Tension stress-strain relation

The bending modulus k is measured by techniques such as monitoring the thermal undulations of membranes [89-94], probing the low-tension stress-strain relation [95], X-ray scattering [96-99], neutron spin echo measurements [100-102] (note however the caveats raised by Watson and Brown [103]), or pulling thin... 

When we proceed to large elastic strains, the problem becomes more complex. The stress-strain relation in uniaxial tension becomes nonlinear. It could be linearized by a proper choice of the... 

Finally, to enable a systematic correspondence between the basic shear-stress-strain relation and a tensile (or compressive) stress-strain relation we note that the expressions for the mean normal stress ffm in tension and compression are... 

Incorporation of the hardening component of the flow stress into the plastic-resistance component then gives the total stress strain relation beyond the flow state for which hardening by molecular alignment becomes prominent. This gives for uniaxial behavior of tension and compression [Pg.263]

Erom a practical viewpoint, Eq. (29.4) can be used to describe the stress-strain relation of a material if vi/(A) is known. m/(A) can be obtained in the laboratory in various ways, such as pure shear experiments as described by Valanis and Landel [60], by torsional measurements as described by Kearsley and Zapas [62] and by a combination of tension and compression experiments as also described by Kearsley and Zapas [62]. Treloar and co-workers [63] have also shown that the VL function description of the mechanical response of rubber is a very good one. The reader is referred to the original literature for these methods. 

In Sect. 1.2 above, the stress-strain relation in uniaxial tension tests was given in Eq. (1.5), indicating a Hookean behavior. This section now considers linear elastic solids, as described by Hooke, according to which (Ty is linearly proportional to the strain, y. Each stress component is expected to depend linearly on each strain component. For example, the Cn may be expressed as follows ... 

The mechanical role of the interface is visible in the overall behaviour of the cement-based composites. Both components - aggregate grains and hardened cement paste - behave separately as linear elastic and brittle bodies in a relatively large domain of applied stresses. However, these two components form a material with a non-linear and inelastic stress-strain relation almost at the beginning of the stress-strain cnrve. This is cansed by the microcracks, which appear and develop nnder tension, shearing and bending mostly in the system of interfaces. 

In a composite element subjected to tension and reinforced with aligned fibres the stress-strain relation is initially linear. Its slope is given by elastic modulus determined by the law of mixtures (cf. Sections 2.5 and 8.3) =... 

The stress-strain relation is a graphical or analytical representation of the relation between stress a and strain s when the material is loaded. The stress-strain relation is determined by measurement on well-defined specimens subjected to tension or compression in special test machines. 

In classical structural analysis materials are assumed to behave in a linear manner. This behavior is commonly described by a linear stress-strain relation obtained by means of a tension test, as shown in Fig. 1. Stress ct, defined as the applied tensile force P divided by the cross-sectional area A, is then proportional to strain s,... 

In case the applied load is increased beyond a certain level, the stress-strain relation is no longer proportional. The material behavior then becomes nonlinear, as illustrated in Fig. 2, which provides an out-of-scale description of a tension test for a steel specimen by means of the stress-strain curve, frequently referred also as the material s constitutive law. The green lines denote unloading and may coincide with the corresponding loading curve, in which case the... 

There exists periodically in the chain a double bond. In the process of vulcanization crosslinks are formed between sulphur and the highly reactive double bonds in the polyisoprene chain. As a result the mechanical behavior of the network material system can be calculated using the present formulation. For simplicity neglecting the effort of the side chains only the principal rubber chain and the crosslink need be considered in the analysis. For comparison the one-dimensional stress-strain relation in tension is calculated as this type of force-extension relationships is widely reported in the literature. For example, the shape of the force extension curve (conventional or nominal stress-strain relationship) for pure-gum GR-S rubber at 2 C is somewhat like a reversed "S" as reported inC 5 J. 

Flexural behaviour modelled by stress-strain relations in tension... 

In the [ 45]j tensile test (ASTM D 3518,1991) shown in Fig 3.22, a uniaxial tension is applied to a ( 45°) laminate symmetric about the mid-plane to measure the strains in the longitudinal and transverse directions, and Ey. This can be accomplished by instrumenting the specimen with longitudinal and transverse element strain gauges. Therefore, the shear stress-strain relationships can be calculated from the tabulated values of and Ey, corresponding to particular values of longitudinal load, (or stress relations derived from laminated plate theory (Petit, 1969 Rosen, 1972) ... 

The current method of determining the energy properties of polyurethane is the Dynamic Thermal Mechanical Analyzer (DTMA). This instrument applies a cyclic stress/strain to a sample of polyurethane in a tension, compression, or twisting mode. The frequency of application can be adjusted. The sample is maintained in a temperature-controlled environment. The temperature is ramped up over the desired temperature range. The storage modulus of the polyurethane can be determined over the whole range of temperatures. Another important property closely related to the resilience, namely tan delta (8), can also be obtained. Tan (8) is defined in the simplest terms as the viscous modulus divided by the elastic modulus. 

We have presented information on the elastic and viscous stress-strain behaviors for a variety of different ECMs in preparation for relating changes in external loading and mechanochemical transduction processes. In order to determine the exact external loading in each tissue that stimulates mechanochemical transduction processes we must take into account the balance between passive loading incorporated into the collagen network in the tissue and active loading applied externally. Inasmuch as the passive load is different for each tissue and is also a function of age (the tension in tissues decreases with age), the net load experienced at the cellular level is difficult to calculate. 

The deformation dependence of the confining potential [Eq. (7.62)] results in a non-classical stress strain dependence of the non-affine tube model. The prediction of this model for the stress-elongation relation in tension is qualitatively similar to the Mooney-Rivlin equation [Eg. (7.59)]... 

The equilibrium small-strain elastic behavior of an "incompressible" rubbery network polymer can be specified by a single number—either the shear modulus or the Young s modulus (which for an incompressible elastomer is equal to 3. This modulus being known, the stress-strain behavior in uniaxial tension, biaxial tension, shear, or compression can be calculated in a simple manner. (If compressibility is taken into account, two moduli are required and the bulk modulus. ) The relation between elastic properties and molecular architecture becomes a simple relation between two numbers the shear modulus and the cross-link density (or the... 

With this development for the rising part of the stress-strain curve in simple shear, a full comparison of the flow model with experiments becomes possible, but a direct comparison of the shear model with tensile experiments must still await relating the basic kinetic law in simple shear to that in tension, where mean normal stresses are present and must be taken into account. 

Elastic modulus is a quantitative measure of the stiffness or rigidity of a material. For example, for homogeneous isotropic substances in tension, the strain (e) is related to the applied stress (o) by the equation E = o/e, where E is defined as the elastic modulus. A similar definition of shear modulus (g) applies when the strain is shear. 

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