(FEA) is a powerful tool for modeling soft robots. It breaks complex systems into smaller, manageable parts, allowing engineers to simulate and analyze the behavior of soft materials under various conditions.
In soft robotics, FEA helps optimize designs, predict performance, and solve challenges unique to flexible structures. From material modeling to contact simulation, FEA provides valuable insights for creating more efficient and effective soft robotic systems.
Basics of finite element analysis
Finite element analysis (FEA) is a numerical method for solving complex engineering problems by dividing the problem domain into smaller, simpler parts called finite elements
FEA is particularly useful in soft robotics due to the complex geometries, nonlinear material properties, and large deformations encountered in soft robotic systems
The method involves discretizing the problem domain, defining material properties and , solving the governing equations, and the results
Discretization in finite element analysis
Nodes and elements
Top images from around the web for Nodes and elements
Frontiers | Application strategy of finite element analysis in artificial knee arthroplasty View original
Is this image relevant?
Frontiers | Dynamic Morphological Computation Through Damping Design of Soft Continuum Robots ... View original
Is this image relevant?
Frontiers | Integrating Soft Robotics with the Robot Operating System: A Hybrid Pick and Place ... View original
Is this image relevant?
Frontiers | Application strategy of finite element analysis in artificial knee arthroplasty View original
Is this image relevant?
Frontiers | Dynamic Morphological Computation Through Damping Design of Soft Continuum Robots ... View original
Is this image relevant?
1 of 3
Top images from around the web for Nodes and elements
Frontiers | Application strategy of finite element analysis in artificial knee arthroplasty View original
Is this image relevant?
Frontiers | Dynamic Morphological Computation Through Damping Design of Soft Continuum Robots ... View original
Is this image relevant?
Frontiers | Integrating Soft Robotics with the Robot Operating System: A Hybrid Pick and Place ... View original
Is this image relevant?
Frontiers | Application strategy of finite element analysis in artificial knee arthroplasty View original
Is this image relevant?
Frontiers | Dynamic Morphological Computation Through Damping Design of Soft Continuum Robots ... View original
Is this image relevant?
1 of 3
Discretization involves dividing the problem domain into smaller, simpler parts called finite elements
Nodes are points in the domain where the degrees of freedom (displacements, rotations, etc.) are defined
Elements are the basic building blocks of the discretized domain and are defined by connecting nodes
Common element types in soft robotics include tetrahedra, hexahedra, and shell elements
Mesh generation techniques
is the process of creating a finite element mesh from the geometry of the problem domain
Structured meshes have a regular pattern and are easier to generate but may not conform well to complex geometries
Unstructured meshes have an irregular pattern and can conform better to complex geometries but are more difficult to generate
Adaptive meshing techniques can refine the mesh in regions of high stress or strain gradients to improve accuracy
Material properties for soft robots
Hyperelastic material models
Hyperelastic materials exhibit nonlinear stress-strain behavior and are commonly used in soft robotics (silicone rubber, elastomers)
describe the nonlinear relationship between stress and strain (Neo-Hookean, Mooney-Rivlin, Ogden)
Material parameters for these models are typically obtained from experimental testing (uniaxial tension, biaxial tension, shear)
Viscoelastic material models
Viscoelastic materials exhibit time-dependent behavior, with stress depending on both strain and strain rate
capture the time-dependent response of soft materials (Prony series, fractional derivative models)
Viscoelastic effects can be important in soft robotics applications involving cyclic loading or prolonged deformation
Boundary conditions and loads
Displacement boundary conditions
specify the displacements or rotations at specific nodes in the finite element model
Essential boundary conditions are imposed directly on the degrees of freedom (fixed , symmetry conditions)
are imposed through the weak form of the governing equations (contact conditions, free surfaces)
Force and pressure loads
Force and represent the external forces acting on the soft robotic system
Point loads are concentrated forces applied at specific nodes in the model
Distributed loads are forces or pressures applied over a surface or volume of the model (pressure in a soft actuator, gravity)
Time-varying loads can be used to simulate dynamic loading conditions (impact, cyclic loading)
Finite element formulation
Weak form of governing equations
The weak form is an integral statement of the governing equations that allows for the incorporation of natural boundary conditions
The principle of virtual work is used to derive the weak form by multiplying the strong form equations by a virtual displacement and integrating over the domain
The weak form reduces the continuity requirements on the solution and allows for the use of piecewise polynomial approximations
Shape functions and interpolation
are used to interpolate the solution variables (displacements, pressures) within each element
Lagrange polynomials are commonly used shape functions that satisfy the nodal property
