τ − x is the displacement vector with respect to the t + ρ t˙b t = 0 , t nt | t, v t t t for τ t
x, τ
= 0 , with b tx, τ = bχ
tx, τ , τ S
tx, τ F
T tx, τ = F
tx, τ S
T tx, τ
10 When the reference configuration is considered to be a natural one, we add the initial conditions S X, 0 = 0, FX, 0 = I, PX, 0 = I, α X, 0 = 0, for every X ∈ and the following boundary conditions on ∂ t : S tx, τ nt
| Ŵ 1t = ˆS tx, τ , χ
t x, τ − x | Ŵ 2t = ˆU tx, τ
11 Here ∂ t ≡ Ŵ 1t S Ŵ 2t denotes the boundary of the thredimensional domain t , nt is the unit external normal at Ŵ 1t , while χ tx, τ − x is the displacement vector with respect to the
configuration at time t. ˆS t and ˆ U t , the surface loading and the displacement vector are time dependent, τ, prescribed functions, with respect to the fixed at time t configuration. The rate quasi-static boundary value problem at time t, involves the time differentiation, i.e. with respect to τ, of the equilibrium equations, 10, ∀ x ∈ t , and of the boundary condition 11, when τ = t div ˙ S t x, t + ρx, t˙b tx, t = 0 ,
˙S t x, t nt | Ŵ 1t = ˙ˆS t x, t, vx, t | Ŵ 2t = ˙ˆU tx, t
12 using the notation ˙b tx, t for
∂ ∂τ b tx, τ
| τ =t . At a generic stage of the process the current values, i.e. at the time t, of F, T, Y, and the set of all material particles, in which the stress reached the current yield surface p t = χ p , t , with p ≡ {X ∈ | FCX, t, YX, t = 0} are known for all x ∈ t , with the current deformed domain t also determined. The set of kinematically admissible at time t velocity fields is denoted by V ad t ≡ {v : t −→ R 3 | v | Ŵ 2t = ˙ˆU t }. and the set of all admissible plastic multiplier Mt ≡ {δ : t −→ R ≥0 | δx, t
= 0, if x ∈ t \ p t , δx, t ≥ 0,
Parts
» Directory UMM :Journals:Journal_of_mathematics:OTHER:
» Introduction Binz - S. Pods - W. Schempp NATURAL MICROSTRUCTURES ASSOCIATED WITH
» The complex line bundle associated with a singularity free gradient field in Euclidean space
» The natural principal bundle P
» Two examples Binz - S. Pods - W. Schempp NATURAL MICROSTRUCTURES ASSOCIATED WITH
» Horizontal and periodic lifts of β
» Continua with microstructures Binz - D. Socolescu MEDIA WITH MICROSTRUCTURES AND
» Discrete systems with microstructures
» The interaction form and its virtual work
» Thermodynamical setting Binz - D. Socolescu MEDIA WITH MICROSTRUCTURES AND
» Introduction Bortoloni - P. Cermelli
» Kinematics Let y be a deformation and F = ∇y its deformation gradient. If F
» Continuous and discrete elastic systems
» Trusses Braun COMPATIBILITY CONDITIONS FOR DISCRETE
» Displacement and strain Equilibrium condition
» Compatibility in plane elasticity Compatibility condition for a plane truss
» Conclusion Braun COMPATIBILITY CONDITIONS FOR DISCRETE
» General remarks on continuum models
» Embedding of S O3 in R Embedding a subgroup of S O3 into Sym
» Conclusion Brocato - G. Capriz
» Introduction Carpinteri - B. Chiaia - P. Cornetti
» Damage mechanics of materials with heterogeneous microstructure
» Fractional calculus, local fractional calculus and fractal functions
» Kinematic and static equations for fractal media
» Conclusions Carpinteri - B. Chiaia - P. Cornetti
» τ = detF τ Ty, τ F τ τ = FX, τ FX, t τ t , τ t = T t and t ≡ τ | t .
» τ − x is the displacement vector with respect to the t + ρ t˙b t = 0 , t nt | t, v t t t for τ t
» Composite materials U − ˜U] − f · U − ˜U + 8
» α = ˆ αG Yt = ϕP Yt − ∂ Normality rules in large-deformation plasticity Mech. Mat. 5 1986, 29–34.
» Introduction Engelbrecht - M. Vendelin
» Structural hierarchy and hierarchy of waves.
» Example: cardiac muscle contraction
» Introduction The generalized Cosserat medium
» The liquid-crystal-like model Epstein
» Material vs. dislocation-based crystal plasticity
» Elastoplastic model of twinning
» Twinning and untwinning under cyclic loading Twinning modes at a crack tip
» Multiple twinning in a zinc coating
» Discussion: The pros and the cons of the model
» Introduction F. Ganghoffer NEW CONCEPTS IN NONLOCAL CONTINUUM MECHANICS
» Path integral formulation of nonlocal mechanics
» Geometrisation of the interaction
» Introduction G ¨umbel - W. Muschik
» t 2̺ t State space t , 22 Reversible part t , v t , 1 , t , t .
Show more