Static Analysis : Virtual Work Equation

1. Static Analysis: Virtual Work Equation

2. Objectives Section II – Static Analysis Module 1 - Virtual Work Equation Page 2 The objective of this module is to develop the rate of virtual work equation that forms the basis for finite element methods used in solid mechanics • This equation will be developed using the differential equation for equilibrium written in terms of the Cauchy stress tensor. • It will then be transformed into an integral equation of equilibrium using the principle of virtual work. • The divergence theorem will be used to transform the integral equation into a more usable form. • The rate of virtual work equation will then be presented in terms of work conjugate stress and strain tensors that are based on a known reference configuration.

3. Configurations Section II – Static Analysis Module 1 - Virtual Work Equation Page 3 • Different configurations that a body passes through while deforming under load are important to the development of the equations used in finite element analysis. • All of the configurations that a real body passes through are in equilibrium (i.e. satisfy Newton’s 2nd Law). • However, some of the configurations that are computed by a finite element program are not in equilibrium. • These non-equilibrium configurations are encountered during the process of searching for a configuration that satisfies the equations of equilibrium.

4. Original Configurations Section II – Static Analysis Module 1 - Virtual Work Equation Page 4 y • The original configuration represents the state the body is in prior to the application of any external disturbances. • The original configuration is in equilibrium. x Original Configuration z

5. Reference Configurations Section II – Static Analysis Module 1 - Virtual Work Equation Page 5 y • It is necessary to use configurations having a known surface area and volume during the solution process. • Everything about a reference configuration is known • Stress distributions, • Temperature distributions, • Position, • Surface area and volume. • The original configuration is a reference configuration. x Original Configuration z

6. Deformed Configurations Section II – Static Analysis Module 1 - Virtual Work Equation Page 6 y • As the body is subjected to external disturbances, it moves through various deformed configurations until it reaches its final configuration. • Each of the deformed configurations satisfies the equations of equilibrium to within a convergence tolerance. • A deformed configuration can also be used as a reference configuration. Deformed Configurations x z Original Configuration

7. Desired Configuration Section II – Static Analysis Module 1 - Virtual Work Equation Page 7 y • During the process of computing the response of a body to external disturbances, there arises the need to determine a new deformed configuration. • The shape, position, and stresses in this new configuration are not known, and it is the purpose of the analysis to determine them. Desired configuration Previously determined Deformed Configuration x z

8. Non-equilibrium Configurations Section II – Static Analysis Module 1 - Virtual Work Equation Page 8 • During the process of determining a desired configuration, a numerical algorithm may develop an estimate of the desired configuration that does not satisfy equilibrium. • The numerical algorithm must have the capability to detect when the equations of equilibrium are not satisfied and a method for converging to the correct solution. Computed Configuration y Desired configuration True path Previously determined configuration x z

9. Equilibrium at a Point Section II – Static Analysis Module 1 - Virtual Work Equation Page 9 • The equations of equilibrium are the fundamental equations used by finite element programs. • The equations for equilibrium at a point are: Cartesian Components of Cauchy Stress Tensor Cartesian components of the body force per unit volume

10. Equilibrium at a Point Section II – Static Analysis Module 1 - Virtual Work Equation Page 10 • This equation must be satisfied in the each configuration. • The Cauchy stress components can be thought of as true stress components that are based on the area of a differential element in the desired configuration. • The components are defined with respect to the current configuration base vectors.

11. Rate of Virtual Work at a Point Section II – Static Analysis Module 1 - Virtual Work Equation Page 11 • The rate of virtual work at a point is obtained by multiplying the differential equation of equilibrium by a virtual velocity, • There are no limitations on the virtual velocity except that it satisfy the boundary conditions acting on the body and that it be constant ( ). Cartesian Components of the Virtual Velocity There are no changes in the external loads or internal stresses when a virtual velocity is applied.

