By Michel Fremond
This publication offers novel insights into primary topics in strong mechanics: digital paintings and form swap. the writer explains how the main of digital paintings represents a device for research of the mechanical results of the evolution of the form of a process, the way it will be utilized to observations and experiments, and the way it can be tailored to provide predictive theories of various phenomena. The e-book is split into 3 components. the 1st relates the main of digital paintings to what we become aware of with our eyes, the second one demonstrates its flexibility at the foundation of many examples, and the 3rd applies the primary to foretell the movement of solids with huge deformations. Examples of either traditional and strange form adjustments are awarded, and equations of movement, a few of that are totally new, are derived for soft and non-smooth motions linked to, for example, platforms of disks, platforms of balls, classical and non-classical small deformation theories, platforms concerning quantity and floor harm, platforms with interactions at a distance (e.g., solids strengthened via fibers), platforms related to porosity, collisions, and fracturing of solids.
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Extra info for Virtual Work and Shape Change in Solid Mechanics
Note that we have D1 (V ) + D2 (V ) = (ω2 − ω1 )e3 × (G 2 − G 1 ). 2) A possible set of velocities of deformation for the system is ˆ ) = (D1 (V ), D2 (V ), Dr (V )). 2) we have D1 (V ) + D2 (V ) = (ω2 − ω1 )e3 × (G 2 − G 1 ) = −Dr (V )e3 × (G 2 − G 1 ). © Springer International Publishing Switzerland 2017 M. 1007/978-3-319-40682-4_6 15 16 6 What We See: The Velocities of Deformation Thus we have the kinematic compatibility relationship D1 (V ) + D2 (V ) + Dr (V )e3 × (G 2 − G 1 ) = 0. The sum of velocities of deformation D1 (V ) and D2 (V ) is a function of Dr (V ).
With actual velocities − → U = ( U 1, 1, − → U 2, 2, − → U 3, 3 ). 2 The Velocities of Deformation The velocities of deformation are Dr 12 (V ) = ω1 − ω2 , Dr 23 (V ) = ω2 − ω3 , Dr 31 (V ) = ω3 − ω1 , © Springer International Publishing Switzerland 2017 M. 1007/978-3-319-40682-4_13 41 42 13 Three Disks in a Plane which are angular velocities of deformation. They are related by the kinematic compatibility relationship Dr 12 (V ) + Dr 23 (V ) + Dr 31 (V ) = 0. 1) There is no reason to eliminate one of them, thus we keep all of them to have a systematic presentation.
The virtual strain rates E are null. 2 The Powers The external forces are elements of the dual space V ∗ < f, V > = f · V dB + B(t) ∂B(t) g · Vd − B(t) σ ext S ˜:EdB + B(t) Mext ˆ: dB, 2 ext where σ ext is an external S is a symmetric stress, for instance a prestress and M moment. The internal forces f are elements of the dual space D∗ of D int ∗ f = (σ, σ int S ,M ) ∈ D . They are defined by their power − << f, D(V ) >> = − =− B(t) B(t) σ : grad V dB + σ int S ˜:E − Mint ˆ: dB 2 int << (grad V , E, ), (σ, σ int S , M ) >>l dB =− B(t) << D(V ), f >>l dB.
Virtual Work and Shape Change in Solid Mechanics by Michel Fremond