Higher-order shape functions can improve accuracy but increase computational cost
Element stiffness matrix assembly
The element relates the nodal displacements to the nodal forces for a single element
The element stiffness matrix is obtained by evaluating the weak form integral over the element domain using the shape functions
The global stiffness matrix is assembled by summing the contributions from all elements in the mesh
The global system of equations is solved to obtain the nodal displacements
Nonlinear finite element analysis
Sources of nonlinearity in soft robotics
Geometric nonlinearity arises from large deformations and rotations, which are common in soft robotic systems
Material nonlinearity occurs due to the nonlinear stress-strain behavior of hyperelastic and viscoelastic materials
Contact nonlinearity results from the changing contact conditions between soft robot components and the environment
Newton-Raphson method
The is an iterative technique for solving nonlinear systems of equations
The method involves linearizing the nonlinear equations about the current solution and solving for the incremental displacements
The solution is updated iteratively until convergence is achieved based on a specified tolerance
Arc-length methods
are used to trace the nonlinear load-displacement path and overcome limit points and snap-through behavior
The method involves controlling the incremental load factor and the incremental displacement norm simultaneously
Arc-length methods are particularly useful for modeling the large deformations and instabilities encountered in soft robotics
Contact modeling for soft interfaces
Contact detection algorithms
are used to identify the regions of contact between soft robot components and the environment
Node-to-surface and surface-to-surface contact formulations are commonly used in soft robotics
Efficient contact detection is crucial for modeling the complex interactions between soft robots and their surroundings
Penalty vs Lagrange multiplier methods
enforce contact constraints by adding a penalty term to the weak form, which introduces a small penetration between contact surfaces
enforce contact constraints exactly by introducing additional unknowns (Lagrange multipliers) to the system of equations
Penalty methods are simpler to implement but may suffer from contact penetration, while Lagrange multiplier methods are more accurate but computationally expensive
Finite element software for soft robotics
Commercial vs open-source software
Commercial finite element software packages (, , COMSOL) offer a wide range of features and technical support but can be expensive
Open-source finite element software (FEniCS, deal.II, FreeFEM) provide flexibility and customization options but may have a steeper learning curve
The choice of software depends on the specific requirements of the soft robotics application and the available resources
Pre-processing and post-processing tools
tools are used to create the finite element model, including geometry creation, mesh generation, and material property assignment (ANSYS SpaceClaim, ABAQUS CAE, Gmsh)
Post-processing tools are used to visualize and analyze the results of the finite element simulation (ParaView, EnSight, Tecplot)
Efficient pre- and post-processing workflows are essential for the rapid design and optimization of soft robotic systems
Verification and validation
Mesh convergence studies
are used to assess the accuracy and reliability of the finite element solution
The mesh is refined systematically, and the solution is compared across different mesh resolutions
Convergence is achieved when further mesh refinement does not significantly change the solution
Comparison with analytical solutions
Analytical solutions provide a benchmark for verifying the accuracy of the finite element implementation
Simple test cases with known analytical solutions (uniaxial tension, pure bending) can be used to verify the correctness of the finite element formulation and material models
helps build confidence in the finite element model before applying it to more complex soft robotics problems
Experimental validation techniques
involves comparing the finite element predictions with experimental measurements on physical soft robotic systems
Common validation techniques include displacement and strain field measurements using digital image correlation (DIC), force-displacement measurements using load cells, and pressure measurements using pressure sensors
Experimental validation is crucial for assessing the predictive capabilities of the finite element model and identifying areas for improvement
Applications of FEA in soft robotics
Design optimization of soft actuators
FEA can be used to optimize the design of soft actuators by exploring different geometries, materials, and actuation strategies
Objective functions for optimization can include maximizing force output, minimizing energy consumption, or achieving a desired deformation profile
Optimization techniques such as topology optimization and parameter optimization can be coupled with FEA to identify optimal designs
Modeling of soft grippers and manipulators
FEA can simulate the grasping and manipulation performance of soft grippers and manipulators
Contact modeling is essential for predicting the interaction forces between the soft gripper and the grasped object
FEA can help optimize the gripper design for specific tasks (delicate grasping, conformable grasping) and environments
Simulation of soft robot locomotion
FEA can be used to model the locomotion of soft robots, such as crawling, walking, or swimming robots
