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The paper involves the study on the elastic and elasto-plastic stres wave propagation in the1-D and 2-D solid media. The Smoth Particle Hydrodynamics equations governing the elastic and elasto-plastic large deformation dynamic response of solid structures are presented. The proposed additionalstress points are introduced in the formulation to mitigate the tensile instability inherent in the SPHapproach. Both incremental rate approach and leap-frog algorithm for time integration are introduced andthe new solution algorithm is developed and implemented. Two examples on stress wave propagation inaluminium bar and 2-D elasto-plastic steel plate are included. Results from the proposed SPH approachare compared with available analytical values and finite element solutions. The comparison illustrates thatthe stres wave propagation problems can be effectively solved by the proposed SPH method. The studyshows that the SPH simulation is a reliable and robust tool and can be used with confidence to treattransient dynamics such as linear and non-linear transient stress wave propagation problems.