Virtual power and the concept of stress in VS .NET Drawer QR Code JIS X 0510 in VS .NET Virtual power and the concept of stress

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2.2 Virtual power and the concept of stress use .net qr code iso/iec18004 creation torender denso qr bar code on .net Modified Plessey linear integral maps over virtual velocity elds v de ned as v Pe (v ) = v Pa (v ) = f v dv + t v ds u v dv. Momentum balance (Newton s second law) is equivalent to the principle of virtual power. Principle of virtual power For every virtual velocity eld v the following equality holds: v v v Pe (v ) + Pi (v ) = Pa (v ). (2.

1). for a body in static equi visual .net QR librium inertial forces vanish, because u = 0, and the principle of virtual power is expressed as: v v Pe (v ) + Pi (v ) = 0 (2.2).

v The virtual power of the internal forces Pi (v ) remains to be de ned by specifying explicitly the nature of the linear map over the space of virtual velocity elds. A rational condition should be imposed that this power (and hence the underlying internal forces) should be independent of the choice of Galilean frame of reference used by the observer. Hence virtual power of internal forces should vanish when applied to virtual velocity elds corresponding to rigid body motion: v Pi (v ) = 0 v v V0 .

(2.3). In this way the form of the VS .NET qr bidimensional barcode vector subspace of virtual velocity elds V derived from the kinematic analysis of the previous chapter results in certain conditions on the system of internal forces Pi . This approach can be used to develop internal force models for ideal uids, solid bodies, columns, beams, plates, etc.

A detailed discussion of this approach can be found in Salencon (2001). Ideal uid Select the space V of virtual velocity elds to be continuous differentiable vector elds over . De ne the virtual power as the following linear map: v Pi (v ) = p div v dv.

(2.4). Here p is a scalar eld tha t characterises internal forces. The scalar p is dual to the rate of volume change de ned by div v and represents therefore the internal pressure eld within the body. It is easy to check that this form of internal virtual power vanishes for virtual velocities corresponding to rigid body motion.

The virtual power expression (2.4) can also be rewritten by the application of the Stokes theorem in the equivalent form v Pi (v ) = grad p v dv +. p n v ds,. (2.5). Dynamics and statics: stresses and equilibrium where n is the outward unit normal to the boundary . The principle of virtual power becomes ( grad p + f ) v dv +. t (t + p n ) v ds = u v dv. (2.6). Because the above equality holds for an arbitrary virtual velocity eld v , this leads to the following equilibrium equations in the local form: for each interior point x : grad p + f = u for each boundary point x : t = +p n . The condition on the boundary shows that the ideal uid model only allows boundary loading in the form of normal surface tractions. This anomaly was rst observed by d Alembert and has since been known as the d Alembert paradox (Truesdell, 1968).

Continuum solid Once again we start our consideration with the space V of virtual velocity elds given by continuous differentiable vector elds over . The linear map representing virtual power is in this case given by v Pi (v ) = : grad v dv, (2.7).

where is a symmetric tens qr barcode for .NET or eld, referred to as the stress eld. In granular materials such as sand, the internal work is done not only due to particle deformation, but also due to the mutual rotation of particles.

The stress tensor in this case may not be symmetric. The model that arises if this is taken into consideration is generally known as a Cosserat material. One must consider the possibility of the existence of an additional external force producing a distribution of body moments.

This, for example, is the case if particles forming the body are magnetic and the body is subject to an externally applied magnetic eld. In classical continuum mechanics, the symmetry of stresses is assumed, = T , implyv ing that : grad v = : [v ]. Thus the symmetric stress tensor is dual to the small strain tensor and does not act on rotations.

As for an ideal uid, one can readily check that the internal virtual power de ned in (2.7) vanishes for virtual velocities corresponding to rigid body motion. Applying the Stokes theorem to (2.

7) as before, one obtains v Pi (v ) = v div : [v ] dv . v n ds,. (2.8). where n is the outward unit normal to the boundary . The principle of virtual power becomes (div + f ) v dv +.
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