Strain relations

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Drucker, D.C., A More Fundamental Approach to Stress-Strain Relations, in Proc. First National Congr. Appl. Mech., ASME, Chicago, June 1951, pp. 487-491. 

Stress, C7 Fig. 2.41 The standard linear solid Stress-Strain Relations As shown earlier the stress-strain relations are (2.44)... 

If Cm -I- 3Cii > 0, a centered simple wave will be produced by impact loading, and a record of this waveform suffices to determine the entire uniaxial stress-strain relation over the range of strains encountered. Vitreous silica is a material responding in this manner, and its coefficients have been determined by Barker and Hollenbach [70B01] (see also [72G02]) on the basis of a simple-wave analysis. 

Equations (2.9) and (2.10) are representative of all isotropic, homogeneous solids, regardless of the stress-strain relations of a solid. What is strongly materials specific and uncertain is the appropriate value for shear stress, particularly if materials are in an inelastic condition or anisotropic, inhomogeneous properties are involved. The limiting shear stress controlled by strength is termed r. ... 

With the foregoing reduction from 36 to 21 independent constants, the stress-strain relations are... 

If there is one plane of material property symmetry, the stress-strain relations reduce to... 

If there are two orthogonal planes of material property symmetry for a material, symmetry will exist relative to a third mutually orthogonal plane. The stress-strain relations in coordinates aligned with principal material directions are... 

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 ... 

Figure 2-5 Physical Significance of the Anisotropic Stress-Strain Relations...

STRESS-STRAIN RELATIONS FOR PLANE STRESS IN AN ORTHOTROPIC MATERIAL... 

The strain-stress relations in Equation (2.61) can be inverted to obtain the stress-strain relations... 

However, as mentioned previously, orthotropic laminae are often constructed in such a manner that the principal material coordinates do not coincide with the natural coordinates of the body. This statement is not to be interpreted as meaning that the material itself is no longer orthotropic instead, we are just looking at an orthotropic material in an unnatural manner, i.e., in a coordinate system that is oriented at some angle to the principal material coordinate system. Then, the basic question is given the stress-strain relations In the principal material coordinates, what are the stress-strain relations in x-y coordinates ... 

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. 

As an alternative to the foregoing procedure, we can express the strains in terms of the stresses in body coordinates by either (1) inversion of the stress-strain relations in Equation (2.84) or (2) transformation of the strain-stress relations in principal material coordinates from Equation (2.61),... 

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... 

However, if the specimen is short and wide, the end restraint of 0 and y = Yxy = 0 leads to a stress-strain relation... 

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 basis for the determination of an upper bound on the apparent Young s modulus is the principle of minimum potential energy which can be stated as Let the displacements be specified over the surface of the body except where the corresponding traction is 2ero. Let e, Tjy, be any compatible state of strain that satisfies the specified displacement boundary conditions, l.e., an admissible-strain tieldr Let U be the strain energy of the strain state TetcTby use of the stress-strain relations... 

Alternatively, the x-direction strain at buckling can be calculated from Equation (3.117) and the uniaxial stress-strain relation as... 

Substitute the deflection function, Equation (3.95), the shear strain expression, Equation (3.130), and the stress-strain relation. Equation (3.131), in Equation (3.132) to get... 

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