Fig. 3: Determination of shear stress at TBLM interfaces with the shearing-boundary model.

a Typical force-indentation depth (F-δ) curves of MoS₂@MoS₂ (blue dots) and MoS₂@SiO₂ (red dots) in logarithmic coordinates. The inset shows a schematic diagram of MoS₂@MoS₂ and MoS₂@SiO₂. b, c Histograms of the in-plane elastic modulus (E^2D) and pretension (σ₀^2D) of all samples measured over 1-μm-diameter and 1.5-μm-diameter holes. Here, MoS₂@MoS₂ samples exhibit smaller E^2D and larger σ₀^2D than MoS₂@SiO₂ samples. The dashed lines represent the fitting curves based on Gaussian distribution. d Schematic illustrations of the nanoindentation process on a 2D membrane and the proposed theoretical model that considers an interfacial shear zone between the tested membrane and the substrate. The in-plane equilibrium analysis for a representative element is illustrated in the top part. N_r and N_θ are the radial and circumferential stress resultants, respectively. r is the distance between the selected element and the center of the hole. dθ is the angle of the sector. a is the radius of the hole. e Comparison between one typical experimental curve and fitted curves based on Eq. (2). The inset is the fitting curve based on τ = 2.795 MPa, suggesting that when the interlayer shear stress τ = 2.795 MPa, the fitting error reaches a minimum. f Distribution of fitted shear stresses that satisfies a logarithmic normal distribution. The inset shows a contour map of the nonlinear fitting error to obtain the closest shear stress of this sample.