A shearable and thickness stretchable finite strain beam model for soft structures

Abstract

Soft materials and structures have recently attracted lots of research interests as they provide paramount potential applications in diverse fields including soft robotics, wearable devices, stretchable electronics and biomedical engineering. In a previous work, an Euler–Bernoulli finite strain beam model with thickness stretching effect was proposed for soft thin structures subject to stiff constraint in the width direction. By extending that model to account for the transverse shear effect, a Timoshenko-type finite strain beam model within the plane-strain context is developed in the present work. With some kinematic hypotheses, the finite deformation of the beam is analyzed, constitutive equations are deduced from the theory of finite elasticity, and by employing the standard variational method, the equilibrium equations and associated boundary conditions are derived. In the limit of infinitesimal strain, the new model degenerates to the classical extensible and shearable elastica model. The corresponding incremental equilibrium equations and associated boundary conditions are also obtained. Based on the new beam model, analytical solutions are given for simple deformation modes, including uniaxial tension, simple shear, pure bending, and buckling under an axial load. Furthermore, numerical solution procedures and results are presented for cantilevered beams and simply supported beams with immovable ends. The results are also compared with the previously developed finite strain Euler–Bernoulli beam model to demonstrate the significance of transverse shear effect for soft beams with a small length-to-thickness ratio. The developed beam model will contribute to the design and analysis of soft robots and soft devices.

Appendices

Appendix 1

Following Simo’s method [37], the direct notation is adopted here to facilitate the derivation of the constitutive equations for the present plane-strain beam model. The position of any material point in the deformed configuration is denoted by ${\mathbf{x}} = {\mathbf{x}}\left( {X,Z} \right)$$= {\mathbf{x}}_{0} \left( X \right) + z^{*} \left( Z \right){\mathbf{e}}_{\varphi } \left( X \right)$, where ${\mathbf{x}}_{0}$ is the corresponding point on the geometrical mid plane, ${\mathbf{e}}_{\varphi }$ is the unit vector reflecting the rotation of the deformed cross-section, and the possible dependence of $z^{*}$ on $X$ is neglected as argued in the kinematic analysis in Sect. 2.1. Since ${\mathbf{e}}_{\varphi }$ is a unit vector, we have ${{\partial {\mathbf{e}}_{\varphi } } \mathord{\left/ {\vphantom {{\partial {\mathbf{e}}_{\varphi } } {\partial X}}} \right. \kern-0pt} {\partial X}} = {\varvec{\upomega}} \times {\mathbf{e}}_{\varphi } , \, {\dot{\mathbf{e}}}_{\varphi } = {\mathbf{w}} \times {\mathbf{e}}_{\varphi } ,$ where ${\varvec{\upomega}}$ is the rotation rate of ${\mathbf{e}}_{\varphi }$ along the arc-length, i.e., the nominal curvature, and ${\mathbf{w}}$ the spin rate (per unit time interval) at a fixed cross-section with the over-dot denoting the material time derivative.

The deformation gradient and the first Piola–Kirchhoff stress tensor are given respectively by:

$${\mathbf{F}} = \frac{{\partial {\mathbf{x}}}}{{\partial X_{I} }} \otimes {\mathbf{E}}_{I} ,\quad {\mathbf{P}} = {\mathbf{T}}_{I} \otimes {\mathbf{E}}_{I} ,$$

where ${\mathbf{E}}_{I} \left( {I = 1,2} \right)$ is the unit basis vector in the reference configuration, $X_{I}$$\left( {I = 1,2} \right)$ represent the reference coordinates $\left( {X,Z} \right)$, ${\mathbf{T}}_{I} \left( {I = 1,2} \right)$ are the stress vectors (per unit reference area) in the current configuration. By the balance law of angular momentum, we have ${\mathbf{F}}^{T} {\mathbf{P}} = {\mathbf{P}}^{T} {\mathbf{F}}$, or equivalently, ${{\partial {\mathbf{x}}} \mathord{\left/ {\vphantom {{\partial {\mathbf{x}}} {\partial X_{I} }}} \right. \kern-0pt} {\partial X_{I} }} \times {\mathbf{T}}_{I} = {\mathbf{0}}$.

