Study On Growth-induced Deformations Of Multi-layered Soft Material Plates Based On A Novel Finite-strain Plate Theory

The growth-induced deformations of soft biological tissues with multi-layered plate forms are commonly observed in nature.For example,the growth of animal skins,plant leaves and organs,which can realize many complex biological functions.In the field of engineering,the soft material structures with multi-layered plate forms are widely used in the structural design of intelligent devices due to the advantages of simple for fabrication,good stability,controllable large growth-induced deformations(expansion,contraction,etc.).During the growth process,soft material samples usually exhibit various and interesting evolutions on the global configura-tion and surface morphology.To have a better understand of the potential mechanisms of these phenomena,researchers have conducted comprehensive studies from biology,medicine,and physics fields.With going deep into of research,it can be found that the formation and evolution of appearance features(such as large deformation,global and local buckling,etc.)during the growth process are closely related to mechanical factors.It can not only reveal the potential mechanism of mechanical response,predict the evolution direction of surface morphology,but also achieve precise control of the growth-induced deformation of multi-layered soft intelligent devices from the mechanical perspective,which has important practical significance and ap-plication value.Based on this,a systematic study on the growth-induced deformations of soft material samples with multi-layered plate forms is conducted.The innovative achievements are listed as follows:1)A general multi-layered finite-strain plate theory with growth is established.Firstly,starting from the total energy functional of the plate,the 3D governing equations of the multi-layered plate model can be obtained by calculating the variation with respect to each indepen-dent variable.To obtain the 2D vector plate equation,a series expansion–truncation scheme is introduced to reduce the dimension of the 3D governing equations.By solving the algebraic equations,the iterative relations of the unknowns in each layer are derived.Further consider-ing the boundary condition of upper surface,the 2D vector plate equation with the asymptotic order of O(h~2)is established,which is suitable for studying the mechanical response during the growth process.To ensure the completeness of the plate equation system,the displacement,traction and bending moment boundary conditions are proposed on the lateral surface of the plate.2)A systematic analytical study is carried out on the growth-induced deformation of typical soft material samples with multi-layered plate forms.Firstly,based on differential geometry the-ory,the plate equations for growth-induced plane-strain deformation of multi-layered plate are derived and solved.By studying some typical examples,the validity of the analytical solution is evaluated,and the influence of material parameters and geometric parameters on the mechanical response during the growth process of multi-layered plate is discussed.Then,by using linear bifurcation analysis,the growth-induced instabilities of a three-layered plate(plane strain)is investigated,the relations between the critical growth values and the buckling mode,material parameters,and geometric parameters are also revealed.Finally,the configuration evolutions of the growth-induced axisymmetric deformations of a bilayer disk are analyzed by proposing the asymptotic orders of the growth functions and the displacement components.3)The shape control problem of some typical soft material samples with multi-layer plate forms is studied.First,an analytical formula for shape control is derived for the bilayer soft plate with growth-induced plane-strain and axisymmetric deformations.Through numerical simula-tion,it can be found that the growth function determined by the analytical formula can generate the target shapes accurately.Furthermore,the effects of layer number in shape-programming of multi-layered is analyzed.The residual stresses in the grown configuration decrease signifi-cantly when the layer number increases.4)An asymptotic and simplified finite-strain plate theory with growth is established.Start-ing from the single-layered plate theory,the simplification method of the finite strain plate the-ory is studied systematically.Based on the asymptotic analysis,an asymptotic and simplified single-layered finite strain plate theory with growth is established and the weak form is derived.By adopting the similar scheme,a systematic asymptotic analysis for the complete multi-layered plate equation is conducted,and the asymptotic order of each terms is identified through specify-ing the asymptotic orders of the growth functions and the surface tractions.Then,an asymptotic and simplified multi-layered finite-strain plate theory with growth is established by ignoring some high-order terms.To facilitate the numerical calculation,the weak form and constraint equation of the simplified multi-layered plate equation are derived.From the process of asymp-totic analysis,it can be found that the simplified finite-strain plate theory isu niformly-valid for different magnitudes of growth functions and surface loads.5)The numerical calculation of the asymptotic simplified finite-strain plate theory is real-ized through implementing the weak form of the plate equation into a finite element software.By studying some growth-induced deformations of typical soft material samples with plate forms,it can be found that the simplified 2D plate theory not only can attains high accuracy,but also shows obvious advantages in the aspects of computational efficiency,convergence rate and sta-bility.