The interaction between the soft robot and the environment (ground, water) can be modeled using contact algorithms and techniques
FEA can help optimize the locomotion performance by exploring different gait patterns, body morphologies, and control strategies
Soft robots often interact with fluids, such as in underwater applications or pneumatic actuation
Fluid-structure interaction (FSI) modeling involves coupling the finite element model of the soft robot with a computational fluid dynamics (CFD) model of the surrounding fluid
FSI modeling can capture the complex interactions between the soft robot and the fluid, such as the deformation of the robot due to fluid forces and the effect of the robot's motion on the fluid flow
Reduced-order modeling techniques
aim to reduce the computational cost of FEA by reducing the number of degrees of freedom in the model
Proper Orthogonal Decomposition (POD) and Reduced Basis (RB) methods are commonly used to construct reduced-order models from high-fidelity FEA simulations
Reduced-order models can enable real-time simulation and control of soft robots by providing fast and accurate approximations of the full-order model
Uncertainty quantification and sensitivity analysis
Soft robotic systems are subject to various sources of uncertainty, such as material property variations, manufacturing imperfections, and sensor noise
(UQ) techniques, such as Monte Carlo methods and polynomial chaos expansions, can be used to propagate uncertainties through the FEA model and quantify their impact on the robot's performance
can identify the most influential input parameters on the robot's behavior, guiding design decisions and control strategies
UQ and sensitivity analysis can help design soft robots that are robust to uncertainties and can operate reliably in real-world conditions
Key Terms to Review (48)
Abaqus: Abaqus is a software suite for finite element analysis and computer-aided engineering, widely used for simulating the behavior of materials and structures under various conditions. It provides powerful tools for modeling, analyzing, and visualizing complex mechanical systems, making it an essential resource in both academic research and industrial applications. Abaqus is particularly valued for its ability to handle non-linear problems and multi-physics simulations, which are critical in fields such as soft robotics.
Ansys: Ansys is a powerful software suite used for engineering simulation that enables users to perform finite element analysis, computational fluid dynamics, and multiphysics modeling. It allows engineers to analyze complex systems and predict their behavior under various conditions, integrating multiple physical phenomena into a single model. This capability is essential for optimizing designs, improving product performance, and reducing the need for physical prototypes.
Arc-length methods: Arc-length methods are numerical techniques used in finite element analysis to trace the equilibrium paths of structures under various loading conditions. These methods allow for the determination of load-displacement relationships in systems that exhibit nonlinear behavior, especially during the transition between stable and unstable configurations. By following the arc-length path, engineers can better predict buckling and other critical responses in structures, making them essential for analyzing complex problems.
Boundary Conditions: Boundary conditions are constraints that define how a physical system behaves at its boundaries. They are essential in modeling and analyzing the behavior of materials and structures, particularly in continuum mechanics and finite element analysis, as they influence the overall response of the system under various loading scenarios. Correctly defining these conditions is crucial for obtaining accurate solutions in simulations and understanding how the system interacts with its environment.
Commercial Software: Commercial software refers to software that is developed for sale or for a specific commercial purpose, usually with the intent of generating profit. This type of software is typically produced by companies and is often subject to licensing agreements that restrict how the software can be used and distributed, ensuring that developers receive compensation for their work.
Comparison with analytical solutions: Comparison with analytical solutions involves evaluating numerical results obtained through computational methods against exact solutions derived from mathematical analysis. This process helps validate the accuracy and reliability of numerical models by identifying discrepancies and confirming that the simulations are effectively representing the physical behavior of the system being studied.
Contact detection algorithms: Contact detection algorithms are computational methods used to determine when two or more bodies in a simulation come into contact or interact with each other. These algorithms are essential for accurately modeling the physical behaviors of soft robotics, as they help predict and manage interactions between deformable materials and rigid objects, enabling realistic simulations and applications.
Convergence testing: Convergence testing is a method used in numerical analysis to determine whether a given sequence or series approaches a specific value or limit as more terms are added. It is essential in ensuring that a numerical solution accurately represents the underlying mathematical model, especially when using techniques like finite element analysis. The importance of convergence testing lies in validating the reliability and accuracy of computational results.
