A New and Efficient Approach to Determining the Rate of the Second-Order Symmetric Tensor and Its Isotropic Function
Source abstract
Abstract
When the rate of a symmetric second-order symmetric tensor is discussed, the spin of the principal axis is involved. This paper proposes a method to establish the basis-free expression of the spin in terms of tensor and its rate by making use of the tensor function representation theorem. The proposed method is simple and the expression of the spin established is compact. To obtain the rate of the isotropic function of a second-order symmetric tensor, the fourth-order tangent tensor needs to be derived, which is the derivative of the tensor function to the second-order tensor. By decomposing the second-order symmetric tensor space into two orthogonal subspaces, the closed-form fourth-order tangent tensor is decomposed into two parts, which are linear mappings in these two orthogonal subspaces, respectively. These two linear mappings are derived in an extremely simple way. Finally, the method proposed in this paper is applied to obtain the expression of the relationship between material logarithmic strain rate and deformation rate. The whole process is simple and avoids tedious operations.
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Research context
OpenAlex topics: Elasticity and Material Modeling · Mechanical Engineering and Vibrations Research · Vibration Control and Rheological Fluids
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References (24)
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