Linear elastic material model

For the equivalent continuum, the LBNL research team used a linear elastic material model. A rock-mass Young s modulus 14.77 GPa and a rock-mass Poisson s ratio of 0.21 were adopted from CRWMS M O (1999). These elastic parameters, which represent the bulk rock mass (including the effect of fractures) have been estimated using an empirical method based on the Geological Strength Index (GSI). The adopted rock-mass Young s modulus is about 50% lower than the Young s modulus of intact rock determined on core samples from the site. 

Handbook solution (using a linear elastic material model) Fast and easy to perform Simple geometries only Does not consider nonlinearity material behavior Only for very small strains May give inaccurate results... 

The linear elastic material model can be used to describe the deformation of IPMCs. The constitutive relation of Hooke s Law can be used to relate stress and strain in the polymer as ... 

Fiber-reinforced composite materials such as boron-epoxy and graphite-epoxy are usually treated as linear elastic materials because the essentially linear elastic fibers provide the majority of the strength and stiffness. Refinement of that approximation requires consideration of some form of plasticity, viscoelasticity, or both (viscoplasticity). Very little work has been done to implement those models or idealizations of composite material behavior in structural applications. 

Mathematical modelling of the compression of single particles 2.5.2.7 Hertz model. The mechanics of a sphere made of a linear elastic material compressed between two flat rigid surfaces have been modelled for the case of small deformations, normally less then 10% strain (Hertz, 1882). Hertz theory provides a relationship between the force F and displacement hp as follows ... 

Tatara analysis. For larger deformations (up to 60%) of soft solid spheres made from linear and non-linear elastic materials, Tatara (1991, 1993) proposed a model to describe the relationship between the force and displacement. For the case of linear elasticity... 

The Compliant Joint Model (CJM) was chosen for the mechanical calculations for this analysis. The CJM uses an equivalent continuum approach to model the behavior of jointed media. An equivalent continuum approach captures the average response of a jointed rock mass by distributing the response of the individual joints throughout the rock mass. The CJM in JAS3D can model up to four joint sets of arbitrary orientation, with the fractures in each set assumed to be parallel and evenly spaced. The intact rock between joints is treated as an isotropic linear-elastic material. More detailed descriptions of the CJM model can be found in Chen (1991). 

This balance between interfacial propagation and bulk deformation has been described for linear elastic materials [56] and results from the competition between two mechanisms the velocity of propagation of an interfacial crack, which is controlled by the critical energy release rate Gc, and the bulk deformation, which is controlled by the cavitation stress and hence essentially by the elastic modulus E or G. In the hnear elastic model, the key parameter is the ratio GJE, which represents the distance over which an elastic layer needs to be deformed before being fuUy detached from the hard surface. This model has been verified experimentally for elastic gels [57]. 

To measure load distribution, standard aerospace bolts were fitted with strain gauges. Both shear and axial load could be measured. A three-dimensional finite element model with linear elastic material properties was developed for calculation of load distribution prior to initiation of material failure and comparison with instm-mented bolt results. Model details are similar to those of the single-bolt model above, with a full contact analysis being performed for all bolts, washers and holes. 

To specify material properties, anisotropy, and non linear behavior of textile structures have to be considered. These assumptions make the assessments quite complex. However, previous studies showed that the simplifying linear elastic material could lead to reliable results in the modeling of yam pullout [1, 5, 10]. 

The gross-section strain in section BE is denoted by egmss- The width of the region is denoted by W2a. The material is first modelled as a linear elastic material. The characteristic tensile strength and design tensile strain of the material are denoted by... 

There are three main types of failure by crack propagation as shown in Figure 10.3 1 (opening), II (sliding), and III (tearing). In real elements made with composite materials and subjected to external load, all modes appear in different combinations and it is not completely clear how to deal with that situation. The models proposed at present seem to still be dependent on the scale if in macro scale the tests executed according to these three modes are possible the obtained results characterize only the effects in the same scale with mean values of some correcting parameters. On lower scales probably all modes appear simultaneously or at least Modes I and II act conjointly. The mixed-mode fracture parameters may be calculated for linear elastic materials within the frames of LEFM, but their direct application to the cement-based composites is questioned. The considerations below are partly based on the paper by Brandt and Prokopski (1990). 

At present there are numerous computer programs available for analyzing bonded joints. However, most of these computer codes incorporate linearly elastic material behavior, and some allow for nonlinearly elastic and plastic behavior. Computer programs which incorporate viscoelastic material behavior are quite often limited to the simple spring-dashpot type of model for linear materials. Such inaccurate modeling of the constitutive behavior of the structure can seriously compromise the accuracy of the analytical predictions. 

The FEA model is shown in Figs. 59.19 and 59.20. Solid structural elements are used to model the PWB, solder joints, and package details. Based on the premise that at high strain rates, the solder material becomes brittle and behaves almost like a linear elastic material, the solder joint elements are modeled as linear elastic. Contact elements are used at the interface between the PWB and anvils to simulate contact between the PWB and the anvils as the assembly is deformed. The solder joint geometry is modeled as closely as possible to the solder joints actual shape after reflow. 

The previous sections give a brief review of some elementary concepts of solid mechanics which are often used to determine basic properties of most engineering materials. However, these approaches are sometimes not adequate and more advanced concepts from the theory of elasticity or the theory of plasticity are needed. Herein, a brief discussion is given of some of the more exact modeling approaches for linear elastic materials. Even these methods need to be modified for viscoelastic materials but this section will only give some of the basic elasticity concepts. 

