NON-LINEARITY

Simplification of problems to linear approach saves costs and time. On the other hand, nonlinear approach, cost exponentially higher than linear ones due to their complexity. Some industries such as aerospace justify advantages of nonlinear approaches in order to obtain more precision, accuracy and safety.

In incremental nonlinear solution, first of all, constitutive equations are not simplified to linear terms. In another word, when independent variables in an equation are power of two or higher, their effects are taken into consideration, even though the result may not differ greatly from linear approach. Secondly, the effect of changes of geometry, boundary condition and properties are taken into account to update the configuration in time intervals. For example, any change in geometry of structure of a material changes its stiffness coefficient. Consequently, deformation under external force reorganizes the material molecular arrangement resulting change of material properties. In addition, deformation of the object under external force causes change of force orientation or direction. Based on the mentioned concepts, two methods are common in nonlinear approach to take into account the changes of the configuration of the material. The first method compares the deformed configuration with the undeformed initial configuration that is called Total Lagrangian. In the second method, the new configuration is compared to the last updated one that is known as Updated Lagrangian. A very simple representation of a nonlinear approach is demonstrated in figure below. As it is shown, the geometry of the object changes while the force is increasing and as a result, the stiffness of the object which is a function of geometry changes within the force. By comparing a small element on the object with respect to the deformation rate, the shape of the element shows considerable changes. For more information refer to “Method and system for real time characterization of soft materials and biological tissues based on nonlinear properties, Patent No. 10126219”.