Spatial form

Sustainable Development and Industrial Ecology

Dr.Salah M. El-Haggar PE, PhD, in Sustainable Industrial Design and Waste Management, 2007

Regional strategies barrier

Often geographic regions may provide a sensible basis for implementing IE. Industries tend to form spatial clusters in specific geographic regions based on factors such as access to raw materials, convenient transportation, technical expertise, and markets. This is particularly true for “heavy” industries requiring large resource inputs and generating extensive waste quantities. Furthermore, the industries supporting large industrial complexes tend to be located within reasonable proximity to their principal customers. Due to the unique character of different regions this work could proceed in the form of case studies of regions containing a concentration of industries in a particular sector.
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Constitutive Models of Soft and Hard Living Tissues

Jean-François Ganghoffer, in Multiscale Biomechanics, 2018

3.13 Variational principles and hints to numerical solution schemes

We first summarize the field equations required to formulate a boundary value problem (BVP in short) in finite elasticity; those equations can be written either in Ω₀, or in Ω_t, and both formulations can be related by pull-back and push-forward operations.
Eulerian description: the strong form of the initial boundary-value problem is written
{divσ+ρf=ρdvdt=ρüinΩtσ=σTu=udonSut=σ.n=tdonStu(x,0)=u0(X)˙u(x,0)=˙u0(X)ü
with f (x,t) body forces per unit mass in the actual configuration, and u^d, t^d imposed displacement and traction vectors (they are data) on two complementary subparts of the external boundary of Ω_t, so that S_u ∪ S_t = ∂Ω_t, S_u ∩ S_t = ∅. Kinematic boundary conditions (resp. static boundary conditions) are sometimes referred to as Dirichlet conditions (resp. Neumann conditions). The density ρ = ρ (x, t) is related to the initial density through the Jacobian (itself depending on the displacement field) ρ = ρ₀ / det (F). This BVP has to be complemented by a constitutive relation expressing Cauchy stress in terms of a conjugated strain tensor. Since the BVP is of second order in space and time, we need two initial conditions which are the last two equations.
Observe that this BVP is formulated over the deformed configuration (the spatial divergence is involved), thus it is written in terms of the actual position x(X, t), which is precisely the unknown principal. This is a difficulty of nonlinear mechanics and a remedy consists in writing the same BVP in the fixed reference configuration.
Note that initial and boundary conditions have to be compatible so that the following conditions are satisfied: