In solid mechanics, a reinforced solid is a brittle material that is reinforced by ductile bars or fibres. A common application is reinforced concrete. When the concrete cracks the tensile force in a crack is not carried any more by the concrete but by the steel reinforcing bars only. The reinforced concrete will continue to carry the load provided that sufficient reinforcement is present. A typical design problem is to find the smallest amount of reinforcement that can carry the stresses on a small cube (Fig. 1). This can be formulated as an optimization problem.
Optimization problem The reinforcement is directed in the x, y and z direction. The reinforcement ratio is defined in a cross-section of a reinforcing bar as the reinforcement area A r {\displaystyle A_{r}} over the total area A {\displaystyle A} , which is the brittle material area plus the reinforcement area.
ρ x {\displaystyle \rho _{x}} = A r x {\displaystyle A_{rx}} / A x {\displaystyle A_{x}}
ρ y {\displaystyle \rho _{y}} = A r y {\displaystyle A_{ry}} / A y {\displaystyle A_{y}}
ρ z {\displaystyle \rho _{z}} = A r z {\displaystyle A_{rz}} / A z {\displaystyle A_{z}}
In case of reinforced concrete the reinforcement ratios are usually between 0.1% and 2%. The yield stress of the reinforcement is denoted by f y {\displaystyle f_{y}} . The stress tensor of the brittle material is
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