Stress-strain relations stiffnesses

If at every point of a material there is one plane in which the mechanical properties are equal in all directions, then the material is called transversely isotropic. If, for example, the 1-2 plane is the plane of isotropy, then the 1 and 2 subscripts on the stiffnesses are interchangeable. The stress-strain relations have only five independent constants ... 

Note that the transformed reduced stiffness matrix Qy has terms in all nine positions in contrast to the presence of zeros in the reduced stiffness matrix Qy. However, there are still only four independent material constants because the lamina is orthotropic. In the general case with body coordinates x and y, there is coupling between shear strain and normal stresses and between shear stress and normal strains, i.e., shear-extension coupling exists. Thus, in body coordinates, even an orthotropic lamina appears to be anisotropic. However, because such a lamina does have orthotropic characteristics in principal material coordinates, it is called a generally orthotropic lamina because it can be represented by the stress-strain relations in Equation (2.84). That is, a generally orthotropic lamina is an orthotropic lamina whose principai material axes are not aligned with the natural body axes. 

A key element in the experimental determination of the stiffness and strength characteristics of a lamina is the imposition of a uniform stress state in the specimen. Such loading is relatively easy for isotropic materials. However, for composite materials, the orthotropy introduces coupling between normal stresses and shear strains and between shear stresses and normal and shear strains when loaded in non-principal material coordinates for which the stress-strain relations are given in Equation (2.88). Thus, special care must be taken to ensure obtaining... 

In Section 2.2, the stress-strain relations (generalized Hooke s law) for anisotropic and orthotropic as well as isotropic materials are discussed. These relations have two commonly accepted manners of expression compliances and stiffnesses as coefficients (elastic constants) of the stress-strain relations. The most attractive form of the stress-strain relations for orthotropic materials involves the engineering constants described in Section 2.3. The engineering constants are particularly helpful in describing composite material behavior because they are defined by the use of very obvious and simple physical measurements. Restrictions in the form of bounds are derived for the elastic constants in Section 2.4. These restrictions are useful in understanding the unusual behavior of composite materials relative to conventional isotropic materials. Attention is focused in Section 2.5 on stress-strain relations for an orthotropic material under plane stress conditions, the most common use of a composite lamina. These stress-strain relations are transformed in Section 2.6 to coordinate systems that are not aligned with the principal material... 

The stress-strain relations in arbitrary in-plane coordinates, namely Equation (4.5), are useful in the definition of the laminate stiffnesses because of the arbitrary orientation of the constituent laminae. Both Equations (4.4) and (4.5) can be thought of as stress-strain relations for the k layer of a multilayered laminate. Thus, Equation (4.5) can be written as... 

Once the deflections are known, the stresses are straightforwardly obtained by substitution in the stress-strain relations. Equation (4.16), after the strains are found from Equation (4.12). Note that the solution in Equation (5.31) is expressed in terms of only the laminate stiffnesses D., Di2. D22. and Dgg. This solution will not be plotted here, but will be used as a baseline solution in the following subsections and plotted there in comparison with more complicated results. 

Figure 4 illustrates the typical volume dilatation-strain behavior along with its first and second derivatives. Clearly these measures are realistic in that the derivatives do take on the character of cumulative and instantaneous frequency distributions. Similar models can be constructed to relate the loss in stiffness to the number of vacuoles that have formed resulting in very simple but accurate stress-strain relations (1). 

Wright et al. (23-24) discussed the elastic response of TPV using the micro-cellular modeling, in which three types of deformation such as elastic and plastic deformation of PP, elastic deformation of EPDM, and localized elastic and plastic rotation about PP junction points were considered. Their microcellular model suggests that EPDM domains are surrounded by the PP struts and the struts are connected with hinges. They were successful to elucidate the permanent deformation and stiffness and stress-strain relation of TPV. 

For all materials (other than fabrics, for which the concept is not relevant) the basic parameter is a measure of stiffness or modulus derived from the stress-strain curve. As with tensile tests, because the stress -strain relation is generally not linear, care must be taken to compare only measures of stiffness defined in the same way. With rigid foams and plastics there are additionally measures of yield or strength. 

