Maximum-strain theory

Figure 12 Stress-strain curve of randomly oriented short glass-fiber/epoxy composites with the maximum strain theory prediction.

Maximum strain theory may be modified to predict the strength of randomly oriented short-fiber composites (22). llie Halpin-Tsai equations (14) have established relations for the stiffness of an oriented short-fiber ply from the matrix and fiber properties. These nations show that the longitudinal stiffness of an oriented short-fiber composite is a s itive function of the aspect ratio. 

Pile rhombus A-B-C-D represents the maximum-strain theory. By referring to Eqs. 9.87 and 9.88. in which/, is taken as equal lt> zero for IwaMlirnensional stress, it may lx s< cu that if the stress s/ and fy. have the same sign, then... 

Redesign the vessel in the example design for the same condit ions but on the ba.sis of the maximum-strain theory. 

The maximum strain theory is similar to the maximum stress theory. Associated with each strain component, relative to the material symmetry axes, e, Cy, or there is an ultimate strain or an arbitrary proportional limit, Xg, T or S, respectively. The maximum strain theory can be expressed in terms of the following inequalities ... 

The maximum strain theory can be determined by assuming that the material is linearly elastic up to the ultimate failure. The ultimate strains in Eqs. 2-29, 2-30, and 2-31 as well as 2-32 and 2-33 can be related directly to the strengths as follows ... 

Figure 2.29 shows the comparison between the maximum strain theory and the same experimental data shown in Fig. 2.28. The formats are similar. Fig. 2.28 shows a comparison between the maximum work theory of the same experimental data as shown in Figs 2.29 and 2.30.

Maximum strain theory in three dimensional. Includes nonlinear shear stress (strain) effect. 

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

Maximum strain energy theory, which postulates that failure will occur in a complex stress system when the total strain energy per unit volume reaches the value at which failure occurs in simple tension. 

A macroscopic theory of strength is based on a phenomenological approach. No direct reference to the mode of deformation and fi acture is made. Essentially, this approach employs the mathematical theories of elasticity and tries to establish a yield or failure criterion. Among the most popular strength theories are those based on maximum stress, maximum strain, and maximum work. 

The material will have a given strength expressed as stress or strain, beyond which it fails. In order to postulate the failure, it is necessary to have a failure criterion with an associate theory to be able to effect a satisfactory design. Such theories include maximum stress, maximum strain, Tsai-Hill (based on deviatoric strain energy theory) and Tsai-Wu (based on interactive polynomial theory). The Tsai-Wu theory is the most commonly used. 

Based on a Tsai review, it shows that the maximum work theory is more accurate than the maximum stress and strain theories. The maximum work theory encompasses the following additional features. 

There is a continuous decrease as the angles 0 and a deviate from 0°. There is no rise in axial strength, as indicated by the maximum stress and strain theories. 

The uniaxial strength is plotted on a logarithmic scale and an error of a factor of 2 exists in the strength prediction of the maximum stress and strain theories in the range of 30°. 

A fundamental difference between the maximum work and the other theories lies in the question of interaction among the failure modes. The maximum stress and strain theories assume that there is no interaction among the three failure modes (axial, transverse, and shear failures). 

For over three decades, there has been a continuous effort to develop a more universal failure criterion for unidirectional fiber composites and their laminates. A recent FAA publication lists 21 of these theories. The simplest choices for failure criteria are maximum stress or maximum strain. With the maximum stress theory, the ply stresses, in-plane tensile, out-of-plane tensile, and shear are calculated for each individual ply using lamination theory and compared with the allowables. When one of these stresses equals the allowable stress, the ply is considered to have failed. Other theories use more complicated (e.g., quadratic) parameters, which allow for interaction of these stresses in the failure process. 

Of the many theories developed to predict elastic failure, the three most commonly used are the maximum principal stress theory, the maximum shear stress theory, and the distortion energy theory. The maximum (principal) stress theory considers failure to occur when any one of the three principal stresses has reached a stress equal to the elastic limit as determined from a uniaxial tension or compression test. The maximum shear stress theory (also called the Tresca criterion) considers failure to occur when the maximum shear stress equals the shear stress at the elastic limit as determined from a pure shear test. The maximum shear stress is defined as one-half the algebraic difference between the largest and smallest of the three principal stresses. The distortion energy theory (also called the maximum strain energy theory, the octahedral shear theory, and the von Mises criterion) considers failure to have occurred when the distortion energy accumulated in the part under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension or compression test. 

The maximum elastic strain theory (St. Venant s theory) states that inception of failure is due if the largest local strain, 3, within the material exceeds somewhere a critical value e. The failure criterion, therefore, is derived as... 

The term in parentheses is an orientation-dependent and minor kinetic contribution derived from the assumption of successive failure of many elements (in the technically relevant range of values from 40 to 200 kJ/mole the term in parentheses assumes values between 1.17 and 1.0). If the nature of the breaking elements is not changed during sample treatment or fracture development, then P can be considered to be constant. The major effect on 7 is derived from the local stress concentration 0/ 0 which here is equal to the stiffness ratio E/E. This theory essentially predicts, therefore, that the increase in strength is equal to the increase in stiffness it is based on the presumption that the elements break indeed at a critical local strain (kinetic version of St. Venant s criterion of maximum strain). A different interpretation of the strength of oriented samples which is based on the rotation of flaws has been presented in an earlier section of this chapter. 

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