Design optimization of soft actuators: Design optimization of soft actuators refers to the process of improving the performance, efficiency, and functionality of flexible, adaptable mechanisms used to produce motion or force in soft robotics. This involves adjusting various parameters such as shape, material properties, and actuation methods to achieve desired behaviors while minimizing resource consumption and maximizing effectiveness. Key aspects include understanding the mechanical behavior of materials, predicting actuator performance, and utilizing computational tools for analysis.
Displacement: Displacement refers to the change in position of an object or point from its original location to a new location. In engineering and finite element analysis, displacement is crucial as it describes how a structure deforms under load, helping engineers understand stress distributions and material behavior under various conditions.
Displacement boundary conditions: Displacement boundary conditions refer to the constraints applied in finite element analysis (FEA) that specify the allowable displacements of a structure or component at its boundaries. These conditions are crucial for simulating how structures will respond under various loading scenarios, ensuring that the analysis accurately represents real-world behaviors. By setting specific displacements at boundaries, engineers can control the movement and deformation of structures, which directly impacts the results of stress, strain, and overall structural integrity assessments.
Dynamic analysis: Dynamic analysis refers to the study of how structures or systems behave under time-dependent loads and conditions, focusing on the response of materials and systems to dynamic forces like vibrations, impact, and motion. This type of analysis is crucial for understanding the behavior of soft robotic structures when they are subjected to varying operational conditions, ensuring their safety, stability, and performance in real-world applications.
Element stiffness matrix assembly: Element stiffness matrix assembly is the process of combining individual stiffness matrices of finite elements into a global stiffness matrix for a structure or system in finite element analysis. This process is crucial as it enables the overall behavior of a complex structure to be analyzed by breaking it down into simpler, manageable elements, each represented by their own stiffness characteristics. The assembly ensures that interactions between elements are properly accounted for, allowing for accurate calculations of deformations and responses under various loading conditions.
Element type: Element type refers to the classification of finite elements in finite element analysis, determining how the element behaves and interacts with its surrounding environment. Different element types are used for various applications, and they can be linear or nonlinear, 1D, 2D, or 3D, each with specific properties that affect the analysis results. Choosing the appropriate element type is crucial for accurate modeling and simulation in engineering problems.
Experimental validation: Experimental validation is the process of confirming that a model or simulation accurately reflects real-world behavior through empirical testing. This process is crucial in ensuring that theoretical predictions made by simulations align with observable data, thus providing confidence in the model's accuracy and applicability. Successful experimental validation can involve adjusting model parameters, refining assumptions, and ultimately enhancing the fidelity of computational tools.
Experimental validation techniques: Experimental validation techniques refer to methods and processes used to confirm the accuracy, reliability, and effectiveness of models and simulations through real-world experimentation. These techniques ensure that theoretical models align with physical behaviors and characteristics, providing essential feedback for design and development.
Finite Element Analysis: Finite Element Analysis (FEA) is a numerical method used to solve complex engineering problems by breaking down structures into smaller, manageable pieces called finite elements. This technique allows engineers and researchers to assess the mechanical behavior of materials under various conditions, including stress, strain, and temperature changes, which is crucial in understanding how materials will perform in real-world applications. FEA connects to essential concepts like mechanical properties, the behavior of materials at a continuum level, dynamics specific to soft robotics, the interactions in multiphysics systems, and innovative applications such as drug delivery systems.
Fluid-structure interaction: Fluid-structure interaction (FSI) refers to the complex interplay between fluid dynamics and solid mechanics, where the motion of a fluid influences the behavior of a solid structure, and vice versa. This interaction is critical in understanding how soft materials deform under fluid forces, enabling the design and analysis of various applications, especially in soft robotics, where flexibility and adaptability are key. By modeling FSI, engineers can create more effective simulations that combine both fluid and structural behaviors.
Force Loads: Force loads refer to the external forces acting on a structure or component that can induce stress, strain, and deformation. Understanding these loads is crucial in the design and analysis of materials and structures to ensure they can withstand applied forces without failure.
Hyperelastic material models: Hyperelastic material models are mathematical representations used to describe the behavior of elastomeric materials under large deformations. These models are crucial for accurately simulating the mechanical response of materials in various conditions, making them essential in applications such as soft robotics, where flexibility and elasticity play vital roles. By utilizing these models, one can predict how materials will behave under different forces and environmental conditions, facilitating better design and analysis in engineering.
Interpolation: Interpolation is a mathematical method used to estimate unknown values that fall within the range of known data points. It involves constructing new data points within the bounds of a discrete set of known values, allowing for smoother transitions and better predictions in models. This technique is particularly valuable in numerical analysis, where accurate estimations are critical for simulations and visualizations.