For single crystals with transverse dimensions large enough to permit a plane wave condition to be attained, the results are unambiguous and virtually free from theoretical assumptions. Five independent elastic constants (stiffnesses or compliances) are required to describe the linear elastic stress-strain relations for hexagonal materials. Only three independent constants are required for cubic (y-Ce, Eu, Yb) materials. Since there are no single crystal elastic constant data for the cubic rare earth metals, this discussion will concentrate on the relationships for hexagonal symmetry. 

When designing a fill, the stiffness of a soil is a means to calculate the deformation (Serviceability Limit State, SLS). The stress-strain relation is important to select the correct stiffness at a certain mobilised stress level. 

Here, T is such that it represents the convolution term used in the stress-strain relation, which after applying finite elements becomes the stiffness-displacement product given by... 

The matrices [5] and [Q] appearing in Equations 8.42 and 8.43 are called the reduced compliance and stiffness matrices, while [5 ] and Q in Equations 8.54 and 8.55 are known as the transformed reduced compliance and stillness matrices. In general, the lamina mechanical properties, from which compliance and stiffness can be calculated, are determined experimentally in principal material directions and provided to the designer in a material specification sheet by the manufacturer. Thus, a method is needed for transforming stress-strain relations from off-axis to the principal material coordinate systan. 

The purpose of introducing the concepts developed in Equation 8.56 through Equation 8.63 has been to provide the tools needed to present a straightforward derivation of the transformed reduced compliance and stiffness matrices [5] and [Q based on matrix algebra. The derivation makes use of the following sequence of operations to obtain stress-strain relations in the reference (x, y) coordinate system ... 

The preceding stress-strain and strain-stress relations are the basis for stiffness and stress analysis of an individual lamina subjected to forces in its own plane. Thus, the relations are indispensable in laminate analysis. 

First, the stress-strain behavior of an individual lamina is reviewed in Section 4.2.1, and expressed in equation form for the k " lamina of a laminate. Then, the variations of stress and strain through the thicyiess of the laminate are determined in Section 4.2.2. Finally, the relation of the laminate forces and moments to the strains and curvatures is found in Section 4.2.3 where the laminate stiffnesses are the link from the... 

The stiffness and compliances in stress-strain and strain-stress relations are fourth-order tensors because they relate two second-order tensors ... 

Finally, the modulus of elasticity E (Young s modulus), which is a measure of the stiffness of the polymer, can be calculated from the stress-strain diagram. According to Hooke s law there is a linear relation between the stress o and the strain e ... 

The volume increase depends on the filler fraction and on the applied strain. This is confiimed in practice. Debonding correlates with loss of stiffness. The first part of the stress-strain curve (elastic stage) is related to the strains beyond which debonding occurs. In glass bead filled polypropylene, this strain was 0.7%. ... 

Tensile properties that are related to fiber stiffness can be used to measure the T of almost all fibers. The elastic modulus, that is, the sl pe of the Hookean region of the fiber stress-strain curve, is a measure of the fiber stiffness and can be used for T determination since, by definition, a glass is stlffer than rubber (Figure 6). Since the transition from a glassy to a rubbery state Involves a reduction in stiffness, the temperature at which the modulus is abruptly lowered is taken as... 

Stiffness n. (1) The load per unit area required to elongate the fihn 1% from the first point in the stress-strain curve where the slope becomes constant. (2) A term relating to the abfiity of a material to resist bending while under stress. Resistance to the bending is called flexural stiffness, and... 

The stiffness properties k( and force-displacement relationships of the uniaxial elements are defined according to constitutive stress-strain relationships implemented in the model for concrete and steel (Fig. 20.2) and the tributary area assigned to each uniaxial element. The reinforcing steel stress-strain behavior implemented in the wall model is the well-known nonlinear relationship of Menegotto and Pinto (1973) (Fig. 20.2b). The hysteretic constitutive relation developed by Chang and Mander (1994) (Fig. 20.2a) is used as the basis for the relation implemented for concrete because it is a general model that provides the... 

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