Relationships for the approximation of transient direct and inverse problems with asymptotic kernels

Many transient problems in mechanics ran be resolved through the use of con volution relationships provided the unit impulse or step kernels are known. Unfortunately, solution of the resulting equations for either the direct o r inverse problem often requires the use of specialized numerical methods. To help overcome this potential shortcoming, approximate rules for direct a nd inverse Laplace transformation were used to modify the step-function bas ed convolution relationships to an algebraically solvable and relatively si mple form. The resulting relationships can be applied as a first-order appr oximation to problems in viscoelasticity and heat flow provided the kernel is of an asymptotic exponential form, materials properties do not vary with temperature or strain, and the underlying excitations are not overly oscil latory in nature. Under these provisions, reasonable agreement was seen bet ween the derived relationships and direct solutions for test case studies i nvolving thermal diffusion of a planar slab and the Voigt/Kelvin viscoelast ic model of a spring and dashpot in parallel. Within the accuracy confines of a non-adaptive inverse analysis utilizing a single response history, the method was also shown to be capable of producing reasonable estimates of t he underlying excitations for both test cases.