page_title: Energy release rate (fracture mechanics)
section_title: 8-node isoparametric element
hierarchical_section_title: Energy release rate (fracture mechanics) / Computational methods in fracture mechanics / 2-D crack tip singular elements / 8-node isoparametric element
caption_attribution_description: English: fdgsdfgs
context_page_description: The energy release rate, is the rate at which energy is transformed as a material undergoes fracture and has units of energy-per-unit-area. Mathematically, the energy release rate is expressed as the decrease in total potential energy per increase in fracture surface area. Various energy balances can be constructed relating the energy released during fracture to the energy of the resulting new surface, as well as other dissipative processes such as plasticity and heat generation. The energy release rate is central to the field of fracture mechanics when solving problems and estimating material properties related to fracture and fatigue.
context_section_description: The 8-node quadratic element are described by Figure 5 in both parent space with local coordinates  and  and by the mapped element in physical/global space by  and  The parent element is mapped from the local space to the physical space by the shape functions  and the degree of freedom coordinates  The crack tip is located at  or  
  In a similar way, displacements (defined as ) can also be mapped.
  A property of shape functions in the finite element method is compact support, specifically the Kronecker delta property (i.e.  at node  and zero at all other nodes). This results in the following shape functions for the 8-node quadratic elements:
  When considering a line in front of the crack that is co-linear with the - axis (i.e. )  all basis functions are zero except for 
  Calculating the normal strain involves using the chain rule to take the derivative of displacement with respect to 
  
  If the nodes are spaced evenly on the rectangular element then the strain will not contain the singularity. By moving nodes 5 and 8 position to a quarter of the length   of the element closer to the crack tip as seen in figure 5, the mapping from  becomes:
  Solving for  and taking the derivative results in:
  Plugging this result into the equation for strain the final result is obtained:
  By moving the mid-nodes to a quarter position results in the correct  crack tip singularity.
caption: Figure 5: Mapped and Parent Element of the 8-node isoparametric finite element. Note the quarter point location of the nodes in the mapped element.
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