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4.7 Multigeneration Solids
This type of material  implements a mechanism for multigenerational interstitial growth of solids whereby each growth generation has a distinct reference configuration determined at the time of its deposition. Therefore, the solid matrix of a growing material consists of a multiplicity of intermingled porous bodies, each representing a generation , all of which are constrained to move together in the current configuration . The deformation gradient of each generation is . The first generation ( is assumed to be present at time , therefore its reference configuration is and its deformation gradient is equivalent to . Each generation's reference configuration has a one-to-one mapping with the master reference configuration , which is that of the first generation. This mapping is postulated based on a constitutive assumption with regard to that generation's state of stress at the time of its deposition. In the current implementation, the newly deposited generation is assumed to be in a stress-free state, even though the underlying material is in a loaded configuration. Therefore, the mapping between generation and the first generation is simply . In other words, when generation first comes into existence, its reference configuration is the current configuration at time . Note that is a time-invariant (though not necessarily homogeneous) quantity that is determined uniquely at the birth of a generation.
The state of stress in a multigeneration solid is given by where is the state of stress in the generation , as would be evaluated from a strain energy density function whose reference configuration is . In the above equation, and the factor ensures that the strain energy density of each generation is properly normalized the volume of the material in the master reference configuration , when summing up the stresses in all the generations.
Multigeneration solids typically exhibit residual stresses when is inhomogeneous.