Lagrange Multiplier Methods: Lagrange multiplier methods are mathematical techniques used to find the local maxima and minima of a function subject to equality constraints. This approach allows for the optimization of a function while adhering to certain conditions, making it especially useful in fields like engineering, physics, and economics. By introducing additional variables (the Lagrange multipliers), these methods transform constrained problems into simpler unconstrained problems.
Linear Static Analysis: Linear static analysis is a method used to determine the response of structures under static loads while assuming that material behavior is linear and elastic. This analysis helps in predicting how a structure will behave when subjected to forces like weight or pressure without considering dynamic effects or changes over time.
Mesh convergence studies: Mesh convergence studies are analytical procedures used in finite element analysis to determine the effect of mesh size on the accuracy of numerical results. By systematically refining the mesh, or grid, used in simulations, engineers can assess how changes in element size impact the solution's precision and stability. This process helps ensure that the results are reliable and not artifacts of an inadequate mesh.
Mesh generation: Mesh generation is the process of creating a discrete representation of a geometric object, dividing it into smaller elements for numerical analysis. This technique is crucial in finite element analysis, as it allows complex shapes to be approximated and analyzed efficiently. By transforming continuous geometries into a mesh of elements, engineers can apply mathematical models to predict behavior under various conditions, making mesh generation a fundamental step in simulations.
Modeling of soft grippers and manipulators: Modeling of soft grippers and manipulators involves creating mathematical and computational representations that simulate the behavior, performance, and interaction of these flexible robotic systems. This process is crucial for understanding how these devices operate under various conditions, enabling improvements in design and functionality. Accurate modeling also assists in predicting how the soft structures will respond to forces, which is essential for effective control and operation.
Multiphysics coupling: Multiphysics coupling refers to the integration of different physical phenomena within a single computational framework, allowing for the interaction between multiple physical fields, such as fluid dynamics, heat transfer, and structural mechanics. This approach enables a more accurate representation of complex systems, where changes in one domain can significantly affect the others. It is essential for modeling real-world applications where several physical processes occur simultaneously.
Natural Boundary Conditions: Natural boundary conditions refer to the constraints applied to a finite element analysis problem that account for the physical behavior of the system at its boundaries without imposing specific values. These conditions help in accurately representing how a material interacts with its environment by allowing for the natural response of the system, such as stress or strain, to dictate the behavior at the boundaries.
Newton-Raphson Method: The Newton-Raphson method is an iterative numerical technique used to find approximate solutions to real-valued equations, particularly for finding roots of nonlinear equations. It is especially useful in finite element analysis for solving nonlinear systems of equations, providing fast convergence when initial guesses are close to the actual root. This method leverages the derivative of the function to iteratively refine estimates until the desired accuracy is achieved.
Nonlinear finite element analysis: Nonlinear finite element analysis is a computational technique used to evaluate the behavior of structures and materials that exhibit nonlinear responses under applied loads. Unlike linear analysis, which assumes proportionality between load and response, nonlinear analysis accounts for factors such as material plasticity, large deformations, and boundary conditions that change during loading. This method is crucial for accurately predicting how structures will perform in real-world scenarios, especially in fields like soft robotics where materials often behave in a nonlinear manner.
Open-source software: Open-source software refers to computer software whose source code is made available to the public for modification and enhancement. This means that anyone can inspect, modify, and distribute the software, promoting collaborative development and transparency. Open-source software fosters innovation and allows developers to build upon each other's work, making it a powerful tool for both individual programmers and organizations.
Penalty Methods: Penalty methods are techniques used in numerical analysis and optimization that incorporate penalty functions into the formulation of a problem to enforce constraints. These methods adjust the objective function by adding a penalty term, which discourages violations of constraints while allowing for the exploration of feasible solutions. This approach is especially useful in finite element analysis, where it helps to manage constraints like boundary conditions or material limits effectively.
Post-processing: Post-processing refers to the steps taken after the initial data collection or simulation to analyze, refine, and visualize the results. This stage is crucial as it transforms raw data into meaningful insights, enabling better understanding and interpretation of the results achieved from computational models.
Pre-processing: Pre-processing refers to the set of operations applied to raw data before it is used for analysis or modeling. This stage is crucial as it involves cleaning, transforming, and organizing data to ensure accuracy and enhance the quality of results in numerical simulations like finite element analysis.
