Vector, strain

In general, the strain vector e is written using a differential operator matrix [A] and the displacement vector U ... 

For the calculation of the thermally induced stress, thermally induced strain is additionally considered. The total strain vector e is written using the elastic strain vector eel and thermally induced strain vector eth as... 

Fig. 11-14. Decomposition of strain vector into two components in a dynamic experiment.

Tlie strain vector OB can be resolved into vector OE along the direction of OA and OF perpendicular to OA. Then the projection OH of OE on the vertical axis is the magnitude of the strain which is in phase with the stress at any lime. Similarly, projection 01 of vector OF is the magnitude of the strain which is 90 (one-quarter cycle) out of phase with the stress. The stress can be similarly resolved into two components with one along the direction of OB and one leading the strain vector by n/2 rad. 

When the stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

The magnitude of the stress at any time is represented by the projection OC of the vector OA on the vertical axis. Vector OA rotates with a frequency w equal to that of the sinusoidally varying stress. The length of OA is the stress amplitude (maximum stress) involved in the experiment. The strain is represented by the proj ection OD of vector OB on the vertical axis. The strain vector OB rotates in the same direction as OA with frequency w but it lags OA by an angle 6. The loss tangent (discussed later) is defined as tan d. 

The strain vector OB can be resolved into vector OE along the direction of OA and OF perpendicular to OA. Then the projection OH of OE on the vertical axis is the magnitude of the strain which is in phase... 

If, on the other hand, the strain vector is resolved into its components, the ratio of the in-phase strain to stress amplitude (maximum stress) is the storage compliance J (oj) and the ratio of the out-of-phase strain to the stress amplitude is the loss compliance... 

P(l) When lap and strap joints are loaded simultaneously by more than one of the above listed loadings, the joint resistance shall be determined using the maximum strain failure criterion applied to the resultant shear strain vector in the adhesive. 

Strain Vector Promoter Source of LDH pH control LA produced (g/L) Yield Reference... 

Removing the rigid body component from F , it is possible to construct the strain tensor = F — A , which conjugated to the asymmetric First Piola Kirchhoff (FPK) stress tensor P = Pi toi referred to the curved reference beam, Simo (1985). P, is the FPK stress vector acting on the deformed face in the current beam corresponding to the normal ioi in the curved reference configuration. The spatial strain vector acting on the current beam cross section is obtained as e = e ioi-... 

Fig. 22. First and second row Temporal development of the components of the strain vector at the upper surface and at (x,y) = (L/2,lmm) and (x,y) = (L/2,10mm). Third row cross-sectional distribution of the stress components at the upper surface and at x = LI2.

OC strain vector (Voigt notation, TYli sbp direction... 

Phase angle, phase shift between stress and strain vectors Tensile strain [—]... 

Figure 10.2 Stress and strain vectors at low and high temperatures.

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