With the assumed kinematics, the deformation gradient and its material time derivative can be written, respectively, as:

$${\mathbf{F}} = \frac{{\partial {\mathbf{x}}_{0} }}{\partial X} \otimes {\mathbf{E}}_{X} + z^{*} \frac{{\partial {\mathbf{e}}_{\varphi } }}{\partial X} \otimes {\mathbf{E}}_{X} + \lambda_{Z} {\mathbf{e}}_{\varphi } \otimes {\mathbf{E}}_{Z} ,$$

$$\begin{aligned} {\dot{\mathbf{F}}} & = \frac{{\partial {\dot{\mathbf{x}}}_{0} }}{\partial X} \otimes {\mathbf{E}}_{X} + \dot{z}^{*} \left( {{\varvec{\upomega}} \times {\mathbf{e}}_{\varphi } } \right) \otimes {\mathbf{E}}_{X} + z^{*} \left( {{\dot{\mathbf{\omega }}} \times {\mathbf{e}}_{\varphi } } \right) \otimes {\mathbf{E}}_{X} + z^{*} {\varvec{\upomega}} \times \left( {{\mathbf{w}} \times {\mathbf{e}}_{\varphi } } \right) \otimes {\mathbf{E}}_{X} \\ & \quad + \dot{\lambda }_{Z} {\mathbf{e}}_{\varphi } \otimes {\mathbf{E}}_{Z} + \lambda_{Z} \left( {{\mathbf{w}} \times {\mathbf{e}}_{\varphi } } \right) \otimes {\mathbf{E}}_{Z} . \\ \end{aligned}$$

Hence, the stress power per unit reference volume can be obtained:

$$\begin{aligned} {\mathbf{P}} \cdot {\dot{\mathbf{F}}} & = {\mathbf{T}}_{1} \cdot \frac{{\partial {\dot{\mathbf{x}}}_{0} }}{\partial X} + \dot{z}^{*} {\mathbf{T}}_{1} \cdot \left( {{\varvec{\upomega}} \times {\mathbf{e}}_{\varphi } } \right) + \left[ {\left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right) \times {\mathbf{T}}_{1} } \right] \cdot {\dot{\mathbf{\omega }}} + {\mathbf{T}}_{1} \cdot \left[ {{\varvec{\upomega}} \times \left\{ {{\mathbf{w}} \times \left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right)} \right\}} \right] \\ & \quad + \dot{\lambda }_{Z} {\mathbf{e}}_{\varphi } \cdot {\mathbf{T}}_{2} + \lambda_{Z} \left( {{\mathbf{e}}_{\varphi } \times {\mathbf{T}}_{2} } \right) \cdot {\mathbf{w}}. \\ \end{aligned}$$

Using the balance law of angular momentum, we have:

$$\begin{aligned} \left( {{\mathbf{e}}_{\varphi } \times {\mathbf{T}}_{2} } \right) \cdot {\mathbf{w}} & = {\mathbf{w}} \cdot \left( {\frac{{\partial {\mathbf{x}}}}{\partial Z}\frac{1}{{\lambda_{Z} }} \times {\mathbf{T}}_{2} } \right) = - {\mathbf{w}} \cdot \left( {\frac{{\partial {\mathbf{x}}}}{\partial X}\frac{1}{{\lambda_{Z} }} \times {\mathbf{T}}_{1} } \right) = - \frac{1}{{\lambda_{Z} }}{\mathbf{w}} \cdot \left[ {\left\{ {\frac{{\partial {\mathbf{x}}_{0} }}{\partial X} + {\varvec{\upomega}} \times \left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right)} \right\} \times {\mathbf{T}}_{1} } \right] \\ & = - \frac{1}{{\lambda_{Z} }}{\mathbf{T}}_{1} \cdot \left[ {{\mathbf{w}} \times \frac{{\partial {\mathbf{x}}_{0} }}{\partial X} + {\mathbf{w}} \times \left\{ {{\varvec{\upomega}} \times \left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right)} \right\}} \right]. \\ \end{aligned}$$

Considering that ${\varvec{\upomega}} \times \left\{ {{\mathbf{w}} \times \left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right)} \right\} - {\mathbf{w}} \times \left\{ {{\varvec{\upomega}} \times \left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right)} \right\} = \left( {{\varvec{\upomega}} \times {\mathbf{w}}} \right) \times \left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right)$, we obtain:

$$\int_{V} {{\mathbf{P}} \cdot {\dot{\mathbf{F}}}{\text{d}}V} = \int_{V} {\left\{ {{\mathbf{T}}_{1} \cdot \left( {\frac{{\partial {\dot{\mathbf{x}}}_{0} }}{\partial X} - {\mathbf{w}} \times \frac{{\partial {\mathbf{x}}_{0} }}{\partial X}} \right) + \left[ {\left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right) \times {\mathbf{T}}_{1} } \right] \cdot \left( {{\dot{\mathbf{\omega }}} - {\mathbf{w}} \times {\varvec{\upomega}}} \right) + \dot{\lambda }_{Z} {\mathbf{e}}_{\varphi } \cdot {\mathbf{T}}_{2} + \dot{z}^{*} {\mathbf{T}}_{1} \cdot \left( {{\varvec{\upomega}} \times {\mathbf{e}}_{\varphi } } \right)} \right\}{\text{d}}V}$$

