A Method of Analysis for Compressible Viscoelastic Solids

1972 ◽  
Vol 45 (6) ◽  
pp. 1669-1679
Author(s):  
E. C. Ting
Keyword(s):  
Stress Analysis ◽  
Simple Procedure ◽  
Viscoelastic Solids ◽  
Incompressible Materials ◽  
Incompressible Solids ◽  
Viscoelastic Characteristics ◽  
Compressible Materials

Abstract Most of the solutions in the development of methods of viscoelastic stress analysis have dealt with incompressible materials or materials with restrictions in dilatation. It is influenced in part by the increasing complexity due to the additional operator which represents the viscoelastic characteristics in dilatation. A simple procedure of solution is suggested in this paper which shows that a certain class of problems for compressible materials can be solved with the similar simplicity as the analysis of corresponding incompressible solids. Examples are given for problems with spherical boundaries.

On the Linear Constitutive Equations of Transversely Isotropic Incompressible Elastic Materials

1972 ◽  
Vol 45 (4) ◽  
pp. 1104-1110
Author(s):  
H. Demiray ◽  
M. Levinson
Keyword(s):  
Stress Concentration ◽  
Stress Analysis ◽  
Circular Hole ◽  
Elastic Material ◽  
Constitutive Equations ◽  
Transversely Isotropic ◽  
Elastic Materials ◽  
Rubber Composites ◽  
Limiting Case ◽  
Compressible Materials

Abstract The linear constitutive equations of a transversely isotropic, incompressible, elastic material are derived in this paper as a limiting case of the constitutive equations for the corresponding compressible materials. These equations should be appropriate for the stress analysis of products fabricated from certain laminated, reinforced rubber composites. Some details of a problem concerned with the stress concentration around a circular hole are also given.

Generalized Plane Strain of Pressurized Orthotropic Tubes

1965 ◽  
Vol 87 (3) ◽  
pp. 337-343 ◽  
Author(s):  
Bernard W. Shaffer
Keyword(s):  
Plane Strain ◽  
The Other ◽  
Cylindrical Coordinates ◽  
Incompressible Materials ◽  
Hooke's Law ◽  
Poisson's Ratios ◽  
Poisson’S Ratios ◽  
Moduli Of Elasticity ◽  
Compressible Materials ◽  
Generalized Plane Strain

The generalized Hooke’s law in cylindrical coordinates is presented in terms of directional moduli of elasticity and Poisson’s ratios. It is used in deriving a solution for long pressurized orthotropic tubes with closed ends. Two sets of equations are found; one set is applicable to the study of compressible materials and the other to incompressible materials. The distinction is also described in terms of the moduli of elasticity and Poisson’s ratios.

scholarly journals Locking-Proof Tetrahedra

2021 ◽  
Vol 40 (2) ◽  
pp. 1-17
Author(s):  
Mihai Frâncu ◽  
Arni Asgeirsson ◽  
Kenny Erleben ◽  
Mads J. L. Rønnow
Keyword(s):  
Poisson Ratio ◽  
Constitutive Law ◽  
Numerical Accuracy ◽  
Good Convergence ◽  
Mixed Fem ◽  
Incompressible Materials ◽  
Volume Preservation ◽  
Incompressible Solids ◽  
Standard Finite Element Method ◽  
Full Volume

The simulation of incompressible materials suffers from locking when using the standard finite element method (FEM) and coarse linear tetrahedral meshes. Locking increases as the Poisson ratio gets close to 0.5 and often lower Poisson ratio values are used to reduce locking, affecting volume preservation. We propose a novel mixed FEM approach to simulating incompressible solids that alleviates the locking problem for tetrahedra. Our method uses linear shape functions for both displacements and pressure, and adds one scalar per node. It can accommodate nonlinear isotropic materials described by a Young’s modulus and any Poisson ratio value by enforcing a volumetric constitutive law. The most realistic such material is Neo-Hookean, and we focus on adapting it to our method. For , we can obtain full volume preservation up to any desired numerical accuracy. We show that standard Neo-Hookean simulations using tetrahedra are often locking, which, in turn, affects accuracy. We show that our method gives better results and that our Newton solver is more robust. As an alternative, we propose a dual ascent solver that is simple and has a good convergence rate. We validate these results using numerical experiments and quantitative analysis.

