Hygrothermoelasticity

The interdependence of heat and moisture in solids has been discussed in [1–3]. It was demonstrated [3] that the effective diffusion constants measured in typical experiments [4] are equal to the diffusion coefficients of the material only in the case of very weak coupling. The processes of heat and moisture transfer can be coupled depending on the specified environmental conditions. An initially uniform dry solid can be suddenly immersed in a body of water of the same temperature which is kept constant. The solid will absorb moisture at a rate which is initially proportional to the square root of time [2, 3, 5]. The slope of the moisture content versus square root of time curve can then be used to compute the effective diffusion coefficient. It depends [3] on the diffusion coefficients for both heat and moisture and on the coupling parameters associated with the heat and moisture flow rate. In addition to this reinterpretation of experimental results of classical tests, there is the related diffusion of heat in the same test. For conditions in which moisture diffuses into the solid, heat has also diffused into the solid at a rate porportional to the square root of time initially resulting in an increase in temperature of the solid [2]. Ultimately, as the diffusion of moisture slows down, the flow of heat is reversed and the temperature of the solid decreases to its initial value. These are similar reciprocal effects for the case in which a slab is suddenly subjected to a change in surface temperature with no change in the surface moisture concentration [2, 3]. Experimental results on coupling of heat and moisture in textile materials can be found in [5].

The nonuniform distribution of moisture and temperature causes differential expansion or contraction from one point to another in a solid. This sets up a state of internal stresses even with no application of any external mechanical forces. Under transient conditions, the stresses may undergo reversals changing from tension to compression or vice versa. If the stress or deformation gradients are assumed to have no feedback that is exerting no influence on the diffusion process, then the moisture and temperature distribution can be determined independently from the stress analysis. Moreover, as long as the solid deforms within the linear elastic range, the stress field resulting from the diffusion of moisture and temperature can be superimposed upon that from the field of external loading which is well-known in any textbook on elasticity. This Chapter considers the transient character of hygrothermal stresses in several problems that are solved analytically in closed form.

The geometry and/or boundary conditions for many practical problems of interest may not possess the type of symmetries discussed in Chapters 2 and 3. For the majority of cases, the coupled diffusion equations (2.11) are not amenable to analytical solutions. The numerical technique of finite element has gained wide acceptance in engineering application and will be adopted to solve the system of time dependent equations that govern the coupled phenomenon of heat and moisture diffusion * [1], Approximations made in discretizing the continuum by a finite number of elements is not always clear because the numerical procedure does lead to local violation of conditions in the analytical theory. The selection of element shape and grid pattern leaves much choice to experience and foresight of the analyst. Accuracy of solution must therefore be carefully checked against problems with known results, preferably with similar geometries and boundary conditions. The presence of sharp corners or discontinuities should be treated with the utmost care as they give rise to high stress elevation within a small distance.

Empirical studies [1] in recent time have shown that mechanically applied stress can significantly alter the moisture and temperature distribution in solid media. Such findings are not unexpected and suggest the need to analyze the coupling between mechanical deformation and diffusion due to moisture and temperature. Because of the complexities of the theoretical treatment and procedures involved in determining the physical constants, only a few cases have been treated up to this date. In order to obtain an in-depth insight into the physical phenomenon, it is necessary to rely on the disciplines of mechanics and thermodynamics for analyzing the combined interaction of heat, moisture and mechanical deformation. Based on suitable postulates, a coupled system of governing equations is derived that contain both space coordinates and time as independent variables. Simplification prevails when diffusion and deformation take place quasi-statically. The results also reduced to those obtained earlier when deformation is uncoupled while the classical coupled equations of thermoelasticity are recovered if moisture effects are further neglected. A method of solution is presented that makes use of a hygrothermoelastic potential.

The power of conformai mapping is well recognized for solving plane elasticity and potential flow problems in mechanics. Complex geometric boundaries may be transformed to those that can be described by a single space variable. By applying the concept of holomorphic functions, general form solutions may be written down. For problems involving removable singularities, numerical calculations carried out in the transformed plane are divorced from singularities as they are generally embedded in the mapping function. Such procedures are now well-known, the details of which can be found in [1, 2] for plane elasticity problems. Extension of the method to include thermal behavior is straightforward [3, 4].

Analytical treatment of coupled diffusion and deformation problems becomes unmanageable when the geometries and/or boundary conditions are complex. The finite element technique as presented in Chapter 4 can again be applied. No additional difficulties prevail as the governing equations for deformation coupled with diffusion are similar to those with uncoupled deformation. An iteration process, however, must be adopted such that the temperature, moisture and stress field for each time increment must satisfy the equations in accordance with the specified boundary conditions. Particular choice of the space and time increment for a given system requires special attention in order for the solution to converge. This is resolved by resolving the time portion of the problem in the Laplace transform domain such that the space portion is evaluated by the finite element procedure.

Failure prediction deals with the forecast of conditions involving material damage that tends to decrease the capability of structural members to support load. With the added influence of moisture and temperature, the situation becomes more complicated because their interaction with mechanical stresses must also be considered. This concern arises since almost all materials will degrade to some extent when undergoing diffusion and/or mechanical deformation. The idea that material damage alters the apparent behavior of solids has been widely appreciated by those working in the areas of material characterization and structure design. The rate of damage should preferably be controlled and sufficiently slow to allow a useful lifetime of the component.