The size of the interface-comer fracture process zone is not known, but one can estimate the extent of yielding. Fig. 13 shows three different predictions for the interface-corner yield zone at joint failure. Epoxy yielding is rate- and temperature-dependent and is thought to be a manifestation of stress-dependent, nonlinear viscoelastic material response. A precise estimate of the size of the interface-comer yield zone is, of course, totally dependent on the accuracy of the epoxy constitutive model. This constitutive model must be valid at the extremely high strain and hydrostatic tension levels generated in the region of an interface comer. Unfortunately, accurate epoxy models of this type are not readily available. Nevertheless, simpler material models can be used to provide some insights. The cmdest yield zone prediction shown in Fig. 13 uses a linear-elastic adhesive model to determine when the calculated effective stress exceeds the epoxy s yield... 

These two models are represented schematically in Figure 10. The micromodel can approximately represent the behavior of the larger macro-model if they are connected by the displacement coupling technique. The contact results of the macro-model are not used in the further evaluations, and only the displacement results of the proper surfaces are transferred to the micro-model as boundary conditions. The contact problem is solved again in order to find the contact parameters (location of the contact area and the contact pressure distribution) for the micro-model. In order to solve the contact problem, contact elements were located between the contacting bodies. The contact solution follows a linear elastic material law for each component of the models. 

The micro-contact and the debonding model could be substituted by one model, but this task would require a huge computational effort, i.e., to handle 750 contact elements, non-linear material behavior and a few hundred thousand degrees of freedom (DOF) in one model. For this reason, as an approximation, a linear elastic material law was assumed for the contact problems on both levels. 

Material property parameters form another important part of the model. Fortunately, a continuous filament glass fiber yam can be accurately represented as a purely linear elastic material. Each filament in the yam is represented by a series of interconnected linear elastic circular beam elements, their bending, tension, and torsion forces being transmitted between one another. The element size, 0.2 mm, has been chosen carefully to allow an accurate representation of the fabric geometry, but also to keep the solving time reasonable. Other elements of the... 

Cracks at, or near, interfaces - The above has considered the aspect of cracks located in bulk material, but a second important case is that of cracks at, or very close to, a bimaterial interface. However, an immediate problem arises namely, that when the joint is subjected to solely tensile loads applied normal to the crack, which is located along or parallel to the interface, then these will induce both tensile and shear stresses around the crack tip. Therefore, both Ku and terms are needed to describe the stress field the subscript i indicating a crack at, or near, the interface. Similarly, an applied pure shear stress will also induce both such terms. However, these Kn and Km terms no longer have the clearly defined physical significance, as for the bulk material case and illustrated in Fig. 7.3. Mathematical modelling has shown [21-27] that, for linear-elastic materials, the local stresses ahead of the crack tip at a bimaterial interface are proportional to ... 

One of the advantages of these dynamic soil-structure interaction analyses is that the soil layers are modeled to reflect the idealized site stratigraphy each soil layer can be either modeled as a linear elastic material with strain-compatible shear modulus and damping values or characterized via soil constitutive models that represent soil nonlinearity and hysteretic response at small strains. However, the use of the nonlinear constitutive models requires careful selection of input parameters and thus more advanced testing to define those input parameters. The nonlinear behavior and the frequency content of the free-field environment contribute to the stmctural racking behavior. 

The situation of adhesives is different fi-om that of ordinary plastics because adhesives are often plasticized to increase their ductility and cannot be treated as simple linear elastic materials. In this case, elasto-plastic models should be applied. Goglio, Peroni et al. have investigated the constitutive relation of a ductile epoxy adhesive at high strain rates using the split Hopkinson bar method (Goglio et al. 2008). The adhesives showed large plastic deformation that was much better fitted with a poly-linear model than the Johnson-Gook model and the Cowper-Symonds model. 

A key step in this procedure is prescribing the material or constitutive model, that is, the relationship between the stress the material experiences and the resulting deformation it undergoes. Conventional macro-scale finite element simulations assume that the material can be described by measurable macroscopic material properties. Typical examples for solids include materials that are modeled as elastic, viscoelastic, plastic, and viscoplastic. The presentation in this chapter will be confined to linear elastic materials. 

An arbitrarily-sized finite-element model of the pipe span, made from a linearly-elastic material of arbitrary modulus E, was subjected to an arbitrary mid-span load P to determine the resulting load-point displacement v. This was repeated for a number of crack lengths a. An interesting feature of this geometry is that the crack front length B changes, sometimes discontinuously, with a. 

The procedure to calculate fiber orientation is the same as explained above, but their implementation into explicit solvers and non-linear material models is more complex than it is for quasi-static load-cases and purely elastic material models. The fiber orientation is characterized by a so called orientation distribution function (ODE) that describes the chance of a fiber being oriented into a certain direction. For isotropic, elastic matrix materials an integral of the individual stiffness in every possible direction weighted with the ODE provides the complete information about the anisotropic stiffness of the compound. However, this integral can not be solved in case of plastic deformation as needed for crash-simulation. Therefore it is necessary to approximate and reconstruct the full information of the ODE by a sum of finite, discrete directions with their stiffness, so called grains [10]. Currently these grains are implemented into a material description and different methods of formulation are tested. 

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