Neglecting the last two small terms relevant to the thickness stretching in comparison with the dominant stretching, shearing and bending power, the integral can be rewritten as:

$$\int_{V} {{\mathbf{P}} \cdot {\dot{\mathbf{F}}}{\text{d}}V} = \int_{L} {\left( {{\mathbf{N}} \cdot \mathop {\varvec{\Gamma}}\limits^{\nabla } + {\mathbf{M}} \cdot \mathop {\varvec{\upomega}}\limits^{\nabla } } \right){\text{d}}X} ,$$

where ${\mathbf{N}} = \int_{A} {{\mathbf{T}}_{1} {\text{d}}A}$, ${\mathbf{M}} = \int_{A} {\left( {{\mathbf{x}} - {\mathbf{x}}_{0} } \right) \times {\mathbf{T}}_{1} {\text{d}}A}$ and ${\varvec{\Gamma}} = {{\partial {\mathbf{x}}_{0} } \mathord{\left/ {\vphantom {{\partial {\mathbf{x}}_{0} } {\partial X}}} \right. \kern-0pt} {\partial X}}$ physically means the internal force vector, internal moment vector and the axial stretch vector, respectively, and $\mathop {\left( \cdot \right)}\limits^{\nabla }$ represents the corotational rate (an objective rate) defined by $\mathop {\left( \cdot \right)}\limits^{\nabla } = {{\partial \left( \cdot \right)} \mathord{\left/ {\vphantom {{\partial \left( \cdot \right)} {\partial t}}} \right. \kern-0pt} {\partial t}} - {\mathbf{w}} \times \left( \cdot \right).$

Decomposing the internal force vector into normal and shear components $N_{n}$ and $N_{s}$ on the deformed cross-section, and equivalently, replacing the material-time derivative or objective rates with variations in the above formula, we have:

$$\int_{L} {\delta \phi {\text{d}}X} = \int_{L} {\left( {N_{n} \delta \lambda_{n} + N_{s} \delta \gamma + M\delta \kappa } \right){\text{d}}X} ,$$

where $\lambda_{n} = \lambda \cos \alpha - 1$, $\gamma = \lambda \sin \alpha$, and $\delta \phi = \int_{A} {{\mathbf{P}} \cdot \delta {\mathbf{F}}{\text{d}}A} = \delta \int_{A} {W{\text{d}}A}$. Finally, we derive the constitutive relation (13) by localization.

Appendix 2

The tangent tensile stiffness, shear stiffness, bending stiffness and coupling stiffnesses are defined and also calculated, based on the constitutive Eq. (21)_1–3 for neo-Hookean beams, as follows:

$$\frac{{K_{T} }}{\mu BH} \equiv \frac{1}{\mu BH}\frac{\partial T}{\partial \lambda } = \left( {1 + \frac{3}{{\lambda^{4} \cos^{2} \alpha }}} \right) + \frac{{7\kappa^{2} H^{2} }}{{\lambda^{8} \cos^{6} \alpha }},$$

$$\frac{{K_{S} }}{\mu BH} \equiv \frac{1}{\mu BH}\frac{\partial S}{\lambda \partial \alpha } = \left[ {\frac{{1 + 2\sin^{2} \alpha }}{{\lambda^{4} \cos^{4} \alpha }} + \frac{{\kappa^{2} H^{2} \left( {1 + 6\sin^{2} \alpha } \right)}}{{\lambda^{8} \cos^{8} \alpha }}} \right],$$

$$\frac{{K_{TS} }}{\mu BH} \equiv \frac{1}{\mu BH}\frac{\partial T}{\lambda \partial \alpha } = - \frac{2\tan \alpha }{{\lambda^{4} \cos^{2} \alpha }}\left( {1 + \frac{{3\kappa^{2} H^{2} }}{{\lambda^{4} \cos^{4} \alpha }}} \right),$$

$$\frac{{K_{M} }}{{\mu BH^{3} }} \equiv \frac{1}{{\mu BH^{3} }}\frac{\partial M}{\partial \kappa } = \frac{1}{{3\lambda^{6} \cos^{6} \alpha }},$$

$$\frac{{K_{TM} }}{{\mu BH^{3} }} \equiv \frac{1}{{\mu BH^{3} }}\frac{\partial T}{\partial \kappa } = - \frac{2\kappa }{{\lambda^{7} \cos^{6} \alpha }},$$

$$\frac{{K_{SM} }}{{\mu BH^{3} }} \equiv \frac{1}{{\mu BH^{3} }}\frac{\partial S}{\partial \kappa } = \frac{2\kappa \sin \alpha }{{\lambda^{7} \cos^{7} \alpha }},$$

where $K_{T}$, $K_{S}$ and $K_{M}$ are the stretching stiffness, the shear stiffness and the bending stiffness, and $K_{TS}$, $K_{TM}$ and $K_{SM}$ are the stretching-shear coupling stiffness, the stretching-bending coupling stiffness and the shear-bending coupling stiffness. These formulae clearly show the effect of the mid-plane stretch on the magnitude of each stiffness.