material model to simulate the behavior of wood with any type of knot at the macro-scale

This article presents a three-dimensional wood material model implemented in finite element (FE) software which is capable of predicting the behavior of timber at the macro-scale taking into account the effect of any type of knot. The model was built in Ansys using the ansys parametric design language, such that the whole simulation process, including the creation of geometries, grain deviation, meshing and failure prediction, is automatically calculated by only introducing conventional measuring parameters of knots and locations of the pith. Knots are generated as elliptical, rotated and oblique cones. The local and global grain deviation is estimated by solving a multiphysics problem and equating the fibers of wood to the trajectory of laminar streamlines in 3-D. Wood is considered a transversely isotropic material with anisotropic plasticity in which the failure prediction is given by means of several phenomenological failure criteria. The model was validated with 4-point bending tests of Scots pine specimens showing errors in failure prediction of 5 %, errors in initial fracture location of less than 20 mm and errors of 9 % in photogrammetrically measured displacements of an average of 65 FE nodes, in which the heterogeneity of wood caused variations of up to 6 %.

Simulation of distortional hardening by generalizing a uniaxial model of finite strain viscoplasticity

In this paper we generalize a uniaxial model of finite strain viscoplasticity (proposed by Shutov and Kreißig) using the so-called concept of representative directions. As a result, a new three-dimensional phenomenological material model is obtained. The original model takes the nonlinear isotropic and kinematic hardening into account, but it does not cover the distortional hardening. Using a series of numerical computations we show that the isotropic and kinematic hardening is completely retained during the process of generalization. Moreover, the distortional hardening effects are automatically induced by the concept itself. This is demonstrated by simulating combined tension–torsion tests on thin-walled tubular specimens. Furthermore, the generalized material model is validated by a comparison with real experimental data concerning the shape of the yield surface. A good correspondence between the simulation results and the measurements is observed.

Three-dimensional chemo-thermomechanically coupled simulation of curing adhesives including viscoplasticity and chemical shrinkage

Based on the one-dimensional material model developed by Liebl et al. (Arch Appl Mech, 2011) a three-dimensional viscoelastic-viscoplastic material model for small deformations of curing adhesives on the basis of continuum mechanics is proposed in this contribution. The model describes the most relevant phenomena which occur during curing processes in the automotive industry and includes the effects of temperature and degree of cure on the mechanical properties of the material. Thermal expansion as well as chemical shrinkage are also contained. The yield stress for the viscoplastic part of the model goes back to the work of Schlimmer and Mahnken (Int J Numer Meth Eng 63:1461---1477, 2005), but is formulated in reference to the degree of cure and the temperature. Therefore this model considers chemo-thermomechanical coupling and extends the plasticity approach of Schlimmer and Mahnken, which is devised for cured adhesives, to the whole curing range, from the uncured to the fully cured adhesive. A peculiar focus is hereby laid on epoxy resins used in the automotive industry as structural adhesives.