scholarly journals Large elastic deformations of isotropic materials IV. further developments of the general theory

1948 ◽  
Vol 241 (835) ◽  
pp. 379-397 ◽  
Keyword(s):  
Energy Function ◽  
Equations Of Motion ◽  
Incompressible Material ◽  
Surface Forces ◽  
Stored Energy ◽  
Elastic Deformations ◽  
Incompressible Materials ◽  
Scalar Invariants ◽  
Compressible Materials ◽  
Motion Boundary

The equations of motion, boundary conditions and stress-strain relations for a highly elastic material can be expressed in terms of the stored-energy function. This has been done in part I of this series (Rivlin 1948 a ), for both the cases of compressible and incompressible materials, following the methods given by E. & F. Cosserat for compressible materials. The stored-energy function may be defined for a particular material in terms of the invariants of strain. The form in which the equations of motion, etc., are deduced, in the previous paper, does not permit the evaluation of the forces necessary to produce a specified deformation unless the actual expression for the stored-energy function in terms of the scalar invariants of the strain is introduced. In the present paper, the equations are transformed into forms more suitable for carrying out such an explicit evaluation. As examples, the surface forces necessary to produce simple shear in a cuboid of either compressible or incompressible material and those required to produce simple torsion in a right-circular cylinder of incompressible material are derived.

Stress Analysis for a Nonlinear Viscoelastic Cylinder With Ablating Inner Surface

1970 ◽  
Vol 37 (1) ◽  
pp. 44-47 ◽  
Author(s):  
E. C. Ting
Keyword(s):  
Stress Analysis ◽  
Solid Propellant ◽  
Firing Condition ◽  
Real Situation ◽  
Nonlinear Viscoelastic ◽  
Incompressible Materials ◽  
Inner Surface ◽  
One Step ◽  
Technical Interest ◽  
Short Time

An approximate constitutive equation for nonlinear viscoelastic incompressible materials under small finite deformation and for short time ranges has been derived by Huang and Lee [1]. The resulting equation is applied to solve the problem of a pressurized viscoelastic hollow cylinder bonded to an elastic easing. This problem is of notable technical interest in solid propellant stress analysis, since it is a close model to represent a cylindrical propellant grain in a solid fuel rocket under firing condition. The method used by Huang and Lee is appropriate for numerical calculations when the boundaries of the cylinder are nonablating. To consider one step closer to the real situation, the inner surface is often assumed to be ablating and hence time-dependent. It is then the purpose of the present paper to extend the analysis to a cylinder with moving inner surface.

Analogie zwischen elastischem und plastischem Potential anisotroper Stoffe / Analogy between the Elastic and Plastic Potential of Anisotropic Materials

1977 ◽  
Vol 32 (5) ◽  
pp. 432-436 ◽  
Author(s):  
Josef Betten
Keyword(s):  
Linear Theory ◽  
Physical Interpretation ◽  
Elastic Strain ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Complementary Energy ◽  
Linear Combinations ◽  
Plastic Potential ◽  
Incompressible Materials ◽  
Compressible Materials

Abstract The elastic strain energy of distortion and its complementary energy of incompressible materials can be represented formally by linear combinations of partial plastic potentials of the forms Fρ = (1/ρ)Dijkl...qr σ′ij′σkl ... σqr. Likewise, it is possible to describe the elastic strain energy of deformation and its complementary energy of compressible materials formally by plastic potentials of the forms Fρ = (1/ρ)Dijkl...qr σijσkl ... σqr. In a linear theory, for example, we have F2= ½Dijkl σ′ij σ′kl for incompressible materials and F2 =½Dijkl σ′ij σ′kl for compressible materials.It is necessary to notice, that only a formal analogy exsists between the elastic and plastic po­tential, because the physical interpretation of these potentials is quite different.

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