  Brush-like structures emerge from stretching of long polymer chains, densely
grafted on to the surface of an impermeable substrate. They arise due to the
competition between conformational entropic elasticity of polymer chains and
excluded volume interactions from the intra and interchain monomer repulsions.
Recently, stimuli responsive polymer brush based soft materials have been
developed to produce controllable and reversible large deformations of the host
substrate. To understand these systems, and improve their functional
properties, we study elastic stress distribution and surface stress-curvature
relations of a neutral polymer brush grafted on to an elastic beam, made of a
soft material. In the strongly stretched brush regime, we combine mean field
theory from polymer physics with a continuum mechanics model and show that the
residual stress variation is a quartic function of distance from the grafting
surface, with maximum stress occurring at the grafted surface. Idealizing the
brush as a continuum elastic surface layer with residual stress, we derive a
closed form expression for surface stress and the surface elasticity of the
layer as a function of brush parameters, such as graft density and molecular
weight. The generalized continuum beam model accounts for the Young-Laplace and
Ogden-Steigman curvature elasticity correction terms, and yields a surface
stress-curvature relation, which contains existing relations in the literature
as special cases. Further, we report experiments on a thermoresponsive random
copolymer brush, Poly(N- isopropylacrylamide)-co-Poly(N,N-Dimethylacrylamide)
(PNIPAm-co-PDMA) brush, grafted on one side of a plasticized pPVC thin film.
Estimated surface stress from measured curvature is on the order of $-10~N/m$,
and it decreases gradually, and reversibly, with increasing ambient temperature
from $15~^\circ{}C$ to $55~^\circ{}C$.