Pressure Loads: Pressure loads refer to forces exerted on a structure or material due to fluid pressure acting over a specific area. Understanding pressure loads is crucial in assessing the behavior and performance of materials and structures when they are subjected to various fluid environments, especially in simulations such as finite element analysis, where these loads influence stress distribution and deformation.
Reduced-order modeling techniques: Reduced-order modeling techniques are methods used to simplify complex mathematical models by reducing the number of variables and equations involved while maintaining the essential features of the original system. These techniques aim to capture the dominant behavior of a system in a more computationally efficient way, making it easier to analyze and simulate scenarios in engineering and scientific applications.
Sensitivity Analysis: Sensitivity analysis is a method used to determine how the variation in the output of a model can be attributed to different variations in its inputs. It helps identify which variables have the most influence on outcomes, aiding in the decision-making process and improving the understanding of system behavior under uncertainty.
Shape Functions: Shape functions are mathematical functions used in finite element analysis to interpolate the values of a field variable within an element based on its nodal values. They are crucial for approximating the behavior of structures or materials by transforming the complex geometry into simpler, solvable forms. Shape functions enable the representation of displacement, temperature, and other physical quantities across an element, facilitating the numerical solution of engineering problems.
Shell element: A shell element is a type of finite element used in structural analysis that represents thin structures, like plates or shells, by simplifying the three-dimensional behavior into two dimensions. This element is especially useful in finite element analysis as it captures bending and membrane actions, making it ideal for modeling complex geometries and load conditions while minimizing computational effort.
Simulation of soft robot locomotion: Simulation of soft robot locomotion refers to the computational modeling and analysis used to predict and understand the movement patterns of soft robots in various environments. It involves simulating the mechanical behavior, deformation, and interactions of soft materials to optimize designs and enhance performance in tasks like crawling, swimming, or climbing.
Stiffness matrix: The stiffness matrix is a mathematical representation used in finite element analysis that relates the displacements of a structure to the forces applied to it. It captures how a system responds to external loads, where each entry in the matrix indicates how much force is needed to produce a given displacement in a particular direction. This matrix plays a crucial role in determining the behavior of elastic structures under various loading conditions.
Strain Energy Density: Strain energy density is a measure of the amount of elastic energy stored per unit volume of a material when it is deformed. It helps in understanding how materials respond to external forces, especially in nonlinear elastic regimes, which is crucial for analyzing complex structures and materials during deformation.
Stress analysis: Stress analysis is the process of determining the stresses and strains in materials or structures under external forces. This method helps in evaluating how different forces affect an object's performance, stability, and safety. By examining these stresses, engineers can predict potential failures and ensure designs meet necessary performance criteria.
Thermal analysis: Thermal analysis is a technique used to study the thermal properties of materials by monitoring their physical and chemical changes as a function of temperature. This method helps in understanding how materials respond to heat, which is crucial for applications involving temperature variations, such as soft robotics. By analyzing these responses, researchers can gain insights into material stability, phase transitions, and other important behaviors under thermal stress.
Truss Element: A truss element is a structural component that is primarily designed to resist axial forces, acting either in tension or compression. Trusses are composed of interconnected members and are often used in various engineering applications due to their lightweight yet strong characteristics, making them efficient for load-bearing structures. The behavior of truss elements can be analyzed using methods such as finite element analysis, which helps in predicting how these structures will perform under different loads and conditions.
Uncertainty Quantification: Uncertainty quantification is the process of systematically evaluating and analyzing the uncertainties in mathematical models, simulations, and experimental data. This involves identifying the sources of uncertainty, quantifying their impact on model outputs, and providing a measure of confidence in the results. It is essential for making informed decisions based on simulations and is particularly important in engineering fields where safety and performance are critical.
Viscoelastic material models: Viscoelastic material models describe materials that exhibit both viscous and elastic characteristics when undergoing deformation. These models are essential in understanding how materials respond to applied forces over time, showing time-dependent strain under constant stress, which is particularly relevant in fields like soft robotics where materials must endure complex loading conditions.
Weak form of governing equations: The weak form of governing equations is a mathematical formulation used in the context of finite element analysis that allows for the approximation of solutions to differential equations. Instead of requiring solutions to satisfy the equations at every point, the weak form integrates the governing equations against test functions, making it possible to work with less smooth functions and incorporate boundary conditions more flexibly.