12. New Form for Rate of Virtual Work Section II – Static Analysis Module 1 - Virtual Work Equation Page 12 Rate of Virtual Work at a Point Chain Rule for a Product Rearranging Decompose this term using the chain rule for a product Substitution Rearranging New form for the rate of virtual work at a point.

13. Rate of Virtual Work for a Body Section II – Static Analysis Module 1 - Virtual Work Equation Page 13 • The rate of virtual work for a body is obtained by integrating the rate of virtual work at a point over the volume of the body. • The Cauchy stress formula relates the components of a traction vector acting on a surface to the surface normal vector and the components of the Cauchy stress.

14. Rate of Virtual Work for a Body Section II – Static Analysis Module 1 - Virtual Work Equation Page 14 • The Cauchy stress formula and the Divergence Theorem allows the rate of virtual work for a body to be written as: • This is an integral form of the equations of equilibrium for a body. • It is sometimes called the weak form of the equations of equilibrium.

15. Rate of Deformation and Vorticity Section II – Static Analysis Module 1 - Virtual Work Equation Page 15 • The gradient of the virtual velocity is defined by the equation where is the rate of the virtual deformation gradient.

16. Rate of Deformation Section II – Static Analysis Module 1 - Virtual Work Equation Page 16 • The rate of the virtual deformation can be broken into symmetric and skew-symmetric components • The first term is the virtual rate of deformation. It is important to this development. • The second term is the virtual vorticity that is not related to material constitutive equations and is dropped.

17. Virtual Rate of Deformation Section II – Static Analysis Module 1 - Virtual Work Equation Page 17 • The components of the virtual rate of deformation are • These are also the components of the rate of the small deformation strains due to virtual displacements.

18. Updated Equation: Rate of Virtual Work Section II – Static Analysis Module 1 - Virtual Work Equation Page 18 • Combining the rate of virtual work equation with the rate of the virtual deformation equation yields the equation • The problem with this equation is that volume and surface area in the desired configuration are not known. For infinitesimal deformations and rotations, the original configuration volume and surface can be used with minimal error. • When deformations and/or rotations are large, a Lagrangian statement of the rate of virtual work is required.

19. Lagrangian Statement: Rate of Virtual Work Section II – Static Analysis Module 1 - Virtual Work Equation Page 19 • The Cauchy stress tensor and the rate of deformation are work conjugate quantities referred to the current configuration • The 2nd Piola-Kirchhoff and rate of Green’s strain are work conjugate quantities referred to a reference configuration. • Relations exist between the Cauchy and 2nd Piola-Kirchhoff stress tensors and the rate of deformation, and rate of Green’s strain tensor.

20. Lagrangian Statement: Rate of Virtual Work Section II – Static Analysis Module 1 - Virtual Work Equation Page 20 • The rate of virtual work written in terms of the 2nd Piola-Kirchhoff stress tensor, , and the rate of Green’s strain tensor, , is • One of the important differences in the above equation is that all quantities are defined with respect to a reference configuration that has a known volume and surface area. • This enables the integrations to be carried out for problems involving large deformations.

21. Internal and External Rate of Virtual Work Section II – Static Analysis Module 1 - Virtual Work Equation Page 21 • The external surface tractions or internal body forces can be thought of as external causes of the internal stresses in the body. • The Lagrangian statement of the rate of virtual work can be written as • The last equation can alternately be written as • This form of the rate of virtual work equation will be used to develop a basic solution strategy. • Any body that is in equilibrium satisfies this equation. Internal rate of virtual work External rate of virtual work

22. Module Summary Section II – Static Analysis Module 1 - Virtual Work Equation Page 22 • This module has developed an equation for the rate of virtual work. • The differential equation of equilibrium for stresses at a point was used as a starting point. • This equation was converted to an integral equation that relates the virtual work done by internal stresses to the virtual work done by external surface tractions and body forces. • This virtual work equation is the foundation upon which commercial finite element programs used to solve problems in solid mechanics are based. • Module 2 will use the virtual work equation to develop an incremental and iterative solution strategy.